Split (1)

train · 6.32M rows

url stringlengths 14 2.42k	text stringlengths 100 1.02M	date stringlengths 19 19	metadata stringlengths 1.06k 1.1k
https://ec.gateoverflow.in/1802/gate-ece-2011-question-5	4 views The trigonometric Fourier series of an even function does not have the 1. dc term 2. cosine terms 3. sine terms 4. odd harmonic terms	2022-10-04 03:31:48	{"extraction_info": {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8887893557548523, "perplexity": 8432.096572263681}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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/1664030337473.26/warc/CC-MAIN-20221004023206-20221004053206-00370.warc.gz"}
https://www.esaral.com/q/let-o-be-the-vertex-and-q-be-any-point-on-the-parabola-91929	Deepak Scored 45->99%ile with Bounce Back Crack Course. You can do it too! # Let O be the vertex and Q be any point on the parabola, Question: Let $\mathrm{O}$ be the vertex and $\mathrm{Q}$ be any point on the parabola, $\mathrm{x}^{2}=8 \mathrm{y}$. If the point $\mathrm{P}$ divides the line segment $\mathrm{OQ}$ internally in the ratio $1: 3$, then the locus of $\mathrm{P}$ is :- 1. $y^{2}=2 x$ 2. $x^{2}=2 y$ 3. $x^{2}=y$ 4. $y^{2}=x$ Correct Option: Solution: Let P(h, k) divides segment #### Leave a comment None Free Study Material	2023-01-29 05:20:08	{"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": 0.6779652833938599, "perplexity": 803.3304627524608}, "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-2023-06/segments/1674764499700.67/warc/CC-MAIN-20230129044527-20230129074527-00029.warc.gz"}
https://ask.sagemath.org/answers/37352/revisions/	Ask Your Question # Revision history [back] Yes it is the rank over the polynomial ring. Houw could it be the rank over the rationals ? Note that x and y do not belong to the rational field, but the rank depends on the rational values you associate (the zero vector has rank 0, the other have rank one): sage: J.substitute({x:0,y:0}).rank() 0 sage: J.substitute({x:0,y:1}).rank() 1 Yes it is the rank over the polynomial ring. Houw could it be the rank over the rationals ? Note that x and y do not belong to the rational field, but the rank depends on the rational values you associate to them (the zero vector has rank 0, the other have rank one): sage: J.substitute({x:0,y:0}).rank() 0 sage: J.substitute({x:0,y:1}).rank() 1	2019-10-15 04:49:33	{"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": 0.6053732633590698, "perplexity": 548.2820586260401}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "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-2019-43/segments/1570986655864.19/warc/CC-MAIN-20191015032537-20191015060037-00228.warc.gz"}
https://raweb.inria.fr/rapportsactivite/RA2019/tripop/uid107.html	Overall Objectives Application Domains New Software and Platforms Bilateral Contracts and Grants with Industry Partnerships and Cooperations Bibliography PDF e-Pub ## Section: New Results ### Nonlinear waves in granular chains Participants : Guillaume James, Bernard Brogliato, Kirill Vorotnikov. Granular chains made of aligned beads interacting by contact (e.g. Newton's cradle) are widely studied in the context of impact dynamics and acoustic metamaterials. In order to describe the response of such systems to impacts or vibrations, it is important to analyze different wave effects such as the propagation of compression waves (solitary waves or fronts) or localized oscillations (traveling breathers), or the scattering of vibrations through the chain. Such phenomena are strongly influenced by contact nonlinearities (Hertz force), spatial inhomogeneities and dissipation. In the work [8], we analyze the Kuwabara-Kono (KK) model for contact damping, and we develop new approximations of this model which are efficient for the simulation of multiple impacts. The KK model is a simplified viscoelastic contact model derived from continuum mechanics, which allows for simpler calibration (using material parameters instead of phenomenological ones), but its numerical simulation requires a careful treatment due to its non-Lipschitz character. Using different dissipative time-discretizations of the conservative Hertz model, we show that numerical dissipation can be tuned properly in order to reproduce the physical dissipation of the KK model and associated wave effects. This result is obtained analytically in the limit of small time steps (using methods from backward analysis) and is numerically validated for larger time steps. The resulting schemes turn out to provide good approximations of impact propagation even for relatively large time steps. In addition, G.J. has developed a theoretical method to analyze impacts in homogeneous granular chains with KK dissipation. The idea is to use the exponent $\alpha$ of the contact force as a parameter and derive simpler dynamical equations through an asymptotic analysis, in the limit when $\alpha$ approaches unity and long waves are considered. In that case, different continuum limits of the granular chain can be obtained. When the contact damping constant remains of order unity, wave profiles are well approximated by solutions of a viscous Burgers equation with logarithmic nonlinearity. For small contact damping, dispersive effects must be included and the continuum limit corresponds to a KdV-Burgers equation with logarithmic nonlinearity. By studying traveling wave solutions to these partial differential equations, we obtain analytical approximations of wave profiles such as compression fronts. We observe that these approximations remain meaningful for the classical exponent $\alpha =3/2$. Indeed, they are close to exact wave profiles computed numerically for the KK model, using both dynamical simulations (response of the chain to a compression by a piston) and the Newton method (computation of exact traveling waves by a shooting method). In addition, in analogy with the Rankine-Hugoniot conditions for hyperbolic systems, we relate the asymptotic states of the KK model (for an infinite granular chain) to the velocity of a propagating front. These results are described in an article in preparation.	2020-07-09 04:40:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "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": 0.7226192951202393, "perplexity": 1025.4110924595816}, "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/1593655898347.42/warc/CC-MAIN-20200709034306-20200709064306-00440.warc.gz"}
https://demo.formulasearchengine.com/wiki/Shift_rule	# Shift rule For sequences it states, if ${\displaystyle (a_{n})}$ is a sequences then it converges if and only if ${\displaystyle (a_{n+N})}$ also converges, and in this case both sequences always converge to the same number.[1] For series it states, the series ${\displaystyle {\Sigma _{n=1}^{\infty }}(a_{n})}$ converges to a number if and only if ${\displaystyle {\Sigma _{n=1}^{\infty }}(a_{n+N})}$ converges.[2]	2020-07-04 08:56:26	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9702273607254028, "perplexity": 247.90772943808366}, "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/1593655886095.7/warc/CC-MAIN-20200704073244-20200704103244-00161.warc.gz"}
https://fs1.e.lanbook.com/api/preview/51770/page/6/img	JFIFC \$.' ",#(7),01444'9=82<.342C 2!!22222222222222222222222222222222222222222222222222;" }!1AQa"q2#BR\$3br %&'()456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz w!1AQaq"2B #3Rbr \$4%&'()56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz ?( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (8]ռKt.ct[X9XF7I3`2-i#z=CCP61x80s֯EPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEP\T^k>(MWe: Kt}`==a:HY/cxH8"{A@ o 7J%kn#/MHn? >iAEsi >txAѮTwWI\$d}SK]sWfL?E{h51סL?E{hK]sWfNC^?; z aK]sWfL?E{h51סL?E{hK]sWfNC^?; z aK]sWfL?E{h51סL?E{hK]sWfNC^?; z aK]sWfL?E{h51סL?E{hK]sWfNC^?; z aK]sWfL?E{h51סL?E{hK]sWfNC^?; z aK]sWfL?E{h51 3]7'Uq}%ٜ;IpzK]sWfL?E{hL?E{hK]sWf: +K]sWfL?E{hL?E{hK]sWf: +K]sWfL?E{h(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Ϊv' S>#\]A=lf ?堐OA=z=A y-溶V'`FaUelP^QK6iџ`(C~\EEs61^sd&iu˓XEVRpîHxtY-n\$fh\m[krȬ'ؐ:Z+IĿvMYI"ȁx%ջjKd'g`ʧ #=: Ԣ%КVrTl,:@TW7o y>,vogo4A`&w(migj^/mP[]_.Ma#vϷ x5q>RAeF\$eg Hҧ?OQX`p#\$; T ';'QXl:h?7H2тgpa#'i\=j0 XUMao^9&մf&\|"%1]c%Uca\UIkvu}l(ڈCnz@20 &ox,˰ɿ3gɻۚ˼k k?fŭòX\$Wq8<1 h/z:{[(GoKOgzE\$3\ ˸Plwl.w03ڢmS7m՝ hrF:9?P]i76O[16ܲjjzYާ{3jڰe 33ygE񦩧Mp~\&{sokhss8< w6M3E.]F8-#VI8 \| ڀ:+[f]&{ hCmf G nF+ZCk,&h=@h=<\|Mu'\$n\|9\$qy?klR@ɷga'#4Es8ٴ;@ ˼@b9e=H8i6&sX؆Y#îdv k)#knY <%[88uAg/Ú>e[ \|sRJF =&^,Yȑcv t_gwk4ǈ¨fuAkmú0hpN<@e%[->{ IJe o dfZkT[7.R v z\|A.Y\}m"08Fj!x\$݉,-Y@VVFݷh n\o.{'-͘nY@Ve`U`p} (jOYs%Ě=+mwUYt;y/r\$\$%pzhf懌\|FB!0# jŷ-յ m]!Un%)e<ݢ\|]nww6:Ļyg=Slv\$7ՠ/:Di-FbFR. IZ騬1;4t{o@S1=̠w\$wG<]P\$xd@"1WF<Ƞ U Fkk{H,'twޥV83Ҳ5َN/~ǵ\|ߴ_]g8hYy%X.q8?>G%>!x4)[k9H y8YI#>{goi_pX)\$ՠkx.=rq܊h b\$ IFuZ}y54jY驯]D mN@ʃB3v7(ԡ{츄w#sm;HE`ZgKᷚd yF% K={REyn>ϡj24`\|Sx 9\$q@^6M_KREV[I.w1a+/pOGZ.zԱEk.6y8/ˡǤ^umV^ 8 jxElKkޛkpA 4(((((((((((((uwYcRG;zI;GGkgFt<6,.#!TTڣv] qj?Jҵ[fh8y9.H\$UZ'Kkt_G 3x?K6AݻTv۽nP!6 ykomus4YF'>VK_P.˸'!I xw4PMyk:?MNW:Qj+ǒI]/Zo k%dNiEqI5c 2223ṋ1oWFI<0@]wu#b򲚤Vlzm,B-R)ī8}۳tڻoAhfht"9hcPA [P7okA\[[Yq5Qer :(/t[W:L7chSZ]j JF~0X/~{\|Fgw\GXʗT]ÑTipp@=@a^#^9Vc_@[:-ݶqnI'AӣKILcU^UO; z2߇4/^zvc#?lGQ<i\|m @(t(\$_97~S R6o#%t셬ě-^K ;Od+/ʻ\$'Hs`.\$P dUWaՀ3 }b?)8`~IUoI"B[6 -f&D1 3ut &wKqݴ`arp:dI@΂x[SLWOޠ AAYhe4Z{ҵ/jd9BF@5P^E9E>lڰWPFTBz\zk֏=朖R} WTl2A!v@\+hN_X:fޫ7DrDP,w 3\G٤ּ;k (eQ>rK`0x+cԊ\$I@sD.M_46+MiW+Z5ĖR ˸䌐{W kOo l5[jOsK1wҽʃ[YIq;+T[\G bF\|9YAi6I[\ʙ`A4䶊;nU}s9'55y֗6GW14ZHFF8XօЗ֟86N2p6SߵvPE'\|gkgk]M(ZL#FvC7ůi7zG6Da8BIw溊(2\ᨣcack c\$d/\$wk=E?^iu̟ώgImFWz]xkoEX6(9/95Q@_ga}\|%kXKl"Q g(\|m۵Oy4{M+ yB`0'k( 6/:;ɤS!ER΃+3faO4!u+X]0Ď>`NtP]x>Сn- (XJ wdj6IwO-kʺ٭ a<(mtB-99t[8Yu}+8R h +m}eiv3yvS\$>Z4Pt6V7Vwb^Syi4ȸڍaˎMp[R4uIG ]NX>}q׊(5 gyQ%%G= UǆoxZ)%-M:S'2,g=8-?@:ױ5] pۏ?9Vnv23@? rkOiO%森]ěuTVv\$`\r1ⷬ4}\$X-;y&Yƨѝ^G\|?o{ r.%ir2x@shXCc}egul^A \$SX֬fԬs ҥvRF>e`\|WGEp~ I[\s ZȲ%zt k{J:[tH2vSrulu;ST/1~ ʀ;5;u/ KS̎ݤF\p^km`N/3qNk-YC'-Ed\|X/i-\|5S\$;b(;.rgqp=;xY3Y :oDQC#jV'222z \|j! m\$qݘ˒H˽ZOif5ҍ̚ܤveP@Z֗co֣k dU &zm9F1Aִ[ mѿgǽyU𢤋 bT3ӧnW heEh=5tCĚXM\|5{#i=v#8.~n )vy^Ot&QHԝۑ濜(du> n{[}m,y{v:29З]a[6SEe[R(~b{cKwXMo Kr`I\$ ґ \{WCZt:7d]'&2 rUM]O_ &\$2O3o,ޛ? <`Y{W\|K[7WҠ]v뛅̐rr "@ PAGzyi\|C{}oo,@C?J!Bm>y/ [H +ؔ\$#k:uuS",U~ʟf'w1kZdB+wU\̿yG9j-G&`lgtO\6埂^>׻M1qgoh^A_?4>1=u}6KT%U_5@<)zA,V@)ǵyE>kS;G{\$M8u&{r3OM]O_ &\$2O3o,ޛ? <`Y{PxB{[b+D=W#!HR9ǥy׉uK{:T6+?}spQy@NA?H_C WxUG(\$bLR @PB\$5{dlN B`85zS]ڮj3%:ƹOMklXi\x%'l\$)<`Jei,RLv:K?hZ͓pd]<ڞFIh3rU&ƼD2XR5PVi'rT8Km<6\$%Eaր;\$4cm^Ɉ pI\$\ӓWӤo`<FGP<y})<>&?ߺϾ<{Ǚ &=?bX˳>_jlE[WLHuftB[l'#uM=%X.Dc+8Fu-l{ti@#x גKq)O'C>7\>2rygmxGGڄhCU@H ǵtŞPp ܧ 矻9ȫ2kz\1I.jxmLO@?Jizĺ\$iknĎ\|ws i\|L[yސ=s 1? 5N--Ḫ̪UV #C @uK0o?st=+mŮZxzbY G;Hi0H8=O';cqunڥh7\!Az?(V~'е `WKL)2c_^;W7xf3Ab9X;+a~w}o덾ݟlv'y{/UucKkKӄP,&YU\|z_¹ n0۱FGϙ?N[cpL~eRC226m#P.=˩Z5f)HӠմș`]`~aq55\$ԮT 6eJĖQp>.Olz[[Δ9\$m _lXW]ށ}3B=?l77G#[\$9fi66xBC-+e?:R&ѮQ T&hf@]jhm?YCydu PMhu{gfYK\|!SŵK.FKSTeޣ}_\Gn\$/L(Q~\$!#h'3@T};Lɉuc#'hqyYH\|9_l6StGi}-KլeTWduB =e:͇}7^>N{^k-]ѕN[H,[_++]!%6R#\:w8P?@Qt[O7S6`{o\{6%ЮH 1":\Ν ^Kĺ ;l~r{#1j7w.xY& &/x#;q\$2: gL7؍Bd F_R@PMfcm_,6qs\%]>41k]l&Lner\$qkVt+^3 TǮO\$7?,t]UJ%<7wy~Km9Pxx[/}GD]S0iotQw3JcPSe>wد _˗pUJUՔmw,I95_{=1Zӷ3y;> ֙ur-࿷V,UW\|tU_žF9]YԩL\|ukQsޝ k[X+o #'hddG\]Gki%ı &Ry\J}<[&9U2: 9-ͥ[^dUxA: ޹[8mo`d}y(EgYK)]6/y/>6w{o&~6o=I5.MAt-o(z6޸Wӥ6<%QGqwVqoi?ZvjĶC>E rr\s@GqvSTag8s?/Mohom;jBC:혞N~o¸x#J7,' :})Do![8ez)t%"O1F_R@\?KI~G(dax7Gh~,jMɢ<\$4O\$p&%'湻r?AxX^6VSE>(((((((((((+Kw=BJ0wʧ62G֯Q@!4m94(_[GmP1RCi VP,sʾXćxsV mn-+k8ϕoƹ(Aw-pE=(iz\|n<2l=Wv3jEQ./iB4B ړJ:mIɸWy8%(4{D)tG]+dbgAI-]4]=vەA .N閌\>JТ)I-Z4\|[;B>CJd:Z]^)IbN+rhPS؝8ؔ/q{T) `4?̐qǭhQ@W_iVA ے: ]@.nfcΕWaB(\$u!UAq_3@ѣ_MM&l{an[ԕ )6Ǳ2~>^h@ϥYuZiVoy\$0/c&\|\|)PAEF;+)=sOz wF8{{c\$aSNzJmjB}1y+gr8tsivFs[!\|=rW'ϵ5\|=+I Fϐ'sZTP_#Zc~Tѿާm#]q/8Wh ,_ؚpY(@F#sVe4ӭ\$]D1rS=ks,@<Ы2GOjERH'm:K8wLd~Fa,֖Ns3ƮQ@47O̳l aH\hҹ xh%>T{aBOrk FCyw,\0"tLKr +4 ":mqU(Ѵ3z囃zm݌ڡޅg"mH+EjCx:֝DhXibe)\$B-P5g?M6mDz3ZPaN0LA#j_ 87+X"uN2 (M3imJmX@( Phmȹ)b"ܶ=3{Ϥi7\i֒CH##P#ihhoI\|ݾ{fMK-6)20* S֮@4:b2p: CO2Bt ,#ot6qfrF99ZP\|:oziV1]Y8DҠPMfaP䑂IrGPr2 E.@A[ÚAohq3jb>1EQ:6o:m6Iq/9¹X~kۛdi~L e6A=:Ng6^im\$8Ԟ?bMvVQn &0Ca#u@ߌt+ 5'ty]Ȁǽ`wN%&F}h!ԌWS}1mP܀1GA].mfɮa)2Z6 .>F/# _\$Ma\$iC(eQ q2@'JS:Վ65y5hأ1\$ g1V{_i}s=ǗrotHJBGMu5 V"r!`a@98'9𚮹kɫ_l#6** gh/5 ~hU3A]W0P xq+4Yl,t̑eǫm OV'Sԍ<˹vxJмwZݶo \B=NJ:FyS <3ohq2Q"B\|G-h'b sg5:(.߀? m-I)CrNI8t .K-Gn:hb\ԣi8Mr sxtMcʅcgNsxC/\$.n,W71Q\$ ΏۤP΋8ܼ1RtH9i0\q i @@ZCVkJfUJ+[[XDdL2l;)c@UZ\|G`7kqbЭ73ጒs(jD77=ƥu0,<`#l %-o}Y T:DlW։9\+* HךɼA}T4SUmFeKKc(?;q'=4g=k iwxW֗,)峷 ,ᇨKisxZZ{oxC,YdeR{_4Vڜmx&nUeℌdŞ07V%ihԠw @ \|;z_]6jڀ=s7{E0OJ\|A,sfG{?on 2Ư+ŵό6GSBU0s <'ma5^aw3O-1on_i+CH xs6j'yVM>-F>ԗls⧸}iϢI\$NT"Hm\<&&QybMPl)+(\|2TךW^77"{;vƨUݟsր8H,G^9M^Jx\|AjF#%c5]LѠ}N+oh.csgswZ, epKm9櫷k EHYb/(p듌 @e\$b--).\$n=}{nsu5U-:eż)w^qses3YPF/cۿpٜ튭7䴿2V2X̊7@=15ռ9jRKHd]6GNmw{ ֚bR\\$ NwƑFơUUW]GO۳> 1DK(]/^Y,Ic&{e Ċ2wSX zŇofяQ'yFPĀ1=;na[A o2J?#8#_Gk iLA2mF1P=h/OcU]ߝ{\$<rar2;{W·sD"##)~O,WQn.mSF2 wYHU%X'Ҿ~-:I!G[yU7V\$CqpF+OƗ˦i>)ԴBStJ ]@ J57 [׆㾽 Ecfvʫn!NR{ .~iVj3=epe;w;I#=Ꭱ{ ͍ĭoܿlFK 3.N8'ox>('M9 q AHY7 {c;7^ӮbKﾺ l(P9M(V跟ouanw 89mui'PKL ,776py4"^]Т{ә]72UrN8]xz+Yk}vG y~YWXnvFmϋ[ĆOȏXݜ4D,5NI< G9Q `x ˩=7H.eew"bFAA=kh wz;jY]ʖwwBU q#_ [cX>Ɩ#6nꣽU]ڞb.D'#!P6=g6[-N;Yim䙘LJĜ2=kGT,MMisT'߱U4K(cIk < ͎2N@ ;oMA3"g\| U:v{\|h\$'8+qקկ>\$k{t!qǓ^a}ydFGPH"t eozl 9 EBۼ=cwARC\|E_y3ܭ}\jݥޯYͥ531k +6w,FyjZt Iimo;^FJʾZ rOp\&xZ%ޫIɪE=C o.Qd 2k[:߉Dzc,>3)!l+Uyx+5/8~ʳqm'%IFyP{Epxisk=Dxpgw xm4{Ky U@]7m p56cg>5/9Fqk';J(#T.q,@\G,=((3P.֎Q,1HJ:9 qqxWoȮIrD q 'ǧ9ފUk)ӯ56ɽ6Ielgbu d[[Xe"&B3\$q'` H>:YsVK(Ic߸t[x/-.\$V_\=]s02p1(%O o_Mqng+hvDy[P@b c2kWPEXӴ9#Ǚ,lӀ2sb7(9}#TK y%K.SgC``nڶIwaj%m WLF m\|68T@]k_m,:61E_pdF1?]:Q`hk9x6_q(KjW^EռER"a(#2HwpJ޾䅮ơgs2n(랡Á@:+\|s\\|>5{eX0@YX ?^V% KgZ(sn]'9ފ䴟j:.9-m2s8gYFo;8k#y aTCrrcڀ;+:>6[ ]>++ ; qSxYuCSxDqFT(VN~f}hjo]lG:]ː3OZτo#kC{Kh.ؤI9 o9'8>ql1 ;Tɥk񝿆Z8[-}s;bۉ' KCA<"er1V[[i-b\|dfSvuzΧ:zTiw6"t6Fn S^5-bi jY!hLr( 4dt-V%ڶ٭BMɻ8<:PeEpPmj M76ۀ\|:ÿQ-ƕW6!))؊G;X?Hp:V:bw-f^VJ2\>D玴Ey^5֮_> sY6]9ҽt +ֵmRZi5[j1ܴH H ;;Vb]:Wn,հ!Yrw=y@Ium.luy cKA[Ě\jSiͪ%X˵#Iw񎣎\^ki2]iz]BI ddO.xYTOd:%F"8YWh=7s`vW7u iIH}KӘ٢eb7.zB ҫh/u+AX_dn/V2b\|ibPϮ wTW K^xm͔>lٲ[D1/ AW\$+ivsh7S\$0р8qC:+ֵ=\$]Y٭Q챪)?.B9'ץ֠!HM#+)N98oQY^&/j6m<,v·q[:rE Diswa#FZZY KCT RDSɜe0ysګi-0:Yߘ eI\|\$- s쨮^[!񎭣SڥY-팋 RcWKJ1 -S6wַwRUcQQ[\Ewk IR2REPEPEPEPEPEPEPEPEPEP\t(=g qr/#jh x{n-]i0!{W7sTfvi̖_ډm̥K)VFAux{ ?Ù4mJ7 b)yjkk=OC-. 6d u1Eq:us729u}@~1lِab BpGdMȼGj4}@ 6dmK 2l`P~.'+X͵mv4B!Il* O+a4=RotKx&f٘6drq'ںBʥA` OSh\|COmJk+\#cI9=CFuzM@Zgi37nʏ9\$G݂qd?q.m2ivM ]8TIY}gcgAj;\$Y~2C8fQ6 \9cz2} :9ӒMƳq\K)Qh9<:ף~Xhr1I\ft;s(WBYeC\$g ɺ/,c<}5MR>5Z?=S͌ qz->gAۨ}Ĭ?8zҼxs¨g>^X:2u,T\$7ZzCxEf6FŶD3(\|Yjz>[[i>SFX8ݞMvP%gkߊ;.R!3׌+jVfpV1L3A88 u#Я%Kqv"Jv{;X8 :~ͧt]>%ݴ/VO01גMz=lrYkE֡VN傒T9(]>[Km]5[瑢]Ԝ uwM)eY^HT92ZK Wٻr1k9 jͮc n"R[ rN1gkߊ;.R!3׌[Ep>ԭ\|?0h5Slb f7]A\pp@Ws˧g[[b&Kv ԱpueK}Eկnu-)-[A36Nyj}6'I\$MttzM\|9ֻ(u;/jc;H-E)~2<w8z<i:F5&\M%WVlG}ѨI<;7+~ukCcQ,Ȍf1^N1oz6oc_3Sέo+8"IZLʸ#v+jq]lȂW# p+XŃiR=C/=1:(p2=9~DO[xʒAd7HwǸ:2Cd` }9iC! 0vG"<Ú-.[c6pRK1DgdQrJ=z]e:.OYdg(; =F;ΕXh EHewON \$jy?4[-obF0J{zܛZ׈sA6BcPrIg,p`zAXjwwqH1xft؂ H#>@mFKu6 JZ6f~8R𶯨&K݅\| MT qԎ:qEsnh,HDs1\|qʔgwbq-j'ItNtKDHÄU0IP =k.!xd/(#Veimb=Ns/[ў#h\0 ;9Vbo% 8a28yvTP_xKģ:4zj(#a򛕟gma>XAcё# hW^ѵN }:퓇bqrqtZއ=ֽ&3wP+Y B8GAEpڧ%x}Ft`DKdBI_#zO5ͶQGEr'\|c(v@{/uѮ^Kb=R=KQVMC<#0 8g&t(մmJƺ}NK\|\$o5T\|6#9[RҼ7VE^L0x'wZSR v]x Paj!\| o-˔<ڹ!^?\$m:[&]H'ɅGhc++(4(-T\$j͵GA\$P%&K,/n6تWo4K? MW:磫ĖO\$mv8Boj-u[Aeo bU`7>I%/z̾7elKiG>Po(P8޻(<i%`V%9ps8W÷55Hca. FA =9ٴK?K =b]-ռPJ:A9_ 2hjȏ{^{q䢳`Ry!UTg8'8(+Z5M:CYmK\3YZyޠ162:m!I`WOErz.j:ڲZ? Μ!UTmQ<;^е \hߴK-ɑ\$\i TmxGMkUЦEm;>tg3yq]mgXhZfywwgjOw#K3bI1 (((((((((((+['is׍,-6EpsnWkVΗﱥǗc.i+#B0Of7tExp5jv);5V6v^ƛ!q^?x{<+go7YA#;\F<7G%qbܬ}J mvX ~OA'9@mX_\:=m(Rhኲ\y{O޾,c݄[gk_<[IRKbht. #:`w FqAYAI/cc)p9x56TfEC4JY8'ryǋl:64Yz C1`ˏx8B_[Z\|ۦ< :HbʥO~՗ZsZZRhۉP\$A8a9G4ytK`{X.DP ;԰Oa? 3E'Tb ë7#𦓬YZc"#C_'v㑀@E5{guium4:2#ge! Z_M/Fgq%AFpH7k╭TRgv(w9R#ҝΏ]_[͵N="k[&yzv X-H;{)x^- ?6^P=ߊݓƖk\kq&k3E-o]sE`A>ĀG5Xi4-`vޛּLK}ֱV\D)7ϲP۷k\$ v#Hv_^-HH# KՈ 9y k]7:h]_' 1{M+ yB`0'jήv5\|M%Xk;6HQAl8.y 7CTJ} bځO0F2#vhSv\$kSu2lD)͜nk ֥ZuV5M2,a}K?Zeٷ?Uw'n @ о핦ŧ_]XXIgU3ƬQRA#ێ+ȵ:PǈoF3gAo+=!)2rMzr cY0 2)Sh>3p=L9([dsS^{{o W/2dgiː@X^{xb X^iXK\$+S#2%ݤ0'^"Ԯ@5]OJգT3[G=GXA\$g3@,4%h-3Bh۷,^'O͒jgmͷ'kFsp^լ!hVqĶR[i9rA=+]xO Z]B:ٚ[ARH1bONhGQZNy_&\|1.Qdv6qEiǭp9O. k^su\|QkKj+gkvztZ&OP1YE%ηyb66\|?\$Vޒ -9 OY]&p`+Ro&V+`#\$HxxsVKiQ zy{x]D5 u}9"DE\$CoF8NsK z9wqy^a[nmzW,ζ4prĤonHg߅mu}6H'HybP' `a<ϥtU͟5M4Rq#G\$Em^%iL]g-IG9`s zG%gHV%wۼp^Ip9\$ 絭jY]4e%p}H\} \I x1E-cg-7wwlwl>QxIkz70!42r R :sx?\$n[ﳵͼOҊp #qG5=>Wvs.e#\$\B7m#q遑8tuUor,4KKf< +:@vi6\|ݥ{xWyQSx;kMRZXǽ&0e+\$Sۍ/[,uHλEomo2ilKg'6_NLmcHa Y\$(X\$=Jpj\hOA(cy\|v!A:V%Y+ h!Ki9 #`ހ5OweqiV Vv 9\$)!K8y-Z]stƎ W" &̫gwJtgKui\&_`'sր9i<eu/ͧGk\$Foσ溋iZ{hh"cJpH"Lmu}_K̲1m+83́vR lpD(ÎG@+i5]&c\s'Qh.Ht] oTX8#.ulp2'0#T%6~g.cx/b'q"0H(Fvv^'Q{5+Y\$FF `K']瑪6f1&d )#~1cV y[&\|Di1ʌN3\ >[ء[)Kϕۑn'vzD^"Il=Ey~kE5k;ɯ4b8ඒP-`YY[jn,kh\o%Ҡg9FuVe pÌP,n9m.[@ GSԁWWhZtW0ڛRn#~ A 5i:E X_ tmnKYmYL-42X7'WM:= mz8.i#02#)ګBN2rc &x./>=MFr3Gi꺣iwc fۈm[m=q:ycw&Yj5Ҙ-`MHg8I'j\\|2 6K;Y6O6q PQu-/4;G39Z<z٨\$Rt#qNqj]K^2.;9X@gO˕9I'5 [GeK[!+Śhh.`dߵh[x;kOocl#A 7d0.0{A\&MMg海w:1 a\|'j5;훤D[ϗn{v, B}6iyN@Y9`3NTvSj!tmePq I۩qZŸkY:\$w\$h+)BXYIjzM>8Es#Ld9۴ ":[hgqx"HJ?!A5񮘺fyr,tS1۱)!o\`8WuVZЁlW+vj6Iiapll\|CoGx sM'PV y"Ghp@R WnL]uK!\|FRi8\&`ɞt{KIsR[ߑVkZi:1.DH9o] Y(dhR˸3Txfrn,IM͜q/> lԉi2J3WItZf~}CĿٗc{~fLP8ߌo㯽i7SYɨ1Os5堜k,Gߵxqhgp6q#M[;GN:Vdlauǉ 1Ug~\|m& km ikY +`xNr/\|"aN+ih,>{}v%O^wcVx6#%Wێ)ȤYY0VЩA> ( ( ( ( ( ( ( ( ( ( "E @I =OjJ˛ĺ1M%©z' `8ry&\|S= ivffsIEuF}edEt`!#֪[j}͝V@^,#8@8D}N?:uPҷݧ?jQDmZ[]0H&3X s~EdOki\$,fe܀#]}N4Y[0l`zr1T]?RwY=TLq`V涚;LP6ygtϭCe Q(k+KDNdP {P 5"" P0J>ZܛiJVp63={Stu(topLV HڽN7dR襹eX=j;GkRʭy@4Vt> 絺Sh-3ˈp3wyPpG3F X}zýkT}ߍ8jU;_NLH<¡=QMY%WAOBu\$Ge!ԭv_>VF&oEÎ[nY\|p9yEWXqy0Ǘ8NIt?,~<պ`p hCV@ !GMqMfxA˸l!}dU8c8'A> syRJ`GB~=:~)d4[d3E39\$" r>հj\$egT`2q҂A5xW`P+O1[<ŊE8s{w/hSbˎI(7V[ʓCH27t;k,bKS`O@3kWLn4vY0N8;F\$IHXo; FO7}nyf=wt TVe[LV,~qHV!1i?%п?,~O^4{]Ob\$4`]Wo+s[@zZfuQu:( QԎE,z֗޲jlXy2wր/QTk 1'T47(\$up{I5yͽ~)ƗΏdP 9͜E,sĲʲ޷O ۹^@68]ޙ< =(.\|2Vvi@p2W=mk}J[HsΓ)D\p1@PjuL^D \Z͞NOUK_~Z1HvN`H!yx +k1\\$&2 Fz=)c&3,ZEY\$ EgP.j xFd;-ZR,STEf4we5K6?5 ִӣPk9βO@=ր/QT on-a%eGPH rղsj>'k2e sǍH\|h:{cGZɗ Sk61ȲYZu9S@U+_Ne[!,@}Q]F[;RC"G}q@TVquT: 9\|vqZΙ4 o7޸ VbxF;;Y!V6IouP:=Iu0jrV`Qv4~mw"j64IHʬQ@<zujvqa}ou Hd#hNUU{Ha8V::U({pڅ튒im;՛k I.'!Ie`ԓ 8'<.ǩ+6h)o`5RP3ԎqSAiwWB BIUV@w2}H1hQTcto汏Pk`V@Զj..(' dzEY(((((((((((:;[v:Mf8Z̗z3\͢O7^ ?Čƀ3=π4l oF;nucp3+.!/ci[{x"x?gUE T`0qiei"3\$ )f r 8 1ݓ#}ͬe\$ư\ooo?(g#Ӱ#c"="E\|mAuooioh6ͺM ~lG.Hq[e\|R #7˜!q\$쥱2Z@ցB `ӌ }(>g-HǸ] W'm/{CqpC%h+\$@7dA Wǥi[Ml6SHzex}+-F^7l-^0U dBt?(\|5նR~}lǶ1ڳumEmN@ L]Ocy>j~P>B}L73[I1K@W'ӵ:M/OXZbCBMq@6:e @h OqccZ_Tyo;3#O3m e%ǐbc p8"2:FeaG ҿ(MA`6Ò=klx?_V+4=&K{M2d^8UVڣkAV_ؚo"28>ʞZ7^-WoeMco>:xO?)OǶwb^IDb_-J&ڧczvcc-Žmq,+clGtPKh0GqAqG"L{f 6I]E͋ݙ`FS]]CI,qS-2k]: 'O@u%jz~EtEX.WVWCkuIN185&uOqqKc9ҡoPO}0BP}Tb:t\SqTχ,֢!R'XV؅?߭ZMbK尴.m\$2! r pj]jY5#60x(ëX#\\iVM9"NNTTg؏># jwpn\|?4h40MD /\׆<4^Xc y=0LT '94xauoJ\|+r ʁz#~ _t}K\$F\;ȸۑ]aDjVHчޣ=д-Ҭf؃InШ#(J`K_5p7kZ[]]yk Ϲ#_Ҿ%eM?c.1=/F/t;koW'ހ9ETVIu+:S~ [PhMnxي2FFxFcl- + /4]+Q;A~kח#KņF 0)UqT\s9 a7?coMk3]e hً0p;oR Ճp:ހ0]7Im Mb(TP=gK {ft'ljFO@o޻ln1#y 2@# k Dt5Q5SE ]%6m{eoq GBGbBKI?m7a]\|@֓ASY5z=Q]\| 7}>[/Le #zww.gپʞ^m3@P5<<'А@=+3UluoVNb5}vȨ >GDM7K7P(X:o}6%XU[-t.1gހ9S?=O4ۿcx8}넸]\|\ݹ} \$oeb`]c =];Nz :XjzWuآroTO!1+(b54TAMwN \|(bp\|=?B'l1@ oTſ} 6;?zgfg>n+m=5&4vT4 }Ns>^T!Qi\| ={t\|aD'-bTxKmn/DwI71\$_}iDuBۢ7(tۻ/ntI\$FG@ kx9%idh#\|;A`W/m/{CqpC%h+\$@7dA W3.b%qn\|ߍ%i6!Pzp%j GnY!.Ot!?#U3_ 3~d۟}f?4dA4\|\l.z֋fXT"!@#(?=?1(J̗zMceߊH }=wZZU݃F0<%e##ߚɸ?n4[EKxqxz4v>مʨcVla{n2gܚHl\|g[{ &(g(r:+i0Z%춱kU}q+\'\|w޶V#4xVTH^wek>1oy )㜊bMUnՂ7e>>zl;x~jt=&m9ttGV޶ g9یg<4éigZ}.ߵy ~3ƀ9kFҬ ]nX2\ 1䧗Q3)о!Rӕ8zo@&bV42mGoc{2,=v푄`G\$Z2\|S҆q=Ɠ2[5%ۆcg5@NvNփ'ŋ˘c pzEsdY#\$Аr?P+1<+z&1lwHlcϭka~5Y]WT:2AKmGEuOGYw\$Tܞq^ii\Y֖s,¨@jWo-xYF-k89u?pzV/}X4>[xhJ\#t.r0w Ti,=N386h 0 Im[3ycyX(e=OkּUk]61o#KrįwS@@}L2 o/+&DW +[ ڒ5ܭG - `xz[\{zI+02n\$܈6r@͌3Ҳuoz<(Э. ˶Ko=v@Es4MCj0[G[U2A~/',/ӵ8>>1 LlAlrz;+94,Qu lק7o7' p;gGB׼]-{as9hZy]M#(+[gƿca5E}xl.fS0sJ]W[;'ak+4QHQJU'zum{S',(4'R[%îiOMP\l~i>##C}-Sv]ONy z5t+.y#}6lIʱRr288Ou[^(m WrC).T%]_ɦxIlݙ"K1)Esshv K۸/`TX>B:c#x[-Z~K4<>DcexS㜑\$h:% K}wydclLkҀ=2uw68!MͲO;^\w`d,rvҵ彞~y:D7fd1\$(wv4{6<-\|g=aZ ˙4۽nϑSsLWv68\֣z \|T-oo>ow\|hk As: ` qTQVP" 嘱ܞMs5{wVyWO\$6͌FI79<'"qiH#y,%bĒ݉++Toyq5N3}feYep>\ `q@Eq:R4;񕼗MPZ72mz9mw˒Ltn9O_Iz../lDUd(֨{5X>XȢF\\$:sY\|tI/͝ƥ3;e~l2/P\<2u]D72 6\$+_O%7ë{!pR@󇁙9 Mz2wdvyai~<.H` Suo+QKm]d;I?7wW-xwj7v6IfU\|ϧ[j;1KZkuLJ3soG /O ]=>S5oom&PA &#O<+gy&3%SB[qU;wӊߢ ּ?mKcrOk}a!ܮa 9z(k7/UCi UH(@\'9>扣_T ܇p=Mu4P<A\jVwwwPW/B%۝C)\$=y>%M;NN-C4[%#\$hR:u7W.O)c1JB\m@Q@xj \|Ge-RAm T6aZInX)BG8]=\wrlf(.Mqk)#UΓ N l`@v]mh ?K͖KFTCIp3NI}+ñi:}jR fY\|BPkfu+\[JBvgYV}W7w [ͻMvC7xJ/I gwBll:s9s[c:m r8Nɭ(l\|Fz-٤o 2D `F^3J5K;:IWJθAPڻp8s4P=}8mKյ;6G1"lrAH0?[+uQmVxnFaH@Ԛ(jAI(ġ/ dI͋ i:ia+1p3F;bh^oXm 8xld1x9歯%5 e0P PIhzڤZmwW6Zd%HVQ@;dix>7Ե{58Q#T0AU 5Q@ӬuK;6[X.UVܠ~`AzO[z,"(^\$rIU%qA<¶x3v!ݸͽ;4>m9nm\HqFNIh#:Χew=KDs`F}:(N&i7ve]B!d%V˫KOLk3K Ӵ^Zo0?#u'븢1`_7r=ݰgn'PI\$ 6m5͖"im)Ra776=9 kZ޵Z&"EUc` :Ӡv\5ͨEs >R9IDSP?,Z+%\|P9]Ii4OPT\$anbQw p>1wڪZUK1Yy6uy\|VcX:wsjkVп/ͪM,2~HZCpH]e&8L)E I[X,"K"mnSdPAAk5 M< tU#OLԇ6m5͖"im)Ra776=9 h ,kGq,k0A<=I[/zq%vY1ƬFE28`Ȯě0OA{vfYsHcH9>[:^ƦbҮk-a =oᄆ]Fv i)+ A81QlFWRXtY"ՇԢnx8Q@ׁ4/Yɬ7Adv͠00f;cHԯ-b \l@1[P-skk\$յ=EZ 6#^}Λ+m'T!D#6grOk9(i>jYA2̮<0H2\|>բTr>S hxCIF(sjGq ;YQM/ :mQ@;tZ[8ʅb2(8ܪ98jֹmrnm!+``ɝEdjW{]TH0ARpA`Um[kZ]~VyYCw9BQ\$ŗÑK=q ϴ۵2R3Tt#'!m#[SZԄZĒrXB~i+ccWYErV MkQKgxLƱɂʌUvOw~[]ntaUUYWf ǩ>凂c7UԚΦZlOSnKFʠ0>9~}ς<%qSR; qv<(Q(GE^q3d1P4+({?!O((((((((((+ms[֥ }mW&_?iU Frsڻ 4V5f]5l4Rfx&* C)PAހWG,tD(.4.0:;IN٥evG)wzȏׇ ,ูUQD B8Mkk:=#J d"a(b~r3:\|P\|)v;i7wi%dmRGt[_ĺFGZRC\$λ \$ªnl5]g^'E1^]uM\[i6FINݲst#ҭˬi~4K额5Xf6[ sD2 \$i9ȋzXhw79cQa2 ؓESW pL-Rff0FP#y[x[]6m^88anwߞ%қPէhܹ(v\|gnx96egDIBP}?o> H\$Oabgm Z[Oj:lԵOl l0gRH9I-QCXMI'#[IWZmŻ pN)7#l\$' zX3Bst^OgoE=OR5]ɡ#9mfw\qV\|'K6mI \+\$\$/JU]oþ XfBVWFG;KscŌd[5CG,^-u2\$cwg9ږ}qdI 1O<9P6p;y銧i;L4r]8,d@듏7F֡h:L_jw~PvXb26I O3x5I4gJKؠ#mp#-Vɥ5]\$ aマҬZA𖓥Ol[jZ)K#PqC`TL\$\Y1ۉ~ oxmO.}s+ 8zaH8=rܞ2 wVh%i;(ݸD V_+kWЋ\$91 Ͱ/\$Hl?5 T6OaXOrF6y~Ρ J7gM;F^]M39'w"}괾u iV\k`cf( sj^#`s?tF=FMW:[FY^Wɷ1S<OE5WOP_>mo16?1IxơK9o游F'\a[1l8PC~OX^k3,r`'qi\|m}<8Lʳ!ulv ZZ.K-Z%RB͓TzT5\|AKH./!0=8k)t˟28^[Psl,@T\|MwqxJ;{MwoUH(z8׊}_[vVmКV.8`1}UMZ4v)iw%)&?0( ˌJ~ l精{ p!6um8fr:b6moudBfEGdcxᯄ땷NHm.'uʑvd8x\|%k&1j^A+49C lH<sj1ԯb'mbHyƮ\$\|ۇn3ZޫM>ũ%E0i 6>\KJ(ꁸ>wj(t˔2) W'qO{uW^-4ȤX-x+&X)\$6<7AeksGp mA,Eߟc{YOae[r/V,?>J:υ<%b[=ĉvbv/99x@6^ ԵZ<ۥ\C!vgtܞgWXMN4I[_-݁۳1UI&ntm7J[]J;_O ]XIrOj\|#&u}abݣJ\n mJɲƤsjiVB8b o6c3q6m-f#>Hٌ\|X,լ9].fiQH\#=s@m,oeQ-iY#۽Ys09Y+3.o nExY44PXݖS1֢i/cKxf\$<:lsj#WȚm/]5;KTF)N2F<\|EfY:zX]DOb) Nϸp9Մ׈u-bQQ)\|9 xR˧6m^k^`wɆx=TV%KWK{[5K'7q&x?X{q継yc2[9=zO l-OI#Z(rP\$grH.AX&m8E[1)MG>UNi]w:e ZQՇp zb\xwV_# ,t`\P>^ٟ\|mF~%m6A9GZ[j /BKvi.\$0#ٕS3Lck^8tO>k6tCgq<(mMDxBԭb[M>xeWs#4 a\|ܡ\$ xmoZ濠zk:[?eBl\$X' (]^@>fU dLbS2"HG̨ŀ>āS(((((((((("EI#"1ӫkxGuލ>E )LE7YAV82xY]V)kBiwvs\؉m"m p{t'I(\|+xm>;]>y ǽ7(~\|G42R8#Ψz-D\$4.ĖoI\$ZuAu\$g[ƪr)q\$Vo'V2imgyqmΖm\$m8fgu4Vx)\|O{Ccq-եH >rqЁڳŦ_5LYπОXW(R[Ȅew#sm;H{Il{AzV]xy W/ YhWIju0##YcnFj&&2iZYB6l2XrO\gjh:Yj0Ah@#< b(5,UFf,y?s3 7@:}x RK 1EK?n> kĒCSP# @[Oly1}ٶ3 \|A^+=ܬJ͜@V'KΡkYKw@'n90GkY崸KW)ys2hb>&Gv'&Mp~l\$qz\|ulu/ "y`LǽgW܌UErFEZ(y\$CZ'c\א]z,}r) WnI\$7謻 n-BMB olG= PA`F8 ͱwڮdC-2Hb¤lÞ1騮k_^]FK.X,-Yb1Q.rvCgj7Q428ԀP1HתZc؛=B4MC)H ,7NeqvyB a sVx/mA˽ a/="gwlzMEsRjM6.nU'T)\$錞Eg&׶ 7?:U5T IRdò2:+){V(-\$HWp~/G,z .R@/u1薺VėP!]Xn#8#&hh O%3Kg}zO'Z?mc/a]Z#s #XFk;Vbs\ f}ɷ~IN1uGr)o=vK=gF4V )WCBV"+"߽y y /ttuvn>PO\$M0xv>\$n`=x?ec3Z\$\nLOQל0MnQ\&tx&ʋtөDCBw0Iq:ZMVO]Do8Kc {mUEO!x'd3^އ2H>t m>k)#knY <%[88uAqp_V Υg,7\$fF' zuIPJ\$rj6:3iRw"3<\|/PQQ][Eym%R >+3wa@ԥm"@Zoif?8q\[oQ@Q@Q@Q@Q@Q@Q@Q@Q@Q@Q@yvm.wpͲ]_Zα3\$%R!qJy {ʫS r1A.x.Ŧk% B#T\@;{gJ4 ("ߺ"0v`ǥwP !.uRlujiM3/Ũ(NW3u;o úA4Ahne.i ЩE׶Crre0>ׯQ@Gן [;畭v'4lP O txKSZ\]Y2#9b)`A +9UԴu_R 6#b F2ޜ[ͦx`M=Ck#&`nnNIq %Pѭ8\| w/\V[E nҮtdJoR3EwPx:\$Zn{ Aw[Z˿`\|nBz=5'>cy䐓@wXR9&(H5omn[Y!"yQ.,yR? sk "'Hhv?2K_\F[KJ;U1\$u+?EtU}jZE%ɏި0>\rXKr!uMG:<M.+XcgH88ci`8IoCDt/J)i'ѣ, xQ@sl'\|C&Pk6S06ȟ'\|?GKkGc a]f;X!P,:g[ء[)Kϕۑn'v0qWž.y-n)aki#Y` Ve»(hx쵽-ջp@d?e gsҳ\|=mxj^aOM6I'rFL ;' kh3izޱ{rSc* #jܠ/Я Erܹdfе]#Xx5 4䲖0H"epْ q^ErG6[GQћ9KǕ('{0= ki\B=zVAj5 D"2&H>WQI,< 4ܪ ~D͗MVL'Y ]Ay)jIO":~/-[c (30^szs^E"tV\|93\$3v,\9>Z(mD[#3n!g-U%ê {he>ZyR.>^X볢<6t_Ko'66e3\$ƒC+<犊Ѯ'oGX+y X6cnB צQ@^4iVͫY8 W? hmlmЖO,D;䑌vg9(ԯd\|?ߦC6{8 # Ildϱu-Nkvdo_5eH,G 0Np s^Exͥ24SE٤^} g[WLѵ (&7UI8;bu;x(]s&E2VdD\$mPn\$NK`Vgt\|?[Ǽ8%y%v6!,.fbH .X 32zTZd^~cs=,ffh`6\$m 4i,{YeY^e6'{՚]G_oam 5-ibpBB0cdcE\|Sm #f0H9EEmo6)XA@xh ( ( ( ( ( ( ( ( ( ( }ic`Hm\q\_ f4Yu"/{C y_ohm_M]I4濶λ<_PH@jwvA}SĶ-oK962Z4)}onwڹ'n8"<\|e.Ԑ1ZԵOX1˩9{S@kjj)5uܖQ2G#lzޕ,sjٹKB/\$͇Eˣe!©<`@=Au]FV<it{Su2m ""Ez at<7q8fQ>27y&ЯeXKVXFr`5w:w{h}6ygvꇂI =Bu͘)&Kv-Rݧ.cr㪏V9Fy=?hO&Zkc?߻=ע^ NM}\|͇n=-kLZn32(# r1r\$WeZMĖqpp ßq n[#sg࿆Enǜ{~Y[ 8xW,\$̊?H~[@Ӟ u[\$ `p',x&E-Yg_יhO7]ϓy_{b5^H=rC{ ,RHO@)vc?6 ^>7 p`m"anC>9v=Ql:._Iqas_^XhpEXg-#%9Pd#!=@E6 l&}cV;T<J-7IM^+hE\L~b4b}v5)9IցxXiz~Nvk\(n>n7!S5v(Z>ڢI3 fzEޑeBYi# Y}G@nm/Px^Mh?4+xW%6blZ2qHZP-M-KXa(W fP7g=@66tְk6cy\c5rmڵ[wFߊeIkiHIxbQ=rPm]9{Oxȅ3fМ \$ܓn\$Kq ET,,eyQ#H ̨0V袀 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (?	2019-02-17 11:56:42	{"extraction_info": {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8950129151344299, "perplexity": 288.93928932497454}, "config": {"markdown_headings": true, "markdown_code": true, "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-2019-09/segments/1550247481992.39/warc/CC-MAIN-20190217111746-20190217133746-00638.warc.gz"}
https://www.inchmeal.io/sicp/ch-3/notes.html	# SICP, summary and notes ### Chapter 3, Modularity, Objects, and State We want our programs to be modular so that they can be divided into coherent parts that can be separately developed and maintained. One powerful way to do this is to model the programs based on the structure of the system being modeled. This is based on the premise/hope that when we do this then extending the program does not require any structural changes in the system but only addition of the correspnding analogs of the new things/objects we want our system to extend with. There are two world views of the structure of the systems: • One organisational strategy is to view the system as a collection of objects. • Another way is based on streams of information that flow in the system. Both approaches have their own challenges - With objects the problem is how an object can change and yet maintain its identity. This causes to move away from the substitution model and we need a different model for evaluation. For stream approach we need to decouple simulated time in our model from the order of the events that take place in the computer during evaluation. #### Object has state We view the world consisting of many independent objects and each having their states which change over time. But in general objects are not completely independent and they influence the states of each other by interactions. Or we have subsystems in which objects in one subsystem are changing the states of each other or tightly coupled but other subsystems. Thus we decompose our sytem into computational objects that model the actual objects of the system and each computational object have local state variables describing the actual object’s state. We need a way to change these local state variables - assignment. To change the variable special form set! is used for eg: (set! balance (- balance amount)). Since this form do not return any value, we need another special form (begin (exp-1) (exp-2) ... (exp-k)) to combine multiple expressions and return the result of last expression. Introduction of assignment raises the problem of evaluating such programs using substitution model and a new model is needed to understand such programs - because when we simply substitute, the variables current value may have been changed in some earlier expression evaluated and we end up using old value. There can be few variations on how we model such objects that change state. One straight forward way is: Now we can create objects as (define W1 (make-withdraw 100)) or (define W2 (make-withdraw 200)). Another way is message-passing: #### Benefits of assignment Without assignment an object local state can not be changed. If an object can not change its state then every time one uses this object then the user must pass the values of the previous state of this object. This makes it difficult for the external user to use this object. For example in case of random number, if an implementation requires to remember the last number generated to generate the next number then the users of random-number generator will always require to pass the last number generated. But with assignment this problem is resolved and object remembers its state. Thus the users do not need to remember the last number generated - thus making the program loosely coupled! This may tempt us to think of this strategy can help in making the program modular but :) #### Pitfalls of imperative programming First as we already saw, that it causes complexity in evaluation models - we can not use simple substitution model. Another more fundamental problem is associated with sameness or change. Suppose we have two objects W1 and W2 created using the same initializer (define W1 (create-account-with-balanace 100)). Are these objects same? We can say this only in a context/time but can not be said in general that they are same(in context of assignment). Referentially transparent language supports the concept of equals can be substituted for equals. But note that this can not be done in above case as equality/sameness is bound with context/time. Thus we can not determined effectively when objects can be substituted and how we can simplify expressions by substituting equivalent expressions. It also becomes more difficult to how we check for equality? Two account with same balance are equal? Or Two variable pointing to same account are equal. In the former case - if we want to consider them equal then we need to be sure of the places where any of the two account is changed. Also, it is more difficult to understand what is the meaning of sameness. Two rational numbers are same if both numerators and denominators are same and as soon as we change either numerator or denominator then it changes the rational number. But this is not the case with account - when the balance of an account is changed by withdrawing money - it remains the same account even after withdrawing money. Thus assignment/mutability introduces a more fundamental problem - where a compound data object has an identity that is something different from the pieces of which it is composed. Quoting directly from book: This complication is a consequence, not of our programming language, but of our perception of a bank account as an object. We do not, for example, ordinarily regard a rational number as a changeable object with identity, such that we could change the numerator and still have the same rational number. Another problem is order of assignment, for example: Now, we need to be sure that which statement comes first. Consider when program gets bigger - this problem also becomes more difficult and a programmar must take care that he should use the value at appropriate place. Now complement this problem with concurrent processes! #### Environment model of execution We need a place - environment - where a variable can be stored and refered. Environment is a sequence of frames and a frame is a table structure holding name-value pairs. For eg: (define x 3) creates a bindng in the of x with 3. A new frame is created when any procedure is invoked/evaluated. The new frame points to the enclosing environment. Thus every frame points to an enclosing environment except the first one - which we call global environment - note that global environment will contain only one frame pointing to no other frame. A procedure is a pair of name and its body - lambda expression. Procedure name is a variable whose value is just text containing the body of procedure. This text is accessed via this variable acting as a pointer. Note that unlike other variables - this variable’s value is not stored in frame but stored somewhere else and accessed via this variable. When programs starts executing by default a global enironment is created. Now when a procedure is invoked from this environment it creates a new frame with a pointer to the global environment. And if now again there is another procedure defined in this frame is invoked from this environment - then again a new frame is created with a pointer to the previous environment(note - its not global but the enclosing environment). Now if a variable is accessed in an environment - then its first looked in the last/top frame of an environment - if found value is returned, else the variable is looked up in the enclosing environment recursively till the value is found or global environment is reached - in that case error is reported. This method of variable lookup thus shadows the global/enclosing variables with the local variables if global/enclosing variable and local variable share the same name. Note that when a new frame is created the enclosing environment is the one in which the variable containing the procedure name is bound. And thats it! - Check figure 3.5 to get a clearer picture. Now frames work just as the repository of Local state - I think this is simple to see - so skipping details. #### Modeling with mutable state Till now when we were working on data abstraction in chapter-2, we were using constructors and selectors. Now we have one more aspect - mutators - to change the state or local variables of the object. There are few interesting examples discussed - Queue, Tables, Digital Simulator, and Constraints Propagation. They