url stringlengths 16 1.41k | text stringlengths 100 838k | date stringlengths 19 19 | metadata stringlengths 1.06k 1.1k |
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https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/1296/5/e/g/ | Properties
Label 1296.5.e.g Level 1296 Weight 5 Character orbit 1296.e Analytic conductor 133.967 Analytic rank 0 Dimension 8 CM no Inner twists 2
Related objects
Newspace parameters
Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 1296.e (of order $$2$$, degree $$1$$, not minimal)
Newform invariants
Self dual: no Analytic conductor: $$133.967472157$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{20}$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$
$q$-expansion
Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.
$$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + ( 3 - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + ( 3 - \beta_{3} ) q^{7} + \beta_{2} q^{11} + ( 2 + \beta_{3} + \beta_{5} ) q^{13} + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{17} + ( -70 - \beta_{6} ) q^{19} + ( -2 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} - \beta_{7} ) q^{23} + ( -88 + 4 \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{25} + ( -8 \beta_{1} - 5 \beta_{2} + \beta_{4} + \beta_{7} ) q^{29} + ( 46 + 4 \beta_{3} - 3 \beta_{5} + \beta_{6} ) q^{31} + ( 15 \beta_{1} - 6 \beta_{2} + \beta_{4} - 2 \beta_{7} ) q^{35} + ( -5 - 19 \beta_{3} - 5 \beta_{5} - \beta_{6} ) q^{37} + ( 26 \beta_{1} - \beta_{2} - 3 \beta_{4} + 2 \beta_{7} ) q^{41} + ( -15 + 3 \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{43} + ( -11 \beta_{1} - 4 \beta_{2} + 3 \beta_{4} - 4 \beta_{7} ) q^{47} + ( 71 - 19 \beta_{3} - 11 \beta_{5} + 2 \beta_{6} ) q^{49} + ( 61 \beta_{1} - 25 \beta_{2} + 5 \beta_{4} + 5 \beta_{7} ) q^{53} + ( 212 - 34 \beta_{3} - 13 \beta_{5} + \beta_{6} ) q^{55} + ( 63 \beta_{1} + 20 \beta_{2} + \beta_{4} - 7 \beta_{7} ) q^{59} + ( 480 - 53 \beta_{3} + 19 \beta_{5} + 2 \beta_{6} ) q^{61} + ( 106 \beta_{1} + 41 \beta_{2} + 3 \beta_{4} + 9 \beta_{7} ) q^{65} + ( 16 - 96 \beta_{3} + 3 \beta_{5} ) q^{67} + ( -35 \beta_{1} - 17 \beta_{2} - 9 \beta_{4} - 13 \beta_{7} ) q^{71} + ( -953 + 83 \beta_{3} - \beta_{5} - 8 \beta_{6} ) q^{73} + ( -105 \beta_{1} + 18 \beta_{2} - 14 \beta_{4} + 6 \beta_{7} ) q^{77} + ( -568 - 90 \beta_{3} + 3 \beta_{5} - 7 \beta_{6} ) q^{79} + ( 229 \beta_{1} + 12 \beta_{2} + 7 \beta_{4} - 12 \beta_{7} ) q^{83} + ( 731 - 97 \beta_{3} - 7 \beta_{5} + 7 \beta_{6} ) q^{85} + ( -241 \beta_{1} - 19 \beta_{2} + 3 \beta_{4} + 7 \beta_{7} ) q^{89} + ( -1940 + 162 \beta_{3} - 7 \beta_{5} - 7 \beta_{6} ) q^{91} + ( -384 \beta_{1} + 14 \beta_{2} - 20 \beta_{4} - 16 \beta_{7} ) q^{95} + ( -1822 - 17 \beta_{3} + \beta_{5} - 9 \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 26q^{7} + O(q^{10})$$ $$8q + 26q^{7} + 10q^{13} - 562q^{19} - 706q^{25} + 374q^{31} + 16q^{37} - 136q^{43} + 654q^{49} + 1818q^{55} + 3874q^{61} + 308q^{67} - 7802q^{73} - 4390q^{79} + 6084q^{85} - 15830q^{91} - 14564q^{97} + O(q^{100})$$
Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 6 x^{6} + 121 x^{5} + 1104 x^{4} - 1647 x^{3} + 6529 x^{2} + 85254 x + 440076$$:
$$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$17 \nu^{7} - 142 \nu^{6} + 635 \nu^{5} - 1115 \nu^{4} + 17923 \nu^{3} - 159143 \nu^{2} + 464580 \nu - 386178$$$$)/104247$$ $$\beta_{2}$$ $$=$$ $$($$$$53 \nu^{7} - 367 \nu^{6} + 5765 \nu^{5} - 70703 \nu^{4} + 113944 \nu^{3} + 106240 \nu^{2} - 1050579 \nu - 42441945$$$$)/104247$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 5 \nu^{6} + 34 \nu^{5} + 116 \nu^{4} + 197 \nu^{3} - 223 \nu^{2} + 25974 \nu + 60657$$$$)/729$$ $$\beta_{4}$$ $$=$$ $$($$$$118 \nu^{7} + 49 \nu^{6} - 1775 \nu^{5} + 11717 \nu^{4} + 118754 \nu^{3} + 166664 \nu^{2} + 172422 \nu + 2470650$$$$)/34749$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{7} - 38 \nu^{6} + 73 \nu^{5} + 410 \nu^{4} + 428 \nu^{3} - 18433 \nu^{2} + 18954 \nu + 178665$$$$)/729$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{7} + 2 \nu^{6} + 143 \nu^{5} + 661 \nu^{4} + 4441 \nu^{3} + 15841 \nu^{2} + 96174 \nu + 445521$$$$)/729$$ $$\beta_{7}$$ $$=$$ $$($$$$-1052 \nu^{7} - 1433 \nu^{6} + 40423 \nu^{5} - 237610 \nu^{4} - 608323 \nu^{3} + 281948 \nu^{2} + 16720785 \nu - 71046885$$$$)/104247$$
$$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 6 \beta_{4} + 18 \beta_{3} - 3 \beta_{2} + 22 \beta_{1} + 186$$$$)/486$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{7} + 3 \beta_{5} + 11 \beta_{4} - 3 \beta_{3} - \beta_{2} - 116 \beta_{1} - 60$$$$)/162$$ $$\nu^{3}$$ $$=$$ $$($$$$19 \beta_{7} + 24 \beta_{6} + 27 \beta_{5} + 56 \beta_{4} - 228 \beta_{3} - 37 \beta_{2} + 125 \beta_{1} - 7947$$$$)/162$$ $$\nu^{4}$$ $$=$$ $$($$$$14 \beta_{7} + 111 \beta_{6} + 96 \beta_{5} - 64 \beta_{4} - 570 \beta_{3} - 220 \beta_{2} - 219 \beta_{1} - 120225$$$$)/162$$ $$\nu^{5}$$ $$=$$ $$($$$$-217 \beta_{7} + 930 \beta_{6} + 285 \beta_{5} - 2155 \beta_{4} - 2505 \beta_{3} + 515 \beta_{2} + 2208 \beta_{1} - 206598$$$$)/162$$ $$\nu^{6}$$ $$=$$ $$($$$$-4159 \beta_{7} + 651 \beta_{6} - 6591 \beta_{5} - 13543 \beta_{4} + 15891 \beta_{3} + 5291 \beta_{2} + 68865 \beta_{1} + 432933$$$$)/162$$ $$\nu^{7}$$ $$=$$ $$($$$$-26773 \beta_{7} - 19995 \beta_{6} - 32457 \beta_{5} - 47548 \beta_{4} + 237246 \beta_{3} + 67505 \beta_{2} + 53688 \beta_{1} + 13250454$$$$)/162$$
Character values
We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$$.
$$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$
Embeddings
For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.
For more information on an embedded modular form you can click on its label.
Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
−3.41053 − 2.74723i −3.05006 − 3.25531i 4.23522 + 4.06612i 3.72537 + 4.42407i 3.72537 − 4.42407i 4.23522 − 4.06612i −3.05006 + 3.25531i −3.41053 + 2.74723i
0 0 0 40.2664i 0 −14.7738 0 0 0
161.2 0 0 0 31.6564i 0 75.3660 0 0 0
161.3 0 0 0 12.2819i 0 14.2840 0 0 0
161.4 0 0 0 8.86801i 0 −61.8763 0 0 0
161.5 0 0 0 8.86801i 0 −61.8763 0 0 0
161.6 0 0 0 12.2819i 0 14.2840 0 0 0
161.7 0 0 0 31.6564i 0 75.3660 0 0 0
161.8 0 0 0 40.2664i 0 −14.7738 0 0 0
$$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 161.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles
Inner twists
Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
Twists
By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.5.e.g 8
3.b odd 2 1 inner 1296.5.e.g 8
4.b odd 2 1 324.5.c.a 8
9.c even 3 1 144.5.q.c 8
9.c even 3 1 432.5.q.c 8
9.d odd 6 1 144.5.q.c 8
9.d odd 6 1 432.5.q.c 8
12.b even 2 1 324.5.c.a 8
36.f odd 6 1 36.5.g.a 8
36.f odd 6 1 108.5.g.a 8
36.h even 6 1 36.5.g.a 8
36.h even 6 1 108.5.g.a 8
By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.5.g.a 8 36.f odd 6 1
36.5.g.a 8 36.h even 6 1
108.5.g.a 8 36.f odd 6 1
108.5.g.a 8 36.h even 6 1
144.5.q.c 8 9.c even 3 1
144.5.q.c 8 9.d odd 6 1
324.5.c.a 8 4.b odd 2 1
324.5.c.a 8 12.b even 2 1
432.5.q.c 8 9.c even 3 1
432.5.q.c 8 9.d odd 6 1
1296.5.e.g 8 1.a even 1 1 trivial
1296.5.e.g 8 3.b odd 2 1 inner
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 2853 T_{5}^{6} + 2238759 T_{5}^{4} + 403999407 T_{5}^{2} + 19274879556$$ acting on $$S_{5}^{\mathrm{new}}(1296, [\chi])$$.
Hecke characteristic polynomials
$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 2147 T^{2} + 2477509 T^{4} - 2147976218 T^{6} + 1511855308306 T^{8} - 839053210156250 T^{10} + 378037872314453125 T^{12} -$$$$12\!\cdots\!75$$$$T^{14} +$$$$23\!\cdots\!25$$$$T^{16}$$
$7$ $$( 1 - 13 T + 4723 T^{2} - 93076 T^{3} + 12134350 T^{4} - 223475476 T^{5} + 27227155123 T^{6} - 179936733613 T^{7} + 33232930569601 T^{8} )^{2}$$
$11$ $$1 - 67376 T^{2} + 2511150250 T^{4} - 60712318203824 T^{6} + 1049763807787064539 T^{8} -$$$$13\!\cdots\!44$$$$T^{10} +$$$$11\!\cdots\!50$$$$T^{12} -$$$$66\!\cdots\!16$$$$T^{14} +$$$$21\!\cdots\!21$$$$T^{16}$$
$13$ $$( 1 - 5 T + 42079 T^{2} - 3738530 T^{3} + 1399773016 T^{4} - 106776155330 T^{5} + 34325133008959 T^{6} - 116490425612405 T^{7} + 665416609183179841 T^{8} )^{2}$$
$17$ $$1 - 288125 T^{2} + 52320681154 T^{4} - 6759733382202755 T^{6} +$$$$63\!\cdots\!86$$$$T^{8} -$$$$47\!\cdots\!55$$$$T^{10} +$$$$25\!\cdots\!74$$$$T^{12} -$$$$97\!\cdots\!25$$$$T^{14} +$$$$23\!\cdots\!61$$$$T^{16}$$
$19$ $$( 1 + 281 T + 110170 T^{2} - 68843041 T^{3} - 21846246566 T^{4} - 8971693946161 T^{5} + 1871079140226970 T^{6} + 621941492257591241 T^{7} +$$$$28\!\cdots\!81$$$$T^{8} )^{2}$$
$23$ $$1 - 103955 T^{2} + 204060092029 T^{4} - 14985633296690750 T^{6} +$$$$21\!\cdots\!46$$$$T^{8} -$$$$11\!\cdots\!50$$$$T^{10} +$$$$12\!\cdots\!69$$$$T^{12} -$$$$49\!\cdots\!55$$$$T^{14} +$$$$37\!\cdots\!21$$$$T^{16}$$
$29$ $$1 - 3708803 T^{2} + 6181658247973 T^{4} - 6420936064399980986 T^{6} +$$$$50\!\cdots\!30$$$$T^{8} -$$$$32\!\cdots\!46$$$$T^{10} +$$$$15\!\cdots\!33$$$$T^{12} -$$$$46\!\cdots\!43$$$$T^{14} +$$$$62\!\cdots\!41$$$$T^{16}$$
$31$ $$( 1 - 187 T + 2550973 T^{2} - 331939744 T^{3} + 3082958795560 T^{4} - 306553324318624 T^{5} + 2175702008453980093 T^{6} -$$$$14\!\cdots\!07$$$$T^{7} +$$$$72\!\cdots\!81$$$$T^{8} )^{2}$$
$37$ $$( 1 - 8 T + 3611368 T^{2} + 1256575624 T^{3} + 6911619203950 T^{4} + 2355025028051464 T^{5} + 12684855900547773928 T^{6} - 52663616046720282248 T^{7} +$$$$12\!\cdots\!41$$$$T^{8} )^{2}$$
$41$ $$1 - 12649388 T^{2} + 74446575673498 T^{4} -$$$$29\!\cdots\!76$$$$T^{6} +$$$$89\!\cdots\!55$$$$T^{8} -$$$$23\!\cdots\!96$$$$T^{10} +$$$$47\!\cdots\!18$$$$T^{12} -$$$$64\!\cdots\!68$$$$T^{14} +$$$$40\!\cdots\!81$$$$T^{16}$$
$43$ $$( 1 + 68 T + 12955228 T^{2} + 545371076 T^{3} + 65226299853025 T^{4} + 1864515179999876 T^{5} +$$$$15\!\cdots\!28$$$$T^{6} +$$$$27\!\cdots\!68$$$$T^{7} +$$$$13\!\cdots\!01$$$$T^{8} )^{2}$$
$47$ $$1 - 19981955 T^{2} + 227810016714829 T^{4} -$$$$17\!\cdots\!90$$$$T^{6} +$$$$10\!\cdots\!46$$$$T^{8} -$$$$42\!\cdots\!90$$$$T^{10} +$$$$12\!\cdots\!09$$$$T^{12} -$$$$26\!\cdots\!55$$$$T^{14} +$$$$32\!\cdots\!41$$$$T^{16}$$
$53$ $$1 - 5145920 T^{2} + 115452291970684 T^{4} - 84051566001475463360 T^{6} +$$$$67\!\cdots\!26$$$$T^{8} -$$$$52\!\cdots\!60$$$$T^{10} +$$$$44\!\cdots\!64$$$$T^{12} -$$$$12\!\cdots\!20$$$$T^{14} +$$$$15\!\cdots\!41$$$$T^{16}$$
$59$ $$1 - 31340696 T^{2} + 646808461063090 T^{4} -$$$$10\!\cdots\!44$$$$T^{6} +$$$$14\!\cdots\!19$$$$T^{8} -$$$$15\!\cdots\!24$$$$T^{10} +$$$$13\!\cdots\!90$$$$T^{12} -$$$$99\!\cdots\!56$$$$T^{14} +$$$$46\!\cdots\!81$$$$T^{16}$$
$61$ $$( 1 - 1937 T + 14281603 T^{2} + 60991858006 T^{3} - 94182960283700 T^{4} + 844483568245653046 T^{5} +$$$$27\!\cdots\!43$$$$T^{6} -$$$$51\!\cdots\!77$$$$T^{7} +$$$$36\!\cdots\!61$$$$T^{8} )^{2}$$
$67$ $$( 1 - 154 T + 33859570 T^{2} + 10195927724 T^{3} + 608674748042419 T^{4} + 205459373273578604 T^{5} +$$$$13\!\cdots\!70$$$$T^{6} -$$$$12\!\cdots\!94$$$$T^{7} +$$$$16\!\cdots\!81$$$$T^{8} )^{2}$$
$71$ $$1 - 68871716 T^{2} + 3244147638477940 T^{4} -$$$$11\!\cdots\!24$$$$T^{6} +$$$$30\!\cdots\!74$$$$T^{8} -$$$$73\!\cdots\!64$$$$T^{10} +$$$$13\!\cdots\!40$$$$T^{12} -$$$$18\!\cdots\!96$$$$T^{14} +$$$$17\!\cdots\!41$$$$T^{16}$$
$73$ $$( 1 + 3901 T + 59309470 T^{2} + 292589317519 T^{3} + 2279602007321194 T^{4} + 8309021952930084079 T^{5} +$$$$47\!\cdots\!70$$$$T^{6} +$$$$89\!\cdots\!21$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8} )^{2}$$
$79$ $$( 1 + 2195 T + 92542939 T^{2} + 195388180760 T^{3} + 5006999286743146 T^{4} + 7610385467044641560 T^{5} +$$$$14\!\cdots\!79$$$$T^{6} +$$$$12\!\cdots\!95$$$$T^{7} +$$$$23\!\cdots\!21$$$$T^{8} )^{2}$$
$83$ $$1 - 81188291 T^{2} + 8014262915096365 T^{4} -$$$$34\!\cdots\!54$$$$T^{6} +$$$$22\!\cdots\!34$$$$T^{8} -$$$$78\!\cdots\!14$$$$T^{10} +$$$$40\!\cdots\!65$$$$T^{12} -$$$$92\!\cdots\!11$$$$T^{14} +$$$$25\!\cdots\!61$$$$T^{16}$$
$89$ $$1 - 294759296 T^{2} + 46567064448316540 T^{4} -$$$$48\!\cdots\!04$$$$T^{6} +$$$$35\!\cdots\!14$$$$T^{8} -$$$$19\!\cdots\!24$$$$T^{10} +$$$$72\!\cdots\!40$$$$T^{12} -$$$$17\!\cdots\!36$$$$T^{14} +$$$$24\!\cdots\!21$$$$T^{16}$$
$97$ $$( 1 + 7282 T + 336254488 T^{2} + 1720884578884 T^{3} + 43500636050571385 T^{4} +$$$$15\!\cdots\!04$$$$T^{5} +$$$$26\!\cdots\!68$$$$T^{6} +$$$$50\!\cdots\!62$$$$T^{7} +$$$$61\!\cdots\!21$$$$T^{8} )^{2}$$ | 2020-08-15 05:42:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9660796523094177, "perplexity": 6318.276001975172}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439740679.96/warc/CC-MAIN-20200815035250-20200815065250-00120.warc.gz"} |
https://labs.tib.eu/arxiv/?author=N.%20Kondrashova | • ### Measurement of the cross-section ratio sigma_{psi(2S)}/sigma_{J/psi(1S)} in deep inelastic exclusive ep scattering at HERA(1606.08652)
June 29, 2016 hep-ex
The exclusive deep inelastic electroproduction of $\psi(2S)$ and $J/\psi(1S)$ at an $ep$ centre-of-mass energy of 317 GeV has been studied with the ZEUS detector at HERA in the kinematic range $2 < Q^2 < 80$ GeV$^2$, $30 < W < 210$ GeV and $|t| < 1$ GeV$^2$, where $Q^2$ is the photon virtuality, $W$ is the photon-proton centre-of-mass energy and $t$ is the squared four-momentum transfer at the proton vertex. The data for $2 < Q^2 < 5$ GeV$^2$ were taken in the HERA I running period and correspond to an integrated luminosity of 114 pb$^{-1}$. The data for $5 < Q^2 < 80$ GeV$^2$ are from both HERA I and HERA II periods and correspond to an integrated luminosity of 468 pb$^{-1}$. The decay modes analysed were $\mu^+\mu^-$ and $J/\psi(1S) \pi^+\pi^-$ for the $\psi(2S)$ and $\mu^+\mu^-$ for the $J/\psi(1S)$. The cross-section ratio $\sigma_{\psi(2S)}/\sigma_{J/\psi(1S)}$ has been measured as a function of $Q^2, W$ and $t$. The results are compared to predictions of QCD-inspired models of exclusive vector-meson production.
• ### Measurement of the cross-section ratio sigma_{psi(2S)}/sigma_{J/psi(1S)} in deep inelastic exclusive ep scattering at HERA(1601.03699)
Jan. 14, 2016 hep-ex
The exclusive deep inelastic electroproduction of $\psi(2S)$ and $J/\psi(1S)$ at an $ep$ centre-of-mass energy of 317 GeV has been studied with the ZEUS detector at HERA in the kinematic range $2 < Q^2 < 80$ GeV$^2$, $30 < W < 210$ GeV and $|t| < 1$ GeV$^2$, where $Q^2$ is the photon virtuality, $W$ is the photon-proton centre-of-mass energy and $t$ is the squared four-momentum transfer at the proton vertex. The data for $2 < Q^2 < 5$ GeV$^2$ were taken in the HERA I running period and correspond to an integrated luminosity of 114 pb$^{-1}$. The data for $5 < Q^2 < 80$ GeV$^2$ are from both HERA I and HERA II periods and correspond to an integrated luminosity of 468 pb$^{-1}$. The decay modes analysed were $\mu^+\mu^-$ and $J/\psi(1S) \,\pi^+\pi^-$ for the $\psi(2S)$ and $\mu^+\mu^-$ for the $J/\psi(1S)$. The cross-section ratio $\sigma_{\psi(2S)}/\sigma_{J/\psi(1S)}$ has been measured as a function of $Q^2, W$ and $t$. The results are compared to predictions of QCD-inspired models of exclusive vector-meson production.
• ### Production of exclusive dijets in diffractive deep inelastic scattering at HERA(1505.05783)
Dec. 17, 2015 hep-ex
Production of exclusive dijets in diffractive deep inelastic $e^\pm p$ scattering has been measured with the ZEUS detector at HERA using an integrated luminosity of 372 pb$^{-1}$. The measurement was performed for $\gamma^*-p$ centre-of-mass energies in the range $90 < W < 250$ GeV and for photon virtualities $Q^2 > 25$ GeV$^2$. Energy and transverse-energy flows around the jet axis are presented. The cross section is presented as a function of $\beta$ and $\phi$, where $\beta=x/x_{\rm I\!P}$, $x$ is the Bjorken variable and $x_{\rm I\!P}$ is the proton fractional longitudinal momentum loss. The angle $\phi$ is defined by the $\gamma^*-$dijet plane and the $\gamma^*-e^\pm$ plane in the rest frame of the diffractive final state. The $\phi$ cross section is measured in bins of $\beta$. The results are compared to predictions from models based on different assumptions about the nature of the diffractive exchange.
• ### Measurement of beauty and charm production in deep inelastic scattering at HERA and measurement of the beauty-quark mass(1405.6915)
Oct. 21, 2014 hep-ex
The production of beauty and charm quarks in ep interactions has been studied with the ZEUS detector at HERA for exchanged four-momentum squared 5 < Q^2 < 1000 GeV^2 using an integrated luminosity of 354 pb^{-1}. The beauty and charm content in events with at least one jet have been extracted using the invariant mass of charged tracks associated with secondary vertices and the decay-length significance of these vertices. Differential cross sections as a function of Q^2, Bjorken x, jet transverse energy and pseudorapidity were measured and compared with next-to-leading-order QCD calculations. The beauty and charm contributions to the proton structure functions were extracted from the double-differential cross section as a function of x and Q^2. The running beauty-quark mass, m_b at the scale m_b, was determined from a QCD fit at next-to-leading order to HERA data for the first time and found to be 4.07 \pm 0.14 (fit} ^{+0.01}_{-0.07} (mod.) ^{+0.05}_{-0.00} (param.) ^{+0.08}_{-0.05} (theo) GeV.
• ### Further studies of the photoproduction of isolated photons with a jet at HERA(1405.7127)
Sept. 12, 2014 hep-ex
In this extended analysis using the ZEUS detector at HERA, the photoproduction of isolated photons together with a jet is measured for different ranges of the fractional photon energy, $x_\gamma^{\mathrm{meas}}$, contributing to the photon-jet final state. Cross sections are evaluated in the photon transverse-energy and pseudorapidity ranges $6 < E_T^{\gamma} < 15$ GeV and $-0.7 < \eta^{\gamma} < 0.9$, and for jet transverse-energy and pseudorapidity ranges $4 < E_T^{\rm jet} < 35$ GeV and $-1.5 < \eta^{\rm jet} < 1.8$, for an integrated luminosity of 374 $\mathrm{pb}^{-1}$. The kinematic observables studied comprise the transverse energy and pseudorapidity of the photon and the jet, the azimuthal difference between them, the fraction of proton energy taking part in the interaction, and the difference between the pseudorapidities of the photon and the jet. Higher-order theoretical calculations are compared to the results.
• ### Measurement of D* photoproduction at three different centre-of-mass energies at HERA(1405.5068)
Sept. 11, 2014 hep-ex
The photoproduction of $D^{*\pm}$ mesons has been measured with the ZEUS detector at HERA at three different ep centre-of-mass energies, $\sqrt{s}$, of 318, 251 and 225 GeV. For each data set, $D^*$ mesons were required to have transverse momentum, $p_T^{D^*}$, and pseudorapidity, $\eta^{D^*}$, in the ranges $1.9 < p_T^{D^*} < 20$ GeV and $|\eta^{D^*}|<1.6$. The events were required to have a virtuality of the incoming photon, $Q^2$, of less than 1 GeV$^2$. The dependence on $\sqrt{s}$ was studied by normalising to the high-statistics measurement at $\sqrt{s} =318$ GeV. This led to the cancellation of a number of systematic effects both in data and theory. Predictions from next-to-leading-order QCD describe the $\sqrt{s}$ dependence of the data well.
• Measurements of neutral current cross sections for deep inelastic scattering in e+p collisions at HERA with a longitudinally polarised positron beam are presented. The single-differential cross-sections d(sigma)/dQ2, d(sigma)/dx and d(sigma)/dy and the reduced cross-section were measured in the kinematic region Q2 > 185 GeV2 and y < 0.9, where Q2 is the four-momentum transfer squared, x the Bjorken scaling variable, and y the inelasticity of the interaction. The measurements were performed separately for positively and negatively polarised positron beams. The measurements are based on an integrated luminosity of 135.5 pb-1 collected with the ZEUS detector in 2006 and 2007 at a centre-of-mass energy of 318 GeV. The structure functions F3 and F3(gamma)Z were determined by combining the e+p results presented in this paper with previously published e-p neutral current results. The asymmetry parameter A+ is used to demonstrate the parity violation predicted in electroweak interactions. The measurements are well described by the predictions of the Standard Model.
• ### Deep inelastic cross-section measurements at large y with the ZEUS detector at HERA(1404.6376)
April 25, 2014 hep-ex
The reduced cross sections for $e^{+}p$ deep inelastic scattering have been measured with the ZEUS detector at HERA at three different centre-of-mass energies, $318$, $251$ and $225$ GeV. The cross sections, measured double differentially in Bjorken $x$ and the virtuality, $Q^2$, were obtained in the region $0.13\ \leq\ y\ \leq\ 0.75$, where $y$ denotes the inelasticity and $5\ \leq\ Q^2\ \leq\ 110$ GeV$^2$. The proton structure functions $F_2$ and $F_L$ were extracted from the measured cross sections.
• ### Measurement of neutral current e+/-p cross sections at high Bjorken x with the ZEUS detector(1312.4438)
April 4, 2014 hep-ex
The neutral current e+/-p cross section has been measured up to values of Bjorken x of approximately 1 with the ZEUS detector at HERA using an integrated luminosity of 187 inv. pb of e-p and 142 inv. pb of e+p collisions at sqrt(s) = 318GeV. Differential cross sections in x and Q2, the exchanged boson virtuality, are presented for Q2 geq 725GeV2. An improved reconstruction method and greatly increased amount of data allows a finer binning in the high-x region of the neutral current cross section and leads to a measurement with much improved precision compared to a similar earlier analysis. The measurements are compared to Standard Model expectations based on a variety of recent parton distribution functions.
• The production of Z0 bosons in the reaction ep -> eZ0p*, where p* stands for a proton or a low-mass nucleon resonance, has been studied in ep collisions at HERA using the ZEUS detector. The analysis is based on a data sample collected between 1996 and 2007, amounting to 496 pb-1 of integrated luminosity. The Z0 was measured in the hadronic decay mode. The elasticity of the events was ensured by a cut on eta_max < 3.0, where eta_max is the maximum pseudorapidity of energy deposits in the calorimeter defined with respect to the proton beam direction. A signal was observed at the Z0 mass. The cross section of the reaction ep -> eZ0p* was measured to be sigma(ep -> eZ0p*) = 0.13 +/- 0.06 (stat.) +/- 0.01 (syst.) pb, in agreement with the Standard Model prediction of 0.16 pb. This is the first measurement of Z0 production in ep collisions.
• Inclusive-jet cross sections have been measured in the reaction ep->e+jet+X for photon virtuality Q2 < 1 GeV2 and gamma-p centre-of-mass energies in the region 142 < W(gamma-p) < 293 GeV with the ZEUS detector at HERA using an integrated luminosity of 300 pb-1. Jets were identified using the kT, anti-kT or SIScone jet algorithms in the laboratory frame. Single-differential cross sections are presented as functions of the jet transverse energy, ETjet, and pseudorapidity, etajet, for jets with ETjet > 17 GeV and -1 < etajet < 2.5. In addition, measurements of double-differential inclusive-jet cross sections are presented as functions of ETjet in different regions of etajet. Next-to-leading-order QCD calculations give a good description of the measurements, except for jets with low ETjet and high etajet. The influence of non-perturbative effects not related to hadronisation was studied. Measurements of the ratios of cross sections using different jet algorithms are also presented; the measured ratios are well described by calculations including up to O(alphas2) terms. Values of alphas(Mz) were extracted from the measurements and the energy-scale dependence of the coupling was determined. The value of alphas(Mz) extracted from the measurements based on the kT jet algorithm is alphas(Mz) = 0.1206 +0.0023 -0.0022 (exp.) +0.0042 -0.0035 (th.); the results from the anti-kT and SIScone algorithms are compatible with this value and have a similar precision. | 2021-04-18 05:31:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8194133639335632, "perplexity": 1733.6412151666932}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038468066.58/warc/CC-MAIN-20210418043500-20210418073500-00397.warc.gz"} |
https://stats.stackexchange.com/questions/152923/what-does-special-case-of-gmm-mean | # What does “special case of GMM” mean?
I was researching purchasing this text book: http://www.amazon.com/dp/0691010188/ref=wl_it_dp_o_pd_nS_ttl?_encoding=UTF8&colid=2QTISO1Y8TYVW&coliid=I3FUEFWL47AC4L
In its description it talks about how the book teaches estimation techniques as special cases of GMM. This is not the first time I have heard that phrase before. What exactly does it mean?
For example, what does OLS as a special case of GMM mean? I was under the impression GMM was its own estimate technique.
For a weighting matrix $\hat W$, regressor matrix $Z$ (following the somewhat unusual notation used in the book by Hayashi you are referring to) and instrument matrix $X$, the GMM estimator to estimate $\delta$ in a linear model $y=Z\delta+\epsilon$ can be written as $$\widehat{\delta}(\widehat{W})=(Z'X\widehat{W}X'Z)^{-1}Z'X\widehat{W}X'y.$$ If you assume your regressors to be "valid" (aka exogenous, uncorrelated with the error term), the $Z$ may be a subset of the set of instruments $X$, or they may even coincide with $X$, $Z=X$.
Consider the latter case first. GMM becomes (notice the matrices in the inverse then are square so that the inverse can be written as the product of the inverses, in reverse order) \begin{align*} \widehat{\delta}(\widehat{W})&=(Z'Z\widehat{W}Z'Z)^{-1}Z'Z\widehat{W}Z'y\\ &=(Z'Z)^{-1}\widehat{W}^{-1}(Z'Z)^{-1}Z'Z\widehat{W}Z'y\\ &=(Z'Z)^{-1}Z'y, \end{align*} which is the OLS estimator.
For the first case, we get equality for the particular choice of weighting matrix (optimal under conditional homoskedasticity) $\hat W=(X'X)^{-1}$:
GMM then becomes \begin{align*} \widehat{\delta}((X'X)^{-1})&=(Z'X(X'X)^{-1}X'Z)^{-1}Z'X(X'X)^{-1}X'y\\ &:=(Z'P_XZ)^{-1}Z'P_Xy, \end{align*} known as two-stage least squares. Now, if $Z\subset X$, projecting $Z$ on $X$ via $P_XZ$ will simply give $Z$ again (you explain something by itself and other instruments, and the best way to explain a variable is by itself - you will get perfect fit). Hence, GMM will again reduce to OLS.
It's generalized method of moments, for which Hansen got Economic Nobel a couple of years ago. It's very popular in econometrics.
You can show that OLS estimates are the same as GMM estimates under certain conditions. It's similar to MLE and OLS being the same under certain conditions (normal errors).
• I edited my post to be more clear. I understand what GMM is but I am not sure what OLS as a special case of GMM means. – Michael May 18 '15 at 23:23
• @Michael, GMM will produce the same estimate as OLS – Aksakal May 18 '15 at 23:47 | 2021-04-20 11:12:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9023758769035339, "perplexity": 637.8523075154234}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039388763.75/warc/CC-MAIN-20210420091336-20210420121336-00340.warc.gz"} |
https://hal.inria.fr/inria-00074107 | HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information
# An Output-Sensitive Convex Hull Algorithm for Planar Objects
1 PRISME - Geometry, Algorithms and Robotics
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : A set of planar objects is said to be of type $m$ if the convex hull of any two objects has its size bounded by $2m$. In this paper, we present an algorithm based on the marriage-before-conquest paradigm to compute the convex hull of a set of $n$ planar convex objects of fixed type $m$. The algorithm is \mbox{output-sensitive}, i.e. its time complexity depends on the size $h$ of the computed convex hull. The main ingredient of this algorithm is a linear method to find a {\em bridge}, i.e. a facet of the convex hull intersected by a given line. We obtain an $O(n\beta (h,m)\log h)$-time convex hull algorithm for planar objects. Here $\beta (h,2)=O(1)$ and $\beta (h,m)$ is an extremely slowly growing function. As a direct consequence, we can compute in optimal $\Theta (n\log h)$ time the convex hull of disks, convex homothets, non-overlapping objects. The method described in this paper also applies to compute lower envelopes of functions. In particular, we obtain an optimal $\Theta (n\log h)$-time algorithm to compute the upper envelope of line segments.
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https://hal.inria.fr/inria-00074107
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Submitted on : Wednesday, May 24, 2006 - 2:32:18 PM
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### Identifiers
• HAL Id : inria-00074107, version 1
### Citation
Franck Nielsen, Mariette Yvinec. An Output-Sensitive Convex Hull Algorithm for Planar Objects. RR-2575, INRIA. 1995. ⟨inria-00074107⟩
Record views | 2022-05-29 06:42:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5756003260612488, "perplexity": 1001.0038490309903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663039492.94/warc/CC-MAIN-20220529041832-20220529071832-00386.warc.gz"} |
https://plainmath.net/20589/write-expression-difference-logarithms-express-powers-factors-equal | Question
# Write the expression as a sum and/or difference of logarithms. Express powers as factors. \ln \frac{2x\sqrt{1+8x}}{(x-3)^{13}}, x>3 \ln \frac{2x\sqrt{1+8x}}{(x-3)^{13}}=
Factors and multiples
Write the expression as a sum and/or difference of logarithms. Express powers as factors.
$$\displaystyle{\ln{{\frac{{{2}{x}\sqrt{{{1}+{8}{x}}}}}{{{\left({x}-{3}\right)}^{{{13}}}}}}}},{x}{>}{3}$$
$$\displaystyle{\ln{{\frac{{{2}{x}\sqrt{{{1}+{8}{x}}}}}{{{\left({x}-{3}\right)}^{{{13}}}}}}}}=$$ (Simplify your answer.)