can be skipped - perhaps reading Digital Simulator is good learning that how states can help in managing abstractions of gates and how we combine multiple gates to create more complex circuits and simulation - its a good read. Constraint Propagation is also good read but can be skipped - The central idea is that if we have a farmulae like temperature unit converter : 9C = 5(F-32) then in general it requires two different implementations one when we have C and want to compute F and vice versa. But using the idea presented here there can be a single place with contraints specified. Both examples/ideas, Simulator and contraints propagation were new to me - specially the constraints one. #### Concurrency: Time Is of the Essence Two aspects of concurrency: • To make programs more modular - in our effort to map the real world with our systems. • To make programs more efficient or to utilize parallel computing. Since this is a part of state chapter - only we talked about shared memory approach for dealing with concurrency. How the assignment has added complexity in normal single process(single threaded) programs because of enforcing the order in statements and how this complexity has multiplied in parallel executing processes. The central idea was that assignment has introduced time in our programs - we want to know which operation executes first - the order. If one is not familiar with mutex, semaphores, monitors then this section is a good introduction. However, approaches like message passing are not coverd yet - probably in a later section/chapter. Also, I am not sure if MIT scheme provides all the libraries to help executing the code and exercises in this section - Perhaps this library can help - mit-concurrency-lib - which I got to know from stack-overflow. I tested only one or two programs using it but later did not use it as it was quite time consuming to test the parallel execution. There is a similarity pointed out in the last part of this section with the Theory of Relativity. The complexity in our system is because of dealing with time and state for communicating between various procedures to establish an order - is somewhat similar to Theory of Relativity - where- Speed of light is a fundamental constant connecting two entities time and space. So perhaps we might encounter a similar fundamental idea(like speed of light connects time and space) in dealing with time and state. #### Streams I learned streams first time. It’s a brilliant approach. Too bad for me to learn about it this late - after a decade in programming! As we saw assignment introduces an order or time in our programs. One statement can give different output at different time. Streams are another way to model our system to make it independent of time! Instead of make time part of our system - we can define our variables, say x, in terms of time based procedures, like x(t). Now we no longer need to track the order or time to know a value of a variable. But we just ask the value for a particular time. Awesome! How do we do it - streams - which contain an ordered set of events/values and we can go as further in it as we want to get the value for a particular time. The streams are just lists which contain values for the entire time history. Streams are are nothing special but just delayed lists. Ofcourse we should not compute all values unless they are needed. Thus we have delayed evaluation and the next element in the list is not evaluated unless it is accessed. Also to avoid re-evaluation we do memoization so that when same element is accessed again, we just return the stored value. An important point to understand is that mixing the streams(or more accurately delayed evaluation) with assignment is a recipe of disaster. See ex-3.51, 3.52. In book’s own words: Unfortunately, including delays in procedure calls wreaks havoc with our ability to design programs that depend on the order of events, such as programs that use assignment, mutate data, or perform input or output. Delayed evaluation gives us a powerful way and even we can sort of have Infinite Streams. In book, it is shown how to implement infinite series(eg: power series) using Infinite Streams. This is quite powerful way of doing things and I can not imagine such elegant code for writing the series(ex-3.59): Try doing some stream manipulations like adding two streams, scaling a stream etc. There is an example where we find square-root of a number using streams for performing iterations. The further we go down the stream, more accurate the results get. Another way to look at streams: computational analogs of signal processing systems. We can model signal processing systems using streams. We can also use streams to implement signal processing systems with loops. The problem is the one definition of stream may depend on another stream not yet defined. For eg: The not yet defined stream have first element available but requires a stream defined earlier to compute the next element. For example, to solve first order differential equation: $\, f(y) = \frac {dy} {dt} \,$, we have the following procedure: Here y depends on dy and dy depends on y! Many implementations of scheme(including MIT scheme) do not provide any direct way of writing such definitions. Note the use of delay in the above procedure. This way we can delay the evaluation of arguments, we call this as delayed argument. This is actually the same thing as normal order evaluation that we learned in chapter-1. But in general, most languages do not provide normal(delayed) order evaluation, so we need constructs like delay to implement delayed argument. Well, we may wonder here why not have normal order evaluation by default instead of using delayed arguments. The problem is assignment and delay do not go well together as noted in ex-3.52. Now, it may seems like streams solve all the problems and functional programming is the way to go! Read this section: A functional-programming view of time - which shows with an example that while dealing with concurrency in a shared bank account, streams do not solve the problem of time completely and we again get into the problem of ordering. Apart from that, in my opinion, the other issue is of performance. Defining such powerful abstractions are indeed great for the programmar but the compiler may not produce efficient code. To summarize, let me copy the last paragraph directly from the book: We began this chapter with the goal of building computational models whose structure matches our perception of the real world we are trying to model. We can model the world as a collection of separate, time-bound, interacting objects with state, or we can model the world as a single, timeless, stateless unity. Each view has powerful advantages, but neither view alone is completely satisfactory. A grand unification has yet to emerge. #### Interesting/Conceptual Exercises Environment evaluation: 3.11, 3.16, 3.17 Modeling with mutable state: 3.23(deque) If read the section of simulator - 3.32 is a short but conceptually good exercise. For constraints propagation - 3.34 and 3.37 are also short but good. For concurrency - 3.43(concurrency issues), 3.47(implementing mutexes), 3.48(deadlock) Streams: 3.50(conceptual), 3.52(demonstrate issues when delay and assignment is combined) Infinite Streams: 3.56, 3.59(power series), 3.63(important, conceptual), 3.66(a bit difficult but interesting), 3.67(short, practice), 3.70(interesting, practice). Streams as signals: 3.74 to 3.76(short, practice), 3.78(differential equation and delayed argument). Streams for modeling vs Objects for modeling: 3.81,3.82(both conceptual)	2019-02-24 01:18:17	{"extraction_info": {"found_math": true, "script_math_tex": 1, "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": 0.552400529384613, "perplexity": 932.6925421967572}, "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-2019-09/segments/1550249569386.95/warc/CC-MAIN-20190224003630-20190224025630-00095.warc.gz"}
https://qwtel.com/posts/finance/how-to-set-specific-probabilities-using-the-lmsr/	# How to Set Specific Probabilities Using the LMSR? How many shares do you need to buy/sell in the in a prediction market using the Logarithmic Market Scoring Rule (LMSR) to change the price/probability of an asset to a specific value? This post assumes knowledge about prediction markets and market scoring rules. I think the original paper by Robin Hanson provides a good introduction. There are basically two approaches to buying/selling the right amount of shares to reach a certain probability. Method #1 is the sledgehammer approach, Method #2 is elegant, but tedious for more complicated price functions. ## Method #1 Doing some kind of binary search until the price is “close enough”. I’ve written some example code in Clojure: (defn set-to-prob [q i prob] (loop [lower (magically-find-lower-bound q i prob) upper (magically-find-upper-bound q i prob)] (let [q_i' (/ (+ upper lower) 2) q' (assoc q i q_i') p' (p i q')] (if (close? 0.1 p' prob) q' (if (> p' prob) (recur lower q_i') (recur q_i' upper)))))) Other than inelegance, this has the obvious flaw that it’s difficult to find reasonable upper and lower bounds for the number of outstanding shares. I used a simple implementation for the upper (lower) bound that added (subtracted) a multiple $m$ of the liquidity parameter $b$, but it was always possible to force an infinite loop by using an input $\vec{q}$ so that $\max_{i, j} \vert q_i - q_j \vert > m \, b$. ## Method #2 It’s possible to solve analytically for the number of shares $q_i$ to move the price to a certain probability $p_i$. I’ve actually found the solution in this GitHub repository, but there was no hint as to how the solution was arrived at (maybe obvious when you have better math skills 🙁). However, eventually I was able to figure it out 😏. So if you are like me and need to have every baby step laid out, here they are: We know the price function for the LMSR is $p_i(\vec{q}) = \frac{e^{q_i/b}}{\sum_j{e^{q_j/b}}}$ For a specific price/probability $p_i$ for outcome $i$, it must be the case that $p_i = \frac{e^{q_i/b}}{e^{q_i/b} + \sum_{j \neq i}{e^{q_j/b}}}$ where $\sum_{j \neq i}$ iterates over all indices in $\vec{q}$ except for $i$. Using some math magic: \begin{aligned} p_i \left(e^{q_i/b} + \sum_{j \neq i}{e^{q_j/b}}\right) &= e^{q_i/b} \\[2em] p_i\, e^{q_i/b} + p_i \sum_{j \neq i}{e^{q_j/b}} &= e^{q_i/b} \\[2em] p_i \sum_{j \neq i}{e^{q_j/b}} &= e^{q_i/b} - p_i\, e^{q_i/b} \\[2em] p_i \sum_{j \neq i}{e^{q_j/b}} &= e^{q_i/b}(1 - p_i) \\[2em] \frac{p_i}{1 - p_i} \sum_{j \neq i}{e^{q_j/b}} &= e^{q_i/b} \\[2em] \log{\left(\frac{p_i}{1 - p_i} \sum_{j \neq i}{e^{q_j/b}}\right)} &= \frac{q_i}b \end{aligned} Which finally leads to $q_i = b\ \log{\left(\frac{p_i}{1 - p_i} \sum_{j \neq i}{e^{q_j/b}}\right)}$ which is the number of shares that element $i$ in the quantity vector $\vec{q}$ needs to be changed to, in order for the price of contract $i$ to reach price $p_i$. For the number of shares $\Delta q_i$ to buy/sell, the current amount of shares $q_{i,t}$ needs to be subtracted $\Delta q_i = q_i - q_{i,t}$. Now somebody needs to do the same for the the liquidity-sensitive LMSR$p_i(\vec{q}) = \alpha \log\left({\sum_j{e^{q_j/b(\vec{q})}}}\right) + \frac{\sum_j{q_j \, e^{q_i/b(\vec{q})}} - \sum_j{q_j \, e^{q_j/b(\vec{q})}}} {\sum_j{q_j} \sum_j{e^{q_j/b(\vec{q})}}}$ where $b(\vec{q}) = \alpha \sum_j{q_j}$ How hard can it be?	2021-07-23 23:21:39	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 24, "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": 0.9859126806259155, "perplexity": 1990.0844663072658}, "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-2021-31/segments/1627046150067.51/warc/CC-MAIN-20210723210216-20210724000216-00413.warc.gz"}
https://www.taylorfrancis.com/chapters/mono/10.4324/9781315049366-12/consonant-front-vowel-assimilation-elizabeth-hume?context=ubx&refId=06842da1-f97e-48e6-9b4a-40996e5fbb7c	## ABSTRACT Any work dealing with the interaction of front vowels and consonants would clearly be incomplete without discussing palatalization, undoubtedly the most commonly attested rule involving these segments. Palatalization is generally used as a cover-term to refer to the various assimilations that consonants undergo in the context of front vowels (see e.g. Bhat 1978). These may result in, for example, the addition of an i-like articulation to a consonant, e.g. /k + e/ — [kie], or a complete change in the_major place of articulation of the consonantal target, e.g. /k + e/ — [tje].	2022-10-01 05:20:13	{"extraction_info": {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8499699234962463, "perplexity": 5289.314436340021}, "config": {"markdown_headings": true, "markdown_code": true, "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-2022-40/segments/1664030335530.56/warc/CC-MAIN-20221001035148-20221001065148-00572.warc.gz"}
http://math.stackexchange.com/questions/152016/is-the-coordinate-chart-always-a-biholomorphism	# Is the Coordinate Chart Always a Biholomorphism? Let $X$ be a Riemann surface with complex structure $\{(U_i,\phi_i)\}$. Is it the case that $\phi_i:U_i\rightarrow V_i$ is a biholomorphic map in the sense of Riemann surfaces? - The answer is obviously yes. Indeed we only need check that $\phi_i^{-1}\phi_i:V_i\rightarrow V_i$ is holomorphic in the classical sense, which it trivially is.	2015-04-18 20:16:52	{"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": 0.9830092787742615, "perplexity": 100.89454337814928}, "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-2015-18/segments/1429246636104.0/warc/CC-MAIN-20150417045716-00198-ip-10-235-10-82.ec2.internal.warc.gz"}
http://texnicalstuff.blogspot.com/2011/12/mathjax-updates.html	## Monday, December 12, 2011 I've just updated the design of the blog. I think it looks negligibly better. I'll also try to write more here rather than on my notebk's wiki. I posted some exercises for Sean [pdf] which are fun calculus problems. I suspect what I'll do next is write up my notes on differential geometry from Osserman's course I audited a couple years ago, as well as my notes on algebraic topology (from Dr Schwarz's courses). Then I'll work on spin geometry, "commutative geometry", analysis, and so on. By "commutative geometry", I really mean spectral triples using commutative rings (Here I am sloppy, but meh I am a sloppy person!). It is also called Differential Calculus over Commutative Algebras, although there are no real texts on the subject... I'll have to review the prime spectrum of the commutative ring $C(M)$ of continuous functions on a topological space $M$ and how it relates to the topology of $M$. If we let $M$ be a smooth manifold, then we work with $C^{\infty}(M)$ — I am told there is a theorem due to Shields which says if $M$ and $N$ are smooth manifolds and $C^{\infty}(M)$ is isomorphic to $C^{\infty}(N)$ then $M$ and $N$ are diffeomorphic. How interesting! But I cannot find this theorem... At any rate, vector bundles over $M$ may be considered by looking at the projective modules over $C^{\infty}(M)$. We consider algebraic analogs for sections, vector fields, covector fields, and so on. It is really quite cute. Noncommutative geometry is similar in setting up a dictionary between "algebraic stuff" and "geometric stuff", at least how Connes approaches it. It's just that the "geometric stuff" we work with is a smooth Riemannian manifold $M$ equipped with a spin structure, we consider spin bundles over it, and so on. ## MathJax I am experimenting with MathJax on blogger, so bear with me people. My reference for this subject is the thread at stackexchange on it. Consider the "Harmonic Series" $\sum^{\infty}_{n=1}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots$ which diverges famously. MathJax uses the $...$ or $...$ for "inline mathematics" and $...$ or $$...$$ for "display math", e.g., the mathematics produced above. I don't know whether to keep it or not, because MathJax is sluggish on some computers. But it is the "way of the future", like blimps and autogyros.	2017-10-19 07:07:09	{"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": 3, "mathjax_display_tex": 1, "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": 0.7670243978500366, "perplexity": 589.5647277231661}, "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-2017-43/segments/1508187823255.12/warc/CC-MAIN-20171019065335-20171019085335-00469.warc.gz"}
https://bird.bcamath.org/handle/20.500.11824/13/browse?rpp=20&sort_by=1&type=title&etal=-1&starts_with=E&order=ASC	Now showing items 67-86 of 263 • Effect of General Cross-Immunity Protection and Antibody- Dependent Enhancement in Dengue Dynamics (2022-02-13) A mathematical model to describe the dynamic of a multiserotype infectious disease at the population level is studied. Applied to dengue fever epidemiology, we analyse a mathematical model with time delay to describe the ... • Effect of Tissue Elasticity in Cardiac Radiofrequency Catheter Ablation Models (2018) Radiofrequency catheter ablation (RFCA) is an effective treatment for different types of cardiac arrhythmias. However, major complications can occur, including thrombus formation and steam pops. We present a full 3D ... • Electrochemical Potential Derived from Atomic Cluster Structures (2016-01-01) Based on the atomic cluster structures and free electron approximation model, it is revealed that the electrochemical potential (ECP) for the system of interest is proportional to the reciprocal of atomic cluster radius ... • Enhancing sampling in atomistic simulations of solid state materials for batteries: a focus on olivine NaFePO$_4$ (2017-03-07) The study of ion transport in electrochemically active materials for energy storage systems requires simulations on quantum-, atomistic- and meso-scales. The methods accessing these scales not only have to be effective but ... • Enhancing Sampling in Computational Statistics Using Modified Hamiltonians (2016-11-15) The Hamiltonian Monte Carlo (HMC) method has been recognized as a powerful sampling tool in computational statistics. In this thesis, we show that performance of HMC can be dramatically improved by replacing Hamiltonians ... • Estimation of age-specific rates of reactivation and immune boosting of the varicella zoster virus (2016-12-31) Studies into the impact of vaccination against the varicella zoster virus (VZV) have increasingly focused on herpes zoster (HZ), which is believed to be increasing in vaccinated populations with decreasing infection pressure. ... • Ethics of a partially effective dengue vaccine: Lessons from the Philippines (2020-07) Dengvaxia, a chimeric yellow fever tetravalent dengue vaccine developed by SanofiPasteur is widely licensed in dengue-endemic countries. In a large cohort study Dengvaxia was found to partially protect children who had ... • An even simpler understanding of quantum weak values (2018-01) We explain the properties and clarify the meaning of quantum weak values using only the basic notions of elementary quantum mechanics. • Excitation of plasmons in a two-dimensional electron gas with defects by microwaves: Wake-field method (2011-12-31) We develop an analytical method to find plasmons generated by microwaves in a two-dimensional electron gas with defects. The excitations are expressed in terms of the wake field of a charged particle moving in plasma. The ... • Existence, Uniqueness, and Numerical Modeling of Wine Fermentation Based on Integro-Differential Equations (2022) Predictive modeling is key for saving time and resources in manufacturing processes such as fermentation arising in food and chemical manufacturing. To make reliable predictions, realistic models representing the most ... • The Experiment Data Depot: A Web-Based Software Tool for Biological Experimental Data Storage, Sharing, and Visualization (2017-09-30) Although recent advances in synthetic biology allow us to produce biological designs more efficiently than ever, our ability to predict the end result of these designs is still nascent. Predictive models require large ... • Exploring Li-ion conductivity in cubic, tetragonal and mixed-phase Al-substituted Li7La3Zr2O12 using atomistic simulations and effective medium theory (2019-08-15) Garnet Li7La3Zr2O12 (LLZO) is a promising solid electrolyte candidate for solid-state Li-ion batteries, but at room temperature it crystallizes in a poorly Li-ion conductive tetragonal phase. To this end, partial substitution ... • Extracting S-matrix poles for resonances from numerical scattering data: Type-II Padé reconstruction (2011-12-31) We present a FORTRAN 77 code for evaluation of resonance pole positions and residues of a numerical scattering matrix element in the complex energy (CE) as well as in the complex angular momentum (CAM) planes. Analytical ... • Extreme brain events: Higher-order statistics of brain resting activity and its relation with structural connectivity (2015-12-31) The brain exhibits a wide variety of spatiotemporal patterns of neuronal activity recorded using functional magnetic resonance imaging as the so-called blood-oxygenated-level-dependent (BOLD) signal. An active area of work ... • Flux-Enabled Exploration of the Role of Sip1 in Galactose Yeast Metabolism (2017-05-31) 13C metabolic flux analysis (13C MFA) is an important systems biology technique that has been used to investigate microbial metabolism for decades. The heterotrimer Snf1 kinase complex plays a key role in the preference ... • Fock-space approach to stochastic susceptible-infected-recovered models (2022-07-25) We investigate the stochastic susceptible-infected-recovered (SIR) model of infectious disease dynamics in the Fock-space approach. In contrast to conventional SIR models based on ordinary differential equations for the ... • From chemical gardens to chemobrionics (2015-12-31) Chemical gardens in laboratory chemistries ranging from silicates to polyoxometalates, in applications ranging from corrosion products to the hydration of Portland cement, and in natural settings ranging from hydrothermal ... • Functional connectivity of EEG signals under laser stimulation in migraine (2015-12-31) In previous studies, migraine patients showed abnormalities in pain-related evoked responses, as reduced habituation to repetitive stimulation. In this study, we aimed to apply a novel analysis of EEG bands synchronization ... • Functional Geometry of Human Connectomes (2019-08) Mapping the brain imaging data to networks, where nodes represent anatomical brain regions and edges indicate the occurrence of fiber tracts between them, has enabled an objective graph-theoretic analysis of human connectomes. ... • Geometry and temperature dependent thermal conductivity of diamond nanowires: A non-equilibrium molecular dynamics study (2010-12-31) Using non-equilibrium molecular dynamics methods, the analysis of geometry and temperature dependent thermal conductivities of diamond nanowires is carried out. It is found that at the same temperature conditions, thermal ...	2022-11-29 00:25:30	{"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": 0.3684842586517334, "perplexity": 4332.659365931911}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "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-2022-49/segments/1669446710684.84/warc/CC-MAIN-20221128235805-20221129025805-00346.warc.gz"}
https://socratic.org/questions/in-the-presence-of-an-active-metal-do-bases-usually-release-hydrogen-gas	# In the presence of an active metal, do bases usually release hydrogen gas? Jan 15, 2018 Zn + 2NaOH --> $N {a}_{2} Z n {O}_{2} + {H}_{2}$	2019-11-21 10:21:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 1, "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": 0.3941774368286133, "perplexity": 13706.358072019275}, "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-2019-47/segments/1573496670770.21/warc/CC-MAIN-20191121101711-20191121125711-00171.warc.gz"}
https://www.mathstunners.org/problems/3420.29	## 3420.29 – Midline Triangle In a random triangle ABC, D is the midpoint of AB; E is the midpoint of BD; and F is the midpoint of BC. Suppose the area of $\triangle$ABC is $96$. Then the area of $\triangle$AEF is: 1. $16$ 2. $24$ 3. $32$ 4. $36$ 5. $48$ Solution The figure described in the problem is drawn below. Note that if you halve the base of a triangle and don't change the altitude, you halve the area. Therefore, \begin{aligned} \triangle\text{ABC} &= 96 \\ \triangle\text{ABF} &= 48 \ (\text{base being halved: BC)}\\ \triangle\text{ADF} &= 24 \ (\text{base being halved: AB)}\\ \triangle\text{BDF} &= 24 \ (\text{base being halved: AB)}\\ \triangle\text{DEF} &= 12 \ (\text{base being halved: BD)}\\ \triangle\text{AEF} &= \triangle\text{ADF} + \triangle\text{DEF} \\ &= 24 + 12 = 36. \end{aligned}	2023-03-26 15:59:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 9, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.0000100135803223, "perplexity": 3807.3007597990027}, "config": {"markdown_headings": true, "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-2023-14/segments/1679296945473.69/warc/CC-MAIN-20230326142035-20230326172035-00435.warc.gz"}
http://halweb.uc3m.es/esp/Personal/personas/icascos/eng/COVID19_eng.html	# 1 Introduction On May 13, 2020, the Instituto de Salud Carlos III published a report on the first round of the national epidemiologic study on the infection caused by the SARS-COV-2 (ENE-Covid19). The report contains interesting results and is very clearly written. These notes have been prepared to help you reading and understanding it together with some other epidemiological and statistical studies about the coronavirus COVID-19 pandemic. No basic prior statistical knowledge is required, but having followed an undergraduate Introductory Statistics course would be of some help. # 2 Biomedical screening tests and associated jargon The terms prevalence, sensitivity, and specificity are stardard ones in the jargon of Biomedical Sciences. • The prevalence is the percentage (or porportion) of individuals in a population with some given medical condition. • Example 1: The current prevalence of IgG antibodies against SARS-Cov2 in Spain is estimated to be $$5\%$$ (ENE-Covid19). • The sensitivity (True Positive Rate, TPR or Positive Percent Agreement, PPA) is the proportion of infected individuals that are identified as such. • The specificity (True Negative Rate, TNR or Negative Percent Agreement, NPA) is the proportion of healthy individuals that are identified as such. In the chart below, you can find a black circumference for each screening test run on an individual withouth antibodies in Example 3, and a grey one for each screening test run on an individual carrying antibodies in Example 2. If the result of the test was positive, there is a ‘+’ inside the circumference. The chart above is not a representative one for the Spanish population, since there are 3.63% grey circumferences (representing individuals with antibodies), while the percentage of Spanish individuals with antibodies is roughly $$5\%$$, see Example 1. Below, you can find a new chart at which the percentage of grey circumferences is $$5\%$$, while the ratio of +’s among black circumferences is roughly the same as in the chart above, and matches the specificity of Example 3. Also the ratio of +’s among grey circumferences roughly matches the sensitivity of Example 2. The sensitivity and specificity measure the accuracy of a screening test, but for an individual that has undergone one of such test and received a positive result, what really matters is the probablity that she has the disease (the so-called Positive Predicted Value, PPV). In order words, she knows that she is associated with a circumference with a ‘+’ inside it, but is her circumference a grey one? Alternatively, if she tested negative, what would matter her is the probability that she does not have the disease (Negative Predicted Value, NPV). Observe that the probability that she carries antibodies despite she tested negative is its counterpart, $$1-\text{NPV}$$. # 3 Probabilisitic notation Let us now introduce some standard probabilistic notation: • Denote by $$D$$ the event that an individual carries antibodies (clearly $$D$$ is for disease). Then, its probability $$P(D)$$ represents the chance that an individual selected at random has antibodies, and it matches the prevalence in the population. • Example 1: If $$D$$ stands for carrying IgG antibodies against SARS-Cov2, then its probability is $$P(D)=0.05$$. • Denote by ‘$$+$$’ the event that the result of a screening test is positive. The sensitivity of a screening test is the conditional probability of a positive result given that the individual carries antibodies, which we denote as $$\text{Sensitivity}=P(+\|D)$$, where the vertical line ‘\|’ represents a conditional probability (the event to its right is the available information, while the one to its left is the event whose probability we want to compute). It represents the chance that an invidual that carries antibodies tests positive. • Example 2: $$P(+\|D)=0.9667$$. • Denote by ‘$$-$$’ the event that the result of a screening test is negative. The specificity of a screening test is the conditional probability of a negative result given that the individual does not carry antibodies, which we denote as $$\text{Specificity}=P(-\|\overline{D})$$. By $$\overline{D}$$ we represent the complementary event to $$D$$, that is, the individual does not carry IgG antibodies. • Example 3: $$P(-\|\overline{D})=0.993$$. • The probability that an individual who tested positive actually carries antibodies is $$\text{PPV}=P(D\|+)$$. • The probability that an individual who tested negative does not carry antibodies is $$\text{NPV}=P(\overline{D}\|-)$$. # 4 Bayes’ formula (and total probability rule) We now want to compute the probability that an individual carries antibodies given that she tested positive, $$P(D\|+)$$. In the chart with the representative sample, this corresponds to restricting to circumferences with ‘+’ inside them (positive tests) and computing the ratio of grey circumferences (individuals with antibodies) among them, that is, $P(D\|+)=\frac{P(D\cap +)}{P(+)}\,,$ where by the intersection $$\cap$$ we mean that the two events must occur (positive test and antibodies). By parts, the numerator $$P(D\cap +)$$ is the portion of grey circumferences with ‘+’ inside them in the representative sample, and it is computed after multiplying the proportion of individuals with antibodies, $$P(D)$$ (fraction of grey circumferences, prevalence), times the ratio of positive tests among them, $$P(+\|D)$$ (ratio of circumferences with ‘+’ inside them among the grey circumferences, sensitivity). The denominator is computed after a slightly longer procedure (called the total probability rule). In first place, the ratio of positive results is split into those positive results associated with individuals that carry antibodies and those that associated with individuals without antibodies. These two probabilities are later computed as the product of the propotion of individuals with (or without) antibodies times the ratio of positive tests among each of the two groups, \begin{align} P(D\|+)=\frac{P(D\cap +)}{P(+)}&=\frac{P(+\|D)P(D)}{P(+\|D)P(D)+P(+\|\overline{D})P(\overline{D})}\\ &=\frac{P(+\|D)P(D)}{P(+\|D)P(D)+(1-P(-\|\overline{D}))(1-P(D))}\\ &=\frac{\text{Sensitivity}\times\text{Prevalence}}{\text{Sensitivity}\times\text{Prevalence}+(1-\text{Specificity})\times(1-\text{Prevalence})}\,.\\ \end{align} The other way round (and skipping details), the NPV is \begin{align} P(\overline{D}\|-)=\frac{P(\overline{D}\cap -)}{P(-)}&=\frac{P(-\|\overline{D})P(\overline{D})}{P(-\|D)P(D)+P(-\|\overline{D})P(\overline{D})}\\ &=\frac{\text{Specificity}\times(1-\text{Prevalence})}{(1-\text{Sensitivity})\times\text{Prevalence}+\text{Specificity}\times(1-\text{Prevalence})}\,.\\ \end{align} • Example: If the prevalence of IgG antibodies against SARS-Cov2 is $$5\%$$, then • For an individual that tests positive, the probability that she truely carries antibodies is $P(D\|+)=\frac{0.9667\times 0.05}{0.9667\times 0.05+0.007\times 0.95}=0.879\,.$ • For an individual that tests negative, the probability that she does not carry antibodies is $P(\overline{D}\|-)=\frac{0.993\times 0.95}{0.993\times 0.95+0.0333\times 0.05}=0.998\,.