2021-08-04
Step 1
The given expression is
$$\displaystyle{\ln{{\frac{{{2}{x}\sqrt{{{1}+{8}{x}}}}}{{{\left({x}-{3}\right)}^{{{13}}}}}}}},{x}{>}{3}$$
Step 2
Using the formula
$$\displaystyle{\ln{{\left({\frac{{{a}}}{{{b}}}}\right)}}}={\ln{{a}}}-{\ln{{b}}}$$
$$\displaystyle{\ln{{\left({a}{b}\right)}}}={\ln{{a}}}+{\ln{{b}}}$$
$$\displaystyle{{\ln{{a}}}^{{{n}}}=}{n}{\ln{{a}}}$$
Step 3
On simplifying with the help of given formula
$$\displaystyle{\ln{{\frac{{{2}{x}\sqrt{{{1}+{8}{x}}}}}{{{\left({x}-{3}\right)}^{{{13}}}}}}}}={\ln{{\frac{{{2}{x}{\left({1}+{8}{x}\right)}^{{{\frac{{{1}}}{{{2}}}}}}}}{{{\left({x}-{3}\right)}^{{{13}}}}}}}}$$
$$\displaystyle={\ln{{2}}}{x}{\left({1}+{8}{x}\right)}^{{{\frac{{{1}}}{{{2}}}}}}-{{\ln{{\left({x}-{3}\right)}}}^{{{13}}}}$$
$$\displaystyle={\ln{{2}}}{x}+{{\ln{{\left({1}+{8}{x}\right)}}}^{{{\frac{{{1}}}{{{2}}}}}}-}{{\ln{{\left({x}-{3}\right)}}}^{{{13}}}}$$
$$\displaystyle={\ln{{2}}}{x}+{{\ln{{\left({1}+{8}{x}\right)}}}^{{{\frac{{{1}}}{{{2}}}}}}-}{{\ln{{\left({x}-{3}\right)}}}^{{{13}}}}$$
$$\displaystyle={\ln{{2}}}+{\ln{{x}}}+{\frac{{{1}}}{{{2}}}}{\ln{{\left({1}+{8}{x}\right)}}}-{13}{\ln{{\left({x}-{3}\right)}}}$$
$$\displaystyle\Rightarrow{\ln{{\frac{{{2}{x}\sqrt{{{1}+{8}{x}}}}}{{{\left({x}-{3}\right)}^{{{13}}}}}}}}={\ln{{2}}}+{\ln{{x}}}+{\frac{{{1}}}{{{2}}}}{\ln{{\left({1}+{8}{x}\right)}}}-{13}{\ln{{\left({x}-{3}\right)}}}$$ | 2021-09-23 15:41:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9471043944358826, "perplexity": 9310.508730995807}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057424.99/warc/CC-MAIN-20210923135058-20210923165058-00090.warc.gz"} |
http://openstudy.com/updates/4d9741a75ca48b0b28bd87bb | ## anonymous 5 years ago An airplane flies 1200 miles into the wind in 3 hours. The return trip takes 2 hours. Find the speed of the airplane without a wind and the speed of wind.
1. anonymous
from what is given in your question, we have the following two equations: $\nu_{plane}-\nu_{air}={1200 miles \over 3 hours}=400mph \rightarrow(1)$ $v_{plane}+v_{air}={1200 miles \over 2 hours}=600mph$ solve the two equation to find both velocities
2. anonymous
adding equation (1) and (2), we get: $2\nu_{plane}=1000 mph \implies \nu_{plane}=500mph$ substitute in either equation (1) by v_plane=500, you get: $500-\nu_{air}=400mph \implies \nu_{air}=100mph$
3. anonymous
does that make sense to you?
4. anonymous
Try www.aceyourcollegeclasses.com
5. anonymous
yes it does. thank you very much
6. anonymous
Try www.aceyourcollegeclasses.com
7. anonymous
you're welcome | 2017-01-18 06:09:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8243313431739807, "perplexity": 1448.7207317641064}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280239.54/warc/CC-MAIN-20170116095120-00346-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://economics.stackexchange.com/questions/1859/oil-as-a-function-of-gdp/1860 | # Oil as a function of GDP
In Mankiw, N. (2012). Principles of Macroeconomics (6th ed.)., on pg 456 Mankiw says "The amount of oil used to produce a unit of real GDP has declined about 40 percent since the OPEC shocks of the 1970s."
How would one measure that?
That's unfortunately just kind of dropped at the end of a chapter as the second to last sentence in a five paragraph side note about OPEC, and I've been curious for years now if that reduction was because the mix of energy supply changed (rise in natural gas consumption?), or production is simply more energy efficient, or what.
Can someone clarify the claim, the process, or point toward primary sources (US EIA maybe?), rereading this book has reminded me how frustrating that bit was.
I don't think it's any more complicated than looking at the ratio of global oil consumption and global Gross Domestic Product. The OECD, World Bank, IMF and others produce estimates of each.
There is some fuel-switching: for example, the world burns much less electricity from oil than it used to: in 1973 (the first oil crisis started around October that year) 251 million tonnes were used for electricity; by 2000, this had dropped to 110; by 2012, to 70 (table 7A on IV.81 of the IEA Electricity Information 2014).
There's several decades of energy efficiency, kickstarted in large part by the oil shocks of the 1970s.
The share of services (relative to physical goods) within global GDP has increased; IIRC services require much lower energy inputs than goods, on average, even though transport is one of the services (I'll check this and edit later - please nudge me in comments if I haven't after a week or so). The World Bank only gives services as a share of global GDP from 1990 onwards: this share grew from 60.7% in 1990 to 70.2% in 2011.
The combination of energy efficiency and the increase in the share of services (relative to physical goods) within global GDP, means that energy consumption per unit of global GDP has fallen: J. Bradford DeLong gives global GDP at \$12 trillion for 1970 and \$41 trillion for 2000 (preferred measure, 1990 international dollars). The BP Statistical Review of World Energy 2014 gives 1970 energy consumption at 4.9 billion tonnes of oil equivalent (btoe) and 9.3 btoe for 2000. So energy per unit global GDP almost halved (44% reduction) over those thirty years 1970-2000.
Understanding an individual country's changes are harder, because we then have to unpick whether oil intensity of GDP has dropped because of energy-efficiency & fuel-switching, or because oil-intensive goods are now imported rather than produced domestically: for further reading on this, see papers by John Barrett and by Dieter Helm.
• I was kind of hoping you'd answer. :) Looking at it as world makes more sense. I had always assumed he meant the domestic (US) ratio because the excerpt is about the effects of the oil shocks on the US economy. But i guess it was left intentionally vague because global seems much easier to quantify, especially for what was essentially a throw away line. – Jason Nichols Jan 2 '15 at 15:39
• Oh, thank you! Yes, quantifying the national picture is harder because of the offshoring of the manufacture of oil-intensive / energy-intensive goods. – 410 gone Jan 2 '15 at 16:12 | 2020-09-27 01:12:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3861733078956604, "perplexity": 3197.942654344694}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400249545.55/warc/CC-MAIN-20200926231818-20200927021818-00121.warc.gz"} |
https://aviation.stackexchange.com/questions/62031/how-does-the-advancing-wing-in-a-flat-spin-create-nose-thrust | # How does the advancing wing in a flat spin create nose thrust?
How does the advancing wing in a flat spin create nose thrust?
I imagine this like some object spinning in a liquid. I would think that the motion of each wing would only create drag that acts as torque opposing the motion of the spin in the geographic plane of the spin that would tend to make the spin slow down rather than be maintained.
A very good question, considering so many people do not draw the thrust vector for gliders.
The wing generates thrust by converting energy from falling to forward motion. From scratch building gliders (and reading) one comes to realize that a falling parachute (straight down) only uses drag to reach a constant rate of descent.
A glider, with its center or gravity off set from its center of pressure (including tail!!!) will start to move sideways (glidogenesis) and generate lift with its wing. This motion, known as "thrust" in this example, uses altitude as fuel and falling as its engine.
Notice both wings may be "thrusting" as it corkscrews down, but the outside wing, with its lower AOA, generates more lift/thrust and much less drag, sustaining the spin.
Once yawing is stopped, the wings equalize, and can be unstalled by lowering angle of attack.
• I see. So it would be better to envision the airplane like a falling rotor blade being struck from the bottom and pushed forward since the plane is tilted nose-down. – Ryan Mortensen Apr 6 '19 at 4:31
• From the auto rotation point of view, yes. You can make a business envelope glide by offsetting drag from falling and center of gravity, and nose down also makes it easier to recover. It is powered flight (gravity). – Robert DiGiovanni Apr 6 '19 at 8:28
I think Peter's diagram explains it well, I've repeated it here:
You can see that for the advancing wing (right side of the image) the vector addition of $$V_\infty$$ and $$\omega_z \times y$$ gives a resultant force R, that has a component pointing forward. This is forward thrust and increases the spin. | 2020-01-20 09:44:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.359562486410141, "perplexity": 1662.9620928979125}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250598217.23/warc/CC-MAIN-20200120081337-20200120105337-00270.warc.gz"} |
https://www.imrpress.com/journal/JOMH/18/7/10.31083/j.jomh1807156 | NULL
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Open Access Original Research
The Influence of BMI Levels on the Values of Static and Dynamic Balance for Students (Men) of the Faculty of Physical Education and Sports
Gabriel Murariu2,*,†Ilie Onu3,4,5,†
Show Less
1 Department of Individual Sports and Physical Therapy, Faculty of Physical Education and Sport, “Dunarea de Jos'' University of Galati, 800008 Galati, Romania
2 Department of Chemistry, Physics and the Environment, Faculty of Sciences and Environment, “Dunarea de Jos'' University of Galati, 800008 Galati, Romania
3 Department of Biomedical Sciences, Faculty of Medical Bioengineering, University of Medicine and Pharmacy “Grigore T. Popa'' Iasi, 700454 Iasi, Romania
4 Doctoral School of Faculty of Chemical Engineering and Environmental Protection “Cristofor Simionescu'', Technical University “Gheorghe Asachi'' Iasi, 700050 Iasi, Romania
5 Department of Physiotherapy, Micromedica Clinic, 610119 Piatra Neamt, Romania
*Correspondence: gabriel.murariu@ugal.ro (Gabriel Murariu)
These authors contributed equally.
J. Mens. Health 2022, 18(7), 156; https://doi.org/10.31083/j.jomh1807156
Submitted: 16 April 2022 | Revised: 19 May 2022 | Accepted: 2 June 2022 | Published: 13 July 2022
(This article belongs to the Special Issue Exercise and sports in men: from health to sports performance)
This is an open access article under the CC BY 4.0 license.
Abstract
Background: Postural stability is a factor that conditions the motor performance of athletes and of different categories of the population involved in activities that require physical effort. The aim of the study is to highlight the differences that appear in terms of balance performance for students, depending on their classification on BMI levels. Methods: A group of 109 students from the Faculty of Physical Education and Sports (1st year undergraduate) participated in this study at the end of the academic year (May, 2019), being divided following the anthropometric assessment into 3 groups related to BMI levels (7 cases of underweight, BMI = 17.21 $\pm$ 1.11; 83 cases of normal weight, BMI = 22.29 $\pm$ 1.56; 19 cases of overweight, BMI = 27.97 $\pm$2.89). The research is cross-sectional, using the MANOVA statistical calculation procedure (multivariate and univariate test—with Bonferroni Post Hoc Test and comparison of significance between the mean values of the 3 defined groups, for the 7 applied balance tests). Results: Univariate test results indicate values of F associated with statistically insignificant thresholds (p $>$ 0.05) for most of the tests used, with weak and very weak values of size effect (Ƞ${}^{2}$${}_{\text{p}}$). This aspect is also reinforced by the differences between the averages of the analyzed pairs, where only the statistically significant superiority (p $<$ 0.05) of the underweight over the overweight for the Stork test is noticed. The underweight group achieves slightly superior performance in the assessments for Standing balance test, Stork test, Flamingo test, Walk and turn field sobriety test, and that of the normal weight for Functional reach test and Bass test, the overweight having the poorest results in most tests; the differences between the 3 BMI levels analyzed are insignificant (p $>$ 0.05). Conclusions: Even if underweight and normal weight have better average scores than overweight, the lack of statistical significance of these differences can be explained by student specialization, constant involvement in physical performance, curricular or leisure activities improving performance on balance tests for the overweight category. These results reflect the particularities of the studied group and cannot be generalized for the university population, especially due to the numerically reduced group of underweight people.
Keywords
students
balance tests
BMI levels
postural stability
Figures
Fig. 1.
Share | 2022-09-29 12:01:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 8, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17592500150203705, "perplexity": 5962.580543992767}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335350.36/warc/CC-MAIN-20220929100506-20220929130506-00568.warc.gz"} |
http://tatome.de/zettelkasten/zettelkasten.php?standalone&thoughts=54ddf61862b31 | # Show Thoughts
Yan et al. report an accuracy of auditory localization of $3.4^\circ$ for online learning and $0.9^\circ$ for offline calibration. | 2020-06-02 23:01:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6779256463050842, "perplexity": 2075.510172586393}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347426956.82/warc/CC-MAIN-20200602224517-20200603014517-00389.warc.gz"} |
https://humanreadablemag.com/issues/1/articles/unionizing-for-pleasure-and-profit | Language Feature
# Exploring the power of union types in Scala
Illustrated by Leandro Lassmar
One obvious advantage of typed languages is that they force you to consider all your data: where it came from and where it is going. If you try to force a String down a path it's not meant to go, the program will not compile. You'll need to retreat or change more code. The compiler is equivalent to millions of little unit tests that you don't get to opt out of.
However, sometimes it can get overbearing. You may have a UserId, RequestId, ProductId, and sundry others. You know that in some circumstances all of these types are equivalent (perhaps designing an API structure around them) and in some cases they are not equivalent (putting them in a database).
A traditional approach to solving the problem of shared functionality between disparate types would be method overloading: the same method defined n times, once for each type. However, that's a lot of boilerplate. It's difficult to refactor, error prone if the definitions diverge, and how do you overload a class properly anyway?
Union types can help.
A union type would generally be considered some constructed type T, which, when examined, could resolve to one of several types A, B, C, etc.
You can think of it as the opposite of the tuple, or product. A tuple T is a constructed type, which, when examined, resolves to one of each of several types A, B, C, etc.
More formally, a union type is a coproduct, the dual of a product.
In Scala, you might immediately be drawn to the Either[_, _] type as an example. And it is indeed a poor man's union type---we can do the following:
type T = Either[Int, String]
val t: T = ???
t match {
case Left(_) => "It's an int!"
case Right(_) => "It's ~a boy~ a string!"
}
You see we have a value of type T, which is (roughly) of type Int or String, and upon examination we find out which one it is and get some result back accordingly. This, for future reference, is a fold on the structure of Either.
So, what's wrong with it?
The first, and most major, impedance is that it is quite clearly intended to be limited to just two types. There aren't many places to go once you've used up left and right. Up and down, forward and backward. Spinward, hubward, rimward,
and widdershins, perhaps.
A tuple, for all its faults (and there are many---I imagine I'll be writing about those faults in the future), is at least available in more than just a pair:
type T2 = (Int, String)
type T3 = (Int, String, Boolean)
type T4 = (Int, String, Boolean, Char)
Scala, for reasons lost to the mists of time, lets you go up to a tuple with twenty-two elements and no further.
You could of course do some nesting of an Either, to emulate more options than just left or right:
type Or[A, B, C] = Either[Either[A, B], C]
val t: Or[Int, String, Boolean] = ???
t match {
case Left(Left(_)) => "An integer"
case Left(Right(_)) => "A string"
case Right(_) => "A boolean"
}
It's pretty horrid though. You have to know that Left(Right(_)) is a possibility, but not Right(Left(_)). And imagine further nesting: Where do you put the extra Either? On the outside? Around the C? Around the B?
No, it's a disaster. Your future user (the next coder who has to use an Or7) won't thank you. And God forbid someone comes along and defines Or8 in the wrong way.
Thankfully, Scala does not leave you hanging. With type-level programming---programming that purely manipulates type structures with no values involved at all---you can program in extra rules on top of the compiler to enable new functionality.
## Typeclasses
We're functional developers! Let's write a typeclass.
What do we want out of it? Let's start by saying we want a way of telling whether a type T is an Int or a String.
So:
sealed trait IntOrString[T]
object IntOrString {
implicit val intInstance: IntOrString[Int] = new IntOrString[Int] {}
implicit val stringInstance: IntOrString[String] = new IntOrString[String] {}
}
The trait IntOrString is sealed to prevent a mischievous client from defining an instance of IntOrString[Boolean] in their code. A Boolean is, by definition, not an int or a string.
It looks...ok. You can easily imagine how to extend it to more things: Adding in OrBoolean is as simple as an extra instance (and a rename!).
This is how IntOrString might be used:
def foo[T: IntOrString](t: T): String = t match {
case i: Int => "Integer!"
case s: String => "String!"
}
foo(5) // compiles
foo("bar") // compiles
foo(true) // does not compile!
Now, the pattern match here is not checked for exhaustivity by the compiler. T is a free type, and it is constrained not by the type system but by our implementations on top of it (our typeclass). Therefore the compiler can't help us.
We could add extra structure to IntOrString to extract our value if we wanted to. Perhaps something like this:
sealed trait IntOrString[T] {
def asInt(t: T): Option[Int]
def asString(t: T): Option[String]
}
But these approaches get unwieldy to use, since there will always be an impossible case to handle
(the case where both asInt and asString are both None is impossible, but the compiler does not know this).
It's a trade-off between exhausting boilerplate, and losing exhaustivity matching. Let's choose the latter.
## Going Further
The eagle-eyed among you will notice that above we just defined a union type for Int and String. That is almost useless for our purposes; we want something generic. Such fixed approaches are fine for limited use cases, but if you have more than a few, you'll quickly get sick of them.
What we want to do is abstract over Int and String: the types themselves and the number of them.
### Abstracting the Types
Let's give it a shot. The first thing to do is simply replace Int and String with generic types. We quickly hit a roadblock:
trait XorY[T]
...what are X and Y? They have to be added somewhere. The fact we were using Int or String before---things already known to us at the coding level and known to the compiler at compile time---hid the fact that we need some new structure.
We need a typeclass, which, when given three types, has instances when the first is equal to one of the latter two.
Let's give it a go:
trait IsOneOf[T, X, Y]
object IsOneOf {
def apply[T, X, Y](implicit ev: IsOneOf[T, X, Y]): IsOneOf[T, X, Y] = ev
implicit def xInstance[X, Y]: IsOneOf[X, X, Y] = null
implicit def yInstance[X, Y]: IsOneOf[Y, X, Y] = null
}
This is simply saying IsOneOf[T, X, Y] has two instances, one where T = X and one where T = Y.
We can take it for a spin:
IsOneOf[Int, Int, Boolean] // Compiles
IsOneOf[Boolean, Int, Boolean] // Compiles
IsOneOf[Float, Int, Boolean] // Does not compile, Float is neither Int nor Boolean
You might use it like this:
def foo[T: IsOneOf[*, Int, String]](id: T): String = id match {
case i: Int => "We got an Int"
case i: String => "We got a String"
}
foo(5)
foo("hello")
(Note the use of * in type-position in how we used IsOneOf, which requires the excellent 'kind-projector' compiler plugin. Before scala 2.13, ? was used instead).
Again, note the pattern match is not exhaustive. If you leave out String, it won't warn you.
But, it works! The Int and String we chose are completely arbitrary. They could be any types we wanted at all, for hardly any more work.
### Abstracting the Arity
The above was easy. This next bit could be easy, but deeply unsatisfying.
You'll notice that IsOneOf is hard-coded to two possibilities---T must be one of X or Y. It has two implicit instances.
We could, of course, just define one for three possibiltiies:
trait IsOneOf3[T, X, Y, Z]
or five:
trait IsOneOf5[T, A1, A2, A3, A4, A5]
but...each has to have a separate definition. A separate name, separate implicits (IsOneOfN needs N implicits).
We also have a new special consideration when dealing with higher arities: If T is X or Y, then T is certainly X or Y or Z.