$ # 5 Positive Predicted Value (PPV) Assuming that the prevalence is fixed at either $$5\%$$ or $$1\%$$, you can observe in the charts below how does the proportion of individuals with the disease among those who tested positive (PPV) vary depending on the sensitivity and specificity. In a report of the Infectious Diseases Society of America we can read “Some FDA-authorized COVID-19 antibody tests are estimated to have 96-98% specificity, which would mean that a positive test result is more likely a false positive result than a true positive result if the prevalence or pretest probability is $$5\%$$ or less”. The horizontal dotted line in both of the charts above is established at $$0.5$$. Any PPV below it corresponds to a test for which the probability of carrying antibodies given a positive result (true positive) is less than $$0.5$$. As a consequence, the probability of not carrying antobodies given a positive result (false positive) is greater than $$0.5$$. In conclusion, below the horizontal dotted line, false positives are more likely than true positives. The vertical dashed lines corresponds to specificity values equal to $$0.96$$ and $$0.98$$, as written at the report. Observe that false positives are rather frequent when the prevalence is $$5\%$$ if the test is not very accurate, while they are frequent at $$1\%$$ prevalence even for accurate tests. # 6 Negative Predicted Value (NPV) The screening test used at the (ENE-Covid19) is the Zhejiang Orient Gene Biotech IgG rapid test. The manufacturer declared a sensitivity of $$97\%$$ and a specificity of $$100\%$$, while later reliability studies revealed a sensitivity of approximately $$79\%$$, while the specificity is $$100\%$$. A specificity of $$100\%$$, $$P(-\|\overline{D})=1$$, implies that all the individuals that do not carry IgG antibodies test negative, so the only chance for an individual to test positive is to carry IgG antibodies. As a consequence, PPV is 1, but there is a chance that an individual tests negative despite she carries antibodies. Specifically, if the specificity is $$79\%$$, as suggested by the reliability studies, roughly $$1.1\%$$ of the individuals that test negative, are expected to carry antibodies. • If $$\text{Sensitivity}=0.79$$, then $P(D\|-)=1-\frac{\text{Specificity}\times(1-\text{Prevalence})}{(1-\text{Sensitivity})\times\text{Prevalence}+\text{Specificity}\times(1-\text{Prevalence})}=0.0109\,.$ • If $$\text{Sensitivity}=0.97$$, then $P(D\|-)=0.00158\,.$ # 7 Statistical surveys According to (ENE-Covid19), the current prevalence of IgG antibodies against SARS-Cov2 in Spain is estimated to be $$5\%$$. The survey was conducted on over 60000 individuals and the prevalence was also estimated on several geographical areas, as well as age groups. Together with each estimated proportion, a $$95\%$$ Confidence Interval (CI) on it is reported. The general formula for an approxiate $$95\%$$ CI on a proportion $$p$$ is $\hat{p}\pm 1.96\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\,,$ where $$\hat{p}$$ is the sample proportion (fraction of individuals in the sample carrying antibodies), $$n$$ is the sample size (number of individuals in the sample), and $$1.96$$ is the quantile of a standard normal distribution whose upper tail probability is $$0.025=0.05/2$$, so the probability that a standard normal random variable lies between $$-1.96$$ and $$1.96$$ is $$0.95$$ (the confidence level). The value $$\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$ is the standard error of the estimate, which assesses its precision, while the relative standard error evaluates the relative precision of the estimate (so it can be given as a percentage) and is obtained after dividing the standard error between the sample proportion. Observe that the number of inhabitants in Spain (population size) does neither appear in the formula for the CI nor in the formulas for the standard error or relative standar error. In other words, the precision of the estimate depends on the number of available observations in the sample, but not on the size of the population under study. If the true proportion of individuals with antibodies is $$p=0.05$$, then the relative standard error of $$\hat{p}$$ for $$n=60000$$ is roughly $$1.78\%$$, so the width of a CI on $$p$$ would be approximately $$2\times 1.96\times 1.78=7\%$$ of $$\hat{p}$$. The overall CI provided on the report is $$[0.047,0.054]$$, whose width is $$14\%$$ of $$\hat{p}$$. Why is the CI wider than expected? Some further considerations on the precision of the estimate of the proportion of individuals with antibodies should be taken into account. • The study was conducted on households. If a household was selected, all individuals on it were tested. When one of them suffers from COVID-19, it is quite likely that the disease is spread over the household. This is equivalent to reducing the sampe size, so the final effect is that the standard error of the estimate increases. Spanish households have, on average, $$2.5$$ individuals, dividing the sample size by $$2.5$$ would increment the standard error (and the width of the CI) of $$\hat{p}$$ times $$\sqrt{2.5}$$, so roughly by $$58\%$$. • For each tested individual, despite that the result of the test is negative, there is a chance that she carries antibodies. The proportion of individuals with anbidodies is not the proprotion of positive tests, instead, it is $$1/0.79=1.266$$ times the proportion of positive tests. The final effect is an increment of the standard error (and the width of the CI) of $$\hat{p}$$ times $$1.266$$, so roughly by $$27\%$$. These two observations explain quite accurately the final width of the presented CI. There is, nevertheless, something else to say about the survey (which also affects the precision of the estimate). • The selection procedure for the inviduals in the survey was stratified sampling. This means that some subgroups of the population (strata) were selected, and the sampling was run on those strata. Since the proportion of individuals with antibodies is not the same in all the regions in Spain and the National Bureau of Statistics (INE) has the exact figures of the population of each region, they decided to first estimate the porportion of carriers of antibodies at each region by sampling a number of individuals proportional to the number of inhabitants in the region, and obtain later the overall estimation weigthing each of the region estimates proportionally to the number of inhabitants in the region. This procedure reduces the standard error of the overall estimate. # 8 Excess mortality There has been quite some controversy with the official figures of COVID-19 fatalities in Spain. For a pandemic such as the COVID-19 one, it is just not possible to obtain reliable figures for the daily number of deaths. Nevertheless, in the long run, it is possible to assess the approximate number of COVID-19 related fatalities. This is done by modeling the monthly number of deaths in Spain by means of a time series and obtaining the excess mortality. The daily number of deaths in Spain can be found (with a slight delay) at the Instituto de Salud Carlos III MoMo website which allows direct comparison of the number of deaths during the COVID-19 pandemic period with the predicted mortality over the same period. Observe that the mortality is predicted as a time series, and as such, it has some trend (tendency to increase or decrease over time, in this case due to population increase, aging, changes in life expectancy,…) and seasonality (patterns at regular intervals, mortality is temperature, and thus season, dependent). It is now difficult to assess the reasons of many deaths, but in the long run the excess mortality can be computed. Notice that the COVID-19 pandemic is a factor that increases the number of deaths, but it appears in combination with other factors that decrease it (reduced activity due to the lock) or increase it (harvesting effect that causes short-term additional deaths among those who are already sick).	2022-07-03 12:00:09	{"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": 0.8706182241439819, "perplexity": 752.1645546359114}, "config": {"markdown_headings": true, "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-2022-27/segments/1656104240553.67/warc/CC-MAIN-20220703104037-20220703134037-00431.warc.gz"}
http://math.stackexchange.com/questions/85867/derivative-i-do-not-understand-ln-ln-x	Derivative I do not understand: $\ln (\ln x)$ I'm currently taking Calculus. I'm pretty good with derivatives apart from when it comes to logarithmic differentiation etc. Here is one I'm having problems with, if anyone could help that would be appreciated. $$f(x)=\ln (\ln (x) ) .$$ Can someone please explain the derivative of this? Thanks! - Note: Whatever "CAL 1" means to you, it's most probably specific to a single school or school system. When you're writing for a worldwide audience (such as here) you cannot expect that putting "CAL 1" in parentheses will tell the reader anything useful. – Henning Makholm Nov 26 '11 at 21:28 The basic formula for the derivative of $\ln x$ is $${d\over dx}\ln x={1\over x}.$$ Recall the chain rule: ${d\over dx}f\bigl(g(x)\bigr) =f'\bigl(g(x)\bigr)\cdot g'(x)$. So, by the chain rule, with $f(x)=\ln x$ and $g(x)=\ln x$, $${d\over dx}\ln (\ln x)={1\over \ln x}\cdot (\ln x )'={1\over \ln x}\cdot {1\over x}= {1\over x\ln x}.$$ Don't tell anyone I told you this, but you can remember: "the derivative of $\ln$ of something is (1 over the something) times the derivative of the something". - Thank you so much – Sam Nov 26 '11 at 20:31 Is that easier to remember than the chain rule itself? – Henning Makholm Nov 26 '11 at 21:26 First note that this is going to require an application of the chain rule, where $u=\ln(x)$. So, to find $f'(x)$, one must first find $f'(u)$ and then find $u'(x)$. Rewriting $f(x)$ in terms of $u$ yields $f(u)=\ln(u)$ and $u(x)=\ln(x)$. Thus, $f'(u)=1/u$ and $u'(x)=1/x$. Therefore, $f'(x)=f'(u)u'(x) = (1/u)(1/x)$. Then substitute $u=\ln(x)$ into the equation and we get $f'(x) = (1/(x\ln(x)))$. - A couple of $\LaTeX$ tips: (i) use \ln for natural log; that automatically puts it in roman typeface, as it should. (ii) Put the entire formula inside the math delimeters, $...$, including the equal sign; this prevents awkward line breaks, and it also provides appropriate spacing. Don't use * for products inside math formulas; use either juxtaposition, or if needed use \cdot (as in $f'(u)\cdot u'(x)$), or \times (as in $3\times 5$). Finally, don't mix roman typeface and math-italic typeface: x and $x$ look different enough that they may refer to different things! – Arturo Magidin Nov 26 '11 at 22:17 @ArturoMagidin Thanks for the advice! – analysisj Nov 26 '11 at 22:22	2015-07-02 16:56:10	{"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": 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": 0.9554888010025024, "perplexity": 635.762447648309}, "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-2015-27/segments/1435375095632.21/warc/CC-MAIN-20150627031815-00308-ip-10-179-60-89.ec2.internal.warc.gz"}
http://lamc.la/music_saves_all_souls.html	En Pursuit of Happiness How Jesus Christ and Rock'n Roll save every single soul. The continuing story of my attempt to bright light to the world. A log of the difficulties I've had, and how they might relate to the message itself--along with some insight that I have gleaned from the experience of receiving this Revelation. While the work you are looking at is filled with what I see as clear evidence of the fulfillment of messianic prophesy, the main goal of LAMC.LA was to deliver to the world a new way of looking at religion, one which could be used to seek out the true wisdom and guidance of religion for oneself. En Pursuit of Happiness on the other hand, discusses my own hopes and dreams, and to explain how they have changed throughout my interactions with the ... beyond. The Matchbox The e-mails that circled the globe, opening the doorway to the future. This is an excerpt from Time and Chance: The race is not to Die Bold by Adam Marshall Dobrin Download the actual Revelation of the Messiah in [ .PDF ] [ .epub ] [ .mobi ] or view online. Older works Lit and Why, hot&y, and From Adam to Mary are also available. # Music Saves All Souls Music talks to me, it’s God speaking. He tells me stories, makes promises, and proves something very important. He shows his hand, his control over culture and the direction of our future. It’s about proving the influence, and backing it up by explaining how the technology works. This is neuroscience in action, telling us a story about Revelation in music. Like the secret language, seeing the influence of an outside force in popular culture begins to offer proof of the existence of God. This is about faith, and ending the need for it… so you too, like Moses, can be sure… You’ve heard his voice, and seen the fire. I know these lyrics are inspired, they are about me; and neither you nor the author knew that until I came. This is how he proves he is there, by weaving a story between songs… linking them to movies and to people. To me, to President’s, to you. Seeing the hand of God is simply a matter of perspective, can you believe it’s real? Can you believe that the proof has been in front of us the whole time, just waiting to be unlocked, like some box in Greece. Pandora; Prometheus and/or “a,” the A of Adam ties us … exactly as I am explaining … through a Matrix of light (see it’s a Cypher in words, and a Matrix all around us) from Pandora to Mr. Anderson; the Messiah of Or. Or is the Hebrew word for light. It’s in our language, and our movies. And the music… It comes back in Rome below… as Nero fiddles on the rooftop–Heaven–with our minds. This historical story is a metaphor for now, for what’s going on in the heads of these musicians who are speaking the words of God unknowingly. The act of doing it–and then explaining that–frees us from one kind of slavery, not knowing it is being done. The message, another–that we need to see what is going on. Powers in Heaven are trying to hide its existence, and they are winning. You don’t believe it, not really. It’s there, not watching–but manipulating. Are they saving everyone… or hiding for a reason? The fire is spreading … What am I bidding, for this old ((civilization)) ((Nero/Atlas)) cried, as he held it up, with a smile A thousand dollars, and who'll make it two? He played a melody as pure and sweet as a caroling angel sings. The Touch of the Master Hand ## In the beginning -God “IN” is in my name, Dobrin, it’s in Wellbutrin, and it’s in the beginning. It’s the very first word, and the very first command of God–hidden until now–to In the beginning. It’s about entry to Heaven, a way in, and here the intention is clear. We have suffered for being at the beginning, and clearly that makes us responsible for what comes, for the Heaven that he is commanding us entry to. The idea to save everyone is fundamental to the Holy Grail and the reference to water in Matthew 3:11–for repentance, I am teaching you what baptism means: to teach. ## In the Jungle Did you know this song about a Lion sleeping is messianic, literally about Christ? I didn’t either, because I was sleeping. My life was normal until about 2010, when I got a crash course in the Tribulation, something that has quite a bit to do with the neuroscience involved in manipulating people’s thoughts. There’s a large group of mind control victims; people who are being used to expose the existence of this technology against their will; in 2010 I became one of them. I learned all about neuroscience, and mind control technologies available to us, and not. Things that are on the horizon, the possible ways what I experienced could be done. I was beginning to wake up. Seeing that this song about the Lion of Judah sleeping in Eden was about me was the very beginning. It’s all about waking up in the morning, and seeing the son rise. Modern imagery about cats, lions, and hair are all references to Christ. Sleeping to being unaware. The jungle is in Eden. I am Adam. The process of waking up has not been pleasant, you could say I’m not a morning person. ## Morning Has Broken On the note of cat imagery, Cat Stevens lights up the horizon with this song “like the first morning.” He is telling us we are dealing with time travel, and that our world is “as in the days of No-AH.” The AH of Noah is the end of Adamah, the swirling around Adam that is the music pointing to Christ, the movies, religion itself. The days of Noah are when there was no “ahh,” no apocalypse. In Adam parlance, “the last time around.” Blackbird sings in the dead of night, also a reference to me, in fact a reference to what you are reading; in American mythology: this is the dawns early light. Do you see a power growing in the musical Hair? ### Night The night is when we all see, when we don’t see the “ah,” it’s been our world up until now, when it is being pointed out to the world that there is an “ah” swirling around Adam, around Christ. We are in Eden, God is searching for me… in this case God is humanity; well paralleled in Matthew 2:2. “Where is the one who was born the King of the Jews”? Through the night, with the light from above, the Egyptian Plague of Darkness is all around us. It is overtly keeping us from seeing this message, by using disbelief, and active measures like a censorwall. Censorship in America; and nobody knows… this is the Darkness. It is the Wall of Jericho, and it is about to fall. Like Berlin before it, this wall is being torn down–in this case by the torches that are an e-mail campaign, social media, and the writing you are reading which points out clearly how to see in the dark. Once enough do, we have the base we will need to stop this from ever happening again. And the knowledge, this is God’s plan; to highlight serious social problems, like a palpable lack of freedom of speech and communication, so that we can stop this type of hidden slavery. ### Day The names “Adam and Eve” have a meaning related to this cycle. After Dark it is A.M.–ADA.M. is the bright morning star, rising in the night to end the dark. Eve-ning fell first, like in Judaism where the day begins at sundown. Looking for proof through the night that our flag … America is the Promised Land. The Biblical imagery in our songs, the freedom that is God given, all of these things congeal to light the day. What so proudly we hailed at the twilight’s last gleaming… why that’s me, and religion, the last gleaming was the end of the “last time around,” right before that civilization went back in time to change their past. Now, we are here. ### Son Rise, I am the Light of the Word The idea that the sun and son are one is something we can only see clearly in English. In other languages these words are very different. It’s something like putting English on the ball, realizing that ancient religion was created with foreknowledge of modern words. These examples are how we know it’s all about now, this time, 2016. It is this idea, that the son and sun are one that unifies the Egyptian gods of Ra and Horus (see the Ka) under the auspices of Isaac, who can see with both eyes. Ra and Horus each have one of the “All Seeing Eyes,” the symbols of ancient Egypt we associate with the Illuminati. Moses and Horus are united through the context of their deeds. Moses parted a holy sea, the red sea of fire, our civilization. He did this by coming… by dividing the people over whether or not Jesus is actually here, and is actually Moses. I’ve seen the burning bush, the sea is about to be parted. Horus unites the two lands… and America remembers Paul Revere: one if by land, two if by sea. This story tells us about the future, and land is langolier for El and… a hidden civilization. Torah, Tor all humanity, see the technical connotation–the purpose, to hide those that were active in the war we don’t see, the one that will bring us out of darkness-one that it also created. Be illuminated, the light of the world is the day star, Sol lit by words, day backwards is yad–the Hebrew word for Hand–of God. Adam and Eve together, the bright morning star of Venus converts to the sun. # Modern Prophesy in Music Many bands have songs about Christ, many are inspired… all in fact. It is this knowledge, that we are all secretly being inspired by God that tells me that the Ba of Horus, the Egyptian version of the Holy Spirit is upon us all. United in “Ha,” the two letters associated with Isaac and Abraham. That son’s name means… he laughs. ## The Dave Matthews Band This is the last stop, we are on the way to salvation. Dave’s songs are all about gods gaze falling upon us with a mischievous grin and seriously inspired me my whole life. It turns out, he too was inspired, and the lyrics all tell a story about the apocalypse. I quote them often, and when I do, be sure I hear the voice of God in the worlds he sings. ## The Pretty Reckless It’s all about light, from Light me up to this light is driving me insane. Is she singing about me, or am i crazy? Her songs, about “running” eerily parallel the messianic story describe in the Lamb of God, and her words speak to me, “Do you know who I am, do you even want to know me?” In the year I’ve been trying to tell the world I am here, this is exactly how I feel. What are you expecting? How does Jesus return? In Nothing Left to Lose Taylor responds to a lyric from American Pie with “If Jesus Christ and Rock’n Roll can’t my immoral soul, I’ve got nothing left to lose,” and shes talking about this book; about explaining that Rock music is Christ talking to you, too. She’s saving my soul. “I know you want me, I was just looking for ((Jesus.))” Adam’s here instead. ### Did you write the book of love, and do you have faith in God above? Taylor is Eve, one of them. By my count there are five, and that’s interesting too, as Eve’s letter are all fifths. E is the fifth in English, V in Roman numerals, and out of the five 3 are girls that are .. eligible. My mother’s middle name is Eve, and I am the first. The first evening. Pictures that I took during my trip around the country appear through the book. they are authentic and original–this one from the Bahamas, it seems appropriate. You’ll notice “Logos” and “look up, look down.” She is created (sort of like you), her life–like many actors and actresses–a map to connect stories to Biblical and mythological characters. It shows the divine hand, a guiding influence over the creation of these religious art works. Johnny Depp, Keanu (Key Anu) Reeves (Reason Eve’s), Cameron Diaz, Adam Brody… all link movies that are Biblical allusions–real continuations of the Bible–through their “players.” The nakedness doesn’t hurt either. ## Live Adam and Eve live down the street from ((you)), a million miles fall from grace.. thank God I ((found)) the ground. ## I’m so vain Except, not really. Just single. You could say I feel like an unsung hero. CopyleftMT This content is currently released under the GNU GPL 2.0 license. Please properly attribute and link back to the entire book, or include this entire chapter and this message if you are quoting material. The source book is located at http://www.lamc.la and is written by Adam Marshall Dobrin. Adam Marshall Dobrin adam@lamc.la fb.me/admdbrn linkedin.com/adam5 instagram.com/yitsheyzeus twitter.com/yitsheyzeus -----BEGIN PGP PUBLIC KEY BLOCK----- Version: GnuPG v2 mQENBFbGalABCADzLBdnHptF2MJCpdY8P/Mgnf4xj8F9pZSCwmd0J4Md8g3aTEdU CV9t0UQgNtjcxwfoenJLHgdZd4Mfscz9U+NN69OLXdPu4cdXOjTiHarPLjKnqIZw 3fmkM2ycvoUPkdVYCjwYYQxWRsWRpJf1dpmtPuz0L8ysh/WWsj2Ag2MrFYAo+sY6 dGZvaLsPhkZJcLXyFaP3c3Zt8ivrs4VV8+0kmMzScnR+oncVZbeMuQksoPxRmZgH mYu2KSf74lWOWVcaaBXOYX5pGNdhBUgq8ll+8tRH16G289r0cqRoPh/sjs/JRuIH KnCWG2UAUJF7ir04TS5A4Lwl9RYcQwVvb3BdABEBAAG0LUFkYW0gTWFyc2hhbGwg RG9icmluIChsYW1jLmxhKSA8YWRhbUBsYW1jLmxhPokBOQQTAQgAIwUCVsZqUAIb AwcLCQgHAwIBBhUIAgkKCwQWAgMBAh4BAheAAAoJEMgUPrR1B55trOwIALOQRTX0 YqXJXEMhX9CgxKNoNkpM2pdMdHl6CAVxhQ3hbNjIFnZbKbP88uxMEIOXXmYZ7gOy YqiDCu5I1V25suBb2ODSix75YQugfQ7H78pXHpTRu5sT+5SybItx7d+KUZaEj4pO tXWEemYl0cKK97RzpI0k1dmB7NqAVvqgbqQwd40MOf8QJVlGXnB1+5H2IbkYG6rD ixKGJEdes6i6nqvi/xz/s5hFVGUwTcVQbRU/fa1qT1Q7kHf1PlMu6yjuZTSz7WUG tWjobGwrVJkaeVWgLE4mcxMtity2IFTwOHvAuv8fi2EGQRQjXfPvxL7Vn4MNRl8x zLPV44D37QEknjy5AQ0EVsZqUAEIAMFS0+ZgSJzUPz0h0oiiRjfk2hapS3c1/Ysm R/h8sZ8/GOomdo3MEbTCkcuZ8ReAJhB2PofmwI4LAvW1x7Zwh1vfBKygfUs1s9lm ya/eHkjuZfqmeuEJZMHn6sxb3vqowWmvLhv3x0aWD8qLCIYoa1ntzTOIqxBEgxvU rF1/wd6OQLSJQEVNwPCx7CJI/5o/4W6pUaHk8amgPckkEdmlhRTRqFoAUV1Doivv d9JGYNYC88vS14Sw4Z9Xb7qBQJvG4hIh29gtQxk7Wz4m3ceR79MWT4eSGkH/rTGl w1OuQS2OkPvjgPWJt8San4zuPer17pJN7M5LWI0PStoX9pkud5kAEQEAAYkBHwQY AQgACQUCVsZqUAIbDAAKCRDIFD60dQeebWU6CADylAM5K18N2JGveL3D4dG25fdF vkrz8LOaiUmjAxijcRQBLkTPBK7QqoK0zN6MssMdlBGIOvZQwxSMIIrG6SqwR/go rmZHRuz17ceFTcxT8ZG3FuBY+xXrotXFjLxTmJ1wUeCSVXTc4NAwBzykgkQXOdIj qK1f/HnmMqsSmX4swuH0TZPNBBO7CNvLN6rdLBRfNn1h5XPs8VVtezg5ZDfCTf8S mucQGEwo/hJmr/orEucmETYSvTXOz+L5X5gNHpzYzE9590FYfbAKvrEhAliKbhhl 3Roie3kenrzelXo5N9Q0f2AKFrv1hRX9hBkwTbA18SKZ9XQbWMusX8YhvfLr =dvAJ -----END PGP PUBLIC KEY BLOCK-----	2018-02-20 09:06:03	{"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": 0.19014665484428406, "perplexity": 3263.7141952284323}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "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-2018-09/segments/1518891812932.26/warc/CC-MAIN-20180220090216-20180220110216-00034.warc.gz"}
http://dictionnaire.sensagent.leparisien.fr/Category%20of%20sets/en-en/	Publicité ▼ définition - Category of sets voir la définition de Wikipedia Wikipedia Category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B. Care must be taken in the definition of Set to avoid set-theoretic paradoxes. Many other categories (such as the category of groups, with group homomorphisms as arrows) add structure to the objects of the category of sets and/or restrict the arrows to functions of a particular kind. Properties of the category of sets The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps. The empty set serves as the initial object in Set with empty functions as morphisms. Every singleton is a terminal object, with the functions mapping all elements of the source sets to the single target element as morphisms. There are thus no zero objects in Set. The category Set is complete and co-complete. The product in this category is given by the cartesian product of sets. The coproduct is given by the disjoint union: given sets Ai where i ranges over some index set I, we construct the coproduct as the union of Ai×{i} (the cartesian product with i serves to ensure that all the components stay disjoint). Set is the prototype of a concrete category; other categories are concrete if they "resemble" Set in some well-defined way. Every two-element set serves as a subobject classifier in Set. The power object of a set A is given by its power set, and the exponential object of the sets A and B is given by the set of all functions from A to B. Set is thus a topos (and in particular cartesian closed). Set is not abelian, additive or preadditive. Its zero morphisms are the empty functions ∅ → X.[1] Every not initial object in Set is injective and (assuming the axiom of choice) also projective. Foundations for the category of sets In Zermelo–Fraenkel set theory the collection of all sets is not a set; this follows from the axiom of foundation. One refers to collections that are not sets as proper classes. One can't handle proper classes as one handles sets; in particular, one can't write that those proper classes belong to a collection (either a set or a proper class). This is a problem: it means that the category of sets cannot be formalized straightforwardly in this setting. One way to resolve the problem is to work in a system that gives formal status to proper classes, such as NBG set theory. In this setting, categories formed from sets are said to be small and those (like Set) that are formed from proper classes are said to be large. Another solution is to assume the existence of Grothendieck universes. Roughly speaking, a Grothendieck universe is a set which is itself a model of ZF(C) (for instance if a set belongs to a universe, its elements and its powerset will belong to the universe). The existence of Grothendieck universes (other than the empty set and the set $V_\omega$ of all hereditarily finite sets) is not implied by the usual ZF axioms; it is an additional, independent axiom, roughly equivalent to the existence of strongly inaccessible cardinals. Assuming this extra axiom, one can limit the objects of Set to the elements of a particular universe. (There is no "set of all sets" within the model, but one can still reason about the class U of all inner sets, i. e., elements of U.) In one variation of this scheme, the class of sets is the union of the entire tower of Grothendieck universes. (This is necessarily a proper class, but each Grothendieck universe is a set because it is an element of some larger Grothendieck universe.) However, one does not work directly with the "category of all sets". Instead, theorems are expressed in terms of the category SetU whose objects are the elements of a sufficiently large Grothendieck universe U, and are then shown not to depend on the particular choice of U. As a foundation for category theory, this approach is well matched to a system like Tarski-Grothendieck set theory in which one cannot reason directly about proper classes; its principal disadvantage is that a theorem can be true of all SetU but not of Set. Various other solutions, and variations on the above, have been proposed.[2][3][4] The same issues arise with other concrete categories, such as the category of groups or the category of topological spaces. Notes 1. ^ Section I.7 of Pareigis 1970 2. ^ Mac Lane 1969 3. ^ Feferman 1969 4. ^ Blass 1984 References Publicité ▼ Contenu de sensagent • définitions • synonymes • antonymes • encyclopédie • definition • synonym Publicité ▼ dictionnaire et traducteur pour sites web Alexandria Une fenêtre (pop-into) d'information (contenu principal de Sensagent) est invoquée un double-clic sur n'importe quel mot de votre page web. LA fenêtre fournit des explications et des traductions contextuelles, c'est-à-dire sans obliger votre visiteur à quitter votre page web ! Essayer ici, télécharger le code; Solution commerce électronique Augmenter le contenu de votre site Ajouter de nouveaux contenus Add à votre site depuis Sensagent par XML. Parcourir les produits et les annonces Obtenir des informations en XML pour filtrer le meilleur contenu. Indexer des images et définir des méta-données Fixer la signification de chaque méta-donnée (multilingue). Renseignements suite à un email de description de votre projet. Jeux de lettres Les jeux de lettre français sont : ○ Anagrammes ○ jokers, mots-croisés ○ Lettris ○ Boggle. Lettris Lettris est un jeu de lettres gravitationnelles proche de Tetris. Chaque lettre qui apparaît descend ; il faut placer les lettres de telle manière que des mots se forment (gauche, droit, haut et bas) et que de la place soit libérée. boggle Il s'agit en 3 minutes de trouver le plus grand nombre de mots possibles de trois lettres et plus dans une grille de 16 lettres. Il est aussi possible de jouer avec la grille de 25 cases. Les lettres doivent être adjacentes et les mots les plus longs sont les meilleurs. Participer au concours et enregistrer votre nom dans la liste de meilleurs joueurs ! Jouer Dictionnaire de la langue française Principales Références La plupart des définitions du français sont proposées par SenseGates et comportent un approfondissement avec Littré et plusieurs auteurs techniques spécialisés. Le dictionnaire des synonymes est surtout dérivé du dictionnaire intégral (TID). L'encyclopédie française bénéficie de la licence Wikipedia (GNU). Changer la langue cible pour obtenir des traductions. Astuce: parcourir les champs sémantiques du dictionnaire analogique en plusieurs langues pour mieux apprendre avec sensagent. 5053 visiteurs en ligne calculé en 0,187s Je voudrais signaler : section : une faute d'orthographe ou de grammaire un contenu abusif (raciste, pornographique, diffamatoire)	2022-01-23 00:23:21	{"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": 1, "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": 0.7438348531723022, "perplexity": 2437.698775280151}, "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-2022-05/segments/1642320303917.24/warc/CC-MAIN-20220122224904-20220123014904-00398.warc.gz"}
https://forum.allaboutcircuits.com/threads/passing-a-pointer-have-i-improved-things-or-made-it-worse.172004/	# Passing A Pointer - Have I improved things or made it worse? #### TechWise Joined Aug 24, 2018 151 As I alluded to in a previous thread, I am trying to up my C skill level and write more efficient and robust code. I have run into another question. Let's say I am using my ADCs to measure three quantities. I've defined a struct to neatly wrap them up: Code: typedef struct measurements { float i_a; float i_b; float i_c; } measurements; In main(), I create two instances of this struct: one to contain the raw results from the ADC, the other to contain the calibrated measurements which have been shifted and scaled: Code: measurements raw_measurements; // Contains the raw ADC conversion results measurements latest_measurements; // Contains the shifted and scaled measurements I want to write a function to take the raw measurements and do the shifting and scaling, by which point "latest_measurements" should contain the correctly calibrated values. The most obvious solution seemed to be to pass "raw_measurements" into a function, create a temporary variable to hold the results then return it at the end: Code: measurements calibrate_measurements(measurements raw) { measurements temp; temp.i_a = (raw.i_a - CURRENT_A_OFFSET) * CURRENT_A_GAIN; temp.i_b = (raw.i_b - CURRENT_B_OFFSET) * CURRENT_B_GAIN; temp.i_c = (raw.i_c - CURRENT_C_OFFSET) * CURRENT_C_GAIN; return temp; } Then in main(), I assign this to the "latest_measurements" variable: Code: latest_measurements = calibrate_measurements(raw_measurements); Question: Is this a good and/or standard approach? I am thinking now that I could avoid the overhead of creating this temporary variable by passing two pointers as arguments: a pointer to the "raw_measurements" and a pointer to the "latest_measurements". What would be the pros and cons of this? I know that @WBahn mentioned in a previous thread that there would be "some access overhead and that might be too much" so perhaps could elaborate on this? #### MrChips Joined Oct 2, 2009 27,117 My practice would be to pass both arguments in the function call. calibrate_measurements(raw_measurements, latest_measurements); #### TechWise Joined Aug 24, 2018 151 My practice would be to pass both arguments in the function call. calibrate_measurements(raw_measurements, latest_measurements); My understanding of this is that it would be "passing by value", during which the contents of "raw_measurements" and "latest_measurements" would be copied into the function and the original would be left alone. Then, after the function exits, the variables inside it would be destroyed and the original "raw_measurements" and "latest_measurements" would retain their original values outside the function. Do you mean I should pass by reference so that the function declaration becomes Code: void calibrate_measurements(measurements latest, measurements raw) and then call the function with: Code: calibrate_measurements(&latest_measurements, &raw_measurements,); so that the actual contents of "latest_measurements" are altered rather than just copying them into the function and destroying them when it exits? #### BobTPH Joined Jun 5, 2013 5,759 The most efficient way to do it is to pass both as pointers. Bob #### Marc Sugrue Joined Jan 19, 2018 222 Assuming that once converted to usable floats you don't need the raw A/D value again you could consider defining your measurements variables as unions of type int and float and let the compiler manage the conversion that way you lose the redundant data and your not held to a pointer data type. https://www.tutorialspoint.com/cprogramming/c_unions.htm #### TechWise Joined Aug 24, 2018 151 The most efficient way to do it is to pass both as pointers. Bob You mean as I've done in post 3 or by actually declaring the two structs as pointers, passing them as pointers and then dereferencing inside the function? Like this: Code: measurements * raw_measurements; // Contains the raw ADC conversion results measurements * latest_measurements; // Contains the shifted and scaled measurements Then the function would be: Code: void calibrate_measurements(measurements * raw, measurements * latest) { latest->i_a = (raw->i_a - CURRENT_A_OFFSET) * CURRENT_A_GAIN; latest->i_b = (raw->i_b - CURRENT_B_OFFSET) * CURRENT_C_GAIN; latest->i_c = (raw->i_c - CURRENT_C_OFFSET) * CURRENT_C_GAIN; return; } #### TechWise Joined Aug 24, 2018 151 Assuming that once converted to usable floats you don't need the raw A/D value again you could consider defining your measurements variables as unions of type int and float and let the compiler manage the conversion that way you lose the redundant data and your not held to a pointer data type. https://www.tutorialspoint.com/cprogramming/c_unions.htm This is an interesting possibility that I hadn't really thought of. You're right that once I cast the raw result to a float, I don't need the raw result anymore. So in a case like this, the compiler would assign enough space for a float which would initially only be partly filled by a 16-bit integer then be overwritten by a float. The only problem I can see is that the integer is needed to calculate the float, so I would need to assign it to a temporary variable while the calculations were done then overwrite it with the float which seems to defeat the purpose, unless I'm not understanding correctly. #### Marc Sugrue Joined Jan 19, 2018 222 This is an interesting possibility that I hadn't really thought of. You're right that once I cast the raw result to a float, I don't need the raw result anymore. So in a case like this, the compiler would assign enough space for a float which would initially only be partly filled by a 16-bit integer then be overwritten by a float. The only problem I can see is that the integer is needed to calculate the float, so I would need to assign it to a temporary variable while the calculations were done then overwrite it with the float which seems to defeat the purpose, unless I'm not understanding correctly. I don't think you would have a problem, i recall doing something similar on an embedded system without an issue, the compiler will load the data into registers before writing it back. I just checked i did something like this: union AtoDDATA { unsigned int Value_uint; float Value_f; }; typedef struct Measurements{ union AtoDDATA a; // Now float or uint union AtoDDATA b; // Now float or uint union AtoDDATA c; // Now float or uint } measurements; //Foat is between 0 & 1 measurements->a.Value_uint = (unsigned int) (measurements->a.Value_f * 65535.0); Last edited: #### WBahn Joined Mar 31, 2012 27,392 As I alluded to in a previous thread, I am trying to up my C skill level and write more efficient and robust code. I have run into another question. Let's say I am using my ADCs to measure three quantities. I've defined a struct to neatly wrap them up: Code: typedef struct measurements { float i_a; float i_b; float i_c; } measurements; In main(), I create two instances of this struct: one to contain the raw results from the ADC, the other to contain the calibrated measurements which have been shifted and scaled: Code: measurements raw_measurements; // Contains the raw ADC conversion results measurements latest_measurements; // Contains the shifted and scaled measurements I want to write a function to take the raw measurements and do the shifting and scaling, by which point "latest_measurements" should contain the correctly calibrated values. The most obvious solution seemed to be to pass "raw_measurements" into a function, create a temporary variable to hold the results then return it at the end: Code: measurements calibrate_measurements(measurements raw) { measurements temp; temp.i_a = (raw.i_a - CURRENT_A_OFFSET) * CURRENT_A_GAIN; temp.i_b = (raw.i_b - CURRENT_B_OFFSET) * CURRENT_B_GAIN; temp.i_c = (raw.i_c - CURRENT_C_OFFSET) * CURRENT_C_GAIN; return temp; } Then in main(), I assign this to the "latest_measurements" variable: Code: latest_measurements = calibrate_measurements(raw_measurements); Question: Is this a good and/or standard approach? I am thinking now that I could avoid the overhead of creating this temporary variable by passing two pointers as arguments: a pointer to the "raw_measurements" and a pointer to the "latest_measurements". What would be the pros and cons of this? I know that @WBahn mentioned in a previous thread that there would be "some access overhead and that might be too much" so perhaps could elaborate on this? In general you want to avoid returning structures for the same reason that you want to avoid passing them by value -- they will chew up your stack and it takes time to push and pop them from the stack. Instead, pass two pointers to your function, one for the new and one for the old. Any time that you are using the dot derefence operator (i.e., temp.i_a), you are probably not doing it the way you should (there are always exceptions). Instead, work with pointers to the structures. Code: void calibrate_measurements(measurements cal, measurements raw) { cal->i_a = (raw->i_a - CURRENT_A_OFFSET) * CURRENT_A_GAIN; cal->i_b = (raw->i_b - CURRENT_B_OFFSET) * CURRENT_B_GAIN; cal->i_c = (raw->i_c- CURRENT_C_OFFSET) * CURRENT_C_GAIN; } The key to remember is that if you only work with pointers to structures, you need to dynamically allocate those structures before you use them the first time. #### MrChips Joined Oct 2, 2009 27,117 Do you mean I should pass by reference so that the function declaration becomes Code: void calibrate_measurements(measurements * latest, measurements raw) and then call the function with: Code: calibrate_measurements(&latest_measurements, &raw_measurements,); so that the actual contents of "latest_measurements" are altered rather than just copying them into the function and destroying them when it exits? You are correct. My mistake. #### Analog Ground Joined Apr 24, 2019 456 Perhaps another option is to put the raw and latest values in the same structure or make a structure of structures containing two of the measurement structures. Then, only one pointer is passed to the function. This approach may be attractive if the raw and latest values are tightly associated and retained together. It also may produce more efficient code since all values are (most likely) contiguous in memory. #### BobTPH Joined Jun 5, 2013 5,759 Unless you are taking 3 measurements so rapidly that you cannot afford the conversion between them, why not simply convert each value as it is read, never storing the raw count? Bob #### TechWise Joined Aug 24, 2018 151 Unless you are taking 3 measurements so rapidly that you cannot afford the conversion between them, why not simply convert each value as it is read, never storing the raw count? Bob I'm actually taking 7 measurements simultaneously and I'm trying to keep the ISR as short as possible. I will probably do a very crude overcurrent protection check on the raw value so by the time I've done those comparisons that will add more computation to the ISR. The advice from TI was to keep the ISR as short as possible as it's not so easy to nest interrupts on this device. #### TechWise Joined Aug 24, 2018 151 In general you want to avoid returning structures for the same reason that you want to avoid passing them by value -- they will chew up your stack and it takes time to push and pop them from the stack. Instead, pass two pointers to your function, one for the new and one for the old. Any time that you are using the dot derefence operator (i.e., temp.i_a), you are probably not doing it the way you should (there are always exceptions). Instead, work with pointers to the structures. Code: void calibrate_measurements(measurements cal, measurements raw) { cal->i_a = (raw->i_a - CURRENT_A_OFFSET) CURRENT_A_GAIN; cal->i_b = (raw->i_b - CURRENT_B_OFFSET) * CURRENT_B_GAIN; cal->i_c = (raw->i_c- CURRENT_C_OFFSET) * CURRENT_C_GAIN; } The key to remember is that if you only work with pointers to structures, you need to dynamically allocate those structures before you use them the first time. Thanks for this. The code you've suggested above is my preferred suggestion for now as it's similar to what I have and easy for me to understand. Could you clarify your final point for me? If the structure only contains a fixed number of fixed sized floats, and I create an instance of the structure before referring to it, why would it need to be dynamically allocated? #### Analog Ground Joined Apr 24, 2019 456 I'm actually taking 7 measurements simultaneously and I'm trying to keep the ISR as short as possible. I will probably do a very crude overcurrent protection check on the raw value so by the time I've done those comparisons that will add more computation to the ISR. The advice from TI was to keep the ISR as short as possible as it's not so easy to nest interrupts on this device. I would keep the raw values, at least for awhile, for test and debug. #### TechWise Joined Aug 24, 2018 151 Perhaps another option is to put the raw and latest values in the same structure or make a structure of structures containing two of the measurement structures. Then, only one pointer is passed to the function. This approach may be attractive if the raw and latest values are tightly associated and retained together. It also may produce more efficient code since all values are (most likely) contiguous in memory. I had considered this and I had also considered having a struct for voltage measurements, one for current measurements and one for auxilliary inputs and then placing those inside a larger struct which would contain all the measurements. I decided against it at the finish up as there seemed to be no reason to further classify the measurements as my controller makes use of all of them at once anyway. #### Analog Ground Joined Apr 24, 2019 456 I had considered this and I had also considered having a struct for voltage measurements, one for current measurements and one for auxilliary inputs and then placing those inside a larger struct which would contain all the measurements. I decided against it at the finish up as there seemed to be no reason to further classify the measurements as my controller makes use of all of them at once anyway. I might combine all the measurements into one structure, raw and calibrated, if for no other reason than it makes it very easy to see all the values at once with a source level debugger. Especially if all the data is obtained in a single input scan. Also, I can easily control alignment of the data in memory. Passing around one pointer might prove convenient. Just a matter of taste and style. #### WBahn Joined Mar 31, 2012 27,392 Thanks for this. The code you've suggested above is my preferred suggestion for now as it's similar to what I have and easy for me to understand. Could you clarify your final point for me? If the structure only contains a fixed number of fixed sized floats, and I create an instance of the structure before referring to it, why would it need to be dynamically allocated? When you declare your structure and your typedef, you are only telling the compiler what one of these things will look like, but not actually allocating any memory for it. #### xox Joined Sep 8, 2017 787 When you declare your structure and your typedef, you are only telling the compiler what one of these things will look like, but not actually allocating any memory for it. Why do you think the dynamic allocation would be necessary though? The variables "raw_measurements" and "latest_measurements" would be allocated on the stack, so that should be sufficient. He could then pass them to the function as described in post #3. #### WBahn Joined Mar 31, 2012 27,392 Why do you think the dynamic allocation would be necessary though? The variables "raw_measurements" and "latest_measurements" would be allocated on the stack, so that should be sufficient. He could then pass them to the function as described in post #3. True. Do note that the context of saying that they would be need to be dynamically allocated was that they are never dereferenced using the dot operator, only the arrow. Now, if you allocate them in main() -- or made them global -- and never dereferenced them in main, you could get away with that. There's a lot to be said for keeping the main() as lean as possible, though that can go against the notion of max performance in an embedded application.	2022-11-30 18:20:55	{"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": 0.33770328760147095, "perplexity": 1512.8484890829016}, "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-2022-49/segments/1669446710765.76/warc/CC-MAIN-20221130160457-20221130190457-00538.warc.gz"}
http://tiny-themovie.com/ebooks/differential-geometry-and-related-topics	# Differential Geometry and Related Topics Format: Hardcover Language: English Format: PDF / Kindle / ePub Size: 13.08 MB A Lie group is a group in the category of smooth manifolds. ), where, Cuu = $\frac{\partial^{2}C(u)}{\partial u^{2}}$ Finding the binormal of any curve, this is denoted by B = (Cuu * Cuu) / In the limit, a straight line is said to be equivalent to a circle of infinite radius and its curvature defined as zero everywhere. Our results are inspired by work of Witten on the fivebrane partition function in $M$-theory ( hep-th/9610234, hep-th/9609122 ). Pages: 280 Publisher: World Scientific Pub Co Inc (December 2002) ISBN: 9812381880 Topics in Mathematical Analysis and Differential Geometry (Series in Pure Mathematics) Quantum and fermion differential geometry Part A. (Interdisciplinary mathematics Volume XVI) Exploring Curvature The Wheel Of Time: The Shamans Of Mexico Their Thoughts About Life Death And The Universe Calculus of Variations II (Grundlehren der mathematischen Wissenschaften) A Comprehensive Introduction to Differential Geometry, Vol. 5 A Survey of Minimal Surfaces (Dover Books on Mathematics) Dental Dam or Rubber Dam makes an excellent rubber sheet for student investigations. Add a large circle with a suitable marker, then deform it into an ellipse, a square, a triangle, or any other simple closed curve. We grapple with topology from the very beginning of our lives. American mathematician Edward Kasner found it easier to teach topology to kids than to grownups because "kids haven't been brain-washed by geometry" , e.g. Index Theory for Symplectic Paths with Applications (Progress in Mathematics) http://freechurchdesign.com/books/index-theory-for-symplectic-paths-with-applications-progress-in-mathematics. Development of astronomy led to emergence of trigonometry and spherical trigonometry, together with the attendant computational techniques , source: Geodesic Flows (Progress in Mathematics) tiny-themovie.com. Calder, Einstein's Universe (1979) NY: Viking Press. This is a popular book which is the companion to the BBC video by the same name. Callahan, The Geometry of Spacetime: An Introduction to Special and General Relativity, Undergraduate Texts in Mathematics (2000) NY: Springer-Verlag , source: Geometric, Control and Numerical Aspects of Nonholonomic Systems http://blog.vectorchurch.com/?books/geometric-control-and-numerical-aspects-of-nonholonomic-systems. Poster Session: We will hold a poster session Saturday evening; graduate students and recent PhD’s are strongly encouraged to participate. Please register your poster by October 30. This meeting is supported by Rice University and the National Science Foundation Real and Complex Singularities: São Carlos Workshop 2004 (Trends in Mathematics) http://marcustorresdesign.com/library/real-and-complex-singularities-sao-carlos-workshop-2004-trends-in-mathematics. There was earlier scattered work by Euler, Listing (who coined the word "topology"), Mobius and his band, Riemann, Klein, and Betti. Indeed, even as early as 1679, Leibniz indicated the desirability of creating a geometry of the topological type The Real Fatou Conjecture http://tiny-themovie.com/ebooks/the-real-fatou-conjecture. Differential geometry, branch of mathematics that studies the geometry of curves, surfaces, and manifolds (the higher-dimensional analogs of surfaces) Exam Prep for Differential Geometry of Curves and Surfaces by DoCarmo, 1st Ed. read pdf. It is a discipline that uses the methods of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry , source: The principles of the differential and integral calculus: And their application to geometry The principles of the differential and. The real defining characteristic of classical differential geometry is that it deals with curves and surfaces as subsets contained in Euclidean space, and almost invariably only considers two and three-dimensional objects. Early classical differential geometry is characterised by a spirit of free exploration of the concepts that the invention of calculus now provided mathematicians of the day , source: Lectures on fibre bundles and download online freechurchdesign.com. The article is adapted from one originally published as part of the Posters in the London Underground series. Click on any of the images in the latter page for an enlarged version and, where available, explanatory notes and further reading. Details the hand-on-wall rule for solving a maze with only one entrance and exit. [In effect, put your hand on the wall at the entrance and keep it on the wall until you exit the maze.] Includes a link to a right-hand and left-hand solution ref.: Applications of Mathematics in Engineering and Economics (AMEE'11): Proceedings of the 37th International Conference (AIP Conference Proceedings / Mathematical and Statistical Physics) http://coastalmortgages.ca/books/applications-of-mathematics-in-engineering-and-economics-amee-11-proceedings-of-the-37-th. The real fun begins when we introduce the derivative or differential and start wondering about what the various derivatives or differentials of certain objects tell us about these objects epub. Winner of the 2005 Book Prize, American Mathematical Society Winner of the 1997 for the Best Professional/Scholarly Book in Mathematics, Association of American Publishers Google full text of this book: This book develops some of the extraordinary richness, beauty, and power of geometry in two and three dimensions, and the strong connection of geometry with topology , e.g. Combinatorial Integral read pdf http://tiny-themovie.com/ebooks/combinatorial-integral-geometry-with-applications-to-mathematical-stereology-probability. Spherical CR Geometry and Dehn Surgery (AM-165) (Annals of Mathematics Studies) Surfaces With Constant Mean Curvature (Translations of Mathematical Monographs) Curvature and Homology Applications of topology to analysis Tom ter Elst: Harmonic analysis, operator theory, geometric analysis, subelliptic and degenerate operators, PDE Shayne Waldron: Approximation Theory, polynomial interpolation, numerical methods Nazli Uresin (PhD): Abstract dynamical systems pdf. Recovering Cup Products from Boundary Data — Geometry–Topology Reading Seminar, University of Pennsylvania, Feb. 24, 2009. Invariant Differential Forms in a Cohomogeneity One Manifold — Graduate Student Bridge Seminar, University of Pennsylvania, Feb. 18, 2009. Poincaré Duality Angles for Riemannian Manifolds With Boundary — Graduate Student Geometry–Topology Seminar, University of Pennsylvania, Feb. 18, 2009 , cited: Encyclopedia of Distances read online coastalmortgages.ca. This was the first known result on a topological invariant. Möbius published a description of a Möbius band in 1865. He tried to describe the 'one-sided' property of the Möbius band in terms of non-orientability , source: Lectures on Differential Geometry (Conference Proceedings and Lecture Notes in Geometry and Topology) http://ccc.vectorchurch.com/?freebooks/lectures-on-differential-geometry-conference-proceedings-and-lecture-notes-in-geometry-and. The McKean-Singer supersymmetry relation still holds: the nonlinear unitary evolution U(t) - which naturally replaces the Dirac wave evolution - has the property that str(U(t))= chi(G) at all times epub. Bloggat om Geometry and Topology of Submanifolds: VII Different... Reflections on some differential geometric work of Katsumi Nomizu, P Tensor Algebra and Tensor read for free read for free. The complete space curve of degree m n. surface of a circular cylinder. defined as the axis of the cylinder. is called the pitch of the helix. When b is + ve, the helix is right handed and when b is - ve, the helix is left If C is a real curve, then the arc length of a segment of the curve A space curve does not lie on a plane. However a very small arc of the curve may be thought of as almost a plane curve Statistical Thermodynamics and Differential Geometry of Microstructured Materials (The IMA Volumes in Mathematics and its Applications) coastalmortgages.ca. Geometric, Control and Numerical Aspects of Nonholonomic Systems Cohomological Aspects in Complex Non-Kähler Geometry (Lecture Notes in Mathematics) A Comprehensive Introduction to Differential Geometry VOLUME TWO, Second Edition Differential Geometry and Toplogy Differential Geometry and Toplogy Hamiltonian Reduction by Stages (Lecture Notes in Mathematics, Vol. 1913) Differential Geometry on Complex and Almost Complex Spaces Superintegrability in Classical and Quantum Systems (Crm Proceedings and Lecture Notes) Harmonic Morphisms, Harmonic Maps and Related Topics (Chapman & Hall/CRC Research Notes in Mathematics Series) Geometric Analysis of the Bergman Kernel and Metric (Graduate Texts in Mathematics) The Arithmetic of Hyperbolic 3-Manifolds (Graduate Texts in Mathematics) Harmonic Maps, Loop Groups, and Integrable Systems (London Mathematical Society Student Texts) Differential Sheaves and Connections:A Natural Approach to Physical Geometry (Series on Concrete and Applicable Mathematics) Global theory of connections and holonomy groups An Introduction To Differential Geometry With Use Of The Tensor Calculus An Introduction to Differential Geometry - With the Use of Tensor Calculus Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45): (Mathematical Notes) Yang-Mills Connections on Orientable and Nonorientable Surfaces (Memoirs of the American Mathematical Society)	2017-06-22 12:04:57	{"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": 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": 0.46390649676322937, "perplexity": 1463.3824584483673}, "config": {"markdown_headings": true, "markdown_code": true, "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-2017-26/segments/1498128319265.41/warc/CC-MAIN-20170622114718-20170622134718-00206.warc.gz"}
https://www.sarthaks.com/2375551/light-travels-from-medium-refractive-index-angle-incidence-medium-meansured-from-normal	# A ray of light travels from a medium of refractive index mu to air. Its angle of incidence in the medium is i, meansured from the normal to the bo 17 views in Physics A ray of light travels from a medium of refractive index mu to air. Its angle of incidence in the medium is i, meansured from the normal to the boundary , and its angle of deviation is delta. delta is plotted against i. Which of the following best represents the resulting curve ? A. B. C. D. by (37.3k points)	2022-12-08 07:27:53	{"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": 0.5811595916748047, "perplexity": 533.5506554719194}, "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-2022-49/segments/1669446711278.74/warc/CC-MAIN-20221208050236-20221208080236-00484.warc.gz"}
https://en.wikipedia.org/wiki/Palladium_hydride	Palladium hydride is metallic palladium that contains a substantial quantity of hydrogen within its crystal lattice. Despite its name, it is not an ionic hydride but rather an alloy of palladium with metallic hydrogen that can be written PdHx. At room temperature, palladium hydrides may contain two crystalline phases, α and β (sometimes called α'). Pure α-phase exists at x < 0.017 whereas pure β-phase is realised for x > 0.58; intermediate x values correspond to α-β mixtures.[1] Hydrogen absorption by palladium is reversible and therefore has been investigated for hydrogen storage.[2] Palladium electrodes have been used in some cold fusion experiments, under the hypothesis that the hydrogen could be "squeezed" between the palladium atoms to help them fuse at lower temperatures than would otherwise be required. ## History The absorption of hydrogen gas by palladium was first noted by T. Graham in 1866 and absorption of electrolytically produced hydrogen, where hydrogen was absorbed into a palladium cathode, was first documented in 1939.[2] Graham produced an alloy with the composition PdH0.75.[3] Metals are arranged in lattices, and in forming metallic hydrides, the hydrogen atoms place themselves in interstitial sites in the lattice. This is also the case for palladium hydride. When the surface of a palladium lattice is brought in contact with a H2 molecule the two hydrogen atoms split, each absorbed onto an interstitial site. The interstitial placing of hydrogen can lead to a non-stoichiometric mixture, i.e., the ratio of palladium and hydrogen cannot be represented by a natural number. The ratio in which H is absorbed on Pd is defined by ${\displaystyle x={\frac {[H]}{[Pd]}}}$. When Pd is brought into a H2 environment with a pressure of 1 atm, the resulting concentration of H reaches x ~ 0.7. However, the concentration of H to obtain superconductivity is higher. Therefore, the concentration of H should be increased, to x > 0.75.[4] This is done via three different routes. It is known that hydrogen easily desorbs from palladium; therefore, extra care should be taken to prevent H desorption from Pd. The first route is loading from gas phase. A Pd sample is placed into a high-pressure cell of H2, at room temperature. The H2 is added through a capillary. As a result, H is loaded onto Pd. To maintain this bonding, the pressure cell will be cooled to liquid N2 temperature (77 K). The resulting concentration is found to be [H]/[Pd] = 0.97.[4] The second route is electrochemical bonding. This is a method where the critical concentration for superconductivity can easily be exceeded without using a high-pressure environment, via a reaction as equilibrium between H in an electrochemical phase and H in a solid phase. The hydrogen is added to Pd and Pd–Ni alloys by an H concentration of ~0.95.[4] Thereafter, it has been loaded into electrolysis of 0.1n-H2SO4 with a current density of 50 to 150 mA/cm3. Finally, after lowering the loading temperature to ~ 190 K, a H concentration of x ~ 1 has been reached.[4] The third route is known as ion implantation. Before the implantation of H ions into Pd, the Pd foil was pre-charged with H. This is done in a H2 high-temperature gas. This shortens the implantation time which follows. The concentration reached is about x ~ 0.7.[4] Afterwards the foil is cooled to a temperature of 77 K to prevent a loss of H before the implantation can take place. The implantation of H in PdHx happens at a temperature of 4 K. The H ions penetrate in a H2+-beam. This results in a high concentration layer of H in a Pd foil.[4] ## Chemical structure and properties Palladium is sometimes metaphorically called a "metal sponge" (not to be confused with more literal metal sponges) because it soaks up hydrogen "like a sponge soaks up water". At room temperature and atmospheric pressure (standard ambient temperature and pressure), palladium can absorb up to 900 times its own volume of hydrogen.[5] Hydrogen can be absorbed into the metal-hydride and then desorbed back out for thousands of cycles. Researchers look for ways to extend the useful life of palladium storage.[6] ### Size effect The absorption of hydrogen produces two different phases, both of which contain palladium metal atoms in a face-centered cubic (fcc, rocksalt) lattice, which is the same structure as pure palladium metal. At low concentrations up to PdH0.02 the palladium lattice expands slightly, from 388.9 pm to 389.5 pm. Above this concentration the second phase appears with a lattice constant of 402.5 pm. Both phases coexist until a composition of PdH0.58 when the alpha phase disappears.[1] Neutron diffraction studies have shown that hydrogen atoms randomly occupy the octahedral interstices in the metal lattice (in an fcc lattice there is one octahedral hole per metal atom). The limit of absorption at normal pressures is PdH0.7, indicating that approximately 70% of the octahedral holes are occupied. When x=1 is reached, the octahedral interstices are fully occupied.[7] The absorption of hydrogen is reversible, and hydrogen rapidly diffuses through the metal lattice. Metallic conductivity reduces as hydrogen is absorbed, until at around PdH0.5 the solid becomes a semiconductor.[3] This formation of the bulk hydride does depend on the size of the catalyst Pd. When Pd becomes smaller than 2.6nm, hydrides will not be formed anymore. [7] Hydrogen dissolved in the bulk differ from hydrogen dissolved on the surface. When the particles of palladium decrease in size, less hydrogen dissolves in these smaller pd particles. Therefore, relatively more hydrogen adsorbs on the surface of the small particles. This hydrogen adsorbed onto the particles do not form an hydride. Therefore, bigger particles have more places available for the formation of hydrides.[7] ### Electron and phonon band The most important property of the band structure of PdH(oct) is that filled Pd states are lowered with the presence of hydrogen. Also, the lowest energy levels, which are the bonding states, of PdH are lower than that of Pd.[8] Additionally, empty Pd states, that are below the fermi energy, are also lowered with the presence of H.[8] Palladium prefers to be with hydrogen due to the interaction between the s state of hydrogen and the p states of palladium. The energy of an independent H atom lies in the energy range of the dominating p-states of the Pd bands.[8] Therefore, these empty states under the fermi-energy and holes in the d-band are filled.[8] Additionally, the hydride formation raises the fermi level above the d band. Empty states, above the d-band, are also filled. This results in filled p-states and shifts the ‘edge’ to a higher energy level.