Implementing these relationships thoroughly in N different traits will get old, fast. We're back to tuples by a different name. You would stop at twenty-two, too, if you had to write all this stuff out.
No, we need a different, more robust approach.
#### Abstracting the Tuple
We will be using HLists.
If you've not come across them before, they are just lists of types, implemented just like List is for values.
Each element of type HList has a head, which is a defined type, and a tail, which is another HList. All except the emptyHListthat is, usually denotedHNil.
We can define our own very simply:
sealed trait HList
trait ::[+H, +T <: HList] extends HList
sealed trait HNil extends HList
You see ::, the cons operation, completely analagous to normal Lists ::. Our definition is a bit simpler than you might find elsewhere, since we're not interested in the value level. The type we're dealing with T is completely free, and the HList is just going to act as information for the compiler. We never will construct a value of type HList---true type-level programming.
Here's how you might define an HList type:
type Possibilities = Int :: String :: HNil
Ideally, we want to be able to define this:
trait IsOneOf[T, L <: HList]
object IsOneOf {
def apply[T, L <: HList](implicit ev: T IsOneOf L): T IsOneOf L = ev
// implicits here
}
// where:
IsOneOf[Int, Int :: String :: Boolean :: HNil] // compiles
IsOneOf[Boolean, Boolean :: HNil] // compiles
IsOneOf[Unit, Int :: Char :: HNil] // does not compile
That syntax is everything we want---dynamic lengths, dynamic types. We just need to implement whatever implicits drive IsOneOf.
Let's start with an IsMemberOf typeclass. The reason why this is separate and not just operating on our original IsOneOf will become clear. The purpose of this IsMemberOf[T, L <: HList] typeclass is to be available if and only if T is an element of L.
Here's the definition:
trait IsMemberOf[T, L <: HList]
How would you write this algorithm at value-level? How would you decide whether 5 was an element of x: List[Int]?
A particularly functional and terse way of writing this value-level algorithm would be:
def isMemberOf(value: Int, list: List[Int]): Boolean = list match {
case Nil => false
case _ :: tail => isMemberOf(value, tail)
}
Three cases to consider:
• If the list is empty, 5 cannot be a member of it.
• If the head of the list is equal to 5, then of course the list contains it.
• Otherwise, just test the tail and recurse.
These are completely analagous to the type-level algorithm:
(a) The false case means there is no implicit avaialable (so a bad example woulnd't compile).
(b) true means there does exist an implicit satisfying that case.
(c) The recursive case is just a recursive implicit call.
Well, let's give the simple case a go:
implicit def headCase[T, Tail <: HList]: IsMemberOf[T, T :: Tail] = null
This reads as "In the case where our HList takes the form T :: SomethingElse, T is a member of it". Read the return type to figure out which run-time case match it corresponds too.
Simple! The recursive case almost writes itself too:
implicit def recursveCase[T, Head, Tail <: HList](
implicit recurse: T IsMemberOf Tail
): IsMemberOf[T, Head :: Tail] = null
This reads as "IfTis a member ofTail, thenTis a member ofHead :: Tail, no matter whatHead is".
These are all we need. You'll notice that neither implicit will satisfy T IsMemberOf HNil (Both require our HList to have a tail)---our third and final requirement.
IsMemberOf[String, Int :: String :: HNil] // compiles
IsMemberOf[Boolean, Int :: String :: HNil] // does not compile
We're so close to being done! We're unfortunately missing one component I mentioned above: subunions.
def foo[T: IsMemberOf[*, Int :: String :: HNil]](value: T): String = bar(value)
def bar[T: IsMemberOf[*, Int :: String :: Boolean :: HNil]]: String = ???
At the moment, this will not compile. When we call bar(value), we have an implicit IsMemberOf[T, Int :: String :: HNil].
But to call bar, we need an implicit IsMemberOf[T, Int :: String :: Boolean :: HNil]. These types aren't equal---we need some more implicit massaging to make them fit into place.
### Subunions
We need a typeclass SubListOf. This typeclass will take two HLists, and be available implicitly if the first is a subHList of the second:
trait IsSubListOf[X <: HList, Y <: HList]
What does SubList here mean? It means simply that every element of X is in Y, somewhere.
Let's again defer to a value-level argument to make it clear how we will proceed. We need a function that takes two lists and says whether one is a "sublist" of the other:
def isSubListOf(x: List[Int], y: List[Int]): Boolean = (x, y) match {
// Remember we wrote isMemberOf above! We can delegate to it here
// The recursive case: The head must be a member, and the tail must be a sublist
case (h :: tail, list) if isMemberOf(h, list) && isSubListOf(tail, list) => true
// The termination case: An empty list is always a sublist
case (Nil, _) => true
// Every other case returns false:
case _ => false
}
This doesn't really look much more complex than IsMemberOf. Let's do the empty-list case:
implicit def hnilCase[Y <: HList]: IsSubListOf[HNil, Y] = null
This simply says HNil is a sublist of Y, for any Y <: HList
And the second case:
implicit def recursiveCase[X <: HList, H, Y <: HList](
implicit member: H IsMemberOf Y,
sublist: X IsSubListOf Y
): IsSubListOf[H :: X, Y] = null
Again, it reads as an ordinary logical statement:
• If your argument is of the form H :: X,
• And H is a member of Y,
• And X is a sublist of Y,
• Then H::X (your original argument) is a sublist of Y.
...and that's all we need. The third case, the false case, is again translated into compile-time recursive implicits by simply not being there.
Here it is working:
type A = Int :: String :: HNil
type B = String :: Boolean :: Int :: HNil
type C = Char :: String :: HNil
IsSubListOf[A, B] // compiles - even though the types are in different orders!
IsSubListOf[A, C] // does not compile, C is missing Int
IsSubListOf[B, A] // does not compile, Boolean is not present in A
### Putting It All Together
We're finally ready to wrap everything up inside IsOneOf:
trait IsOneOf[T, L <: HList]
object IsOneOf {
def apply[T, L <: HList](implicit ev: T IsOneOf L): T IsOneOf L = ev
implicit def isMemberOfInstance[A, L <: HList](implicit m: IsMemberOf[A, L]): IsOneOf[A, L] = null
implicit def isSubListInstance[X, Y <: HList, Z <: HList](
implicit o: IsOneOf[X, Y],
s: IsSubListOf[Y, Z]
): IsOneOf[X, Z] = null
}
A quick step through:
• isMemberOfInstance just delegates to the IsMemberOf typeclass we defined above. It's the simple case where we're just considering simple elements.
• isSubListInstance provides the transitivity: X is one of Y and Y is a sublist of Z, implies X is one of Z.
And, now, everything works beautifully:
def testing1[T: IsOneOf[*, Int :: String :: Boolean :: HNil]](t: T): String = t match {
case i: Int => i.toString
case b: Boolean => b.toString
case s: String => s
}
// These compile:
testing1(5)
testing1(true)
testing1("testing!")
// This fails:
testing1('c')
It even works with multiple unions and their intersections! Observe:
def testing2[
T: IsOneOf[*, Int :: String :: HNil]
: IsOneOf[*, Boolean :: Int :: HNil]
](t: T): String = ???
testing2(17) // compiles
testing2("string") // does not compile: String is not Boolean or Int
testing2(true) // does not compile: Boolean is not Int or String
And, finally, proof that it works for subunions:
// Compiles:
def bar[T: IsOneOf[*, Int :: String :: Boolean :: HNil]](t: T): String = ???
def foo[T: IsOneOf[*, Int :: String :: HNil]](value: T): String = bar(value)
You see bar requires a wider union than foo, and so foo is able to call it successfully. This will
work for unions of any size, as long as foos are strictly contained within bars.
So what have we achieved?
With these capabilities, union intersections and union widening, we have an unboxed, arbitrary arity union type.
You can program your unions in a completely abstract way, coding to a L <: HList rather than a fixed union. The subunion and intersection relations displayed above ensure that you can rely on the compiler machinery to automatically validate all union logic in your program, based entirely on how you define them at the very end of the world---for free.
##### James Phillips
author
James is a functional and scala enthusiast with a keen interest in type-level programming. He runs a consultancy in London and Brighton, and occasionally cycles to Paris.
##### Leandro Lassmar
illustrator
Leandro Lassmar is an illustrator living in Minas Gerais, Brazil.
He worked in animation studios, currently works for magazines, books and advertising.
It won the SND (society of news design) and ÑH - Lo Mejor del Diseño Periodístico awards. | 2020-09-24 23:13:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.31305694580078125, "perplexity": 5813.473286387209}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400221382.33/warc/CC-MAIN-20200924230319-20200925020319-00576.warc.gz"} |
https://byjus.com/maths/real-numbers-for-class-10/ | # Real Numbers For Class 10
A real number includes all the rational numbers (3, 5/9 etc.) and irrational numbers ($\sqrt{3}, \sqrt{7}$ etc.)
Euclid’s Division Lemma-
For a given positive integer a and b, there exist unique integers q and r satisfying $a = bq+ r, 0 \leq r < b$.
Fundamental Theorem of Arithmetic-
Every composite number can be expressed (factorised) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.
Example- $36 = 2^{2} \times 3^{2}$<
#### Practise This Question
Represent the following equations graphically:
x+y=10
10x+y+10y+x=110 | 2018-12-11 07:38:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5211895704269409, "perplexity": 960.1847777932862}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823588.0/warc/CC-MAIN-20181211061718-20181211083218-00420.warc.gz"} |
https://brokenco.de/2017/02/10/being-a-console-luddite.html | Back atom.xml blog In Defense of Being a Console Luddite 10 Feb 2017 opinion
Most people would consider me to be a nerd. I work in the tech industry, my laptop looks quite non-standard (a stickered Thinkpad), and I tend to travel with suitable amount of electronic kit. Within what I would call “the nerd community,” I sometimes get looks as if I’m especially nerdy. I use a tiling window manager on my Linux desktop, I have strong opinions on free and open source software, and above all else, I use a myriad of “super nerdy” console-only applications like mutt and irssi.
Presently I find myself delayed in a foreign airport with a “hostile wifi situation.” That is to say that while technically there is wifi, one must surrender their information to a captive portal which will no doubt result in a plethora of new spam, all for a meager allotment of usage time. Instead I am passing the time, with my Android phone acting as my wireless hotspot, over my “unlimited” 2G data.
You really haven’t experienced the bloat of the internet in 2017 until you have attempted to be productive over a 2G link, with bonus latency between the European and American continents. Even websites I would have assumed were fairly simple, looking at you reddit.com, download excessive amounts of data between loading pages and client/server background-chatter.
As a console luddite however, things aren’t so bad! The benefit of console-based applications is that they tend to be much lighter, not only in CPU and memory consumption, but also in network utilization. The difference between irssi and IRC Cloud, for example, is staggering. With mutt, my mail client of course, I am only downloading the emails themselves rather than the entire interface around the emails like with a web mail client. Even for content which only lives at the other end of an HTTP connection, using the console-based browser w3m results in much lighter page loads and zero on-going data consumption after the page has loaded.
I don’t advocate going to 100% console-based applications however. Chrome, with the Vimium extension, is one of my most heavily used applications. But there are certainly some benefits to maintaining familiarity with console-based applications today.
Recommendations
Below are some recommendations I can make for resource-thrifty console-based tools.
• w3m - For most basic browsing while on low-bandwidth connections. I also find the -dump option to be very useful when inside of tmux for dumping HTML-based test reports or other locally generated HTML files. For most websites, their mobile versions render quite nice in w3m.
• mutt - As my primary email client, mutt allows me to speedily navigate around email via its stellar key bindings, but perhaps most importantly, it allows me to use vim for authoring my emails.
• irssi - For IRC (and also Gitter) chat; very important for actively participating in most free and open source projects.
• newsbeuter - I am apparently one of the few remaining humans who uses RSS/Atom feeds for consuming content. As a console-based news reader, I find newsbeuter to be very user-friendly.
All of these applications have the added benefit of being primarily keyboard-driven, giving them a higher learning curve, but once the basics are mastered it’s quite easy to rapidly context-switch within and between them. A number of console-based tools are also easily incorporated into other scripts. w3m for example is referenced in a few task-specific scripts I keep floating around in ~/bin.
There are downsides to frequently using console-based applications. Other nerds will look down their nose at you whilst complaining about Slack, Firefox, or Chrome consuming heaps of heap. Strangers will come up to you and ask you silly questions like “how do you READ all that!?” And of course, the more comfortable you get with console-based tools, custom scripts, and all the other things you start to use because they make you work faster, the harder it will be for you to ever use a “normal desktop” again.
You may end up being a console luddite like me, but at least you’ll be efficient and productive regardless of the situation you find yourself in. | 2021-05-11 07:36:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25872546434402466, "perplexity": 3423.238140421091}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991904.6/warc/CC-MAIN-20210511060441-20210511090441-00085.warc.gz"} |
http://www.mscroggs.co.uk/puzzles/tags/calculus | mscroggs.co.uk
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# Puzzles
## Archive
Show me a random puzzle
▼ show ▼
## An integral
Source: Alex Bolton (inspired by Book Proofs blog)
What is
$$\int_0^{\frac\pi2}\frac1{1+\tan^a(x)}\,dx?$$
## Find them all
Find all continuous positive functions, $$f$$ on $$[0,1]$$ such that:
$$\int_0^1 f(x) dx=1\\ \mathrm{and }\int_0^1 xf(x) dx=\alpha\\ \mathrm{and }\int_0^1 x^2f(x) dx=\alpha^2$$
## Integrals
$$\int_0^1 1 dx = 1$$
Find $$a_1$$ such that:
$$\int_0^{a_1} x dx = 1$$
Find $$a_2$$ such that:
$$\int_0^{a_2} x^2 dx = 1$$
Find $$a_n$$ such that (for $$n>0$$):
$$\int_0^{a_n} x^n dx = 1$$
## Double derivative
What is
$$\frac{d}{dy}\left(\frac{dy}{dx}\right)$$
when:
(i) $$y=x$$
(ii) $$y=x^2$$
(iii) $$y=x^3$$
(iv) $$y=x^n$$
(v) $$y=e^x$$
(vi) $$y=\sin(x)$$?
## Differentiate this
$$f(x)=e^{x^{ \frac{\ln{\left(\ln{x}\right)}}{ \ln{x}}} }$$
Find $$f'(x)$$.
## x to the power of x again
Let $$y=x^{x^{x^{x^{...}}}}$$ [$$x$$ to the power of ($$x$$ to the power of ($$x$$ to the power of ($$x$$ to the power of ...))) with an infinite number of $$x$$s]. What is $$\frac{dy}{dx}$$? | 2019-03-24 16:44:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9553565979003906, "perplexity": 2804.496970566386}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203462.50/warc/CC-MAIN-20190324145706-20190324171706-00018.warc.gz"} |
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# What is the area of the figure above?
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What is the area of the figure above? [#permalink]
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21 Aug 2019, 07:34
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Difficulty:
55% (hard)
Question Stats:
64% (01:56) correct 36% (01:59) wrong based on 72 sessions
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What is the area of the figure above?
A. $$40\sqrt{2}$$
B. 64
C. 68
D. 81
E. 92
Attachment:
image2264.jpg [ 3.38 KiB | Viewed 1161 times ]
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Re: What is the area of the figure above? [#permalink]
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21 Aug 2019, 07:56
1
What is the area:
A. 402√402
B. 64
C. 68
D. 81
E. 92
draw line parallel from both ends and to the opposite side,
then we have a square at the top and two rectangles of dimentions 2*8
so the triangle we are left with 8,8 ,x this will form a 45-45-90 trianlge and rule x,x,xroot2
or we can also say that the area of the triangle =32
then the area of the two rectangles=2*8*2=32
and the square at the top=2*2=4
so total area=32+32+4=68
Option C is correct.
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triangle.jpg [ 9.75 KiB | Viewed 1026 times ]
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Re: What is the area of the figure above? [#permalink]
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21 Aug 2019, 07:59
1
What is the area of the figure above?
A. 40√2
B. 64
C. 68
D. 81
E. 92
If we divide the figure as
A triangle of area = 8*8/2 = 32
A rectangle with area = 10*2 = 20
A rectangle with area = 8*2 = 16
Adding 3 areas = 32 + 20 + 16 = 68
IMO C
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Re: What is the area of the figure above? [#permalink]
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Updated on: 21 Aug 2019, 08:11
IMO-C
Refer attached Image.
Draw a line joining the vertices B & E
This divides the figure into an isosceles triangle & a trapezium.
From fig, Sides of parallel side of trapezium are 10 sqrt2 & (10 sqrt2 -2 sqrt2 ) and distance between them sqrt2
Area= 1/2 x (10 x 10) + 1/2 x (10 sqrt2 + 8sqrt2 ) x sqrt2 = 68
Attachments
WhatsApp Image 2019-08-21 at 9.29.01 PM.jpeg [ 76.63 KiB | Viewed 1000 times ]
Originally posted by MayankSingh on 21 Aug 2019, 08:04.
Last edited by MayankSingh on 21 Aug 2019, 08:11, edited 1 time in total.
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Re: What is the area of the figure above? [#permalink]
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21 Aug 2019, 08:09
area of the given figure = area of a triangle ABE + area of a trapezium BCDE
= $$\frac{1}{2}$$*10*10 + $$\frac{1}{2}$$(10√2 +8√2 )√2
= 50 + 18
= 68
Attachments
ps.png [ 7.88 KiB | Viewed 1009 times ]
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Re: What is the area of the figure above? [#permalink]
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21 Aug 2019, 08:32
Image
What is the area of the figure above?
A. 402‾√402
B. 64
C. 68
D. 81
E. 92
divide the figure we get 2 rectangles; 10*2 + 8*2 ; 36
and an isosceles ∆ 8:8:8√2 ; area ; 1/2 * 8*8 ; 32
total area ; 36+ 32 ; 68
IMO C
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Re: What is the area of the figure above? [#permalink]
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21 Aug 2019, 08:32
its area of square - area of right angled isoceles triangle of non hypotenuse side of 8
=10 ^2- 1/2 *8*8
=100- 32
=68
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Re: What is the area of the figure above? [#permalink]
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21 Aug 2019, 09:01
This is how I did. Hope it helps.
Posted from my mobile device
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Re: What is the area of the figure above? [#permalink]
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21 Aug 2019, 09:08
2
We can continue the small sides (the length of the side =2) to get the square( the length of the side of square =10.
The area of the square =10*10=100
—> See the picture below.
Triangle ABC is right-angled isosceles triangle.
—> the area of ABC =(8*8)/2=32
—> the area of the figure = 100–32=68
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FDAB4625-FA4B-4D52-B481-160642219A45.jpeg [ 869.06 KiB | Viewed 915 times ]
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Re: What is the area of the figure above? [#permalink]
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21 Aug 2019, 09:47
What is the area of the figure above?
A. 402√402
B. 64
C. 68
D. 81
E. 92
Refer attached snapshot.
Alternative method is slightly longer.
Attachments
File comment: What is the area of the figure above
What is the area of the figure above.jpg [ 1.08 MiB | Viewed 874 times ]
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Re: What is the area of the figure above? [#permalink]
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21 Aug 2019, 10:34
See the image attached below,
We can complete a square since we have 3 angles of 90 degrees, and two sides that are equal. Then we can see the area is a square with length of 10, minus an isosceles right triangle with a side of 8. The triangle is isosceles because both sides are 8. The added outer angle must be 90 because a 4 sided polynomial must have all interior angles adding up to 360.
Therefore the area is 10*10 - 8*8/2 = 100 - 32 = 68.
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completed image.png [ 35.47 KiB | Viewed 834 times ]
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Re: What is the area of the figure above? [#permalink]
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21 Aug 2019, 10:39
Area of the figure = Area of the triangle formed by extending the sides with value 10 and 10 to form one big isosceles triangle - area of 2 small isosceles triangle which have one side as 2.
side of the smaller triangle: one side will be 2 and the hypotenuse = 2*sqrt2
Area of the smaller triangle = (1/2)*2*2 = 2
Side of bigger triangle: Base and height = 10+2=12, 12 and hypotenuse = 12*sqrt2
Area of the bigger triangle = (1/2)*12*12 = 72
Area of the figure = 72 - 2*2 = 68
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Re: What is the area of the figure above? [#permalink]
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21 Aug 2019, 10:45
From the image below, triangle ABE is a right angled isosceles triangle. This means that EB=(10^2 + 10^2)^0.5 = 200^0.5 = 10(2)^0.5
Area of solid ABCDE = Area of Triangle ABE + Area of trapezoid EBCD.
Area of triangle ABE = (10*10)/2 = 50.
Angle AED (90) = angle AEB + angle DEF ....(1)
Since we know that triangle ABE is a right angled isosceles triangle, angle AEB = angle ABE = 45. Hence angle DEF=45, implying triangle EFD is a right angled isosceles triangle. This also means that EF=FD=(2)^0.5
Likewise GB=CG=(2)^0.5
Hence DC = EB-EF-GB = 8(2)^0.5
So, Area of EBCD=
{(10(2)^0.5 + 8(2)^0.5)*(2)^0.5}/2 ={(18)(2)/2}
=18
So Area of ABCDE = 50 + 18 = 68
Posted from my mobile device
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Re: What is the area of the figure above? [#permalink]
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21 Aug 2019, 11:45
Quote:
What is the area of the figure above?
A. 40√2
B. 64
C. 68
D. 81
E. 92
shaded: area of square - area of triangle = 10*10 - (8*8/2) = 68
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Re: What is the area of the figure above? [#permalink]
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21 Aug 2019, 12:00
Just think of it as a square of 10*10 having area as 100.
A right triangle whose 2 sides are 8 and 8 is cut. Area of that triangle is 32.
100-32 is 68.
IMO C
Note : 1min 37sec. But could have solved way quicker if had i realized the shape
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Re: What is the area of the figure above? [#permalink]
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21 Aug 2019, 12:50
As we see in the figure, we could have a squire with sides 10.but the shape is not perfect.
So the area of the shape is less than 100=10*10.
In deed we got a triangle out of the shape whose area is 8*8/2=32
So,100-32=68
Option C
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Re: What is the area of the figure above? [#permalink] 21 Aug 2019, 12:50
Display posts from previous: Sort by | 2020-02-22 13:31:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.807400643825531, "perplexity": 5857.831691868606}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145676.44/warc/CC-MAIN-20200222115524-20200222145524-00129.warc.gz"} |
https://tysonbarrett.com/furniture/reference/table1.html | Produces a descriptive table, stratified by an optional categorical variable, providing means/frequencies and standard deviations/percentages. It is well-formatted for easy transition to academic article or report. Can be used within the piping framework [see library(magrittr)].
table1(
.data,
...,
splitby = NULL,
FUN = NULL,
FUN2 = NULL,
total = FALSE,
second = NULL,
row_wise = FALSE,
test = FALSE,
param = TRUE,
type = "pvalues",
output = "text",
rounding_perc = 1,
digits = 1,
var_names = NULL,
format_number = FALSE,
NAkeep = NULL,
na.rm = TRUE,
booktabs = TRUE,
caption = NULL,
align = NULL,
float = "ht",
export = NULL,
label = NULL
)
Arguments
.data the data.frame that is to be summarized variables in the data set that are to be summarized; unquoted names separated by commas (e.g. age, gender, race) or indices. If indices, it needs to be a single vector (e.g. c(1:5, 8, 9:20) instead of 1:5, 8, 9:20). As it is currently, it CANNOT handle both indices and unquoted names simultaneously. Finally, any empty rows (where the row is NA for each variable selected) will be removed for an accurate n count. the categorical variable to stratify (in formula form splitby = ~gender) or quoted splitby = "gender"; instead, dplyr::group_by(...) can be used within a pipe (this is the default when the data object is a grouped data frame from dplyr::group_by(...)). the function to be applied to summarize the numeric data; default is to report the means and standard deviations a secondary function to be applied to summarize the numeric data; default is to report the medians and 25% and 75% quartiles whether a total (not stratified with the splitby or group_by()) should also be reported in the table a vector or list of quoted continuous variables for which the FUN2 should be applied how to calculate percentages for factor variables when splitby != NULL: if FALSE calculates percentages by variable within groups; if TRUE calculates percentages across groups for one level of the factor variable. logical; if set to TRUE then the appropriate bivariate tests of significance are performed if splitby has more than 1 level. A message is printed when the variances of the continuous variables being tested do not meet the assumption of Homogeneity of Variance (using Breusch-Pagan Test of Heteroskedasticity) and, therefore, the argument var.equal = FALSE is used in the test. logical; if set to TRUE then the appropriate parametric bivariate tests of significance are performed (if test = TRUE). For continuous variables, it is a t-test or ANOVA (depending on the number of levels of the group). If set to FALSE, the Kruskal-Wallis Rank Sum Test is performed for the continuous variables. Either way, the chi-square test of independence is performed for categorical variables. a character vector that renames the header labels (e.g., the blank above the variables, the p-value label, and test value label). what is displayed in the table; a string or a vector of strings. Two main sections can be inputted: 1. if test = TRUE, can write "pvalues", "full", or "stars" and 2. can state "simple" and/or "condense". These are discussed in more depth in the details section below. how the table is output; can be "text" or "text2" for regular console output or any of kable()'s options from knitr (e.g., "latex", "markdown", "pandoc"). A new option, 'latex2', although more limited, allows the variable name to show and has an overall better appearance. the number of digits after the decimal for percentages; default is 1 the number of significant digits for the numerical variables (if using default functions); default is 1. custom variable names to be printed in the table. Variable names can be applied directly in the list of variables. default is FALSE; if TRUE, then the numbers are formatted with commas (e.g., 20,000 instead of 20000) when set to TRUE it also shows how many missing values are in the data for each categorical variable being summarized (deprecated; use na.rm) when set to FALSE it also shows how many missing values are in the data for each categorical variable being summarized when output != "text"; option is passed to knitr::kable when output != "text"; option is passed to knitr::kable when output != "text"; option is passed to knitr::kable the float applied to the table in Latex when output is latex2, default is "ht". character; when given, it exports the table to a CSV file to folder named "table1" in the working directory with the name of the given string (e.g., "myfile" will save to "myfile.csv") for output == "latex2", this provides a table reference label for latex
Value
A table with the number of observations, means/frequencies and standard deviations/percentages is returned. The object is a table1 class object with a print method. Can be printed in LaTex form.
Details
In defining type, 1. options are "pvalues" that display the p-values of the tests, "full" which also shows the test statistics, or "stars" which only displays stars to highlight significance with *** < .001 ** .01 * .05; and 2. "simple" then only percentages are shown for categorical variable and "condense" then continuous variables' means and SD's will be on the same line as the variable name and dichotomous variables only show counts and percentages for the reference category.
Examples
## Fictitious Data ##
library(furniture)
library(dplyr)#>
#> Attaching package: ‘dplyr’#> The following objects are masked from ‘package:stats’:
#>
#> filter, lag#> The following objects are masked from ‘package:base’:
#>
#> intersect, setdiff, setequal, union
x <- runif(1000)
y <- rnorm(1000)
z <- factor(sample(c(0,1), 1000, replace=TRUE))
a <- factor(sample(c(1,2), 1000, replace=TRUE))
df <- data.frame(x, y, z, a)
## Simple
table1(df, x, y, z, a)#> #>
#> ────────────────────────
#> Mean/Count (SD/%)
#> n = 1000
#> x
#> 0.5 (0.3)
#> y
#> 0.1 (1.0)
#> z
#> 0 500 (50%)
#> 1 500 (50%)
#> a
#> 1 478 (47.8%)
#> 2 522 (52.2%)
#> ────────────────────────
## Stratified
## all three below are the same
table1(df, x, y, z,
splitby = ~ a)#> #>
#> ──────────────────────────────
#> a
#> 1 2
#> n = 478 n = 522
#> x
#> 0.5 (0.3) 0.5 (0.3)
#> y
#> 0.1 (1.0) 0.1 (1.0)
#> z
#> 0 247 (51.7%) 253 (48.5%)
#> 1 231 (48.3%) 269 (51.5%)
#> ──────────────────────────────table1(df, x, y, z,
splitby = "a")#> #>
#> ──────────────────────────────
#> a
#> 1 2
#> n = 478 n = 522
#> x
#> 0.5 (0.3) 0.5 (0.3)
#> y
#> 0.1 (1.0) 0.1 (1.0)
#> z
#> 0 247 (51.7%) 253 (48.5%)
#> 1 231 (48.3%) 269 (51.5%)
#> ──────────────────────────────
## With Piping
df %>%
table1(x, y, z,
splitby = ~a)#> #>
#> ──────────────────────────────
#> a
#> 1 2
#> n = 478 n = 522
#> x
#> 0.5 (0.3) 0.5 (0.3)
#> y
#> 0.1 (1.0) 0.1 (1.0)
#> z
#> 0 247 (51.7%) 253 (48.5%)
#> 1 231 (48.3%) 269 (51.5%)
#> ──────────────────────────────
df %>%
group_by(a) %>%
table1(x, y, z)#> Using dplyr::group_by() groups: a#> #>
#> ──────────────────────────────
#> a
#> 1 2
#> n = 478 n = 522
#> x
#> 0.5 (0.3) 0.5 (0.3)
#> y
#> 0.1 (1.0) 0.1 (1.0)
#> z
#> 0 247 (51.7%) 253 (48.5%)
#> 1 231 (48.3%) 269 (51.5%)
#> ──────────────────────────────
## Adjust variables within function and assign name
table1(df,
x2 = ifelse(x > 0, 1, 0), z = z)#> #>
#> ────────────────────────
#> Mean/Count (SD/%)
#> n = 1000
#> x2
#> 1.0 (0.0)
#> z
#> 0 500 (50%)
#> 1 500 (50%)
#> ──────────────────────── | 2020-08-05 18:34:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2995564639568329, "perplexity": 7049.325890356222}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735964.82/warc/CC-MAIN-20200805183003-20200805213003-00234.warc.gz"} |
https://physics.stackexchange.com/questions/593329/does-amplitude-modulation-change-a-photons-frequency-or-the-number-of-photons | # Does amplitude modulation change a photon's frequency or the number of photons?
In the following, we assume that the polarization is aligned such that the scalar treatment of the electric field is justified. Furthermore, we limit the discussion to a fixed coordinate $$x=0$$ to drop the wave vector dependency.
### Classical description
Let's consider a classical electric field
$$E(t)=A(t)\cos\omega_c t \tag{1}$$
with angular carrier frequency $$\omega_c$$ where we modulate the field amplitude with
$$A(t)=A_0\cos\omega_m t \tag{2}$$
where $$\omega_m$$ is the angular modulation frequency. For illustration, we assume $$\omega_m\ll\omega_c$$, for example, $$\omega_c$$ could be in the optical whereas $$\omega_m$$ could be in the low-frequency domain.
In this picture, we can think of the electric wave propagating with frequency $$\omega_c$$ while it's amplitude slowly oscillates with $$\omega_m$$.
Nevertheless, we can also combine eq. (1) with eq. (2) and write
$$E(t) = A_0\cos\omega_m t\cos\omega_c t = \frac{1}{2}A_0\bigl(\cos\omega_+t+\cos\omega_-t\bigr) \tag{3}$$
where we define $$\omega_\pm=\omega_c\pm\omega_m$$.
In this picture, we actually have two waves, one oscillating rapidly with $$\omega_+$$ and one oscillating slowly with $$\omega_-$$.
So far, so good. Although, eq. (1) and eq. (3) suggest a different point of view both are equivalent and should give the same (classical) predictions.
### Quantum description
Now, we turn to the quantum description where we define the electrical field operator to be
$$\hat{E} = i\sum_i\omega_i\left\{\hat{a}_ie^{-i\omega_it}-\text{c.c}\right\} \tag{4}$$
with $$\text{c.c.}$$ referring to the complex conjugate term, and where $$\hat{a}_i$$ is the annihilation operator of mode $$i$$ that satisfies the canonical commutation relation
$$\left[\hat{a}_i,\hat{a}_j^\dagger\right]=\delta_{ij} \tag{5}.$$
We assume a two-mode coherent state $$\vert\alpha_1,\alpha_2\rangle$$ with $$\alpha_i\in\mathbb{C}$$ and calculate the expectation value of the electric field operator for that state
\begin{aligned} \langle\alpha_1,\alpha_2\vert\hat{E}\vert\alpha_1,\alpha_2\rangle &= i\omega_1\left(\alpha_1e^{-i\omega_1t}-\text{c.c.}\right)+ i\omega_2\left(\alpha_2e^{-i\omega_2t}-\text{c.c.}\right)\\ &= -2\omega_1\operatorname{Im}\left\{\alpha_1e^{-i\omega_1t}\right\}+ -2\omega_2\operatorname{Im}\left\{\alpha_2e^{-i\omega_2t}\right\} \end{aligned}\tag{6}.
With the choice $$\omega_1=\omega_+$$ and $$\omega_2=\omega_-$$ as well as $$\alpha_i=-iA_0/(4\omega_i)$$ (upto some unit-preserving factors) we can recover our classical result given in eq. (4).
On the other hand, we should also be able to reproduce eq. (1) by asserting a single-mode coherent state $$\vert\alpha(t)\rangle$$ with $$\alpha(t)\propto A_0\cos(\omega_mt)/\omega$$.
However, this time, we could think of an experiment that distinguishes between these two states!
Recap the photoelectric effect describes electron emission by a photon hitting a metal. The photon energy $$\hbar\omega$$ needs to be greater than the work potential $$W$$ which binds the electron to the metal. Suprisingly, the photoelectric effect is (neglecting multi-photon absorption) independent of the intensity.
Let's assume that we are in possession of a metamaterial where the work potential is tailored to $$\hbar\omega_c$$. In this case, we could distinguish between the single- and two-mode coherent state because the amplitude $$\vert\alpha\vert$$ of the coherent state $$\vert\alpha\rangle$$ fixes the mean of the (Poissonian) photon statistics but the energy, which determines if electron emission occurs, is given by the mode frequency.
If we remind ourselves on how we derived eq. (3) from modes in a confined cavity this also makes sense.
Is it correct to conclude that amplitude modulation of an optical laser signal, shifts the energy of the participating photons?
This is an answer to your classical description of an amplitude modulation of EM wave.
• Most of EM radiation are without modulation. Examples are the radiation from thermic sources (light bulb, sun, radiant heater).
• EM radiation is caused by the relaxation of electrons (and other particles and quasi-particles ) and the emission of photons this time. Another source of EM radiation does not exist. Therefor every EM radiation consists of photons. EM radiation is a stream of photons.
• Every photon has an electric and a magnetic field component and these field components are oscillation during the movement of the photon through space.
Only with two conditions you will be able to measure wave properties on a stream of photons:
• the photons are polarized (their electric and magnetic field components are polarized)
• the radiation is modulated (the best example is the antenna of a communication device)
Your equation (1) describes a modulated EM wave. To be precise, it is a radio wave, with its periodically changing number of emitted photons. Remember, that a wave generator pushes the electrons in the antenna rod fourth and back and the accelerated electrons emit the photons. The electrons get accelerated at any moment all in the same direction and this makes the polarization of the emitted photons.
Equation (2) describes the amplitude modulation of a communication. Clearly the number of photons changes.
One last thing. The photons emitted by the accelerated electrons on the rod have a different wavelength than the carrier frequency. In dependency from the power of the wave generator, the length of the rod, the material of the rod, ... the photons are in the range from infrared to X-rays. It is not for nothing that you should never stand in front of a military radar. However, equation (3) describes the number of photons (in the equivalent of the intensity, named unfortunately amplitude) under the influence of a carrier frequency and a modulation of this frequency.
• I think it a bit problematic to talk about "photons" in a classical picture. You are correct that in the semiclassical picture, the number of photons corresponds to the amplitude. But then again, according to the quantization procedure of the electric field, clearly the photon frequency changes while the number of photons splits evenly across the two frequency modes? Nov 15 '20 at 20:39 | 2022-01-18 16:34:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 32, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9087184071540833, "perplexity": 264.043808574758}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300934.87/warc/CC-MAIN-20220118152809-20220118182809-00386.warc.gz"} |
https://quantum-computing.ibm.com/docs/circ-comp/q-gates | # Quantum instruction glossary¶
This page is a reference that defines the various quantum instructions you can use to manipulate qubits in a quantum circuit. Quantum instructions include quantum gates, such as the Hadamard gate, as well as operations that are not quantum gates, such as the measurement instruction.
To learn more about using quantum gates to create quantum algorithms, see the single- and multi-qubit gates chapter of the IBM textbook, Learn Quantum Computation using Qiskit.
Click on a quantum instruction below to view its definition.
## H gate¶
The H or Hadamard gate rotates the states and to and , respectively. It is useful for making superpositions. As a Clifford gate, it is useful for moving information between the x and z bases.
Composer reference
OpenQASM reference
Bloch sphere rotation
h q[0];
## CX gate¶
The controlled-X gate is also known as the controlled-NOT. It acts on a pair of qubits, with one acting as ‘control’ and the other as ‘target’. It performs an X on the target whenever the control is in state . If the control qubit is in a superposition, this gate creates entanglement.
Composer reference
OpenQASM reference
cx q[0], q[1];
## Id gate¶
The identity gate is actually the absence of a gate. It ensures that nothing is applied to a qubit for one unit of gate time.
Composer reference
OpenQASM reference
id q[0];
## Rx gate¶
The Rx gate requires a single parameter: an angle expressed in radians. On the Bloch sphere, this gate corresponds to rotating the qubit state around the x axis by the given angle.