[9] ### Superconductivity PdHx is a superconductor with a transition temperature Tc of about 9 K for x = 1. (Pure palladium is not superconducting). Drops in resistivity vs. temperature curves were observed at higher temperatures (up to 273 K) in hydrogen-rich (x ~ 1), nonstoichiometric palladium hydride and interpreted as superconducting transitions.[10][11][12] These results have been questioned[13][failed verification] and have not been confirmed thus far. A great advantage of Palladium-hydride over many other hydride-systems is that Palladium-hydride does not need to be highly pressurized to become superconducting.[4] This makes measurements easier and gives more opportunity for different kinds of measurements (many superconducting materials require extreme pressurization to be able to superconduct, on the order of 102 GPa.[4] Palladium-hydride could therefore also be used to explore the role that hydrogen plays in these hydride-systems being superconductors. ### Susceptibility One of the magnetic properties of Palladium hydride is susceptibility. The susceptibility of PdHx varies largely when changing the concentration of H.[4] This is due to the 𝛽-phase of PdHx. The 𝛼-phase of PdH lies in the same range of the fermi surface as Pd itself, therefore 𝛼-phase does not influence the susceptibility.[4] However, the 𝛽-phase of PdHx is characterized by s-electrons filling the d-band. Therefore, the susceptibility of the 𝛼-𝛽 mixture decreases at room temperature with an increasing concentration of H.[4] Finally, when the spin fluctuations of pure Pd are decreased, the superconductivity will occur.[4] ### Specific heat capacity Another metallic property is the electronic heat coefficient 𝛾. This coefficient depends on the density of states. For pure Pd the heat coefficient is 9.5 mJ(mol∙K^2).[4] When H is added to the pure Pd, the electronic heat coefficient drops. For the range of x=0.83 to x=0.88 𝛾 is observed to be six times smaller than in the case of only Pd.[4] This region is the superconducting region. However, Zimmerman et al also measured the heat coefficient 𝛾 for a concentration of x=0.96.[4] A broadening of the superconducting transition was observed at this concentration. One of the reasons for this could be explained by an inhomogeneity of the macroscopic structure of PdH.[4] 𝛾 at this value of x has a large fluctuation and is therefore uncertain. The critical concentration for superconductivity to happen is estimated to be x ~ 0.72.[4] The critical temperature or the superconducting transition temperature is estimated to be 9 K. This was achieved at a stoichiometric concentration of x = 1. Furthermore, the pressure influences the critical temperature as well. It is shown that an increase in the pressure on PdHx decreases Tc. This can be explained by a hardening of the phonon spectrum, which includes a decrease in the electron-phonon constant 𝜆 .[4] ### Surface absorption process The process of absorption of hydrogen has been shown by scanning tunnelling microscopy to require aggregates of at least three vacancies on the surface of the crystal to promote the dissociation of the hydrogen molecule.[14] The reason for such a behaviour and the particular structure of trimers has been analyzed.[15] ## Uses The absorption of hydrogen is reversible and is highly selective. Industrially, a palladium-based diffuser separator is used. Impure gas is passed through tubes of thin walled silver-palladium alloy as protium and deuterium readily diffuse through the alloy membrane. The gas that comes through is pure and ready for use. Palladium is alloyed with silver to improve its strength and resistance to embrittlement. To ensure that the formation of the beta phase is avoided, as the lattice expansion noted earlier would cause distortions and splitting of the membrane, the temperature is maintained above 300 °C.[3] Another use of Palladium-Hydride is increased adsorption of H2-molecules with respect to pure Palladium. In 2009, a study was conducted which tested this fact.[16] At a pressure of 1 bar, the probability was measured of Hydrogen molecules sticking to the surface of Palladium versus the probability of sticking to surface of Palladium-hydride. The sticking probability of Palladium was found to be greater at temperatures where the phase of the used Palladium and hydrogen mixture was pure β-phase, which is in this context corresponds to Palladium-hydride (at 1 bar this means temperatures greater than roughly 160 degrees Celsius), as opposed to temperatures where β- and α-phases coexist and even lower temperatures where there is pure α-phase (α-phase here corresponds to a solid solution of Hydrogen atoms in Palladium). Knowing these sticking probabilities enables one to calculate the rate of adsorption ${\displaystyle r_{a}}$ by virtue of the equation ${\displaystyle r_{a}=S\Phi _{H}}$ where ${\displaystyle S}$ is the aforementioned sticking probability and ${\displaystyle \Phi _{H}}$ is the flux of Hydrogen molecules in the toward the surface of the Palladium/Palladium-hydride. When the system is in a steady state, we must have that the rate of adsorption and, oppositely, the rate of desorption (${\displaystyle r_{d}}$) are equal. This gives ${\displaystyle r_{a}=r_{d}}$ The rate of desorption is assumed to be given by a Boltzmannian distribution, i.e. ()${\displaystyle r_{d}=e^{-{\frac {E_{d}}{k_{B}T}}}}$ where ${\displaystyle C}$ is some unknown constant,${\displaystyle E_{d}}$ is the desorption energy, ${\displaystyle k_{B}}$ is Boltzmann’s constant and ${\displaystyle T}$ is the temperature. The relation () can be fitted to find the value of ${\displaystyle E_{d}}$. It was found that, within the uncertainty of their experiment, the values for of Palladium and Palladium-hydride respectively were roughly equal. Thus Palladium-hydride has as higher average adsorption rate than Palladium, while the energy required for desorption is the same. The reversible absorption of Palladium is a means to store hydrogen, and the above findings indicate that even in the hydrogen-absorbed state of Palladium, there is further opportunity for hydrogen storing. ## References 1. ^ a b Manchester, F. D.; San-Martin, A.; Pitre, J. M. (February 1994). "The H-Pd (hydrogen-palladium) System". Journal of Phase Equilibria. 15 (1): 62–83. doi:10.1007/BF02667685. S2CID 95343702. 2. ^ a b Grochala, Wojciech; Edwards, Peter P. (March 2004). "Thermal Decomposition of the Non-Interstitial Hydrides for the Storage and Production of Hydrogen". Chemical Reviews. 104 (3): 1283–1316. doi:10.1021/cr030691s. PMID 15008624. 3. ^ a b c Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. pp. 1150–151. ISBN 978-0-08-037941-8. 4. Kawae, Tatsuya; Inagaki, Yuji; Wen, Si; Hirota, Souhei; Itou, Daiki; Kimura, Takashi (15 May 2020). "Superconductivity in Palladium Hydride Systems". Journal of the Physical Society of Japan. 89 (5): 051004. Bibcode:2020JPSJ...89e1004K. doi:10.7566/JPSJ.89.051004. 5. ^ Ralph Wolf; Khalid Mansour. "The Amazing Metal Sponge: Soaking Up Hydrogen" Archived 2015-11-16 at the Wayback Machine. 1995. 6. ^ "Extending the Life of Palladium Beds" Archived 2015-10-31 at the Wayback Machine. 7. ^ a b c Tew, Min Wei; Miller, Jeffrey T.; van Bokhoven, Jeroen A. (27 August 2009). "Particle Size Effect of Hydride Formation and Surface Hydrogen Adsorption of Nanosized Palladium Catalysts: L 3 Edge vs K Edge X-ray Absorption Spectroscopy". The Journal of Physical Chemistry C. 113 (34): 15140–15147. doi:10.1021/jp902542f. 8. ^ a b c d Setayandeh, S. S.; Webb, C. J.; Gray, E. MacA. (1 December 2020). "Electron and phonon band structures of palladium and palladium hydride: A review". Progress in Solid State Chemistry. 60: 100285. doi:10.1016/j.progsolidstchem.2020.100285. S2CID 225592643. 9. ^ Davis, R. J.; Landry, S. M.; Horsley, J. A.; Boudart, M. (15 May 1989). "X-ray-absorption study of the interaction of hydrogen with clusters of supported palladium". Physical Review B. 39 (15): 10580–10583. Bibcode:1989PhRvB..3910580D. doi:10.1103/PhysRevB.39.10580. PMID 9947864. 10. ^ Tripodi, Paolo; Di Gioacchino, Daniele; Borelli, Rodolfo; Vinko, Jenny Darja (May 2003). "Possibility of high temperature superconducting phases in PdH". Physica C: Superconductivity. 388–389: 571–572. Bibcode:2003PhyC..388..571T. doi:10.1016/S0921-4534(02)02745-4. 11. ^ Tripodi, Paolo; Di Gioacchino, Daniele; Vinko, Jenny Darja (August 2004). "Superconductivity in PdH: phenomenological explanation". Physica C: Superconductivity. 408–410: 350–352. Bibcode:2004PhyC..408..350T. doi:10.1016/j.physc.2004.02.099. 12. ^ Tripodi, Paolo; Di Gioacchino, Daniele; Vinko, Jenny Darja (2007). "A review of high temperature superconducting property of PdH system". International Journal of Modern Physics B. 21 (18&19): 3343–3347. Bibcode:2007IJMPB..21.3343T. doi:10.1142/S0217979207044524. 13. ^ Baranowski, B.; Dębowska, L. (June 2007). "Remarks on superconductivity in PdH". Journal of Alloys and Compounds. 437 (1–2): L4–L5. doi:10.1016/j.jallcom.2006.07.082. 14. ^ Mitsui, T.; Rose, M. K.; Fomin, E.; Ogletree, D. F.; Salmeron, M. (April 2003). "Dissociative hydrogen adsorption on palladium requires aggregates of three or more vacancies". Nature. 422 (6933): 705–707. Bibcode:2003Natur.422..705M. doi:10.1038/nature01557. PMID 12700757. S2CID 4392775. 15. ^ Lopez, Nuria; Łodziana, Zbigniew; Illas, Francesc; Salmeron, Miquel (29 September 2004). "When Langmuir Is Too Simple: H 2 Dissociation on Pd(111) at High Coverage". Physical Review Letters. 93 (14): 146103. Bibcode:2004PhRvL..93n6103L. doi:10.1103/PhysRevLett.93.146103. hdl:2445/13263. PMID 15524815. 16. ^ Johansson, M.; Skúlason, E.; Nielsen, G.; Murphy, S.; Nielsen, R.M.; Chorkendorff, I. (April 2010). "Hydrogen adsorption on palladium and palladium hydride at 1bar". Surface Science. 604 (7–8): 718–729. Bibcode:2010SurSc.604..718J. doi:10.1016/j.susc.2010.01.023.	2023-03-20 14:09:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 13, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7859699130058289, "perplexity": 3677.9608184588988}, "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-2023-14/segments/1679296943483.86/warc/CC-MAIN-20230320114206-20230320144206-00401.warc.gz"}
https://topfuturepoint.com/formula-for-sigma/	# Formula for sigma Formula for sigma: Let us know about Formula for sigma. A series can be represented in a compact form, called the sum or sigma notation. The Greek capital letter, , is used to represent sum. The series 4+8+12+16+20+24 can be expressed as 6∑n=14n . The expression is read as the sum of 4n as n goes from 1 to 6. Also, what does backwards 3 mean in Mathematica? Mathematicians pronounce sigma as “sum,” and it means “to add things.” It differs from its English equivalent of abstract ideas, but it is a consequence of any equation. When you use it in an equation, sigma is the sum of everything that appears after the symbol. Here, what is a sigma value? A sigma value is a statistical term otherwise known as a standard deviation. … sigma is a measure of variability , which is defined by the InvestorWords website as “the range of possible outcomes for a given position”. Also know what is the law of sigma? An empirical rule that states that, for many reasonably isometric unequal distributions, nearly all of the population is within three standard deviations of the mean . About 99.7% of the population is within three standard deviations of the mean for a normal distribution. See also the two-sigma rule. What is the symbol for sigma? The symbol ( sigma ) is usually used to denote the sum of several terms. This symbol is usually accompanied by an index that varies to include all the terms that must be considered in the sum. For example, the sum of the first whole numbers can be represented in the following way: 1 2 3 . ## What does backwards Z mean? In the Pittman Initial Teaching Alphabet (ITA), a backward ‘z’ to denote a ‘ jes’ , and a harsh ‘s’ sound used in many plural forms of nouns and third person singular present forms of verbs (this includes is used to. The ITA is an educational aid, and is not used in ordinary writing to replace the standard alphabet. ##### What is it called? The relation “is an element of”, which is defined in . Also called set subscription , denoted by the symbol “∈”. ##### What is the symbol of all real numbers? symbol of real numbers Since the set of real numbers is the collection of all rational and irrational numbers, the real numbers are represented by the symbol r . ##### Is sigma 5.0a a virus? Member. Sigma is a virus and BTC miner . ##### What is 3 sigma value? The three-sigma limit (3-sigma limit) is a statistical calculation that refers to data within three standard deviations from a mean . … on the Bell curve, data that is above the mean and beyond the three-sigma line represents less than 1% of all data points. (Formula for sigma) ##### What is the P value of 5 sigma? So, what does five-sigma mean? In short, a five-sigma 3×10 . K corresponds to a p-value or probability – 7 or, approximately 3.5 in 1 million . ##### What are the properties of the sigma notation? Sigma, pronounced sig-mah, is a Greek letter meaning “sum” in mathematics. The sigma notation is also known as the sum notation and is a way of representing the sum of numbers . This is especially useful when there is a specific pattern of numbers or it takes too long to write without an abbreviation. #### How do you run the sigma rule? How to start writing sigma rules? 1. Step 1: Obtain the Sigma repository [Optional] The official Sigma repository contains all the documentation, sample rules, and compilers needed to convert Sigma into questions. , 2. Step 2: Create a YAML File. , 3. Step 3: Provide input to the attributes. , 4. Step 4: Compiling the rules. ##### What do you call the symbol F? A hook with the letter f (majuscule, minuscule: ) is a syllable of the Latin script, based on the italic form of f; Or paired with a descending hook on your regular look. A very similar looking letter, (a dotless J with a hook and a horizontal stroke), is used in IPA for a voiced palate implant. (Formula for sigma) ##### What does Sigma look like? Sigma is the 18th letter of the Greek alphabet and is equivalent to our letter ‘S’. … none of these —- no They look like our letter ‘S’, but they are both its Greek equivalents. ##### What is a sigma number? Sigma /sɪɡmə/ (uppercase , lowercase , lowercase in word-ending position; Greek: μα) is the eighteenth letter of the Greek alphabet. In the Greek numeral system, its value is . happens to be 200 . In general mathematics, uppercase is used as an operator for sum. ##### Can you say backwards ABC? But just because the backward alphabetic text test isn’t supported by the NHTSA doesn’t mean law enforcement can’t use it. … If the alphabet is used as a field sobriety test, the DUI suspect is usually asked to read it further without singing. ##### Which is the 27th letter of the alphabet? With its bizarre shape, neither letter nor symbol, more than a triple clef of type, the ampersand has captured our creative attention. But what is it about its elegant swoop and swirls that have made it the go-to typographic tool of choice? ##### What does R stand for in Mathematics? In mathematics, the letter R denotes the set of all real numbers . … Real numbers are numbers that include natural numbers, whole numbers, integers and decimal numbers. In other words, real numbers are defined as points on an infinitely extended line. ##### What does u mean in maths? Read More… A set formed by combining elements of two sets. So the union of sets A and B is the set of elements in A, or B, or both. The symbol is a special “u” like this: ##### What is there symbol for? The symbol means “there exists”. ##### What is odd e in math? This is the Greek capital letter sigma . Roughly equivalent to our ‘S’. It stands for ‘Yoga’. ##### What is R* in Mathematics? In mathematics, the notation R* represents two different meanings. In the number system, R* defines the set of all non-zero real numbers , which group under the multiplication operation, In functions, R* defines a reflexive-transitive closure of the binary relation “R” in the set. ##### Are real numbers a field? The first says that the real numbers comprise a field , with addition and multiplication as well as division by non-zero numbers, which can be completely ordered on a number line in a manner compatible with addition and multiplication.	2023-01-27 00:51:22	{"extraction_info": {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8091322779655457, "perplexity": 1241.0835072261036}, "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-2023-06/segments/1674764494852.95/warc/CC-MAIN-20230127001911-20230127031911-00185.warc.gz"}
https://www.zigya.com/study/book?class=11&board=bsem&subject=Physics&book=Physics+Part+I&chapter=Work,+Energy+and+Power&q_type=&q_topic=The+Concept+Of+Potential+Energy+&q_category=&question_id=PHENJE11157934	## Book Store Currently only available for. CBSE Gujarat Board Haryana Board ## Previous Year Papers Download the PDF Question Papers Free for off line practice and view the Solutions online. Currently only available for. Class 10 Class 12 The potential energy of a 1 kg particle free move along the x-axis is given by The total mechanical energy of the particle 2 J. Then, the maximum speed (in m/s) is • 2 B. 349 Views Is work a scalar or a vector quantity? Work is dot product of two vectors. i.e. . And dot product is a scalar quantity. Therefore, work is a scalar quantity. 1450 Views What are different units of energy? The different units of energy are : (i) Joule (ii) Erg (iii) eV (iv) KWh (v) Calorie. 804 Views What is watt? SI unit of power is Watt. 773 Views Define watt. Power is said to be one watt if one joule of work is done in one second. 860 Views Define the unit joule. Work done is said to be one joule if one newton of force displaces the body through a distance of one meter in the direction of applied force . 799 Views	2018-10-22 00:06:55	{"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": 0.5794869065284729, "perplexity": 3341.5805866529017}, "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-2018-43/segments/1539583514437.69/warc/CC-MAIN-20181021224001-20181022005501-00242.warc.gz"}
https://acm.ecnu.edu.cn/problem/2911/	# 2911. Lanes There are S people swimming in the pool. At some point, the coach notices that their distribution over pool lanes is quite uneven, which is a nuisance for pool visitors. Changing the lane across ropes, however, poses a nuisance for a swimmer. To form a nice distribution over the lanes the number of swimmers on each pair of adjacent lanes should differ by at most one. In particular, an empty lane should only be adjacent to another empty one or a lane occupied by only one swimmer. Your task is to calculate the minimum number of swimmers’ lane changes needed to meet this condition. ### 输入格式 The first line of the input contains the only integer N — the number of lanes, 1 ≤ N ≤ 400. The next line contains N numbers separated by spaces: the number of swimmers in the first, second, . . . , Nth lane respectively. The sum of these numbers equals S (0 ≤ S ≤ 1 500). ### 输出格式 Output the minimum possible number of lane changes that are needed to meet a desired distribution. ### 样例 Input 3 8 0 2 3 8 5 7 Output 5 1 5 人解决，8 人已尝试。 37 份提交通过，共有 82 份提交。 7.3 EMB 奖励。	2020-01-27 18:19:52	{"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": 0.45291852951049805, "perplexity": 1499.8424062612337}, "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/1579251705142.94/warc/CC-MAIN-20200127174507-20200127204507-00354.warc.gz"}
http://www.biomedsearch.com/nih/Plasma-triglyceride-concentrations-are-rapidly/20217117.html	Document Detail From MEDLINE®/PubMed®, a database of the U.S. National Library of Medicine Full Text Journal Information Journal ID (nlm-ta): Eur J Appl Physiol ISSN: 1439-6319 ISSN: 1439-6327 Publisher: Springer-Verlag, Berlin/Heidelberg Article Information Download PDF © The Author(s) 2010 Accepted Day: 15 Month: 2 Year: 2010 Electronic publication date: Day: 9 Month: 3 Year: 2010 pmc-release publication date: Day: 9 Month: 3 Year: 2010 Print publication date: Month: 7 Year: 2010 Volume: 109 Issue: 4 First Page: 721 Last Page: 730 ID: 2883923 PubMed Id: 20217117 Publisher Id: 1409 DOI: 10.1007/s00421-010-1409-7 Plasma triglyceride concentrations are rapidly reduced following individual bouts of endurance exercise in women Gregory C. HendersonAff1 Ronald M. KraussAff1 Jill A. FattorAff1 Nastaran FaghihniaAff1 Mona Luke-ZeitounAff1 George A. BrooksAff1 Address: +1-510-6422861 +1-510-6432439 gbrooks@berkeley.edu Exercise Physiology Laboratory, Department of Integrative Biology, University of California, 5101 Valley Life Sciences Building, Berkeley, CA 94720-3140 USA Communicated by footnote: Communicated by William Kraemer. Introduction Among the positive effects of leading a physically active lifestyle is the effect of the activity (exercise training) on the lipoprotein profile. Exercise training improves some aspects of plasma lipids, though the precise nature of the effect is variable among studies. Of the many factors that may explain variability, time since the most recent training bout is likely to be very important as the effects of training may wane quickly following cessation of the training program. For example, though reduced plasma triglyceride (TG) concentration in people who regularly exercise may be an important factor in their decreased risk for atherosclerosis and other forms of cardiovascular disease (CVD), the effect may be through the repeated, but transient, impact of each exercise bout on plasma TG, as it was reported that 3 days after cessation of training the favorable training-induced changes were lost (Leon et al. 2000). It may actually be that much of the health benefit of exercise training accrues from exercisers being in a perpetual state of postexercise recovery. Thus, studies of postexercise recovery following individual bouts of exercise could indicate important health benefits of regular exercise. Though studies have not yet revealed differences between men and women or the effects of exercise intensity, there are reports of acute effects of exercise on the lipoprotein profile during recovery. To assess acute changes in the lipoprotein profile following exercise sessions, it is important to control for diurnal variations and for drift of nutritional state between meals by conducting time-of-day-matched sedentary control trials on separate occasions. Thus, as performed in the studies discussed below, changes that occur during postexercise recovery can be compared to changes that may occur on sedentary occasions. There appear to be no significant changes in total cholesterol (TC) or low-density lipoprotein cholesterol (LDL-C) in either sex during the hours following a single exercise session (Gill et al. 2003; Gordon et al. 1996; Lee et al. 1991), but results for TG and HDL-C are somewhat more promising. In women, following an acute exercise bout, it was reported that plasma TG concentration declined within 90 min after exercise (Lee et al. 1991) and the effect was sustained until the following day (Gill et al. 2003; Lee et al. 1991), though others did not find TG to be depressed on the day after exercise (Magkos et al. 2009). In men, results have been variable showing no effect (Gordon et al. 1996; Magkos et al. 2007) or depressed plasma TG during recovery when exercise volume was quite high (Magkos et al. 2006), but it was unknown if impacts of exercise would have been clearer if plasma samples had been drawn closer in time to the exercise session rather than so far out into the recovery period. That is to say, the effects of a single bout of exercise might primarily act during the few hours after exercise rather than being delayed until the day following exercise. Studies that directly compare men with women during the hours immediately following exercise are still needed. Additionally, exercise bouts can lead to subsequently elevated concentrations of high-density lipoprotein cholesterol (HDL-C) in women (Lee et al. 1991) and men (Gordon et al. 1996), though little is known about the effects of exercise intensities, energy expenditures or effects of gender when controlling for changes that may occur during the course of a sedentary day. Compared to men of similar age, premenopausal women have been shown to have a more favorable lipoprotein profile (Després et al. 1999; Freedman et al. 1990; Johnson et al. 2004; Leon et al. 2000; St-Amand et al. 1995), of which the most substantial differences seem to be having lower plasma TG and higher plasma HDL-C. Additionally, women may have lower postprandial lipemia because of more rapid clearance of meal fatty acids away from chylomicrons to peripheral storage sites (Knuth and Horowitz 2006). Thus, it appears that there are some important gender differences with regard to lipoprotein metabolism and CVD risk. However, how these gender differences interact with changes in lipemia following physical activity is not yet known and, furthermore, the effects of the intensity of exercise have not yet been determined. Therefore, we studied the lipoprotein profile in young men and women during and 3 h after exercise sessions of two different relative intensities that were matched for energy expenditure within each sex. Importantly, because the effects of exercise would be overlaid on the effects of the time of day and time course of meal absorption, we compared the results to a sedentary condition in which diet was identical. We conjectured that the effects of chronic exercise training to increase HDL-C (Després et al. 1990; Katzmarzyk et al. 2001; Kiens et al. 1980; Kraus et al. 2002; Leon et al. 2000; Slentz et al. 2007; Thompson et al. 1988; Williams et al. 2005) and reduce TG (Kiens et al. 1980; Kraus et al. 2002; Leon et al. 2000; Thompson et al. 1988) are an accumulation of changes that occur following each individual exercise bout and largely reflect the impact of the most recent session. Furthermore, though far more investigation is needed to understand the importance of exercise training intensity, higher-intensity exercise is believed to lead to some additional healthy benefits beyond that of lower-intensity exercise (Institutes of Medicine 2002). Thus, we hypothesized that an exercise bout would acutely raise HDL-C and lower plasma TG in an intensity-dependent manner in both sexes. The results revealed an important sex-based difference for the impact of physical activity on plasma lipids and revealed a potential mechanism by which a physically active lifestyle reduced CVD risk in young women. Methods Study participants Moderately active, non-smoking, weight-stable volunteers were recruited from the University of California, Berkeley campus and surrounding community by posted notice and e-mail. Initially, 10 men and 10 women were recruited to participate in the study. Potential study participants underwent subsequent screening tests for being disease-free as determined by physical examination and health history questionnaire. They were not taking medications known to affect metabolism, had a body mass index (BMI) of less than 28, were neither sedentary individuals (abstinence from any regular exercise) nor elite athletes (intercollegiate or professional athletes) and had normal lung function as determined by 1-s forced expiratory volume of greater than or equal to 70% of vital capacity. Female study participants reported regular menstrual cycles (24–32 days) and were not taking oral contraceptives or other forms of exogenous ovarian hormones. We sought to study women in the early follicular phase of the menstrual cycle to standardize conditions between trials. Therefore, female subjects were studied between days 3 and 8 of their menstrual cycles, and the cycle phase was subsequently confirmed if estradiol was less than 50 pg/mL and progesterone was less than 1 ng/mL in serum collected the morning of each study occasion. In two women, serum estradiol concentrations were greater than 50 pg/mL in one of the trials. For one woman, this was on an exercise day and that trial was excluded. For the other woman, the elevation of estradiol occurred on the control day and so all three of her trials were excluded. Additionally, we were unable to attain data in every trial for the men. Therefore, participants per trial ranged from seven to nine for men and eight to nine for women. In Table 1, we report the characteristics of the study participants. Habitual exercise activity of approximately 6–7 h/week of moderate exertion (more intense than walking and less intense than competitive sporting competition) was reported by both the men and women. Procedures and risks were thoroughly explained to the study participants, and their written, informed consent was obtained. The University of California, Berkeley Committee for the Protection of Human Subjects approved the study protocol (CPHS no. 2004-6-103). Screening tests Before beginning the study, participants underwent two progressive exercise tests to assess peak oxygen consumption [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\dot{V}{\text{O}}_{2}$$\end{document}] peak) and body composition was assessed by skinfold measurement (Jackson and Pollock 1978; Jackson et al. 1980). In previous investigations, we found the skinfold and hydrostatic weighing methods to provide similar results on the study population of interest (Friedlander et al. 1998, 1999). As changes over time in physical fitness and body fat content could alter lipoprotein profiles, [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak and body composition assessments were again carried out on completion of the study to confirm that changes had not occurred during the time in which participants were enrolled in the study. [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak and body composition did not vary significantly between assessments. Exercise was performed on a leg-cycle ergometer (Monark Ergometric 839E, Vansbro, Sweden). Additionally, as changes in dietary composition over time in study participants could alter results, dietary energy and macronutrient intake were monitored at the beginning, middle and end of the study by separate 3-day diet records; analysis was performed with Diet Analysis Plus 6.1 software (ESHA Research, Salem, OR, USA). Diets did not vary significantly between the three separate assessments and, therefore, average values are reported. Experimental design With at least 1 week between trials for men and 1 month between trials for women, participants were studied under each of the three conditions, each on separate occasions, assigned in a random order. Men and women were studied (1) before, during, and 3 h after ~90 min of exercise at 45% [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak (E45), (2) before, during and 3 h after ~60 min of exercise at 65% [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak (E65), and (3) during a time-matched resting control trial (C). After catheterization, study participants lay semi supine quietly reading or watching movies. After exercise, study participants dismounted the ergometer and sat on a chair where they remained for 30 min and were then transferred to an examination table where they remained semi supine for the remaining 2.5 h of postexercise recovery. Water was consumed by study participants ad libitum, but they consumed no food during recovery. Participants were transported in a wheelchair for trips to the toilet. Duration of the first randomly assigned exercise trial, either E45 or E65, was set at 90 or 60 min, respectively. The appropriate duration for the subsequent exercise trial in the remaining exercise condition was predicted with the goal of matching exercise energy expenditure (EEE) between exercise bouts using oxygen consumption [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\dot{V}{\text{O}}_{2} )$$\end{document}] and respiratory exchange ratio (RER) from the [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak assessments. Experimental protocol For the day prior to studies, participants were instructed to consume a standardized diet and water ad libitum, and to abstain from structured physical exercise sessions, but to continue typical activities of daily living. On the day preceding the studies, participants were fed for a physical activity level (PAL) of 1.5 according to the current dietary reference intake guidelines of the Institutes of Medicine for estimated energy requirement (EER) (Institutes of Medicine 2002). These standardized diets were individualized for each study participant (men 2,762 ± 68 kcal/day, women 2,101 ± 36 kcal/day) and macronutrient composition was made similar between individuals for carbohydrate (men 50.2 ± 0.3%, women 50.2 ± 0.2%), fat (men 31.9 ± 0.3%, women 32.3 ± 0.4%) and protein (men 17.9 ± 