Composer reference
OpenQASM reference
Bloch sphere rotation
rx(pi/2) q[0];
## Ry gate¶
The Ry gate requires a single parameter: an angle expressed in radians. On the Bloch sphere, this gate corresponds to rotating the qubit state around the y axis by the given angle.
Composer reference
OpenQASM reference
Bloch sphere rotation
ry(pi/2) q[0];
## Rz gate¶
The Rz gate requires a single parameter: an angle expressed in radians. On the Bloch sphere, this gate corresponds to rotating the qubit state around the z axis by the given angle.
Composer reference
OpenQASM reference
Bloch sphere rotation
rz(pi/2) q[0];
## X gate¶
The Pauli X gate has the property of flipping the state to , and vice versa. It is equivalent to Rx for the angle .
Composer reference
OpenQASM reference
Bloch sphere rotation
x q[0];
## Y gate¶
The Pauli Y gate is equivalent to Ry for the angle . It is also equivalent to the combined effect of X and Z.
Composer reference
OpenQASM reference
Bloch sphere rotation
y q[0];
## Z gate¶
The Pauli Z gate has the property of flipping the to , and vice versa. It is equivalent to Rz for the angle .
Composer reference
OpenQASM reference
Bloch sphere rotation
z q[0];
## S gate¶
The S gate is equivalent to Rz for the angle . As a Clifford gate, it is useful for moving information between the x and y bases.
Composer reference
OpenQASM reference
Bloch sphere rotation
s q[0];
## Sdg gate¶
The inverse of the S gate. Equivalent to Rz for the angle . As a Clifford gate, it is useful for moving information between the x and y bases.
Composer reference
OpenQASM reference
Bloch sphere rotation
sdg q[0];
## T gate¶
The T gate is equivalent to Rz for the angle . Fault-tolerant quantum computers will compile all quantum programs down to just the T gate and its inverse, as well as the Clifford gates.
Composer reference
OpenQASM reference
Bloch sphere rotation
t q[0];
## Tdg gate¶
The inverse of the T gate, which is equivalent to Rz for the angle . Fault-tolerant quantum computers will compile all quantum programs down to just the T gate and its inverse, as well as the Clifford gates.
Composer reference
OpenQASM reference
Bloch sphere rotation
tdg q[0];
## cH gate¶
The controlled-Hadamard gate, like the controlled-NOT, acts on a control and target qubit. It performs an H on the target whenever the control is in state .
Composer reference
OpenQASM reference
ch q[0], q[1];
## cZ gate¶
The controlled-Z gate, like the controlled-NOT, acts on a control and target qubit. It performs a Z on the target whenever the control is in state .
Composer reference
OpenQASM reference
cz q[0], q[1];
## cRz gate¶
The controlled-Rz gate, like the controlled-NOT, acts on a control and target qubit. It performs a Rz rotation on the target whenever the control is in state .
Composer reference
OpenQASM reference
crz(pi/2) q[0], q[1];
## ccX gate¶
The ccX gate, commonly known as the Toffoli, has two control qubits and one target. At applies an X to the target only when both controls are in state .
Composer reference
OpenQASM reference
ccx q[0], q[1], q[2];
## SWAP gate¶
The SWAP gate simply swaps the states of two qubits.
Composer reference
OpenQASM reference
swap q[0], q[1];
## Barrier operation¶
To make your quantum program more efficient, the compiler will try to combine gates. The barrier is an instruction to the compiler to prevent these combinations being made.
Composer reference
OpenQASM reference
barrier q;
## \left|0\right\rangle operation¶
The reset operation returns a qubit to state , irrespective of its state before the operation was applied. It is not a reversible operation.
Composer reference
OpenQASM reference
reset q[0];
## IF operation¶
The IF operation allows quantum gates to be conditionally applied, depending on the state of a classical register.
Composer reference
OpenQASM reference
if (c==0) x q[0];
## z measurement¶
Measurement in the standard basis, also known as the z basis or computational basis. Can be used to implement any kind of measurement when combined with gates. It is not a reversible operation.
Composer reference
OpenQASM reference
measure q[0];
## U3 gate¶
The three parameters allow the construction of any single-qubit gate. Has a duration of one unit of gate time.
Composer reference
OpenQASM reference
Bloch sphere rotation
u3(pi/2,pi/2,pi/2) q[0];
## U2 gate¶
The two parameters control two different rotations within the gate. Has a duration of one unit of gate time.
Composer reference
OpenQASM reference
Bloch sphere rotation
u2(pi/2,pi/2) q[0];
## U1 gate¶
Equivalent to Rz. This can be implemented by the control software, requiring no actual manipulation of the qubits, and so effectively has a duration of zero.
Composer reference
OpenQASM reference
Bloch sphere rotation
u1(pi/2) q[0]; | 2020-07-09 23:57:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.805421769618988, "perplexity": 4989.286820131592}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655902377.71/warc/CC-MAIN-20200709224746-20200710014746-00194.warc.gz"} |
https://math.stackexchange.com/questions/493692/how-can-i-explain-the-sketch-of-the-proof-of-classification-of-finite-simple-gro/495635?noredirect=1 | # How can I explain the sketch of the proof of classification of finite simple groups to someone who knows only basics of algebra?
This question was popped up in my mind when I read Finnish Wikipedia. How can I explain the sketch of the proof to layman? Is it worth to explain for example Ree groups in the text or just say something general like that kind of groups are important without going to details?
• What are "the basics of Algebra"? – Mark Bennet Sep 14 '13 at 20:59
• Lets say to someone who knows simple groups, Lagrange's theorem and Sylow theorem. – FinnWiki Sep 15 '13 at 6:40
• Do you just want to sketch the result or indeed the proof? – j.p. Sep 15 '13 at 9:58
• I would like to have a sketch of the proof. – FinnWiki Sep 15 '13 at 10:07
• You can take a look either at www.ams.org/notices/199502/solomon.pdf or at the first volume of "The Classification of the Finite Simple Groups" by Daniel Gorenstein, Richard Lyons and Ronald Solomon (which - if I remember correctly - contains a sketch of the proof). For a sketch of the proof you might not need to explain how to construct the families of groups of Lie type, as in the proof all work is "done in characteristic 2". – j.p. Sep 15 '13 at 12:49
## 1 Answer
An outline of the proof of CSFG (Classification of finite simple groups) is given in the "Surveys of the AMS, Volume 172, by Aschbacher, Lyons, Smith, and Solomon. From the summary:
Since “most” finite simple groups $G$ are in fact matrix groups over finite fields, an early result determining the overall shape of the CFSG was the Dichotomy Theorem---which shows that an abstract simple group $G$G (away from cases with 2-subgroups of rank at most 2) is either:
of COMPONENT TYPE (resembling a group over a field of odd order), or of
CHARACTERISTIC 2-TYPE (resem- bling a group in characteristic 2).
The treatment of the “odd case”, namely component type, was based on Aschbacher’s notion of a quasisimple component L of STANDARD FORM, in the centralizer in G of an element t of order 2. The various possible L were treated by Aschbacher and various other researchers. The treatment of the remaining “even case”, namely characteristic 2-type, was obtained via suitable analogies of the above case divisions---but replacing t by an element of ODD prime order p. Here the initial “small” case corresponds to QUASITHIN groups G; namely where the rank of suit- able p-subgroups is at most 2. This situation involves many complications; it was eventually treated in a lengthy work of Asch- bacher and Smith. For the remaining cases involving p-subgroups of rank at least 3, the above Dichotomy is replaced by a Trichotomy---established by Gorenstein and Lyons (with contributions from Asch- bacher). The three branches which emerge are: a (p-component) branch called Standard Type; a (roughly characteristic p-type) branch leading to “GF(2) type”; and a further “disconnected” branch called the Uniqueness Case. The groups of standard type were determined by Gilman and Griess. The groups of GF(2) type were determined by various authors, including Aschbacher, Timmesfeld, and Smith. And the final contradiction in the CFSG (although quasithin groups were chronologically the last to be treated) was established by Aschbacher, who showed that no group can actually satisfy the Uniqueness Case. | 2019-01-24 01:11:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7320371866226196, "perplexity": 449.0297895468068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547584431529.98/warc/CC-MAIN-20190123234228-20190124020228-00362.warc.gz"} |
https://math.stackexchange.com/questions/1147317/prove-this-proposition-concerning-a-theory-with-%E2%88%80%E2%88%83-axiomatization | # Prove this proposition concerning a theory with ∀∃-axiomatization
Setting
A theory $\pmb{T}$ has a $\forall\exists$-axiomatization if it can be axiomatized by sentences of the form $$\forall v_1\ldots \forall v_n \exists w_1 \ldots \exists w_n ~~ \phi(\bar{v},\bar{w})$$ where $\phi$ is a quantifier free $\mathcal{L}$-formula.
Furthermore, suppose whenever $(\mathcal{M}_i : i \in \mathbb{I})$ is a chain of models of $\pmb{T}$, then $$\mathcal{M} = \bigcup \mathcal{M}_i \models \pmb{T}.$$ Let $\Gamma = \{ \phi : \phi \text{ is a$\forall\exists$-sentence and$\pmb{T} \models \phi$}\}$. Let $\mathcal{M} \models \Gamma$.
Finally, suppose there is $\mathcal{N} \models \pmb{T}$ such that if $\psi$ is an $\exists\forall$-sentence and $\mathcal{M} \models \psi$, then $\mathcal{N} \models \psi$.
Now I would like to show that there is $\mathcal{N}' \supseteq \mathcal{M}$ with $\mathcal{N}' \equiv \mathcal{N}$.
Problem
1. First, I am not sure if the assumption is that $\mathcal{M} \subseteq \mathcal{N}'$? I am guessing since you can always create an extension, then $\mathcal{N}'$ is assumed to exist?
2. Suppose my presumption is correct so that $\mathcal{N}'$ exist, but as an extension of $\mathcal{M}$, it does not need to satisfy the same sentences correct? So why is it true that $\mathcal{N}' \equiv \mathcal{N}$?
3. Finally, if I am reading the proposition totally wrong, and instead I just have to show/construct some $\mathcal{N}'$ so that $\mathcal{N}' \equiv \mathcal{N}$, how would I go about doing such a construction?
• Isn't your "furthermore" redundant? A theory axiomatized by $\forall\exists$ sentences is necessarily closed under unions of chains, isn't it? – bof Feb 14 '15 at 3:03
• By "there exists $\mathcal M\subseteq\mathcal N'$" you must mean "there exists $\mathcal N'\supseteq\mathcal M$", right? – bof Feb 14 '15 at 3:06
• I don't see how the theory $\pmb{T}$ has anything to do with your question. It seems you are simply asking, if $\mathcal M$ and $\mathcal N$ are two models such that every $\forall\exists$ sentence which holds in $\mathcal M$ also holds in $\mathcal N$, does it follow that some extension of $\mathcal M$ is elementarily equivalent to $\mathcal N$? Is that right? I don't think you need $\forall\exists$ sentences for that; I believe it's enough if every existential sentence that holds in $\mathcal M$ also holds in $\mathcal N$. – bof Feb 14 '15 at 3:19
• @bof I changed the direction of inclusion and their arguments as you suggested. My problem is that if $\mathcal{N}'$ extends $\mathcal{M}$, then wouldn't it be possible that $\mathcal{N}'$ contain elements that may not satisfy the $\forall$ quantifier in some $\forall\exists$-sentence? – chibro2 Feb 14 '15 at 3:26
Please excuse my barbaric notation and terminology; it has been years since I sat in a logic class. I will sketch a proof of the fact that, if $\mathcal M,\mathcal N$ are models such that every existential sentence which holds in $\mathcal M$ also holds in $\mathcal N$, then there is a model $\mathcal N'$ such that $\mathcal M$ is isomorphic to a submodel of $\mathcal N'$ and $\mathcal N$ is isomorphic to an elementary submodel of $\mathcal N'$.
Let $U$ be the first-order theory of $\mathcal N$ in an expanded language which has a constant symbol for each element of $\mathcal N$. Thus any model of $U$ (more precisely, its reduct to the original language) will contain an isomorphic copy of $\mathcal N$ as an elementary submodel.
Let $Q$ be the quantifier-free theory of $\mathcal M$ in an expanded language which has a constant symbol for each element of $\mathcal M$. Thus any model of $Q$ will contain a submodel isomorphic to $\mathcal M$.
Since any model $\mathcal N'$ of $U\cup Q$ will do what we want, all we have left to show is that $U\cup Q$ is consistent. By the compactness theorem, it will suffice to show that $U\cup Q'$ is consistent for each finite set $Q'\subseteq Q$. This follows from the hypothesis that every existential sentence which holds in $\mathcal M$ also holds in $\mathcal N$. [*]
P.S. By virtue of the Keisler-Shelah theorem (elementarily equivalent models have isomorphic ultrapowers) the statement can be strengthened to read: if every existential sentence which holds in $\mathcal M$ also holds in $\mathcal N$, then $\mathcal M$ is isomorphic to a submodel of an ultrapower of $\mathcal N$.
[*] Consider a finite set $Q'\subseteq Q$. By forming a conjunction, we may assume that $Q'$ consists of a single quantifier sentence $\varphi(c_1,\dots,c_n)$, where $c_1,\dots,c_n$ are constants in the expanded language for $\mathcal M$. Since $\varphi(c_1,\dots,c_n)$ holds in $\mathcal M$, so does the existential semtemce $\exists x_1,\dots,x_n)\varphi(x_1,\dots,x_n)$ of the original language. Hence the sentence $\exists x_1,\dots,x_n)\varphi(x_1,\dots,x_n)$ also holds in $\mathcal N$. Therefore, the constants $c_,\dots,c_n$ can be assigned values in $\mathcal N$ so as to satisfy $\varphi(c_1,\dots,c_n)$, showing that $U\cup Q'$ is consistent.
• could you elaborate on the signficance that $Q$ has to be quantifier free? Especially in relation to the sentence "This follows from the hypothesis that every existential sentence which holds in $\mathcal{M}$ also holds in $\mathcal{N}$"? The hypothesis states that if $\mathcal{M}$ satisfies a quantified sentence, then $\mathcal{N}$ also satisfies it. But $Q$ has no quantified sentences although $U$ does. So can't there be quantified sentences in $U$ that cannot be satisfied by $\mathcal{M}$? – chibro2 Feb 14 '15 at 16:02
• Furthermore, how do you know the quantifier-free sentences in $Q$ and $U$ using the expanded langauge will also be satisfiable? – chibro2 Feb 14 '15 at 16:21
• Finally, if I wanted to prove $\mathcal{N}' \equiv \mathcal{N}$, should I use the original language $\mathcal{L}$ or the extended language containing constants from $\mathcal{M}$ and $\mathcal{N}$? – chibro2 Feb 14 '15 at 17:23
• I added some explanation to my answer. – bof Feb 14 '15 at 22:41
• thank you for answering my question. I asked a continuation of the question above. And I am wondering if you'd like to take a look? math.stackexchange.com/questions/1149697/… – chibro2 Feb 15 '15 at 23:05 | 2020-08-05 08:30:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9091401100158691, "perplexity": 114.60495219720899}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735916.91/warc/CC-MAIN-20200805065524-20200805095524-00584.warc.gz"} |
http://mtosmt.org/issues/mto.12.18.1/mto.12.18.1.dodson.html | Solutions to the “Great Nineteenth-Century Rhythm Problem” in Horowitz’s Recording of the Theme from Schumann’s Kreisleriana, Op. 16, No. 2
Alan Dodson
KEYWORDS: microtiming, phrase rhythm, meter, Schumann, Horowitz
ABSTRACT: This case study on the interpretation of microtiming is framed by the hypothesis that the avoidance of monotony in the case of highly regular phrase/hypermetric structures was not only one of the great compositional problems of the nineteenth century, as William Rothstein has proposed, but also was and remains a problem for performers. This analytical strategy helps to organize and synthesize diverse observations about a complex set of subtle microtiming practices in the titular recording. It is shown that Horowitz introduces subtle variations when materials are repeated, and that he tends to bring out the most salient aspects of Schumann’s own solutions to the “rhythm problem” (metric tensions, contrasts in rhythmic shape), except in cases of phrase linkage. This study suggests that principles of phrase rhythm could make a valuable contribution to the analytical toolbox for performance studies.
Volume 18, Number 1, April 2012
[1] The empirical and statistical tools that music theorists interested in performance have borrowed from our colleagues in other disciplines (see Clarke 2004) offer considerable clarity and precision, and these tools will no doubt continue to evolve in the years ahead. Even at its most precise, however, the analysis of microtiming data tends to produce results that are heterogeneous and resistant to interpretation, and we are still at the stage where new analytical strategies need to be developed and evaluated. In the present study, I am especially interested in exploring ways in which a familiar, speculative (theoretically driven) approach to the analysis of works can feed into the lexicon for the interpretation of microtiming.(1) At the outset I think it is important to mention that although the object of my analysis—the microtiming in a recording—was measured in a very precise way, my interpretation of that object is by no means “scientific” or “objective.” As Marion Guck has demonstrated, “musical analyses typically—necessarily—tell stories of the analyst’s involvement with the work she or he analyzes” (Guck 1994, 218). Accordingly, this case study should be taken as only one of many possible readings of the microtiming in a recording.
[2] In Phrase Rhythm in Tonal Music, William Rothstein points out that a “danger...of too unrelievably duple a hypermetrical pattern, of too consistent and unvarying a phrase structure” was “endemic in nineteenth-century music,” and he refers to this danger, memorably, as the “Great Nineteenth-Century Rhythm Problem” (GNCRP) (Rothstein 1989, 184–85). Rothstein makes frequent reference to the GNCRP in the analytical chapters that comprise the second part of his book. He shows that Romantic-era composers did not necessarily have to retreat to an earlier, more elastic style in order to solve the GNCRP, although some (most notably Mendelssohn) did so. For other composers, such as Chopin and Wagner, the solution to the GNCRP lay not so much in manipulating the phrase lengths as in finding new ways to conceal—and ultimately to transcend—phrase boundaries and thereby to achieve the effect Wagner referred to as “endless melody.” These composers, in other words, solved the GNCRP not by avoiding regular metric and grouping structures, but instead by enlivening them in novel ways so that their effect would not become mechanical and tiresome.
[3] I would like to build on Rothstein’s composer-centered narrative by proposing that the GNCRP was a problem not only for composers, but also for performers. This problem can be solved by performers, I suggest, through effective use of the various expressive means under their control, including subtleties of microtiming. I will try to demonstrate this through a short case study on both the composer’s and the performer’s solutions to the GNCRP in a recording by Vladimir Horowitz of the theme from the second movement of Schumann’s Kreisleriana, op. 16 (“Sehr innig und nicht zu rasch”) (Horowitz 2011, first issued 1969).(2) Some readers will find it self-evident that the manner of performance can mitigate the risk of musical monotony, a danger that is especially acute in the case of works that feature a high level of rhythmic regularity. However, the ramifications of this idea have not yet been explored in the theoretical literature on phrase rhythm or in the empirical literature on microtiming, so a case study seems warranted.
Example 1. Schumann, Kreisleriana, op. 16, no. 2, theme
(click to enlarge and see the rest)
[4] The score, based on the version in the Gesamtausgabe edited by Clara Schumann (Schumann 1887) is provided as Example 1. The rhythmic structure of the theme is very regular; duple hypermeter and four-bar grouping (with one-beat anacrusis) are present throughout, and there are continuous eighth-note subdivisions in all but six of the theme’s thirty-seven measures. Overall, the work has a clear rounded binary form with coda, within which the four-bar groups combine to form three phrases of ever-increasing length:
• A (measures 1–8, repeated)
• B (measures 9–20)
• A’ (measures 21–28) + coda (measures 29–37)
Each of these phrases is supported by a straightforward harmonic progression: I–II–V (measures 1–8), I–IV5–6–VI–(II–VI)– II–V (measures 9–20), I–II–V–I (measures 21–37). From a melodic and motivic point of view, the coda initially appears to be an independent phrase, but it is harmonically static (see the tonic pedal) and, moreover, it completes the harmonic progression of the third phrase, which would otherwise be left hanging on V. Thus, the reprise and coda form a single unit, a point I will discuss in greater detail below (see paragraph 23).
[5] The high degree of variability in the microtiming of Horowitz’s recording presents us with an abundance of information to interpret. I provide two short excerpts from it here (Example 4 below); I encourage readers to listen to the entire recording, which is widely available at university music libraries and can also be purchased from various online vendors, including iTunes and Amazon.com.
[6] I studied the recording using Sonic Visualiser, a program designed by researchers affiliated with the Centre for the History and Analysis of Recorded Music (Cook and Leech-Wilkinson 2009).(3) I began by listening to the recording repeatedly at the normal playback speed and taking note of the most salient expressive details, including many examples of tempo variation and asynchrony. By “tempo variation,” I mean a momentary change in speed and/or quality of motion, and by “asynchrony,” I mean grace notes (which have no specified metrical position) as well as situations in which notes that occupy the same metrical position are executed in a non-simultaneous fashion, namely hand displacements (where the left hand plays marginally before or after the right) and arpeggiations.
[7] I sharpened and refined these preliminary observations through empirical analysis. To track the tempo variations, I first used a tool in Sonic Visualiser that records the counter times of each keystroke as I tapped along with the recording, with playback speed reduced by 50%. Next, I used a tool that automatically identifies the physical onset time of each note in the recording,(4) followed by a tool that selects the physical onset that falls closest to each of my tap times.(5) Finally I made corrections, as needed, using a click track and spectrogram as guides. To make the measurements as accurate as possible, I gradually reduced the playback window to find the cusp at which each event becomes audible, which in every case corresponded to a clear discontinuity in the spectrogram. Making corrections in this way is time-consuming, but it is thought to be the most accurate approach to microtiming analysis. The three-step procedure that I have outlined—tapping, automated onset detection and alignment, and manual correction—is among the most efficient and accurate procedures currently available for the analysis of tempo variations; it combines the advantages of two earlier approaches (one based on tapping, the other on manual onset detection, see Clarke 2004) while mitigating their disadvantages. I used this method to identify all of the note onsets in the recording at the eighth-note level. Subtracting the onset times of successive beats at a given level yields a series of durations. Based on these calculations, I prepared tempo graphs at both the quarter-note (tactus) and eighth-note level. I then measured the asynchronies that were audible at full playback speed by locating the onsets of the relevant events, again using the spectrogram tool in Sonic Visualiser. For arpeggiations, I measured the time between the first and last onsets (i.e., between the bass note and the melody note).(6) For all asynchronies, I used the melody note for purposes of tempo tracking, as is customary in the empirical literature.
[8] Results of the empirical analysis are shown in Examples 4–6, 8–10, 12–13, and 15–16. In each case, the upper graph shows durations at the quarter-note (tactus) level, while the lower graph shows durations at the eighth-note level. Readers unfamiliar with these kinds of graphs may find it helpful to think of a mechanical metronome: the higher a data point’s position on the page, the slower the tempo. A few tactus pulses are unarticulated (e.g., measure 2.2), and in these cases the durations of two successive timespans—those for which the unarticulated pulse serves as endpoint and starting-point—cannot be measured using the method employed in this study.(7) Accordingly, in such instances there are gaps in the graph. Asterisks above the score indicate asynchronies, only some of which are indicated in the score through arpeggiation signs or grace notes. In each case, the discrepancy (in milliseconds) between the first and last onset times is provided.
[9] My illustrations of the microtiming in Horowitz’s recording will take the form of line graphs. Although this format displays the essential information clearly, it does have at least one counterintuitive and potentially confusing feature: the line segments between the data markers might, at first glance, be taken to represent continuous tempo change during the timespan in question—that is, one might assume that an ascending or a descending slope indicates that a tempo change is already underway before the next beat is articulated. This would be a false assumption, because each data marker represents the duration of a metric event. Thus, for instance, descending slope over a barline indicates that the upbeat is longer than the downbeat, not that there is an acceleration from the upbeat to the downbeat. Indeed, if there were such an acceleration then the upbeat would in consequence be shortened, not lengthened. In other words, these graphs represent tempo change as an articulated phenomenon, not a continuous one. Beat-to-beat tempo changes are represented by the relative position of the data markers, but instant-to-instant tempo changes are not represented by the slope of the line segments between them.(8)
[10] As with any empirical approach, the limitations of the tools must be acknowledged. Some measurement error is inevitable because of occasional buzzing or other noise around the time of an onset. There are probably individual differences in how listeners entrain to the music, particularly in phrases that have asynchronies; some listeners may entrain to the first event (e.g., the bottom note of an arpeggiated chord), others to the last event (the top note), and some may experience a blurring of the beat when asynchronies are used (see Yorgason 2009). Most importantly, some tempo changes are too small to be audible; the threshold seems to fall somewhere in the range of 10 to 40 ms (Benadon 2007, [15]), and it appears to be proportional to the basic tempo (Halpern and Darwin 1982). The salience of a tempo change is also affected by its grouping context; tempo changes that adhere to stylistic norms (e.g., deceleration at the end of a phrase) tend to be less noticeable than those that diverge from the norm (Repp 1998). Finally, the contour of the tempo graph does not always give a clear reflection of the qualities of motion that the listener experiences. Beyond the possible confusion arising from the line segments between the data markers (discussed above), it should be noted that some aurally distinct phenomena (e.g., hesitation, tenuto, and ritardando) are indistinguishable from each other on the basis of a tempo graph alone, a point that is seldom acknowledged in the empirical literature (but see Dodson 2002, [3.5] and passim). In an effort to compensate for this limitation, I have added annotations that describe the aural effects of many details, as I experience them. Despite their outwardly scientific appearance, then, such graphs are partly based on personal, subjective judgments. They can nevertheless be valuable in guiding and sharpening the reader’s listening experience, so I will refer to them often in the discussion that follows.
Example 2. Schumann, Kreisleriana, op. 16, no. 2, annotated score of measures 1–8
(click to enlarge)
[11] Before turning to details in the recording, it will be useful to consider Schumann’s solution to the GNCRP in the opening phrase (measures 1–8, see Example 2), and in particular to consider how he uses metrical tension to provide the phrase with inner vitality and shape. The harmony and grouping provide a basic orientation for duple hypermeter from the outset, but phenomenal accents in measures 2 and 4 (i.e., harmonic dissonance, durational accents, dynamic accents, and density accents) project a displacement of that hypermeter strongly. There is, in other words, a conflict between the basic hypermeter and another six-beat periodicity that serves as its “shadow” (to borrow a metaphor from Samarotto 1999).(9) Using Krebs’s labeling system for displacement dissonance, where the first number indicates the periodicity and the second indicates the size of the displacement (Krebs 1999, 35), this situation can be represented as D6+3 (1 = quarter note). These periodicities are labeled within and above the score in Example 2, and in similar examples below, while a harmonic analysis is given below the score. The tension between the hypermeter and its shadow continues in the second half of the phrase. Here the grouping structure leaves no doubt as to the primacy of the odd-strong hypermeter, but the registral accent (F5) and harmonic dissonance ( chord) in measure 6, together with the durationally accented F in measure 8, sustain the hypermeter’s shadow to the end of the phrase, albeit with less intensity than in measures 1–4.
Example 3. Schumann, Kreisleriana, op. 16, no. 2, annotated score of measures 1–4
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[12] In addition to the three- and six-beat metrical layers, a two-beat layer is established in the first half of the phrase through the registral and dynamic accents at measure 1.2 and the various phenomenal accents at measure 2.1 (see Example 3). In retrospect, this duple layer is understood to begin at the first note of the melody. The progression in accentual force from each beat of this layer to the next gives it something of the character of an extended anacrusis—one that culminates, notably, at the point at which the duple layer and the hypermeter’s shadow converge. The conflict between the duple layer and the primary layer (the notated meter) involves both displacement and metric grouping, so it is an example of what Krebs calls a compound metrical dissonance (Krebs 1999, 59–60). Overall, the rhythm of this opening phrase is far from being mechanical and predictable, despite its clear, regular hypermeter. The metrical dissonances I have mentioned enliven it with inner tension and momentum.
Example 4. Schumann/Horowitz, Kreisleriana, op. 16, no. 2, measures 1–8a
(click to open the animation)
[13] Example 4 is an animation that shows the microtiming in the first eight measures from Horowitz’s recording. One of the first things one notices in the recording is that whereas Schumann’s expression markings draw attention to the even-numbered downbeats (the “shadow”), Horowitz instead concentrates on reinforcing the odd-numbered downbeats. Indeed, he all but ignores the crescendo and sf markings in measures 1–4, and he adds a strong dynamic accent at the downbeat of measure 5. In the first half of the phrase, he emphasizes the odd-numbered downbeats mainly by playing the beats that precede them relatively quickly. This inherently increases the upbeats’ anacrustic quality (Butterfield 2006, also Hasty 1997, 164). In contrast to his handling of the first and third downbeats, he hesitates very noticeably before the downbeats of measures 2 and 4, thereby allowing the anacrustic energy to dissipate somewhat. He also hesitates slightly before the melodic peaks at measures 1.2 (second time only, see Example 5) and 3.2 (both times), thereby highlighting the point in the melodic gesture where the duple layer comes into focus. Although Horowitz’s reading of the phrase is far from literal, it arguably does enhance the metric tension latent within the music. As Krebs has pointed out, in the case of metrically dissonant passages, the primary metrical layer is often less salient than the competing layers, and in such cases emphasizing the primary layer can be an effective performance strategy (Krebs 1999, 179). It is possible, then, to read Horowitz’s solution to the GNCRP as an extension of Schumann’s own.
Example 5. Schumann/Horowitz, Kreisleriana, op. 16, no. 2, measures 1–4b(click to enlarge) Example 6. Schumann/Horowitz, Kreisleriana, op. 16, no. 2, measures 5–8b(click to enlarge)
[14] There is more to Horowitz’s solution than this, however. By introducing subtle variations in the second iteration of the phrase, he enlivens the music in ways that have no parallel in the score. One subtle difference is that in the first half of the phrase, he displaces the bass notes by a full eighth-note during the first iteration but plays them decisively on the beat the second time through. Another difference is that there are more asynchronies the second time (compare Example 4 to Examples 5–6); on the first iteration Horowitz adds only one salient asynchrony, at measure 7.1, beyond the two indicated in the score through grace notes (at measures 2.1 and 4.1), but on the second iteration he adds further asynchronies at the downbeats of measures 3, 5, and 6. A third difference is that although he hesitates before measures 1.2 and 3.2 both times, the delay is longer the second time through (average delay: 65 ms first time, 105 ms second time). Similarly, his pre-downbeat hesitations are in most cases longer the second time through. The difference is especially pronounced in the second half of the phrase; the average delay is 240 ms the first time and 375 ms the second time, and the longest hesitation of all (480 ms) sets off the second iteration of measure 6.1, the point where the hypermeter’s shadow begins to recede. In all of these ways, Horowitz subtly reinforces many of the conflicting metric accents during the repetition of the phrase, including those associated with the bar meter, the hypermeter and its shadow, and the duple layer at the beginning of the phrase. Thus, his handling of the repeat not only lends his performance a subtle improvisatory quality that itself helps to sustain the listener’s attention, but also draws that attention more forcefully to musical elements that relate, in some way, to the metrical innovations that lie at the core of Schumann’s own solution to the GNCRP in this phrase.
Example 7. Schumann, Kreisleriana, op. 16, no. 2, annotated score of measures 9–20
(click to enlarge and see the rest)
[15] In the second phrase (measures 9–20, Example 7), Schumann solves the GNCRP by giving each four-bar subphrase a distinct rhythmic shape. In the first subphrase (measures 9–12), the melody (now in the left hand) ascends gradually to A, the accented upper neighbor to the third of the goal harmony, IV. Together with this ascending line, the acceleration in the anacrusis from one measure to the next (duplet eighth notes, then triplet eighths, then sixteenths, and finally a trill) gives measures 9–12 a strongly teleological shape. The accented passing tone E on the downbeat of measure 10 and the accented upper neighbor A on the downbeat of measure 12 weakly sustain the hypermeter’s shadow here. The second subphrase (measures 13–16), in contrast, has the least rhythmic tension of any unit in the entire theme; it consists of a repeated descending gesture, and the registral accents on the odd-numbered downbeat (see the G in the top voice of measures 13 and 15) remove any trace of metrical dissonance. In contrast to this oasis of calm, the third subphrase (measures 17–20) is among the theme’s most dynamic units; the melody now has continuous eighth notes for the first time in the movement, and the hypermeter’s shadow is reinstated through the registral accent at measure 18.1 (where the theme’s highest note, A5, is heard for the first time) and the arrival of the cadence point at measure 20.1 (despite the withheld bass note, F2, a detail I will discuss shortly). Harmonic rhythm also contributes to the contrasts within the second phrase; both the first and third subphrases have a 3+1-measure harmonic rhythm, while the second subphrase has a steady 1-measure harmonic rhythm throughout (see the harmonic analysis in Example 7). Beyond the sphere of rhythm, several other forms of contrast might also be mentioned: the first and third subphrases involve dissonant harmony and tonicization of the goal harmony (I–IV, II–V) while the second has purely consonant, triadic harmony and plagal motion;(10) the first subphrase has melodic activity in the left hand, the second in the right hand, and the third in both hands (see the countermelody in the tenor); the melody ascends in the first subphrase, descends in the second, and moves in an arch shape in the third; and the counterpoint emphasizes oblique motion in the first subphrase (see the static descant), similar motion in the second, and contrary motion in the third. In all of these respects, contrast lies at the heart of Schumann’s solution to the GNCRP within the second phrase.(11)
Example 8. Schumann/Horowitz, Kreisleriana, op. 16, no. 2, measures 9–12
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Example 9. Schumann/Horowitz, Kreisleriana, op. 16, no. 2, measures 13–16
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Example 10. Schumann/Horowitz, Kreisleriana, op. 16, no. 2, measures 17–20
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[16] Horowitz emphasizes many of these contrasts, and his microtiming underscores each subphrase’s distinct quality of motion. In the first subphrase (Example 8), his average tempo (809 ms, equivalent to 74 beats per minute) is considerably faster than in the work’s opening phrase (62 bpm overall), and this tempo change reinforces the dynamic and teleological character of this subphrase. Within each measure, Horowitz accelerates through the second beat and draws out the first and third beats, so that the ascending left-hand melody and the progression of rhythmic values can be heard clearly. This results in a V-shaped tempo profile within each measure. Horowitz’s interpretation of Schumann’s grace notes is also noteworthy; his asynchronies become gradually shorter as the subphrase proceeds, and this conveys a sense of motion towards a goal, namely the point of zero asynchrony at the downbeat of measure 12. Paradoxically, the very lack of an asynchrony at that point gives it a strong sense of arrival and emphasis.(12)
[17] In keeping with the second subphrase’s more relaxed character, Horowitz plays it at a somewhat slower tempo (68 bpm) and adds a total of eight gentle asynchronies, only two of which are indicated in the score (Example 9). The greatest of these asynchronies (333 ms) falls on the downbeat of measure 15 and conveys a subtle sense of intensification at the registral peak of the second two-bar gesture. Horowitz sets off all four downbeats by hesitating slightly before them and by lengthening the first eighth note in each measure slightly; this gives the subphrase a poised, unhurried quality that differs markedly from the musical effect of measures 9–12.
[18] In the third subphrase (Example 10), Horowitz takes an even faster average tempo than in the first (84 bpm, excluding the ritard), and he abandons the V-shaped tempo profile within the measure—a pattern evident in both of the preceding subphrases—in favor of a single tempo arch spanning the entire subphrase. Through inspection of the lower graph, it is also evident that the durations of his eighth notes are much more consistent here than elsewhere in the piece; it is here that Horowitz’s capacity for rhythmic precision and control is demonstrated most clearly. Another notable feature of this subphrase is that the parts are tightly coordinated throughout; there is only one audible asynchrony here, located at the very beginning of the subphrase (at measure 16.3). Taken together, these features give measures 17–20 a stronger sense of continuity and momentum than any other unit in the piece.
[19] Schumann employs an innovative combination of phrase linkage techniques in measure 20, at the boundary of the theme’s second and third phrases. As mentioned above, the bass note (F) is withheld at the downbeat of measure 20, where the cadence point is reached. The result is an unstable chord at a point where a chord is strongly expected based on the stylistic norms associated with Classical rounded binary or small ternary form (Caplin 1998, 75). The instability grows as the measure proceeds, for the phrase’s melodic goal (the F4 on the downbeat) gives way immediately to a lead-in figure that involves chromatic passing tones and (for the first time in the piece) a series of sixteenth notes, and the ritard during this lead-in only adds to the tension. Although the right-hand slur ends before the next phrase begins, the slur in the bass transcends the phrase division, and this bass slur implies some degree of tension and continuity into the third beat and, by extension, into the new phrase. These techniques of phrase linkage weaken the sense of closure normally experienced at cadence points and supersede Classical conventions of formal articulation. Such techniques were integral to some Romantic composers’ solutions to the GNCRP, as Rothstein has amply demonstrated, albeit in reference to the music of Chopin rather than Schumann (Rothstein 1989, 214–248, esp. 233–234).