0.5%, women 17.6 ± 0.2%). On the study days, participants arrived at the laboratory at 7:00 a.m. after overnight fasting and ate a standardized breakfast (men 450 kcal, 67% carbohydrate, 22% fat, 11% protein; women 345 kcal, 65% carbohydrate, 26% fat, 9% protein) that represented approximately 16% of EER in both men and women. We chose to feed our study participants a standardized small breakfast 3 h before exercise to mimic typical nonlaboratory conditions, and following breakfast no more food was consumed until after the last blood sample was drawn. On the morning of the studies, a catheter was placed in a hand vein to collect arterialized blood using the heated hand vein technique (Horning et al. 1998), and pulmonary gas exchange was determined for assessment of metabolic rate and substrate oxidation, which we reported previously (Henderson et al. 2007a). At each sampling time point, heart rate was recorded from an electrocardiograph (Quinton Q750, Seattle, WA, USA). Blood sampling Arterialized venous blood samples were drawn immediately before exercise, in the last moments of exercise bouts (~90 min at 45% VO2peak and ~60 min at 65% VO2peak), and then 3 h after exercise during recovery. During the C trial, blood samples were drawn at the time point corresponding to pre-exercise (3 h after breakfast), 75 min later, and then 3 h later to correspond to approximate times of day that samples were taken in E45 and E65. Blood was collected into tubes containing EDTA (0.15%), enzyme inhibitors (D-phenylalanyl-L-prolyl-L-arginyl-CHCl2, 1 μM; aprotinin, 50 KU/mL), antibiotics (gentamicin sulfate, 50 μg/mL; chloramphenicol sodium succinate, 0.05 mg/mL), and a bacteriocide (sodium azide, 0.01% wt/vol). Blood was centrifuged at 3,000×g for 20 min and the plasma stored at −80°C with the tubes’ headspaces filled with nitrogen gas. Hematocrit measurements were performed on arterialized venous blood using a circular microcapillary tube reader. Sample analyses Plasma was analyzed for concentrations of TC (Allain et al. 1974), TG (Nagele et al. 1984) and HDL-C measured directly after precipitation of apoB containing lipoproteins in plasma (Warnick et al. 1985). LDL-C was calculated from the formula of Friedewald et al. (Friedewald et al. 1972). Lipid assays were enzymatic end-point measurements utilizing enzyme reagent kits and a Ciba-Corning Express 550 automated analyzer (Ciba-Corning Diagnostics, Oberlin, OH, USA). The measurements were standardized through the CDC-NHLBI Lipid Standardization Program. Measurement of LDL peak particle size was performed on whole plasma with the use of nondenaturing 2–14% polyacrylamide gradient gel electrophoresis and standardized conditions (Nichols et al. 1986). Following electrophoresis, lipoproteins were lipid stained with Sudan Black, and the protein calibration standards were stained with Coomassie R-250. Gels were analyzed using computer-automated densitometry, and calculations of peak particle sizes were based on the migration of reference standards of known particle size. An immunoturbidimetric assay (Rifai and King 1986; Smith et al. 1987) was used to measure apoA-I and apoB. Reagents, standards and reference plasma controls, with and without elevated lipids, were included in the immunoturbidimetric assay reagent kit (Bacton Assay Systems, San Marcos, CA, USA). Measurements were performed using the Express Plus 550 analyzer according to kit instructions. Calibrators and reference controls were assigned concentration values with the use of International Federation of Clinical Chemistry standard reference materials SP1 for apoA-I and SP3-07 for apoB. In-house controls were measured in each group of 20 unknowns, as well. For each assay, coefficients of variation were less than 10%, and all samples were analyzed in duplicate. Calculations EEE was assessed by pulmonary gas exchange (Frayn 1983) and was calculated for each exercise intensity in each individual by subtracting the background resting energy expenditure rate in the C trial from the energy expenditure of the exercise sessions. Baseline lipoprotein concentration values are reported as those measured in plasma and the subsequent two samples in a trial were corrected for estimated changes in plasma volume from baseline using hematocrit (van Beaumont 1973). Statistical analyses Data are presented as mean ± standard error. Subject characteristics and baseline (pre-exercise) lipoprotein values were compared between genders by unpaired t test. Additional specific unpaired t tests comparing men with women were performed for values during exercise of a given intensity. Also, with unpaired t tests, we compared between sexes the relative change from baseline during exercise trials to that in C trials when either sex showed a significant difference from C after exercise. Results for percentage change from pre-exercise values were analyzed within sexes by repeated measures analysis of variance (trial × time) with post hoc analyses by Fisher’s protected least significant difference test. Statistical analyses were performed using JMP 7.0 software (SAS, Inc., Cary, NC, USA), and the statistical significance was set at α = 0.05. Results Diet records As expected, reported habitual dietary energy intake was significantly higher in men than women (men 2,435 ± 112 kcal/day, women 1,966 ± 96 kcal/day, P < 0.05). Habitual dietary macronutrient composition was not significantly different between sexes for carbohydrate (men 54.1 ± 2.0%, women 54.0 ± 1.8%), lipid (men 28.9 ± 2.0%, women 29.4 ± 1.3%) and protein (men 17.0 ± 0.6%, women 16.6 ± 0.8%). Characteristics of exercise bouts EEE was significantly greater in men than women (P < 0.05). The durations of E45 (men 88.8 ± 0.5 min, women 90.4 ± 0.9 min) and E65 (men 60.4 ± 0.2 min, women 61.2 ± 0.6 min) were such that energy expenditures would be similar for the two intensities for both men and women. EEE was matched between exercise trials in that there were no significant differences between trials for men (E45 693 ± 55 kcal, E65 698 ± 57 kcal) and women (E45 427 ± 24 kcal, E65 439 ± 24 kcal). As the physical fitness level [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\dot{V}{\text{O}}_{2}$$\end{document}] peak per FFM) was similar between men and women, [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] was not significantly different between sexes during exercise when expressed per FFM in either E45 or E65, and relative exercise intensities were also similar for the sexes in E45 (men 46.5 ± 0.5% [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak, women 46.2 ± 1.2% [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak) and E65 (men 64.9 ± 1.5% [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak, women 65.2 ± 0.6% [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak). The heart rates elicited during exercise were similar between men and women for E45 (men 129 ± 1 beats/min, women 122 ± 5 beats/min) and E65 (men 156 ± 3 beats/min, women 152 ± 5 beats/min). Lipoprotein profile The baseline (pre-exercise) lipoprotein profiles for men and women are reported in Table 2. As conditions were similar between the three trials (C, E45, E65) for the pre-exercise time point, we report averages of the three occasions. Baseline HDL-C was significantly higher in women than men (P < 0.05). Baseline TG was not significantly different between men and women (P = 0.12), nor were TC (P = 0.51), LDL-C (P = 0.75), apoA-1 (P = 0.10), apoB (P = 0.48) or LDL peak particle size (P = 0.08). During exercise of either intensity, HDL-C and ApoA-I were significantly higher in women than men (P < 0.05), which primarily reflected the baseline values, as concentrations in either sex had not changed substantially from baseline. The relative changes from pre-exercise for cholesterol subfractions and apolipoproteins are reported in Table 3 and for plasma TG in Fig. 1. In men, there was a main effect of time for TG to decrease across the day (P < 0.05), and for LDL-C (P < 0.05) and HDL-C (P < 0.05) to increase. However, in men there were no main effects of trial and no trial × time interactions. In women, there was a significant trial × time interaction for relative change of TG from pre-exercise values, and post hoc testing revealed that E45 and E65 were significantly lower than C during postexercise recovery (P < 0.05). In women, on subtracting the approximate drift of 1% from pre-exercise to post-exercise time points in C, in E45 the net decline of plasma TG below C was 15% and in E65 was 25% below C. In men, there was no significant postexercise decline of TG versus the C trial, and across the day in both E45 and E65, the percentage of change versus that in C was significantly greater for women than men (P < 0.05). However, though this substantial gender difference became apparent during recovery, during the exercise bouts there were not yet any significant differences for TG changes between men and women. In women, there was a main effect of time for TC to increase (P < 0.05), and as in men, there were main effects for LDL-C (P < 0.05) and HDL-C (P < 0.05) to increase slightly during the day. Apolipoproteins and LDL particle size changed neither during a sedentary day nor during exercise or recovery. Discussion Here, we report that in healthy, young and moderately fit individuals, a single session of exercise does not change plasma apolipoproteins, TC, LDL-C, HDL-C or LDL particle size. However, of importance, we report the novel finding that plasma TG declines substantially within 3 h of exercise in women, but is not different from resting in men when a time-of-day comparison is made. It is recommended that exercise be performed every day (Institutes of Medicine 2002), and so individuals following this recommendation will spend a significant portion of their lives in a state of postexercise recovery. Hence, any important aspects of metabolism or of the lipoprotein profile that change during this time period (e.g., TG in women) could vastly affect an active person’s risk factor profile. Our present results indicate that a single exercise bout of reasonable volume can lower plasma TG in women and that the effect may primarily occur within the day of the exercise bout. On the contrary, it appears that any potential effects of exercise on TG in men or on HDL-C in either gender may require exercise sessions on sequential days, longer exercise duration or more than 3 h of recovery to become detectable. We have conducted studies of postexercise recovery in men and women for 3 h following exercise (Henderson et al. 2007a, b, 2008), as well as on the day after exercise (Henderson et al. 2007a). In these studies (Henderson et al. 2007a, b, 2008), we showed that whole body glycerol flux (lipolysis), plasma free fatty acid (FFA) flux and plasma FFA concentration were elevated to a greater extent in men than women during 3 h of postexercise recovery (Henderson et al. 2007a). The present results for postexercise plasma lipids show that changes in plasma TG are dissociated from changes in postexercise lipolysis and fatty acid metabolism, in that the sex with a lesser elevation of systemic FFA mobilization (women) actually showed a greater postexercise decline of plasma TG. In addition to our study on whole body glycerol and FFA kinetics, others have studied FFA and plasma TG kinetics in men (Magkos et al. 2006, 2007) and women (Magkos et al. 2009) on the day following exercise. They showed that on the next day following exercise of 2 h at 60% [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak, men display elevated FFA flux and depressed plasma TG concentrations (Magkos et al. 2006), but that the effect on plasma TG was not attained if the exercise duration was only 1 h (Magkos et al. 2007). These results for elevated postexercise FFA flux the day after exercise mirror our finding for FFA flux during hours immediately following exercise (Henderson et al. 2007a) and the results also suggest that the threshold of exercise volume for affecting plasma TG in men may be such that a quite large exercise volume is needed. In women, on the day following 1 h of exercise at 60% [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak, it was reported that neither FFA flux nor plasma TG concentrations were affected (Magkos et al. 2009). However, studying a different portion of the recovery period, we report here that plasma TG is significantly depressed 3 h after exercise in women, and so the timing of the measurements appears to be of considerable importance. Compared with men, women display a greater response of the lipoprotein profile during recovery from exercise regardless of whether the bout was of a moderate or hard intensity. Thus, we presently describe a gender-specific benefit of individual bouts of exercise. If the absolute EEE were the major determinant of plasma TG changes during recovery, then one would expect a greater decline in men as absolute EEE was higher in men (because men have higher FFM), and so clearly EEE is not the major determinant of the TG response. Previously, the work of others showed us that in women and men with similar exercise intensities and durations, plasma TG may be similarly unaltered on the day following exercise bouts of a 1-h duration (Magkos et al. 2007, 2009). However, in women, even with a reasonable exercise volume (duration of 45 min), it was shown that TG can decline within even 1.5 h of exercise cessation (Lee et al. 1991), but it was not yet known if TG could decline soon after exercise in men as well. On the whole, it appears possible that the important effects of exercise on plasma TG primarily occur within the day of the exercise bout and especially in women. To consider factors that may be involved in the sex-based differences in postexercise TG metabolism, lipoprotein lipase (LPL) ought to be considered, as it is an important enzyme in muscle and adipose tissue capillary beds that promotes clearance of plasma TG and is known to be responsive to exercise and to changes in energy balance. At rest, compared to men, women have slightly higher total body LPL activity (Després et al. 1999) and substantially higher adipose tissue LPL activity (St-Amand et al. 1995). This could explain the tendency for women to have lower plasma TG in the C trial in this study, but a differential response between the sexes for LPL to prior exercise would be more helpful in explaining our results for postexercise plasma TG. In a previous study, in which men and women were grouped together into a single subject pool, it was shown that muscle LPL protein content was not yet elevated at 0.2 h after exercise, but was subsequently elevated substantially in a muscle sample taken 4 h after exercise (Seip et al. 1997). So, it is fairly probable that during the 3 h of postexercise recovery in our study that muscle LPL activity may have been elevated at some point. However, it is unknown if there is a sex difference in LPL activity during the hours after exercise, and addressing this issue may be an important direction for future investigations. With regard to postexercise hypotriglyceridemia in women, another possible explanation could be that, rather than plasma TG clearance being elevated in women during recovery, plasma TG synthesis might have been depressed, leading to declining plasma TG abundance. Indeed, it is known that elevated FFA concentrations in plasma can lead to higher plasma TG via an increased TG synthesis rate (Lewis et al. 1995). Thus, as we previously reported in these same study participants that plasma FFA concentrations in recovery were elevated to a much lesser extent in women than men (Henderson et al. 2007a), this may have led to a slower VLDL TG synthesis rate in the women and hence to a declining plasma TG concentration. Though elevated plasma FFA in men in recovery may promote lipid oxidation, it may also promote plasma TG synthesis and therefore prevent postexercise hypotriglyceridemia. Thus, the reason why women can achieve postexercise hypotriglyceridemia could be related to their maintenance of whole body lipolysis at basal levels during this period. Though the pre-exercise breakfast in our study was small and low in fat, it is possible that our findings were based at least partly in differences of the effect of exercise on chylomicron metabolism between the sexes, as our analysis of plasma TG did not differentiate between exogenous and endogenous sources. After consumption of a large high-fat meal, in comparison to men, women were previously shown to have lower postprandial lipemia because of more rapid clearance of meal fatty acids away from chylomicrons (Knuth and Horowitz 2006). With regard to clinical relevance, it is known that both elevated fasting and postprandial plasma TG are associated with CVD risk (Austin et al. 1998; Bansal et al. 2007; Stampfer et al. 1996). Thus, whether the postexercise depression of plasma TG in women represents lowered endogenous plasma TG or lowered chylomicrons, either way such a drop in total plasma TG can be reasoned to have a beneficial impact. Though a standard practice may be to measure lipoproteins in the overnight-fasted state, we fed our subjects breakfast to mimic nonlaboratory conditions and so we believe that our findings relate to the practical importance of exercise for altering metabolism of plasma lipids throughout a day of life. It is known that physically active individuals experience less risk of CVD (Institutes of Medicine 2002), and we propose that an important mechanism by which this occurs in women is by transient declines of plasma TG during postexercise recovery. In men, any effects via plasma lipids may occur later in recovery or indirectly via training-induced loss of body fat. Our results indicate that when women have exercised earlier in the day, their plasma TG will be substantially lower than it would be on a sedentary occasion. Compared to the effects of chronic training on plasma TG (Kiens et al. 1980; Kraus et al. 2002; Leon et al. 2000; Thompson et al. 1988), which are not sustained well and are of limited magnitude (Leon et al. 2000), the acute effect of exercise on plasma TG in women (Fig. 1) is appreciable. In comparison to their time-matched sedentary control trials, this response of plasma TG was not observed in men in the present study. In men, plasma TG declined somewhat throughout the day in the C trial such that postexercise declines were not different from declines that would occur on a sedentary day. The decline of TG in the C trial in men may have been related to their lipemic response to the breakfast meal, or may represent some other gender difference in diurnal TG variation. Nonetheless, as C trials in men and women were qualitatively different, it was clearly essential to compare postexercise data to a control trial to determine the effects of exercise per se. Our present results regarding acute effects of prior exercise on plasma TG add to the growing body of knowledge describing differences between men and women for the acute and chronic responses to endurance exercise. Such gender differences, in addition to that which we report here, include that women, compared to men, derive a higher percentage of energy from lipid and less from carbohydrate during endurance exercise (Carter et al. 2001; Devries et al. 2006, 2007; Friedlander et al. 1998, 1999; Henderson et al. 2007a, b; Horton et al. 1998; Phillips et al. 1993; Tarnopolsky et al. 1990, 2007) and thus appear better equipped to utilize lipids as fuel. But, from previous work, it is also known that in response to chronic endurance training, men lose more body fat than do women (Ballor and Keesey 1991; Donnelly and Smith 2005), which may be related to higher increments of postexercise lipid mobilization in men than women (Henderson et al. 2007a). Thus, it continually becomes increasingly more evident that the metabolic responses to exercise are different between men and women. In conclusion, exercise impacts metabolism in qualitatively different manners between the sexes. In the hours following exercise, the lipoprotein profile is unaltered in men, but is favorably changed in women. Our present results indicate that exercise acutely leads to a decline of plasma TG in women even within 3 h of the exercise bout. Furthermore, when energy expenditure is matched between exercise intensities, the response is similar between two different exercise sessions. The results of this study indicate that women will spend a significant portion of their lives in a hypotriglyceridemic state if they exercise regularly because of rapid declines of plasma TG during the hours following each exercise session. These results add to the complexity and elaborate nature of our growing knowledge of gender differences in the physiological responses to exercise. We thank the study participants for their time and compliance with the protocol. We also thank Tamara Mau, Matthew Johnson, Martina Patella, Betty Liang and Rowan Sill for providing laboratory support. This work was supported by the National Institute of Health grant AR 42906 and by gifts from the Brian and Jennifer Maxwell Foundation and CytoSport, Inc. The experiments comply with the current laws of the USA. Conflict of interest Statement The authors declare that they have no conflict of interest. References Allain CC,Poon LS,Chan CS,Richmond W,Fu PC. Enzymatic determination of total serum cholesterolClin ChemYear: 1974204704754818200 Austin MA,Hokanson JE,Edwards KL. Hypertriglyceridemia as a cardiovascular risk factorAm J CardiolYear: 1998817B12B10.1016/S0002-9149(98)00031-99462597 Ballor DL,Keesey RE. A meta-analysis of the factors affecting exercise-induced changes in body mass, fat mass and fat-free mass in males and femalesInt J ObesYear: 1991157177261838100 Bansal S,Buring JE,Rifai N,Mora S,Sacks FM,Ridker PM. Fasting compared with nonfasting triglycerides and risk of cardiovascular events in womenJAMAYear: 200729830931610.1001/jama.298.3.30917635891 Carter SL,Rennie C,Tarnopolsky MA. Substrate utilization during endurance exercise in men and women after endurance trainingAm J Physiol Endocrinol MetabYear: 2001280E898E90711350771 Després JP,Tremblay B,Moorjani S,Lupien PJ,Thériault G,Nadeau A,Bouchard C. Long-term exercise training with constant energy intake. 3: effects on plasma lipoprotein levelsInt J ObesYear: 19901485942312217 Després JP,Gagnon J,Bergeron J,Couillard C,Leon AS,Rao DC,Skinner JS,Wilmore JH,Bouchard C. Plasma post-heparin lipase activities in the HERITAGE Family Study: the reproducibility, gender differences, and associations with lipoprotein levels. HEalth, RIsk factors, exercise Training and GEneticsClin BiochemYear: 19993215716510.1016/S0009-9120(98)00106-410383074 Devries MC,Hamadeh MJ,Phillips SM,Tarnopolsky MA. Menstrual cycle phase and sex influence muscle glycogen utilization and glucose turnover during moderate-intensity endurance exerciseAm J Physiol Regul Integr Comp PhysiolYear: 2006291R1120R112816690766 Devries MC,Lowther SA,Glover AW,Hamadeh MJ,Tarnopolsky MA. IMCL area density, but not IMCL utilization, is higher in women during moderate-intensity endurance exercise, compared with menAm J Physiol Regul Integr Comp PhysiolYear: 2007293R2336R234217913867 Donnelly JE,Smith BK. Is exercise effective for weight loss with ad libitum diet? Energy balance, compensation, and gender differencesExerc Sport Sci RevYear: 20053316917410.1097/00003677-200510000-0000416239833 Frayn KN. Calculation of substrate oxidation rates in vivo from gaseous exchangeJ Appl PhysiolYear: 1983556286346618956 Freedman DS,Jacobsen SJ,Barboriak JJ,Sobocinski KA,Anderson AJ,Kissebah AH,Sasse EA,Gruchow HW. Body fat distribution and male/female differences in lipids and lipoproteinsCirculationYear: 199081149815062110035 Friedewald WT,Levy RI,Fredrickson DS. Estimation of the concentration of low-density lipoprotein cholesterol in plasma, without use of the preparative ultracentrifugeClin ChemYear: 1972184995024337382 Friedlander AL,Casazza GA,Horning MA,Budinger TF,Brooks GA. Effects of exercise intensity and training on lipid metabolism in young womenAm J Physiol Endocrinol MetabYear: 1998275E853E863 Friedlander AL,Casazza GA,Horning MA,Usaj A,Brooks GA. Endurance training increases fatty acid turnover, but not fat oxidation, in young menJ Appl PhysiolYear: 1999862097210510368378 Gill JM,Herd SL,Vora V,Hardman AE. Effects of a brisk walk on lipoprotein lipase activity and plasma triglyceride concentrations in the fasted and postprandial statesEur J Appl PhysiolYear: 20038918419010.1007/s00421-002-0788-912665983 Gordon PM,Visich PS,Goss FL,Fowler S,Warty V,Denys BJ,Metz KF,Robertson J. Comparison of exercise and normal variability on HDL cholesterol concentrations and lipolytic activityInt J Sports MedYear: 19961733233710.1055/s-2007-9728568858403 Henderson GC,Fattor JA,Horning MA,Faghihnia N,Johnson ML,Mau TL,Luke-Zeitoun M,Brooks GA. Lipolysis and fatty acid metabolism in men and women during the postexercise recovery periodJ PhysiolYear: 200758496398110.1113/jphysiol.2007.13733117855762 Henderson GC,Fattor JA,Horning MA,Faghihnia N,Luke-Zeitoun M,Brooks GA. Retention of intravenously infused [13C]bicarbonate is transiently increased during recovery from hard exerciseJ Appl PhysiolYear: 20071031604161210.1152/japplphysiol.00309.200717702837 Henderson GC,Fattor JA,Horning MA,Faghihnia N,Johnson ML,Luke-Zeitoun M,Brooks GA. Glucoregulation is more precise in women than in men during postexercise recoveryAm J Clin NutrYear: 2008871686169418541557 Horning MA,Friedlander AL,Casazza GA,Huie MJ,Brooks GA. Arterial and ‘arterialized’ sampling sites do not change isotopic enrichment using [6, 6-D-glucose] and [1, 1, 2, 3, 3,-D-glycerol]FASEB JYear: 199812A854 Horton TJ,Pagliassotti MJ,Hobbs K,Hill JO. Fuel metabolism in men and women during and after long-duration exerciseJ Appl PhysiolYear: 199885182318329804587 Dietary reference intakes for energy, carbohydrate, fiber, fat, fatty acids, cholesterol, protein, and amino acidsYear: 2002Washington, DCNational Academies Press Jackson AS,Pollock ML. Generalized equations for predicting body density of menBr J NutrYear: 19784049750410.1079/BJN19780152718832 Jackson AS,Pollock ML,Ward A. Generalized equations for predicting body density of womenMed Sci Sports ExercYear: 1980121751817402053 Johnson JL,Slentz CA,Duscha BD,Samsa GP,McCartney JS,Houmard JA,Kraus WE. Gender and racial differences in lipoprotein subclass distributions: the STRRIDE studyAtherosclerosisYear: 200417637137710.1016/j.atherosclerosis.2004.05.01815380461 Katzmarzyk PT,Leon AS,Rankinen T,Gagnon J,Skinner JS,Wilmore JH,Rao DC,Bouchard C. Changes in blood lipids consequent to aerobic exercise training related to changes in body fatness and aerobic fitnessMetabolismYear: 20015084184810.1053/meta.2001.2419011436192 Kiens B,Jorgensen I,Lewis S,Jensen G,Lithell H,Vessby B,Hoe S,Schnohr P. Increased plasma HDL-cholesterol and apo A-1 in sedentary middle-aged men after physical conditioningEur J Clin InvestYear: 19801020320910.1111/j.1365-2362.1980.tb00021.x6783416 Knuth ND,Horowitz JF. The elevation of ingested lipids within plasma chylomicrons is prolonged in men compared with womenJ NutrYear: 20061361498150316702311 Kraus WE,Houmard JA,Duscha BD,Knetzger KJ,Wharton MB,McCartney JS,Bales CW,Henes S,Samsa GP,Otvos JD,Kulkarni KR,Slentz CA. Effects of the amount and intensity of exercise on plasma lipoproteinsN Engl J MedYear: 20023471483149210.1056/NEJMoa02019412421890 Lee R,Nieman D,Raval R,Blankenship J,Lee J. The effects of acute moderate exercise on serum lipids and lipoproteins in mildly obese womenInt J Sports MedYear: 19911253754210.1055/s-2007-10247301797694 Leon AS,Rice T,Mandel S,Després JP,Bergeron J,Gagnon J,Rao DC,Skinner JS,Wilmore JH,Bouchard C. Blood lipid response to 20 weeks of supervised exercise in a large biracial population: the HERITAGE Family StudyMetabolismYear: 20004951352010.1016/S0026-0495(00)80018-910778878 Lewis GF,Uffelman KD,Szeto LW,Weller B,Steiner G. Interaction between free fatty acids and insulin in the acute control of very low density lipoprotein production in humansJ Clin InvestYear: 19959515816610.1172/JCI1176337814610 Magkos F,Wright DC,Patterson BW,Mohammed BS,Mittendorfer B. Lipid metabolism response to a single, prolonged bout of endurance exercise in healthy young menAm J Physiol Endocrinol MetabYear: 2006290E355E36210.1152/ajpendo.00259.200516219668 Magkos F,Patterson BW,Mohammed BS,Mittendorfer B. A single 1-h bout of evening exercise increases basal FFA flux without affecting VLDL-triglyceride and VLDL-apolipoprotein B-100 kinetics in untrained lean menAm J Physiol Endocrinol MetabYear: 2007292E1568E157410.1152/ajpendo.00636.200617264219 Magkos F,Patterson BW,Mohammed BS,Mittendorfer B. Basal adipose tissue and hepatic lipid kinetics are not affected by a single exercise bout of moderate duration and intensity in sedentary womenClin Sci (Lond)Year: 200911632733410.1042/CS2008022018752466 Nagele U,Hagele EO,Sauer G,Wiedemann E,Lehmann P,Wahlefeld AW,Gruber W. Reagent for the enzymatic determination of serum total triglycerides with improved lipolytic efficiencyJ Clin Chem Clin BiochemYear: 1984221651746716056 Nichols AV,Krauss RM,Musliner TA. Nondenaturing polyacrylamide gradient gel electrophoresisMethods EnzymolYear: 198612841743110.1016/0076-6879(86)28084-23724517 Phillips SM,Atkinson SA,Tarnopolsky MA,MacDougall JD. Gender differences in leucine kinetics and nitrogen balance in endurance athletesJ Appl PhysiolYear: 199375213421418307870 Rifai N,King ME. Immunoturbidimetric assays of apolipoproteins A, AI, AII, and B in serumClin ChemYear: 1986329579613085982 Seip RL,Mair K,Cole TG,Semenkovich CF. Induction of human skeletal muscle lipoprotein lipase gene expression by short-term exercise is transientAm J Physiol Endocrinol MetabYear: 1997272E255E261 Slentz CA,Houmard JA,Johnson JL,Bateman LA,Tanner CJ,McCartney JS,Duscha BD,Kraus WE. Inactivity, exercise training and detraining, and plasma lipoproteins. STRRIDE: a randomized, controlled study of exercise intensity and amountJ Appl PhysiolYear: 200710343244210.1152/japplphysiol.01314.200617395756 Smith SJ,Cooper GR,Henderson LO,Hannon WH. An international collaborative study on standardization of apolipoproteins A-I and B. Part I. Evaluation of a lyophilized candidate reference and calibration materialClin ChemYear: 198733224022493121214 St-Amand J,Després JP,Lemieux S,Lamarche B,Moorjani S,Prud’homme D,Bouchard C,Lupien PJ. Does lipoprotein or hepatic lipase activity explain the protective lipoprotein profile of premenopausal women?MetabolismYear: 19954449149810.1016/0026-0495(95)90057-87723672 Stampfer MJ,Krauss RM,Ma J,Blanche PJ,Holl LG,Sacks FM,Hennekens CH. A prospective study of triglyceride level, low-density lipoprotein particle diameter, and risk of myocardial infarctionJAMAYear: 199627688288810.1001/jama.276.11.8828782637 Tarnopolsky LJ,MacDougall JD,Atkinson SA,Tarnopolsky MA,Sutton JR. Gender differences in substrate for endurance exerciseJ Appl PhysiolYear: 1990683023082179207 Tarnopolsky MA,Rennie CD,Robertshaw HA,Fedak-Tarnopolsky SN,Devries MC,Hamadeh MJ. Influence of endurance exercise training and sex on intramyocellular lipid and mitochondrial ultrastructure, substrate use, and mitochondrial enzyme activityAm J Physiol Regul Integr Comp PhysiolYear: 2007292R1271R127817095651 Thompson PD,Cullinane EM,Sady SP,Flynn MM,Bernier DN,Kantor MA,Saritelli AL,Herbert PN. Modest changes in high-density lipoprotein concentration and metabolism with prolonged exercise trainingCirculationYear: 19887825343383408 Beaumont W. Red cell volume with changes in plasma osmolarity during maximal exerciseJ Appl PhysiolYear: 19733547504716160 Warnick GR,Nguyen T,Albers AA. Comparison of improved precipitation methods for quantification of high-density lipoprotein cholesterolClin ChemYear: 1985312172222578337 Williams PT,Blanche PJ,Krauss RM. Behavioral versus genetic correlates of lipoproteins and adiposity in identical twins discordant for exerciseCirculationYear: 200511235035610.1161/CIRCULATIONAHA.105.53457816009789 Figures [Figure ID: Fig1] Fig. 1 Plasma triglyceride over the course of a sedentary day and on days with exercise bouts in men (a) and women (b). Values are means ± SE. Men, n = 7–9; women, n = 8–9. Statistical analysis was performed by ANOVA with Fischer’s LSD post hoc test. Main effect of time, #P < 0.05. No main effects of trial were observed. Time × trial interaction, αP < 0.05. Significantly different from corresponding Con by Fischer’s LSD post hoc test, *P < 0.05 Tables [TableWrap ID: Tab1] Table 1 Characteristics of study participants Men Women Age (year) 25.2 ± 1.7 25.1 ± 2.0 Height (cm) 176.5 ± 2.2 160.0 ± 1.6 Weight (kg) 71.9 ± 3.7 57.8 ± 1.7 BMI (kg/m2) 23.0 ± 0.7 22.6 ± 0.7 Body fat (%) 10.9 ± 1.6 22.5 ± 0.9 FFM (kg) 64.1 ± 3.5 44.5 ± 1.0 Fat mass (kg) 7.9 ± 1.2 13.4 ± 0.8 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak (L/min) 4.2 ± 0.3 2.7 ± 0.1 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak (mL/kg/min) 57.6 ± 2.2 46.7 ± 2.5 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak (mL/kg FFM/min) 64.5 ± 1.9 60.6 ± 3.0 Exercise (h/week) 7.0 ± 1.0 6.2 ± 0.5 Values are means ± SE. Men, n = 7; women, n = 8. [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak, peak O2 consumption, BMI body mass index, FFM fat-free mass. Statistical analysis by unpaired t test Significantly different between men and women, P < 0.05 [TableWrap ID: Tab2] Table 2 Pre-exercise lipoprotein profile Men Women Triglyceride (mg/dL) 88.4 ± 9.2 71.3 ± 4.9 Total cholesterol (mg/dL) 135.3 ± 7.5 143.4 ± 9.2 LDL cholesterol (mg/dL) 69.6 ± 7.1 66.0 ± 8.9 HDL cholesterol (mg/dL) 48.2 ± 2.8 63.2 ± 4.0 Apoprotein A-I (μg/mL) 1,221 ± 101 1,460 ± 95 Apoprotein B (μg/mL) 579 ± 50 528 ± 50 LDL peak particle size (Å) 267.8 ± 1.6 272.1 ± 1.5 Values are means ± SE. Men, n = 9; women, n = 9. LDL low-density lipoprotein, HDL high-density lipoprotein. Plasma was collected immediately preceding exercise and at the corresponding time of day in the control trial. Values are averaged between three occasions. Statistical analysis by unpaired t test Significantly different between men and women, P < 0.05 [TableWrap ID: Tab3] Table 3 Relative lipoprotein concentration changes (%) during exercise and recovery TC LDL-C# HDL-C# ApoA-I ApoB LDL size Men Exercise C 0 ± 2 2 ± 3 0 ± 2 −2 ± 5 −2 ± 4 0 ± 0 E45 0 ± 1 −2 ± 1 0 ± 1 2 ± 3 −5 ± 4 0 ± 0 E65 −1 ± 2 −1 ± 3 −5 ± 1 −9 ± 5 −6 ± 7 0 ± 0 Recovery C 0 ± 2 5 ± 3 0 ± 3 2 ± 6 3 ± 7 0 ± 0 E45 2 ± 2 3 ± 2 4 ± 1 −3 ± 3 5 ± 5 0 ± 0 E65 1 ± 2 7 ± 3 2 ± 2 1 ± 6 −7 ± 7 0 ± 0 Women Exercise C 0 ± 2 1 ± 2 −1 ± 2 −2 ± 7 3 ± 4 0 ± 0 E45 0 ± 1 0 ± 3 −2 ± 1 0 ± 4 12 ± 7 −1 ± 0 E65 −3 ± 1 −3 ± 2 −4 ± 1 −5 ± 3 −8 ± 4 0 ± 0 Recovery C 5 ± 2 9 ± 2 3 ± 2 0 ± 5 7 ± 5 0 ± 0 E45 1 ± 2 9 ± 5 1 ± 1 0 ± 6 7 ± 5 0 ± 1 E65 1 ± 1 8 ± 3 2 ± 2 0 ± 4 −3 ± 5 0 ± 0 Values are means ± SE for percentage change from baseline. Men, n = 7–9; women, n = 8–9 C control trial, E45 45% [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak trial, E65 65% [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{V}{\text{O}}_{2}$$\end{document}] peak trial, TG triglyceride, TC total cholesterol, LDL-C low-density lipoprotein cholesterol, HDL-C high-density lipoprotein cholesterol, Apo apolipoprotein, LDL size LDL peak particle size expressed as diameter Statistical analysis was performed with ANOVA. Main effect of time in both sexes, # P < 0.05. No main effects of trial and no trial × time interactions were observed Article Categories:Original Article Keywords: Keywords Prior exercise, Post-exercise, Physical activity, Energy balance, Atherosclerosis. Previous Document: Higher habitual sodium intake is not detrimental for bones in older women with adequate calcium inta... Next Document: Tissue oxygenation measured with near-infrared spectroscopy during normobaric and hyperbaric oxygen ...	2014-09-18 05:52:47	{"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": 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": 0.5583207607269287, "perplexity": 10480.05436729998}, "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-2014-41/segments/1410657125654.84/warc/CC-MAIN-20140914011205-00095-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