[20] Horowitz does surprisingly little to emphasize the tensions inherent in measure 20. He signals the arrival of the cadence by adding a slight ritard toward the end of measure 19, followed by an elongation (tenuto accent) at the cadence point itself (measure 20.1). He makes much of the prescribed ritard in beats 1–2 of measure 20, where the tempo profile goes off the scale, but it sounds as though he ignores the continuation of the left-hand slur to beat 3 and the sense of continuity that it implies. This situation could perhaps be read as a decision on Horowitz’s part that bringing out the instability in measure 20 (for instance, by emphasizing the bass C and D and making a legatissimo connection between these notes) would sound overwrought or otherwise unpleasing. Aesthetic judgment is, after all, an essential part of interpretation, and for this reason only some subtleties are brought out by performers, while others are left alone.
Example 11. Schumann, Kreisleriana, op. 16, no. 2, annotated score of measures 21–28
(click to enlarge)
Example 12. Schumann/Horowitz, Kreisleriana, op. 16, no. 2, measures 21–24
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Example 13. Schumann/Horowitz, Kreisleriana, op. 16, no. 2, measures 25–28
(click to enlarge)
[21] The third phrase (measures 21–28) begins with a nearly literal return of material from measures 1–4 (Example 11). The first significant change occurs in measure 24, where a new bass gesture provides added momentum into the material that follows in measures 25–28—material that differs radically from measures 5–8 in ways that add further complexity to the metrical structure. Crescendo markings now point toward the downbeats of measures 26 and 28, thereby reinforcing the hypermeter’s shadow instead of allowing it to recede in the second half of the phrase. Melodic peaks in measures 25 and 27 now precede the shadow by one beat in each case, so that for the first and only time in the theme there are three distinct six-beat periodicities: the hypermetric layer (projected by the odd-numbered downbeats), its shadow (projected by the even-numbered downbeats), and a displacement of the shadow (projected by the registral accents at measures 25.3 and 27.3). The theme’s surface-level harmonic instability also reaches its high point here (see especially the chords at measures 25.1, 25.3, and 27.1, as well as the passing chords at measure 25.2 and measure 27.2), although the functional harmonic progression (I5–6II–V) is straightforward. The hypermeter’s shadow is once again supported by the harmonic rhythm (1+2+1 bars), as shown in Example 11, and by the registral accent in the bass at measure 26.1 (here I am referring to the descent to C2, the lowest note heard thus far in the third phrase). Overall, measures 21–28 are, like measures 1–8, enlivened by metrical dissonances of subtly varying intensities, but now the music is more intricate and dynamic in both its metric and its harmonic dimensions.
[22] Horowitz highlights these changes in a number of ways (Examples 12–13). First of all, he takes a faster basic tempo than before (72 bpm in measures 21–28 [excluding the ritard], compared to 62 bpm in measures 1–8), and he brings out the new bass figure in measure 24 simply by playing it louder (not through microtiming). On a more subtle level, he adds small asynchronies at the downbeats of measures 21 and 23, and the alternation between these and the larger asynchronies at the downbeats of measures 22 and 24 underscores the metrical dissonance. As seen in Example 13, he adds eight further asynchronies in measures 25–28, only four of which are prescribed in the score. Three of his extra asynchronies fall on downbeats, and these once again project a short-long-short pattern that reinforces the metrical dissonance in a subtle way. The remaining asynchrony (found at the end of measure 25), like the four that are indicated in the score, occurs during a crescendo that projects the D6+3 pattern. Also noteworthy is the relatively high degree of continuity in Horowitz’s performance of this passage, especially in measures 26–27, where a single harmony (II) is sustained. (He does not hesitate before the downbeat here.) As in other parts of the recording, Horowitz’s microtiming shows his sensitivity to details of meter, motion quality, and harmonic rhythm.
[23] One challenging point in the analysis of the theme’s form concerns the third phrase’s cadence. (For the broader musical context, see again Example 1.) Is there a half cadence at measure 28.1, followed by a lead-in to the coda, or is there an authentic cadence at measure 29.1 (or perhaps measure 28.3), with a phrase overlap? If these were the only options, then I would choose the first option for at least two reasons: the bass ascends by step from $\stackrel{ˆ}{5}$ to $\stackrel{ˆ}{1}$ in measure 28, a gesture much more characteristic of a lead-in than a cadence, and measures 28.3–29.1 initiate the reprise, so these beats sound much more like a beginning than an ending. This half-cadence reading is not entirely satisfying, however. Its main shortcoming is that it suggests that the tonic chords at measures 28.3 and 29.1, and also the ensuing tonic prolongation in measures 29–37, are entirely separate from the cadential process and, therefore, that the theme as a whole (measures 1–37) essentially ends with a structural half cadence at measure 28.1. (The coda has a pedal bass throughout, so there is no possibility of an authentic cadence after measure 28. Although there is a clear V–I motion in measures 36.3–37.1, even this cannot be regarded as the structural cadence, because a prolongation of the final tonic is established long before this point.) As Poundie Burstein has recently proposed, complex situations like the one found in measure 28 highlight inadequacy of a binary (“black-and-white”) conception of cadence types in tonal music and point to the need for additional categories along a spectrum of cadential possibilities (Burstein 2010). The absence of an “Im Tempo” marking after the ritard in measure 28 enhances the ambiguity somewhat; the ritard heightens the sense of “ending” associated with the cadence, and the return to the main tempo heightens the sense of beginning, but the lack of an “Im Tempo” means that we cannot tell, based on the tempo markings in the score alone, precisely where the feeling of ending ends, where the sense of beginning begins, and whether there is any transitional zone between these diametrically opposed qualities of motion. In this case, the rhythmic and metric aspects of the cadence suggest that it lies closer to the “half cadence” pole of Burstein’s spectrum, while the stylistic conventions associated with the form, and with tonal music in general, push it a few notches in the direction of the “authentic cadence” pole. Regardless of how we decide to interpret the cadence type, its ambiguity (from the vantage of conventional conceptions of cadence) complicates the grouping boundary by blending aspects of continuity and discontinuity, so this can be understood as a further example of linkage technique.(13)
Audio Example 2. Walter Gieseking
Audio Example 3. Murray Perahia
Example 14. Schumann, Kreisleriana, op. 16, no. 2, annotated score of measures 29–37
(click to enlarge)
[24] Amid the play of opposing forces in measure 28, Horowitz once again tips the scales in favor of discontinuity, just as he had done in measure 20. He makes much of the ritard in measure 28.1–28.2 and adds a Luftpause after measure 28.2 (see again Example 13) before returning to the main tempo at measure 28.3. He also makes a clear dynamic contrast (to f, not p as marked) at measure 28.3 (Audio Example 1).These details seem to push measure 28 almost entirely toward the “half cadence” pole. As in the case of measure 20, Horowitz does nothing to highlight the tensions inherent in the linkage. It would be premature to suggest on the basis of just two examples that this represents a basic feature of his performing style (a general aesthetic preference), but perhaps this is a topic worthy of further study. A close comparison to other recordings is also revealing; some other performers, such as Walter Gieseking (Audio Example 2) and Murray Perahia (Audio Example 3), convey a clear sense of continuity into the coda, in terms of both rhythm and dynamics—continuity that traverses the grouping boundary and pushes the effect toward the “authentic cadence” pole.
[25] Harmonic motion ceases once the tonic harmony is reached; as noted above, a pedal bass is present throughout the coda, which lasts from measure 29 until the end of the theme (Example 14). Despite this harmonic stasis, within the coda a sense of directed motion is conveyed through a descent from the registral peaks G5 (measure 30), F5 (measure 32), and E (measure 34) followed by a scalar descent to D3 in measures 34–37. Linear motion to $\stackrel{ˆ}{1}$ occurs twice within measure 35—first at beat 2, where the tenor and alto voices effectively converge on Bb3, and then at beat 3 in the top voice of the right hand. These are points of closure in a weak sense (the tenor line ends at beat 2, and the texture and surface rhythm change at beat 3), but the ensuing sixteenth notes, together with the reactivation of the registral space above B3 and the reintroduction of dissonance and chromaticism in measure 36, override the weak sense of closure in measure 35 and carry the motion forward to measure 37.1. The final melodic gesture (measures 35.3–37) revisits the F–G–F figure from the theme’s first measure in a chromaticized form. Because this concluding figure ends with $\stackrel{ˆ}{5}$ in the highest voice, it too provides only an incomplete sense of closure. (This compositional choice relates to the theme’s larger formal context; the theme we have been examining is the A section of a rondo form, and descent to $\stackrel{ˆ}{1}$ occurs only at the end of the entire movement.) The melodic peaks within the coda (measures 30, 32, 34, and 36), most of which are highlighted in the score through dynamic accents, strongly project the hypermeter’s shadow; once again, metrical dissonance is central to Schumann’s solution to the GNCRP here.
Example 15. Schumann/Horowitz, Kreisleriana, op. 16, no. 2, measures 29–32
(click to enlarge)
Example 16. Schumann/Horowitz, Kreisleriana, op. 16, no. 2, measures 33–37
(click to enlarge)
[26] Horowitz brings out the metric tensions in the coda not so much by reinforcing the main hypermetric layer as by supplying dynamic accents for the melodic peaks that project its shadow. He does, however, bring out the main hypermeter subtly by making much of the asynchronies that are necessitated by the wide spacing in the left hand in most of the odd-numbered measures of the coda (see Examples 15–16). In short, he uses dynamic accents to make one of the six-beat layers conspicuous, and he uses microtiming to draw more subtle attention to the other.
[27] This preliminary study supports the hypothesis that the GNCRP was a problem not only for composers but also for performers and suggests that it can provide a useful focal point in the interpretation of recordings. In the theme we have been examining, Schumann solves the GNCRP by enlivening phrases through metrical tension and directed harmonic/linear motion, and through contrasts in texture and harmonic rhythm between successive subphrases. Beyond the phrase level, he sustains musical tension through linkage techniques and, it might be added, by developing the F5–G5–F5 motive in a variety of subtle ways. (The chromaticized version in measure 36, mentioned above in paragraph 25, is only the most conspicuous variation. For a comprehensive account of the many guises of the F5–G5–F5 motive throughout the movement, see Fisk 1997.) Horowitz, in turn, solves GNCRP partly by bringing out aspects of Schumann’s own solutions (the metric tensions and qualitative contrasts in rhythm), and partly by introducing subtle variations in the repeated materials. He is highly selective; in particular, he makes no attempt to underscore the linkage techniques, and in the case of metric conflicts he often simply reinforces the primary metric layer and the hypermetric (odd-strong) layer.
[28] Further research would be needed in order to determine the scope of the GNCRP’s relevance to the interpretive practices of performers of the past and present, but on the basis of the present study it seems that the GNCRP and other principles of phrase rhythm could make a valuable contribution to the toolbox for performance analysis. This kind of approach is far from being objective, and those who favor a more scientific approach may think that it borders on overinterpretation. For those who can suspend disbelief, however, phrase rhythm could perhaps be a focal point in a sustained study of a given performer’s style, or in a comparison of different recordings of a given work. This is just one of the many ways in which music theory might make a distinctive and valuable contribution to the analysis and interpretation of recordings, a field of study that is now beginning to rise above its necessarily positivistic and empirical roots through the participation of scholars from a wide variety of disciplines.
Alan Dodson
University of British Columbia
School of Music
alan.dodson@ubc.ca
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Carey, Norman. 2007. “An Improbable Intertwining: An Analysis of Schumann’s Kreisleriana I and II, with Recommendations for Piano Practice.” Theory and Practice 32: 19–50.
Carey, Norman. 2007. “An Improbable Intertwining: An Analysis of Schumann’s Kreisleriana I and II, with Recommendations for Piano Practice.” Theory and Practice 32: 19–50.
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Perahia, Murray. 1997. Schumann: Kreisleriana, Piano Sonata No. 1. Recorded 1997. Issued Sony Classical 62768.
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Footnotes
1. In this sense, the study builds on other recent scholarship on microtiming that engages and adapts pre-existing theories of compositional structure (e.g., Butterfield 2006, 2011; Dodson 2002, 2008, 2009).
2. Horowitz made at least two recordings of Kreisleriana aside from the one under consideration here. These were issued in 1985 and 1986, and they differ from the 1969/2011 recording in many details of interpretation. Performance-related issues in other parts of Kreisleriana, including the second intermezzo (but not the theme) from the second movement, are discussed in Carey 2007.
4. The beat-tracking algorithm that Sonic Visualiser employs is an upgrade of the algorithm described in Dixon 2001, which received the highest score of the Music Information Retrieval Evaluation Exchange (MIREX) in its evaluation of audio beat tracking software in 2006. See http://www.music-ir.org/mirex/wiki/2006:Audio_Beat_Tracking_Results (accessed August 11, 2011).
5. The tool that coordinates tapping data and onset data is called TapSnap, and it was developed by Craig Sapp for Nicholas Cook’s project on recordings of Chopin’s Mazurkas. TapSnap is a web-based application; see http://mazurka.org.uk/cgi-bin/tapsnap (accessed October 30, 2011).
6. For a review of recent empirical research on these aspects of microtiming, see Gabrielsson 2003, 226–30. Regarding asynchrony, see also Yorgason 2009.
7. Unarticulated beats can be accommodated by the tap-along method described in Clarke 2004, but that approach is much less accurate than the onset-detection method used here. For a psychological account of metric entrainment that can accommodate missing beats, see London 2004, 20–23.
8. This potentially confusing aspect of the traditional tempo graph is avoided in an alternative approach used in Butterfield 2011.
9. Following Krebs 1999 and London 2004, I do not consider metrical dissonances to be conflicts between autonomous meters. Thus, although I find the “shadow” metaphor evocative and compelling, I find Samarotto’s term “shadow meter” somewhat problematic. To avoid possible confusion, I will instead refer to “the hypermeter’s shadow.” The odd-strong pattern is present throughout the piece, and its stability, together with the general tendency for metrical accents to fall near the beginnings of groups, gives it primacy (Lerdahl and Jackendoff 1983, 72 and 76). In my view, the even-strong pattern introduces a high-level metrical dissonance, not a separate meter.
10. In terms of harmonic functions, IV5–6–VI and II–VI are both S–T progressions. It is in this sense that I regard them as plagal progressions in the tonic key (B major).
11. I do not mean to suggest that the second phrase lacks coherence on an underlying level. One obvious unifying element is the overall harmonic progression from I to V in measures 9 to 20; this progression is initiated by the motion from I to IV in measures 9–12, delayed by the turn to VI in measures 13–16, and then propelled forward again by the II in measures 17–19, yielding an overall progression of harmonic functions that is highly coherent: T–S–(T)–S–D (again, see the harmonic analysis in Example 7). Another possible unifying element, suggested to me by one of the anonymous readers, is the melodic line in measures 9 to 12, which is characterized by ascending stepwise motion from B to G. This melody could be understood to return in varied forms at the beginnings of the remaining subphrases: first in a curtailed retrograde form (measures 13–14), and then in an accelerated and registrally expanded version of the original melody (measure 17).
12. I say “paradoxically” because in most other contexts, it is the presence rather than the absence of an asynchrony that renders an event accented, that is to say, “marked for consciousness” (Cooper and Meyer 1960, 8).
13. As noted previously, this also means that the formal boundary between the reprise and coda is blurred somewhat. For this reason I consider this a case of form-functional fusion, albeit an atypical one. (More typical contexts for fusion are discussed in Caplin 1998, 11, 165–67, 203.)
In this sense, the study builds on other recent scholarship on microtiming that engages and adapts pre-existing theories of compositional structure (e.g., Butterfield 2006, 2011; Dodson 2002, 2008, 2009).
Horowitz made at least two recordings of Kreisleriana aside from the one under consideration here. These were issued in 1985 and 1986, and they differ from the 1969/2011 recording in many details of interpretation. Performance-related issues in other parts of Kreisleriana, including the second intermezzo (but not the theme) from the second movement, are discussed in Carey 2007.
The beat-tracking algorithm that Sonic Visualiser employs is an upgrade of the algorithm described in Dixon 2001, which received the highest score of the Music Information Retrieval Evaluation Exchange (MIREX) in its evaluation of audio beat tracking software in 2006. See http://www.music-ir.org/mirex/wiki/2006:Audio_Beat_Tracking_Results (accessed August 11, 2011).
The tool that coordinates tapping data and onset data is called TapSnap, and it was developed by Craig Sapp for Nicholas Cook’s project on recordings of Chopin’s Mazurkas. TapSnap is a web-based application; see http://mazurka.org.uk/cgi-bin/tapsnap (accessed October 30, 2011).
For a review of recent empirical research on these aspects of microtiming, see Gabrielsson 2003, 226–30. Regarding asynchrony, see also Yorgason 2009.
Unarticulated beats can be accommodated by the tap-along method described in Clarke 2004, but that approach is much less accurate than the onset-detection method used here. For a psychological account of metric entrainment that can accommodate missing beats, see London 2004, 20–23.
This potentially confusing aspect of the traditional tempo graph is avoided in an alternative approach used in Butterfield 2011.
Following Krebs 1999 and London 2004, I do not consider metrical dissonances to be conflicts between autonomous meters. Thus, although I find the “shadow” metaphor evocative and compelling, I find Samarotto’s term “shadow meter” somewhat problematic. To avoid possible confusion, I will instead refer to “the hypermeter’s shadow.” The odd-strong pattern is present throughout the piece, and its stability, together with the general tendency for metrical accents to fall near the beginnings of groups, gives it primacy (Lerdahl and Jackendoff 1983, 72 and 76). In my view, the even-strong pattern introduces a high-level metrical dissonance, not a separate meter.
In terms of harmonic functions, IV5–6–VI and II–VI are both S–T progressions. It is in this sense that I regard them as plagal progressions in the tonic key (B major).
I do not mean to suggest that the second phrase lacks coherence on an underlying level. One obvious unifying element is the overall harmonic progression from I to V in measures 9 to 20; this progression is initiated by the motion from I to IV in measures 9–12, delayed by the turn to VI in measures 13–16, and then propelled forward again by the II in measures 17–19, yielding an overall progression of harmonic functions that is highly coherent: T–S–(T)–S–D (again, see the harmonic analysis in Example 7). Another possible unifying element, suggested to me by one of the anonymous readers, is the melodic line in measures 9 to 12, which is characterized by ascending stepwise motion from B to G. This melody could be understood to return in varied forms at the beginnings of the remaining subphrases: first in a curtailed retrograde form (measures 13–14), and then in an accelerated and registrally expanded version of the original melody (measure 17).
I say “paradoxically” because in most other contexts, it is the presence rather than the absence of an asynchrony that renders an event accented, that is to say, “marked for consciousness” (Cooper and Meyer 1960, 8).
As noted previously, this also means that the formal boundary between the reprise and coda is blurred somewhat. For this reason I consider this a case of form-functional fusion, albeit an atypical one. (More typical contexts for fusion are discussed in Caplin 1998, 11, 165–67, 203.)
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