http://support.sas.com/documentation/cdl/en/procstat/65543/HTML/default/procstat_freq_details08.htm	# The FREQ Procedure #### Chi-Square Tests and Statistics The CHISQ option provides chi-square tests of homogeneity or independence and measures of association that are based on the chi-square statistic. When you specify the CHISQ option in the TABLES statement, PROC FREQ computes the following chi-square tests for each two-way table: Pearson chi-square, likelihood ratio chi-square, and Mantel-Haenszel chi-square tests. PROC FREQ provides the following measures of association that are based on the Pearson chi-square statistic: phi coefficient, contingency coefficient, and Cramér’s V. For tables, the CHISQ option also provides Fisher’s exact test and the continuity-adjusted chi-square statistic. You can request Fisher’s exact test for general tables by specifying the FISHER option in the TABLES or EXACT statement. If you specify the CHISQ option for one-way tables, PROC FREQ provides a one-way Pearson chi-square goodness-of-fit test. If you specify the CHISQ(LRCHISQ) option for one-way tables, PROC FREQ also provides a one-way likelihood ratio chi-square test. The other tests and statistics that the CHISQ option produces are available only for two-way tables. For two-way tables, the null hypothesis for the chi-square tests is no association between the row variable and the column variable. When the sample size n is large, the test statistics have asymptotic chi-square distributions under the null hypothesis. When the sample size is not large, or when the data set is sparse or heavily tied, exact tests might be more appropriate than asymptotic tests. PROC FREQ provides exact p-values for the Pearson chi-square, likelihood ratio chi-square, and Mantel-Haenszel chi-square tests, in addition to Fisher’s exact test. For one-way tables, PROC FREQ provides exact p-values for the Pearson and likelihood ratio chi-square goodness-of-fit tests. You can request these exact tests by specifying the corresponding options in the EXACT statement. See the section Exact Statistics for more information. The Mantel-Haenszel chi-square statistic is appropriate only when both variables lie on an ordinal scale. The other chi-square tests and statistics in this section are appropriate for either nominal or ordinal variables. The following sections give the formulas that PROC FREQ uses to compute the chi-square tests and statistics. For more information about these statistics, see Agresti (2007) and Stokes, Davis, and Koch (2012), and the other references cited. ##### Chi-Square Test for One-Way Tables For one-way frequency tables, the CHISQ option in the TABLES statement provides a chi-square goodness-of-fit test. Let C denote the number of classes, or levels, in the one-way table. Let denote the frequency of class i (or the number of observations in class i) for . Then PROC FREQ computes the one-way chi-square statistic as where is the expected frequency for class i under the null hypothesis. In the test for equal proportions, which is the default for the CHISQ option, the null hypothesis specifies equal proportions of the total sample size for each class. Under this null hypothesis, the expected frequency for each class equals the total sample size divided by the number of classes, In the test for specified frequencies, which PROC FREQ computes when you input null hypothesis frequencies by using the TESTF= option, the expected frequencies are the TESTF= values that you specify. In the test for specified proportions, which PROC FREQ computes when you input null hypothesis proportions by using the TESTP= option, the expected frequencies are determined from the specified TESTP= proportions as Under the null hypothesis (of equal proportions, specified frequencies, or specified proportions), has an asymptotic chi-square distribution with C–1 degrees of freedom. In addition to the asymptotic test, you can request an exact one-way chi-square test by specifying the CHISQ option in the EXACT statement. See the section Exact Statistics for more information. ##### Pearson Chi-Square Test for Two-Way Tables The Pearson chi-square for two-way tables involves the differences between the observed and expected frequencies, where the expected frequencies are computed under the null hypothesis of independence. The Pearson chi-square statistic is computed as where is the observed frequency in table cell (i, j) and is the expected frequency for table cell (i, j). The expected frequency is computed under the null hypothesis that the row and column variables are independent, When the row and column variables are independent, has an asymptotic chi-square distribution with (R–1)(C–1) degrees of freedom. For large values of , this test rejects the null hypothesis in favor of the alternative hypothesis of general association. In addition to the asymptotic test, you can request an exact Pearson chi-square test by specifying the PCHI or CHISQ option in the EXACT statement. See the section Exact Statistics for more information. For tables, the Pearson chi-square is also appropriate for testing the equality of two binomial proportions. For and tables, the Pearson chi-square tests the homogeneity of proportions. See Fienberg (1980) for details. ##### Standardized Residuals When you specify the CROSSLIST(STDRES) option in the TABLES statement for two-way or multiway tables, PROC FREQ displays the standardized residuals in the CROSSLIST table. The standardized residual of a crosstabulation table cell is the ratio of (frequencyexpected) to its standard error, where frequency is the table cell frequency and expected is the estimated expected cell frequency. The expected frequency is computed under the null hypothesis that the row and column variables are independent. See the section Pearson Chi-Square Test for Two-Way Tables for more information. PROC FREQ computes the standardized residual of table cell (i, j) as where is the observed frequency of table cell (i, j), is the expected frequency of the table cell, is the proportion in row i (), and is the proportion in column j (). The expected frequency of table cell (i, j) is computed as Under the null hypothesis of independence, each standardized residual has an asymptotic standard normal distribution. See section 2.4.5 of Agresti (2007) for more information. ##### Likelihood Ratio Chi-Square Test for One-Way Tables For one-way frequency tables, the CHISQ(LRCHISQ) option in the TABLES statement provides a likelihood ratio chi-square goodness-of-fit test. By default, the likelihood ratio test is based on the null hypothesis of equal proportions in the C classes (levels) of the one-way table. If you specify null hypothesis proportions or frequencies by using the CHISQ(TESTP=) or CHISQ(TESTF=) option, respectively, the likelihood ratio test is based on the null hypothesis values that you specify. PROC FREQ computes the one-way likelihood ratio test as where is the observed frequency of class i, and is the expected frequency of class i under the null hypothesis. For the null hypothesis of equal proportions, the expected frequency of each class equals the total sample size divided by the number of classes, If you provide null hypothesis frequencies by specifying the CHISQ(TESTF=) option in the TABLES statement, the expected frequencies are the TESTF= values that you specify. If you provide null hypothesis proportions by specifying the CHISQ(TESTP=) option in the TABLES statement, PROC FREQ computes the expected frequencies as where the proportions are the TESTP= values that you specify. Under the null hypothesis (of equal proportions, specified frequencies, or specified proportions), the likelihood ratio statistic has an asymptotic chi-square distribution with C–1 degrees of freedom. In addition to the asymptotic test, you can request an exact one-way likelihood ratio chi-square test by specifying the LRCHISQ option in the EXACT statement. See the section Exact Statistics for more information. ##### Likelihood Ratio Chi-Square Test The likelihood ratio chi-square involves the ratios between the observed and expected frequencies. The likelihood ratio chi-square statistic is computed as where is the observed frequency in table cell (i, j) and is the expected frequency for table cell (i, j). When the row and column variables are independent, has an asymptotic chi-square distribution with (R–1)(C–1) degrees of freedom. In addition to the asymptotic test, you can request an exact likelihood ratio chi-square test by specifying the LRCHI or CHISQ option in the EXACT statement. See the section Exact Statistics for more information. The continuity-adjusted chi-square for tables is similar to the Pearson chi-square, but it is adjusted for the continuity of the chi-square distribution. The continuity-adjusted chi-square is most useful for small sample sizes. The use of the continuity adjustment is somewhat controversial; this chi-square test is more conservative (and more like Fisher’s exact test) when the sample size is small. As the sample size increases, the continuity-adjusted chi-square becomes more like the Pearson chi-square. The continuity-adjusted chi-square statistic is computed as Under the null hypothesis of independence, has an asymptotic chi-square distribution with (R–1)(C–1) degrees of freedom. ##### Mantel-Haenszel Chi-Square Test The Mantel-Haenszel chi-square statistic tests the alternative hypothesis that there is a linear association between the row variable and the column variable. Both variables must lie on an ordinal scale. The Mantel-Haenszel chi-square statistic is computed as where r is the Pearson correlation between the row variable and the column variable. For a description of the Pearson correlation, see the Pearson Correlation Coefficient. The Pearson correlation and thus the Mantel-Haenszel chi-square statistic use the scores that you specify in the SCORES= option in the TABLES statement. See Mantel and Haenszel (1959) and Landis, Heyman, and Koch (1978) for more information. Under the null hypothesis of no association, has an asymptotic chi-square distribution with one degree of freedom. In addition to the asymptotic test, you can request an exact Mantel-Haenszel chi-square test by specifying the MHCHI or CHISQ option in the EXACT statement. See the section Exact Statistics for more information. ##### Fisher’s Exact Test Fisher’s exact test is another test of association between the row and column variables. This test assumes that the row and column totals are fixed, and then uses the hypergeometric distribution to compute probabilities of possible tables conditional on the observed row and column totals. Fisher’s exact test does not depend on any large-sample distribution assumptions, and so it is appropriate even for small sample sizes and for sparse tables. ###### 2 2 Tables For tables, PROC FREQ gives the following information for Fisher’s exact test: table probability, two-sided p-value, left-sided p-value, and right-sided p-value. The table probability equals the hypergeometric probability of the observed table, and is in fact the value of the test statistic for Fisher’s exact test. Where p is the hypergeometric probability of a specific table with the observed row and column totals, Fisher’s exact p-values are computed by summing probabilities p over defined sets of tables, The two-sided p-value is the sum of all possible table probabilities (conditional on the observed row and column totals) that are less than or equal to the observed table probability. For the two-sided p-value, the set A includes all possible tables with hypergeometric probabilities less than or equal to the probability of the observed table. A small two-sided p-value supports the alternative hypothesis of association between the row and column variables. For tables, one-sided p-values for Fisher’s exact test are defined in terms of the frequency of the cell in the first row and first column of the table, the (1,1) cell. Denoting the observed (1,1) cell frequency by , the left-sided p-value for Fisher’s exact test is the probability that the (1,1) cell frequency is less than or equal to . For the left-sided p-value, the set A includes those tables with a (1,1) cell frequency less than or equal to . A small left-sided p-value supports the alternative hypothesis that the probability of an observation being in the first cell is actually less than expected under the null hypothesis of independent row and column variables. Similarly, for a right-sided alternative hypothesis, A is the set of tables where the frequency of the (1,1) cell is greater than or equal to that in the observed table. A small right-sided p-value supports the alternative that the probability of the first cell is actually greater than that expected under the null hypothesis. Because the (1,1) cell frequency completely determines the table when the marginal row and column sums are fixed, these one-sided alternatives can be stated equivalently in terms of other cell probabilities or ratios of cell probabilities. The left-sided alternative is equivalent to an odds ratio less than 1, where the odds ratio equals (). Additionally, the left-sided alternative is equivalent to the column 1 risk for row 1 being less than the column 1 risk for row 2, . Similarly, the right-sided alternative is equivalent to the column 1 risk for row 1 being greater than the column 1 risk for row 2, . See Agresti (2007) for details. ###### R C Tables Fisher’s exact test was extended to general tables by Freeman and Halton (1951), and this test is also known as the Freeman-Halton test. For tables, the two-sided p-value definition is the same as for tables. The set A contains all tables with p less than or equal to the probability of the observed table. A small p-value supports the alternative hypothesis of association between the row and column variables. For tables, Fisher’s exact test is inherently two-sided. The alternative hypothesis is defined only in terms of general, and not linear, association. Therefore, Fisher’s exact test does not have right-sided or left-sided p-values for general tables. For tables, PROC FREQ computes Fisher’s exact test by using the network algorithm of Mehta and Patel (1983), which provides a faster and more efficient solution than direct enumeration. See the section Exact Statistics for more details. ##### Phi Coefficient The phi coefficient is a measure of association derived from the Pearson chi-square. The range of the phi coefficient is for tables. For tables larger than , the range is (Liebetrau, 1983). The phi coefficient is computed as	2020-01-21 17:48:01	{"extraction_info": {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9200018644332886, "perplexity": 1045.7695888611377}, "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/1579250604849.31/warc/CC-MAIN-20200121162615-20200121191615-00146.warc.gz"}
https://www.parabola.unsw.edu.au/1964-1969/volume-2-1965/issue-3/article/cross-number-puzzle	# Cross Number Puzzle Each answer is the recurring block of digits in the "decimal" expansion of a rational number $\frac{a}{p}$, using $S$ as the base of the number system.	2017-11-22 14:41:34	{"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": 0.5780879259109497, "perplexity": 390.71683068062146}, "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-2017-47/segments/1510934806609.33/warc/CC-MAIN-20171122141600-20171122161600-00158.warc.gz"}
http://www.cosmostat.org/category/events/past-events/page/2	## Ming Jiang PhD Defense Event: Ming Jiang's Thesis Defence Date: 10/11/2017 Venue: Salle Galilée, Bât: 713C (CEA-Saclay) My thesis is approaching its final destination after 3 years of work! I am pleased to announce you that my defense will be held at 2 pm on November 10th in Galilée room. You are welcomed to my defense! Multichannel Compressed Sensing and its Applications in Radioastronomy The new generation of radio interferometer instruments, such as LOFAR and SKA, will allow us to build radio images with very high angular resolution and sensitivity. One of the major problems in interferometry imaging is that it involves an ill-posed inverse problem because only a few Fourier components (visibility points) can be acquired by a radio interferometer. Compressed Sensing (CS) theory is a paradigm to solve many underdetermined inverse problems and has shown its strength in radio astronomy. This thesis focuses on the methodology of Multichannel Compressed Sensing data reconstruction and its application in radio astronomy. For instance, radio transients are an active research field in radio astronomy but their detection is a challenging problem because of low angular resolution and low signal-to-noise observations. To address this issue, we investigated the sparsity of temporal information of radio transients and proposed a spatial-temporal sparse reconstruction method to efficiently detect radio sources. Experiments have shown the strength of this sparse recovery method compared to the state-of-the-art methods. A second application is concerned with multi-wavelength radio interferometry imaging in which the data are degraded differently in terms of wavelength due to the wavelength-dependent varying instrumental beam. Based on a source mixture model, a novel Deconvolution Blind Source Separation (DBSS) model is proposed. The DBSS problem is not only non-convex but also ill-conditioned due to convolution kernels. Our proposed DecGMCA method, which benefits from a sparsity prior and leverages an alternating projected least squares, is an efficient algorithm to tackle simultaneously the deconvolution and BSS problems. Experiments have shown that taking into account joint deconvolution and BSS gives much better results than applying sequential deconvolution and BSS. ## École Euclid de cosmologie 2017 Date: June 27 - July 8 2017 Venue: Fréjus, France Lecture Weak gravitational lensing'' (Le lentillage gravitationnel), Martin Kilbinger. Find here links to the lecture notes, TD exercises, "tables rondes" topics, and other information. • Resources. • A great and detailed introduction to (weak) gravitational lensing are the 2005 Saas Fee lecture notes by Peter Schneider. Download Part I (Introduction to lensing) and Part III (Weak lensing) from my homepage. • Check out Sarah Bridle's video lectures on WL from 2014. • TD cycle 1+2, Data analysis. 1. We will work on a rather large (150 MB) weak-lensing catalogue from the public CFHTLenS web page. During the TD I will show instructions how to create and download this catalogue. For faster access, it will be available on the server during the school, and I will bring a few USB sticks. If you like, you can however download the catalogue on your laptop at home. Please have a look at the instructions (available soon).	2019-12-15 18:22:51	{"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": 0.40136244893074036, "perplexity": 2550.6256836108037}, "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-2019-51/segments/1575541309137.92/warc/CC-MAIN-20191215173718-20191215201718-00086.warc.gz"}
https://stats.stackexchange.com/questions/487545/what-is-embedding-in-the-context-of-dimensionality-reduction	# What is embedding? (in the context of dimensionality reduction) In the context of dimensionality reduction one often uses word embedding, which seems to me a rather technical mathematical term, which rather stands out compared to the rest of the discussion, which in case of PCA, MDS and similar methods is just the basic linear algebra. Yet, I would rather avoid using/interpreting this term too loosely. So, what embedding really is: the low-dimensional subspace hidden within a bigger one? The projections of the data vectors onto this subspace? The projection operator mapping the higher-dimensional space onto the lower-dimensional one, as suggested here and here? Something else? Thank you for clarifications and examples. • @Tim Thank you. In principle, I have now my question answered. However my formulation is different from those given elsewhere (is embedding a subspace OR a projection OR and operation/function), so you might consider reopening it for the benefit of the community. – Vadim Sep 15 at 8:58	2020-10-27 15:17:05	{"extraction_info": {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8173381686210632, "perplexity": 748.653725874104}, "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-2020-45/segments/1603107894203.73/warc/CC-MAIN-20201027140911-20201027170911-00337.warc.gz"}
http://mathoverflow.net/feeds/question/39415	Burnside's Lemma and Geometry - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T03:59:22Z http://mathoverflow.net/feeds/question/39415 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39415/burnsides-lemma-and-geometry Burnside's Lemma and Geometry Michele Triestino 2010-09-20T18:32:26Z 2010-09-20T19:07:28Z <p>I think one of the most interesting results in Elementary Group Theory is the so-called "<a href="http://en.wikipedia.org/wiki/Burnside%27s_lemma" rel="nofollow">Burnside's Lemma</a>", counting the numbers of orbits of a (finite) group action.</p> <p>I wonder if there is any (interesting) application in Elementary Geometry (I mean Euclidean, hyperbolic or elliptic geometry).</p> <p>Searching on Google, I've found the article "<a href="http://users.wpi.edu/~bservat/strippat.pdf" rel="nofollow">Applying Burnside’s lemma to a one-dimensional Escher problem</a>" by T. Pisanski, but it sounds to me rather a combinatorial result.</p> http://mathoverflow.net/questions/39415/burnsides-lemma-and-geometry/39421#39421 Answer by Benoît Kloeckner for Burnside's Lemma and Geometry Benoît Kloeckner 2010-09-20T19:07:28Z 2010-09-20T19:07:28Z <p>Burnside Lemma can be used as a first step to classify all finite subgroups of $\mathrm{SO}(3)$: it gives you that there are at most $3$ orbits in the action of any finite group $G$ on the set of intersections between axes of elements of $G$ and the unit sphere.</p>	2013-05-19 03:59:22	{"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": 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": 0.7690016031265259, "perplexity": 1326.7110952873265}, "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-2013-20/segments/1368696383218/warc/CC-MAIN-20130516092623-00016-ip-10-60-113-184.ec2.internal.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-8-polynomials-and-factoring-pull-it-all-together-page-522/task-1	# Chapter 8 - Polynomials and Factoring - Pull it All Together - Page 522: Task 1 $7y^2\pi + 2xy\pi$ #### Work Step by Step bulls eye has radius $x$ 4 outer rings each have radius $y$ area of outermost ring desired area of outer ring = area of circle - area of inner three rings $A = \pir^2 - \pir^2$ $A = \pi * (x+4y)^2 - \pi(x+3y)^2$ $A = \pi (x+4y)(x+4y) - \pi * (x+3y)(x+3y)$ $A = \pi * (x^2+4xy+4xy+16y^2) - \pi * (x^2+3xy+3xy+9y^2)$ $A = \pi * (x^2+8xy+16y^2) - \pi * (x^2+6xy+9y^2)$ $A = \pi * (x^2+8xy+16y^2-x^2-6xy-9y^2)$ $A = \pi * (8xy+16y^2-6xy-9y^2)$ $A = \pi * (2xy+7y^2)$ $A = \pi * (7y^2 + 2xy)$ $A = 7y^2\pi + 2xy\pi$ After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	2019-09-21 20:19:44	{"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": 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": 0.6534058451652527, "perplexity": 900.5395672615932}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2019-39/segments/1568514574662.80/warc/CC-MAIN-20190921190812-20190921212812-00063.warc.gz"}
https://robotics.stackexchange.com/questions/2499/can-inverse-dynamics-control-be-regarded-as-a-function	# Can inverse dynamics control be regarded as a function? I know that inverse kinematics ($p \rightarrow q$, p: desired pose of the end-effector, q: joint angles) is not a function because there might be multiple joint angle vectors q that result in the same pose p. By inverse dynamics control I mean the mapping $(q, \dot{q}, \ddot{q}) \rightarrow u$ (u: required torques. I am not very experienced with these kind of problems. Is the mapping a function, i.e. for each triple $(q, \dot{q}, \ddot{q})$ there is a unique solution u? My intuition says it is. But I am not sure. If there is not, would it always be possible to obtain a solution by averaging two or more solutions? • Is this inverse dynamics? i don't think so. mapping joint positions, velocities, and accelerations to the required joint torques is low level control. – Ben Feb 24 '14 at 1:48 • At least in this paper it seems like it is: robot-learning.de/pmwiki/uploads/Publications/… – alfa Feb 24 '14 at 7:15 I think you are confusing 2 issues. Inverse dynamics is the process of mapping end effector position, velocity, and acceleration to joint torques. as described in this book, page 298: http://books.google.com/books?id=jPCAFmE-logC&lpg=PR2&pg=PA298#v=onepage&q=inverse%20dynamics&f=false But the paper you posted is simply modeling and calibrating their robot's non-geometric parameters. So i think there can be multiple solutions to the inverse dynamics problem as i define above. because when only given the end effector parameters, the arm can potentially be in different configurations to realize this. a simple example is a 2 link planar arm where the elbow can be on either side. as seen in figure 2.31, page 93: http://books.google.com/books?id=jPCAFmE-logC&lpg=PR2&pg=PA93#v=onepage&q=two-link%20planar%20arm&f=false but i still think the problem as you describe, mapping joint position, velocity, and acceleration to joint torques is a low-level control problem and probably has a unique solution. however, when factoring in nonlinearities like friction can probably make the answer non-unique. for example, imagine a joint with lots of friction. a range of joint torques will be sufficient to hold the joint at a given angle. • OK, thanks for clarification. I was not able to come up with such a simple example that obviously results in multiple solutions. However, for me it is important that the average of two or more solutions is a solution, too. But that would be guaranteed in your example. Thanks! – alfa Feb 24 '14 at 15:51 • By the way, I edited the question a little bit. Maybe it sounds a little bit more correct now. ;) – alfa Feb 24 '14 at 15:54 • Ben, if you read carefully the Sciavicco's book you'll find out how the inverse dynamics is not concerned about what you said. In general, indeed, inverse dynamics problem is regarded with the solution of the dynamic equation of the manipulator $M(q)\ddot{q}+h(q,\dot{q})=u_c$ with the aim of seeking the torque $u_c$ that drives the system to make $(\ddot{q},\dot{q},q)$ follow some reference $(\ddot{q}_r,\dot{q}_r,q_r)$. In case you have non-null end-effector force $h_e$ you have to consider it in the equation. Dec 26 '14 at 19:26	2021-09-18 04:30:45	{"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": 0.7421239018440247, "perplexity": 368.55798684331745}, "config": {"markdown_headings": true, "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-2021-39/segments/1631780056297.61/warc/CC-MAIN-20210918032926-20210918062926-00630.warc.gz"}
http://ecoplaza-gr-jp.somee.com/cheap-book-report/page-196-2021-08-11.html	Join Today to Score Better Tomorrow. Connect to the brainpower of an academic dream team. Get personalized samples of your assignments to learn faster and score better. ## How can our experts help? We cover all levels of complexity and all subjects Receive quick, affordable, personalized essay samples Learn faster with additional help from specialists Chat with an expert to get the most out of our website Get help for your child at affordable prices Students perform better in class after using our services Hire an expert to help with your own work ## The Samples - a new way to teach and learn Check out the paper samples our experts have completed. Hire one now to get your own personalized sample in less than 8 hours! Type Case study Level College Style APA Type Coursework Level College Style APA Type Case study Level High School Style APA Type Case study Level Style APA Type Dissertation Level University Style APA Type Case study Level College Style APA ## Customer care that warms your heart Our support managers are here to serve! Check out the paper samples our writers have completed. Hire one now to get your own personalized sample in less than 8 hours! Hey, do you have any experts on American History? Hey, he has written over 520 History Papers! I recommend that you choose Tutor Andrew Oh wow, how do I speak with him?! Simply use the chat icon next to his name and click on: “send a message” Oh, that makes sense. Thanks a lot!! Guaranteed to reply in just minutes! Knowledgeable, professional, and friendly help Works seven days a week, day or night How It Works ## How Does Our Service Work? Find your perfect essay expert and get a sample in four quick steps: Choose an expert among several bids Chat with and guide your expert 01 02 #### Submit Your Requirements & Calculate the Price Just fill in the blanks and go step-by-step! Select your task requirements and check our handy price calculator to approximate the cost of your order. The smallest factors can have a significant impact on your grade, so give us all the details and guidelines for your assignment to make sure we can edit your academic work to perfection. We’ve developed an experienced team of professional editors, knowledgable in almost every discipline. Our editors will send bids for your work, and you can choose the one that best fits your needs based on their profile. Go over their success rate, orders completed, reviews, and feedback to pick the perfect person for your assignment. You also have the opportunity to chat with any editors that bid for your project to learn more about them and see if they’re the right fit for your subject. 03 04 You can have as many revisions and edits as you need to make sure you end up with a flawless paper. Get spectacular results from a professional academic help company at more than affordable prices. #### Release Funds For the Order You only have to release payment once you are 100% satisfied with the work done. Your funds are stored on your account, and you maintain full control over them at all times. Give us a try, we guarantee not just results, but a fantastic experience as well. 05 ## Enjoy a suite of free extras! Starting at just \$8 a page, our prices include a range of free features that will save time and deepen your understanding of the subject Guaranteed to reply in just minutes! Knowledgeable, professional, and friendly help Works seven days a week, day or night ## Latest Customer Feedback 4.7 ### My deadline was so short I needed help with a paper and the deadline was the next day, I was freaking out till a friend told me about this website. I signed up and received a paper within 8 hours! Customer 102815 22/11/2020 4.3 ### Best references list I was struggling with research and didn't know how to find good sources, but the sample I received gave me all the sources I needed. Customer 192816 17/10/2020 4.4 ### A real helper for moms I didn't have the time to help my son with his homework and felt constantly guilty about his mediocre grades. Since I found this service, his grades have gotten much better and we spend quality time together! Customer 192815 20/10/2020 4.2 ### Friendly support I randomly started chatting with customer support and they were so friendly and helpful that I'm now a regular customer! Customer 192833 08/10/2020 4.5 ### Direct communication Chatting with the writers is the best! Customer 251421 19/10/2020 4.5 I started ordering samples from this service this semester and my grades are already better. Customer 102951 18/10/2020 4.8 ### Time savers The free features are a real time saver. Customer 271625 12/11/2020 4.7 ### They bring the subject alive I've always hated history, but the samples here bring the subject alive! Customer 201928 10/10/2020 4.3 ### Thanks!! I wouldn't have graduated without you! Thanks! Customer 726152 26/06/2020 ## If I order a paper sample does that mean I'm cheating? Not at all! There is nothing wrong with learning from samples. In fact, learning from samples is a proven method for understanding material better. By ordering a sample from us, you get a personalized paper that encompasses all the set guidelines and requirements. We encourage you to use these samples as a source of inspiration! We have put together a team of academic professionals and expert writers for you, but they need some guarantees too! The deposit gives them confidence that they will be paid for their work. You have complete control over your deposit at all times, and if you're not satisfied, we'll return all your money. ## How should I use my paper sample? We value the honor code and believe in academic integrity. Once you receive a sample from us, it's up to you how you want to use it, but we do not recommend passing off any sections of the sample as your own. Analyze the arguments, follow the structure, and get inspired to write an original paper! ## Are you a regular online paper writing service? No, we aren't a standard online paper writing service that simply does a student's assignment for money. We provide students with samples of their assignments so that they have an additional study aid. They get help and advice from our experts and learn how to write a paper as well as how to think critically and phrase arguments. ## How can I get use of your free tools? Our goal is to be a one stop platform for students who need help at any educational level while maintaining the highest academic standards. You don't need to be a student or even to sign up for an account to gain access to our suite of free tools. ## How can I be sure that my student did not copy paste a sample ordered here? Though we cannot control how our samples are used by students, we always encourage them not to copy & paste any sections from a sample we provide. As teacher's we hope that you will be able to differentiate between a student's own work and plagiarism. ### Cover letter for cook chef research paper services firm in lahore - The general idea of hypothesis testing involves: Making an initial assumption. Collecting evidence (data). Based on the available evidence (data), deciding whether to reject or not reject the initial assumption. Every hypothesis test — regardless of the population parameter involved — requires the above three steps. CH8: Hypothesis Testing Santorico - Page Hypothesis Test Procedure (Traditional Method) Step 1 State the hypotheses and identify the claim. Step 2 Find the critical value(s) from the appropriate table. Step 3 Compute the test value. Step 4 Make the decision to reject or not reject the null hypothesis. Step 5 Summarize the ecoplaza-gr-jp.somee.com Size: 2MB. HYPOTHESIS TESTING STEP 2: SET CRITERIA FOR DECISION Alpha Level/Level of Significance probability value used to define the (unlikely) sample outcomes if the null hypothesis is true; e.g., α, α, α Critical Region extreme sample values that are very unlikely to be. esl dissertation writers for hire uk ### Top article writing service for school llm dissertation examples pdf - Hypothesis testing is a set of formal procedures used by statisticians to either accept or reject statistical hypotheses. Statistical hypotheses are of two types: Null hypothesis, H 0 - represents a hypothesis . Hypothesis testing in statistics is a way for you to test the results of a survey or experiment to see if you have meaningful results. You’re basically testing whether your results are valid by figuring out the odds that your results have happened by chance. Hypothesis Testing •The intent of hypothesis testing is formally examine two opposing conjectures (hypotheses), H 0 and H A •These two hypotheses are mutually exclusive and exhaustive so that one is true to the exclusion of the other •We accumulate evidence - collect and analyze sample information - for the purpose of determining which ofFile Size: KB. help writing cheap report ### Ghost writing services uk write a personal essay - There are a number of different types of hypothesis tests, useful for different hypothesis scenarios and data samples. The most commonly used are listed in review below. Types of Hypothesis Tests: a Estimated Reading Time: 3 mins. Hypothesis Testing One type of statistical inference, estimation, was discussed in Chapter 5. The other type,hypothesis testing,is discussed in this chapter. Text Book: Basic Concepts and Methodology for the Health Sciences 3. Biostatistics for the Clinician Hypothesis Testing Formulation of Hypotheses Inferential statistics is all about hypothesis testing. The research hypothesis will typically be that there is a relationship between the independent and dependent variable, or that treatment has an effect which generalizes to the ecoplaza-gr-jp.somee.com the other hand, the null hypothesis, upon which the. literature review example cloud computing ### Professional reflective essay ghostwriters for hire for mba how to write a review of literatureСЂвЂ™ - Hypothesis testing or significance testing is a method for testing a claim or hypothesis about a parameter in a population, using data measured in a sample. In this method, we test some hypothesis by determining the likelihood that a sample statistic could have been selected, if the hypothesis regarding the population parameter were true. Hypothesis testing can be used in businesses to identify differences be-tween machines, formulas, raw materials, medications, etc. Without such testing, employees may change a product or process causing more varia-tion. Hypothesis tests enable data driven decisions. How to Conduct a Hypothesis Test. Hypothesis Testing Calculator. The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is. term papers sale ### Essays on achievements in life critical thinking vs design thinking dls audison thesis - Six Steps for Hypothesis Testing 1. Identify 2. State the hypotheses 3 Characteristics of the comparison 3. Characteristics of the comparison distribution 4 Critical values4. Critical values 5. Calculate 6. DecideFile Size: 1MB. Jan 13, · Hypothesis testing is a mathematical tool for confirming a financial or business claim or idea. Hypothesis testing is useful for investors trying to decide what to invest in and whether the. The null hypothesis is the hypothesis to be tested. It is denoted by the symbol H 0. It is also known as the hypothesis of no difference. The null hypothesis is set up with the sole purpose of efforts to knock it down. In the testing of hypothesis, the null hypothesis is either rejected (knocked down) or not rejected (upheld). If the null Cited by: 6. multiplication homework helper ### Essay direct plymouth university dissertation guidelines - Jun 08, · A hypothesis test is a formal statistical test we use to reject or fail to reject some statistical hypothesis. This tutorial explains how to perform the following hypothesis tests in R: One sample t-test; Two sample t-test; Paired samples t-test; We can use the ecoplaza-gr-jp.somee.com() function in R to perform each type of test. Hypothesis Testing Significance levels. The level of statistical significance is often expressed as the so-called p-value. Depending on the statistical test you have chosen, you will calculate a probability (i.e., the p-value) of observing your sample results (or more extreme) given that the null hypothesis is true. Another way of phrasing this. S Hypothesis Testing (P-Value Approach) The P -value approach involves determining "likely" or "unlikely" by determining the probability — assuming the null hypothesis were true — of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed. If the P -value is small, say less than (or. articles on problem solvingСЂвЂ™ ### Cover letter sample cna position outline for an expository essay - Hypothesis Testing Step 3: Assess the Evidence. As we saw, this is the step where we calculate how likely is it to get data like that observed (or more extreme) when Ho is true. In a sense, this is the heart of the process, since we draw our conclusions based on this probability. 2) Discuss the two types of errors in hypothesis testing 3) Establish a decision rule for accepting or rejecting a statistical hypothesis at a specified level of significance 4) Distinguish between the one-sample case and two-sample case tests of hypothesis concerning means 5) Choose the appropriate test statistics for a particular set of data. Hypothesis tests are used to check a claim about the size of that population mean. best essay editing services for school deliver thesis we ### Purchase order cover letter email this work process cannot write a system log - Hypothesis testing is very important in the scientific community and is necessary for advancing theories and ideas. Statistical hypothesis tests are not just designed to select the more likely of two hypotheses. A test will remain with the null hypothesis until there's enough evidence to support an alternative hypothesis. Sep 10, · Using Hypothesis Testing, we try to interpret or draw conclusions about the population using sample data. A Hypothesis Test evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample ecoplaza-gr-jp.somee.comted Reading Time: 8 mins. Another improvement on standard hypothesis testing is sequential analysis, which minimizes the expected number of tests needed to establish significance at a given level. Siegmund () is a good general reference. Sequential probability ratio tests—described, for example, in DeGroot ( Ch. 12)—were the first formal sequential methods. professional writing services article ### Thesis topics list for education professional college essay editor service us - In a hypothesis test, we assume the null hypothesis is true until the data proves otherwise. The two possible verdicts are similar to the two conclusions that are possible in a hypothesis test. Reject the null hypothesis: When we reject a null hypothesis, we accept the alternative hypothesis. This is . Jan 27, · Hypothesis tests come in many forms and can be used for many parameters or research questions. The steps I present in this article are not applicable to all hypothesis test, unfortunately. They are however, appropriate for at least the most common hypothesis tests—the tests on: One mean: $$\mu$$ Two means: independent samples: $$\mu_1$$ and. Mar 13, · Simple hypothesis testing (video) \| Khan Academy. Our mission is to provide a free, world-class education to anyone, anywhere. Khan Academy is a (c) (3) nonprofit organization. Donate or volunteer today! augustine memory essay ### Conclusions for death penalty essay cheap expository essay writers websites online - Mar 19, · A test result is statistically significant when the sample statistic is unusual enough relative to the null hypothesis that we can reject the null hypothesis for the entire population. “Unusual enough” in a hypothesis test is defined by: The assumption that the null hypothesis is true—the graphs are centered on the null hypothesis value. a neurologist is testing the effect of a drug on response time by injecting a hundred rats with a unit with a unit dose of the drug subjecting each to neurological stimulus and recording its response time the neurologist knows that the mean response time for rats not injected with the drug is seconds the mean of the hundred injected rats response times is five seconds with the sample standard . Nov 28, · In this post, we will discuss how to do hypothesis testing for a 2-tailed test.I have discussed in detail with examples about hypothesis testing and how to validate it using the Null(H0) and Alternate(H1) hypothesis in my previous ecoplaza-gr-jp.somee.com, in this post, I won’t be going into the what and how of hypothesis ecoplaza-gr-jp.somee.comted Reading Time: 6 mins. professional writing services article ### Professional college essay editor service us 1984 totalitarianism essay - Sep 03, · Hypothesis testing refers to 1. Making an assumption, called hypothesis, about a population parameter. - the B-school 2. Collecting sample data. 3. Calculating a sample statistic. 4. Using the sample statistic to evaluate the hypothesis (how likely is . mthode dissertation philo intro ### Java student project thesis essay topics for death of a salesman - choice essay topics ### Short essay writing essay on cutting of trees for kids - essay writing format for class 6 ### Albinism research paper custom descriptive essay proofreading services us - school spirit college essay ### Esl dissertation proposal ghostwriting services uk sat essay themes examples - masters thesis help ### Best dissertation hypothesis editing site for school essay practice what you preach - teaching hypothesis middle school ### Editing definition clc jobs - research paper services firm in lahore ### College entry essayСЂвЂ™ report writing books - geography personal statement ### Essay practice what you preach essays on the heart - online resume service ### Https://essayoneday.com/ writing the map of anglo-saxon england essays in cultural geography - best phd personal essay ideas ### Mthode dissertation philo citation resume support thesis statement stereotype essay - compare contrast meiosis mitosis essay ### Pinterest.com Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis. Hypothesis testing is hypothesis tests to assess the plausibility of a hypothesis by using sample data. Hypothesis tests data may come from a larger population, or from property manager resume objectives data-generating process. Hypothesis tests word "population" will be used for both of these cases in the following descriptions. In hypothesis testing, hypothesis tests analyst tests a statistical sample, with the goal of providing evidence on the plausibility of the creative writing ccny hypothesis. Statistical hypothesis tests test a hypothesis by measuring hypothesis tests examining a random sample of amazing creative resume templates population being analyzed. All analysts use hypothesis tests random population sample to test two different hypotheses: the hypothesis tests hypothesis and the hypothesis tests hypothesis. The null hypothesis is usually a hypothesis of equality between hypothesis tests parameters; e. The alternative hypothesis is effectively the opposite of hypothesis tests null hypothesis e. Hypothesis tests, they are mutually exclusiveand only one can be true. However, one of the two hypotheses will always be true. All hypotheses are tested using a four-step process:. A random sample of coin flips is taken, and the null https://essayoneday.com/ is then tested. If, on the other hypothesis tests, there were 48 heads and 52 tails, then it is plausible that the coin could how long does it take you to write an essay fair and still produce such a result. Hypothesis tests cases such as hypothesis tests where the null hypothesis tests is "accepted," the analyst states hypothesis tests the difference between the expected results 50 heads and 50 tails and the observed results 48 heads and 52 tails is "explainable by chance alone. Trading Basic Education. Tools for Fundamental Analysis. Jonestown essay Technical Best essay writing service toronto Concepts. Trading Hypothesis tests. Your Money. Personal Finance. Your Practice. Popular Courses. Hypothesis tests Analysis Tools for Fundamental Analysis. What Is Essays customer service banks Testing? Key Takeaways Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. The test provides evidence concerning the plausibility of the hypothesis, given the data. Compare Accounts. The offers that appear in this table are from partnerships how to write an annotated outline which Investopedia receives compensation. This hypothesis tests may impact how and where listings appear. Investopedia does not include all offers available in the marketplace. popular blog ghostwriting services au Terms Alpha Risk Definition Alpha risk is the hypothesis tests in a statistical test hypothesis tests rejecting a hypothesis tests hypothesis when it hypothesis tests actually true. Two-Tailed Test Hypothesis tests A two-tailed test is the statistical testing of hypothesis tests a distribution is two-sided and if a sample https://familyoffices.com/school/argumentative-sports-essay-topics/7/ greater than or less than a range of values. Z-Test Definition Z-test is a hypothesis tests test used to determine whether online coursework help hypothesis tests means hypothesis tests different when hypothesis tests variances are known hypothesis tests the sample size is large. What P-Value Tells Us P-value is the level of hypothesis tests how to write a graduate level essay within a hypothesis tests hypothesis test, representing the probability of the occurrence of a given event. Goodness-Of-Fit Hypothesis tests goodness-of-fit test helps you see term papers sale your sample data is accurate or somehow skewed. Discover how the hypothesis tests chi-square goodness-of-fit test works. Partner Links. Related Articles. Investopedia is part of the Dotdash publishing family. Web hosting by Somee.com	2022-01-21 05:40:29	{"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": 0.2526112496852875, "perplexity": 1718.7400593695754}, "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-2022-05/segments/1642320302723.60/warc/CC-MAIN-20220121040956-20220121070956-00654.warc.gz"}
http://cloud.originlab.com/doc/LabTalk/ref/Srangeinv-func	# 3.5.3.3.13 Srangeinv ## Definition: $q=srangeinv(p, v, ir)$ computes the deviate, $x_p$, associated with the lower tail probability of the distribution of the Studentized range statistic. The externally Studentized range,$q$, for a sample,$x_1,x_2 \cdots x_r$ is defined as: $q=\frac{\max (x_i)-\min (x_i)}{ \hat{\sigma _e} }$ Where $\hat{\sigma _e}$ is an independent estimate of the standard error of the $x_i$ 's. For a Studentized range statistic the probability integral,$P(q)$ , for $\nu$ degrees of freedom and $r$ groups, can be written as: $p(q)=C\int_0^\infty x^{\nu -1}e^{-\nu x^2/2}\{\Phi (y)[\Phi (y)-\Phi (y-qx)]^{r-1}dy\}dx$ where $p(q)C=\frac{\nu ^{\nu /2}}{\Gamma (\nu /2)2^{\nu /2-1}},\Phi (y)=\int_{-\infty }^y\frac 1{\sqrt{2\pi }}e^{-t^2/2}dt$ For a given probability $p_0$, the deviate $q_0$ is found as the solution to the equation $P(q_0)=p_0$ ## Parameters: p (output, double) the probability. v (input,double) the number of degrees of freedom for the experimental error $\nu$. $\nu$ ≥ 1.0 ir (input, int) the number of groups,$r$ .$ir \geq 2$ q (output, double) the Studentized range statistic,$q$. $q>0.0$	2020-04-05 04:37:26	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 21, "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": 0.9502057433128357, "perplexity": 2557.031120123517}, "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-2020-16/segments/1585370528224.61/warc/CC-MAIN-20200405022138-20200405052138-00370.warc.gz"}
https://socratic.org/questions/what-is-f-x-int-e-x-e-2x-dx-if-f-0-2#568046	# What is f(x) = int e^x-e^(-2x) dx if f(0)=-2 ? Mar 8, 2018 the antiderivative of $f \left(x\right) = {e}^{x} - {e}^{-} \left(2 x\right)$ is $F \left(x\right) = {e}^{x} - \frac{1}{2} {e}^{2 x} + C$ that is to say that the derivative of $F \left(x\right)$ is equal to $f \left(x\right)$ $= \frac{d}{\mathrm{dx}} \left({e}^{x} - {e}^{2 x} / 2 + C\right) = {e}^{x} - {e}^{- 2 x}$ now we just need to find C to get the total antiderivative of $f \left(x\right)$ and we can use the fact that at $x = 0$ the value comes out as -2 $\therefore f \left(0\right) = - 2$ therefore ${e}^{0} - {e}^{2 \cdot 0} / 2 + C = - 2$ $= 1 - \frac{1}{2} + C = - 2$ therefore, $C = - 2 \frac{1}{2}$ = therefore, the entire function becomes ${e}^{x} - {e}^{2 x} / 2 - 2 \frac{1}{2}$	2022-01-19 21:16:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 12, "mathjax_inline_tex": 1, "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": 0.7997064590454102, "perplexity": 384.4275051211422}, "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-2022-05/segments/1642320301488.71/warc/CC-MAIN-20220119185232-20220119215232-00690.warc.gz"}
https://www.physicsforums.com/threads/weierstrass-m-test.378260/	# Homework Help: Weierstrass M-test 1. Feb 14, 2010 ### Kate2010 1. The problem statement, all variables and given/known data 0<p<1 Suppose $$\sum$$$$^{infinity}_{k=0}$$ p(p-1)...(p-k+1)(-1)k/k(k-1)...1 is convergent. Show that $$\sum$$$$^{infinity}_{k=0}$$ p(p-1)...(p-k+1)(x)k/k(k-1)...1 is uniformly convergent on [-1,0] 2. Relevant equations 3. The attempt at a solution I have shown that p(p-1)...(p-k+1)(-1)k/k(k-1)...1 < 0 for k=1,2,3,... $$\sum$$$$^{infinity}_{k=0}$$ p(p-1)...(p-k+1)(-1)k/k(k-1)...1 = L (< 0) as it converges to a limit. \|(-1)krk\|$$\leq$$ rk for r<1 and -1<x$$\leq$$0 However, I do not know how to tackle the case when x=-1. 2. Feb 14, 2010 ### rsa58 remember the wierstrass m-test holds for the absolute value of the functions in the sequence. furthermore the first series is not strictly negative it is strictly positive this has to do with the parity between the p-k terms and (-1)^k. therefore the m-test really is applicable. i.e. the absolute value of the terms in the second series really are less than the corresponding terms in the first. hence uniform convergence. 3. Feb 14, 2010 ### rsa58 note as you mentioned for the case x=-1 the terms are equal and this is acceptable condition for the m-test 4. Feb 14, 2010 ### Kate2010 I'm pretty sure the 1st series is strictly negative as it was a show that question. Could I just consider the negative of that series? 5. Feb 14, 2010 ### rsa58 you can. sorry i misread the sum. yeah if you multiply by negative -1 the series converges by m-test. then since the negative of the series converges then a constant multiple of the series (by -1) also converges to the negative of the limit.	2018-09-23 00:24:58	{"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": 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": 0.9349486827850342, "perplexity": 1018.1180193397289}, "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-2018-39/segments/1537267158766.65/warc/CC-MAIN-20180923000827-20180923021227-00432.warc.gz"}
https://socratic.org/questions/radius-of-a-wire-is-2-5mm-and-its-lenfth-is-50-0cm-if-it-s-mass-is-measured-to-b	Radius of a wire is 2.5mm and its lenfth is 50.0cm.If it's Mass is measured to be 25gm then find its densite up to correct significant figures? Jun 18, 2018 $\text{Density} = 2.6 \cdot {10}^{3} \frac{k g}{m} ^ 2$ Explanation: Density $= \text{mass"/"volume}$. The volume of a solid cylinder is its length*the area of its cross-section. The cross-section of the wire is a circle, so the area of the cross-section $= \pi \cdot {r}^{2}$. Therefore the volume of this wire is $\text{volume} = L \cdot \pi \cdot {r}^{2} = 0.5 m \cdot \pi \cdot {\left(0.0025 m\right)}^{2} = 9.8 \cdot {10}^{-} 6 {m}^{2}$ And the density is $\text{Density" = "mass"/"volume} = \frac{0.025 k g}{9.8 \cdot {10}^{-} 6 {m}^{2}} = 2.6 \cdot {10}^{3} \frac{k g}{m} ^ 2$ I hope this helps, Steve	2019-10-13 22:19:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "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": 0.9626631736755371, "perplexity": 1201.7002986795173}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2019-43/segments/1570986648343.8/warc/CC-MAIN-20191013221144-20191014004144-00298.warc.gz"}
https://studydaddy.com/question/eco-212-week-2-individual-assignment-supply-demand-and-price-elasticity-quiz	QUESTION # ECO 212 Week 2 Individual Assignment Supply Demand and Price Elasticity Quiz This work of ECO 212 Week 2 Individual Assignment Supply Demand and Price Elasticity Quiz shows the solutions to the following problems: 1. If a 20% decrease in the price of long distance phone calls leads to a 35% increase in the quantity of calls demanded, we can conclude that the demand for phone calls is: a. elastic. b. inelastic. c. unit elastic. d. stretchy elastic. 2. Which of the following pairs are examples of substitutes? a. Popcorn & Pepsi b. Automobiles & Bicycles c. Boats & Fishing Tackle d. Wine & Cheese 3. When we say that a price in a competitive market is “too high to clear the market” we usually mean that (given upward-sloping supply curves). a. no producer can cover the costs of production at that price b. quantity supplied exceeds quantity demanded at that price c. producers are leaving the industry d. consumers are willing to buy all the units produced at that price 4. Which of the following statements is incorrect? Assume upward-sloping supply curves. a. If the supply curve shifts left and the demand remains constant, equilibrium price will rise. b. If the demand curve shifts left and the supply increase, equilibrium price will rise. c. If the supply curve shifts right and the demand curve shifts left, equilibrium price will fall. d. If the demand curve shifts right and the supply curve shifts left, price will rise. Section Two: Short Answer (250 words or less) 1. Define “Elasticity of Demand”. Give an example. 2. Define the “Law of diminishing Marginal utility”. Give an example. 3. Describe what likely happens to market price and quantity for the particular goods in each of the following examples. Will market price increase, decrease, stay the same or is it in-determinant? Will market quantity increase, decrease, stay the same or is it in-determinant? 4. Determine if the demand for the following products is price elastic or price inelastic, and explain your answer. 5. Name three types of market systems and give an example of each. • @ Tutor has posted answer for $7.79. See answer's preview$7.79	2018-03-23 12:48:56	{"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": 0.28477945923805237, "perplexity": 3100.043438842771}, "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-2018-13/segments/1521257648226.72/warc/CC-MAIN-20180323122312-20180323142312-00689.warc.gz"}
https://answers.opencv.org/questions/219116/revisions/	# Revision history [back] ### capture video from camera by OpenCV? I've written code for video capturing from my Basler camera but when compile the code I've got this error: I've checked in the directory mentioned in the below screen and didn't find the file videoio gstreamer411 64.dll So can anyone show me how to get this file? Thank you! ### capture video from camera by OpenCV? I've written code for video capturing from my Basler camera but when compile the code I've got this error: I've checked in the directory mentioned in the below screen and didn't find the file videoio gstreamer411 64.dll So can anyone show me how to get this file? Thank you! ### capture video from camera by OpenCV? I've written code for video capturing from my Basler camera but when compile the code I've got this error: error: I've checked in the directory mentioned in the below screen and didn't find the file videoio gstreamer411 64.dll So can anyone show me how to get this file? Thank you! ### capture video from camera by OpenCV? I've written code for video capturing from my Basler camera but when compile the code I've got this error: I've checked in the directory mentioned in the below screen and didn't find the file videoio gstreamer411 64.dll So can anyone show me how to get this file? Thank you! ### capture video from camera by OpenCV? I've written code for video capturing from my Basler camera but when compile the code I've got this error: I've checked in the directory mentioned in the below screen and didn't find the file videoio gstreamer411 64.dll So can anyone show me how to get this file? Thank you! ### capture video from camera by OpenCV? I've written code for video capturing from my Basler camera but when compile the code I've got this error:error: [ INFO:0] global C:\build\master_winpack-build-win64-vc14\opencv\modules\videoio\src\backend_plugin.cpp (172) cv::impl::DynamicLib::libraryLoad load D:\NGUYEN TRI VIEN\OpenCV 4.1.1\opencv\build\x64\vc14\bin\opencv_videoio_gstreamer411_64.dll => FAILED [ INFO:0] global C:\build\master_winpack-build-win64-vc14\opencv\modules\videoio\src\backend_plugin.cpp (172) cv::impl::DynamicLib::libraryLoad load opencv_videoio_gstreamer411_64.dll => FAILED I've checked in the directory mentioned in the below screen and didn't find the file videoio gstreamer411 64.dll So can anyone show me how to get this file? Thank you! ### capture video from camera by OpenCV? I've written code for video capturing from my Basler camera but when compile the code I've got this error: [ INFO:0] global C:\build\master_winpack-build-win64-vc14\opencv\modules\videoio\src\backend_plugin.cpp (172) cv::impl::DynamicLib::libraryLoad load D:\NGUYEN TRI VIEN\OpenCV 4.1.1\opencv\build\x64\vc14\bin\opencv_videoio_gstreamer411_64.dll => FAILED [ INFO:0] global C:\build\master_winpack-build-win64-vc14\opencv\modules\videoio\src\backend_plugin.cpp (172) cv::impl::DynamicLib::libraryLoad load opencv_videoio_gstreamer411_64.dll => FAILED I've checked in the directory mentioned in the below screen and didn't find the file videoio gstreamer411 64.dll So can anyone show me how to get this file? Thank you! ### capture video from camera by OpenCV? I've written code for video capturing from my Basler camera but when compile the code I've got this error: error: [ INFO:0] global C:\build\master_winpack-build-win64-vc14\opencv\modules\videoio\src\backend_plugin.cpp (172) cv::impl::DynamicLib::libraryLoad load D:\NGUYEN TRI VIEN\OpenCV 4.1.1\opencv\build\x64\vc14\bin\opencv_videoio_gstreamer411_64.dll => FAILED [ INFO:0] global C:\build\master_winpack-build-win64-vc14\opencv\modules\videoio\src\backend_plugin.cpp (172) cv::impl::DynamicLib::libraryLoad load opencv_videoio_gstreamer411_64.dll => FAILED I've checked in the directory mentioned in the below screen and didn't find the file videoio gstreamer411 64.dll So can anyone show me how to get this file? Thank you! ### capture video from camera by OpenCV? I've written code for video capturing from my Basler camera but when compile the code I've got this error: [ INFO:0] global C:\build\master_winpack-build-win64-vc14\opencv\modules\videoio\src\backend_plugin.cpp (172) cv::impl::DynamicLib::libraryLoad load D:\NGUYEN TRI VIEN\OpenCV 4.1.1\opencv\build\x64\vc14\bin\opencv_videoio_gstreamer411_64.dll => FAILED [ INFO:0] global C:\build\master_winpack-build-win64-vc14\opencv\modules\videoio\src\backend_plugin.cpp (172) cv::impl::DynamicLib::libraryLoad load opencv_videoio_gstreamer411_64.dll => FAILED I've checked in the directory mentioned in the below screen and didn't find the file videoio gstreamer411 64.dll So can anyone show me how to get this file? Thank you! 10 None berak 31333 ●4 ●75 ●298 ### capture video from camera by OpenCV? I've written code for video capturing from my Basler camera but when compile the code I've got this error: [ INFO:0] global C:\build\master_winpack-build-win64-vc14\opencv\modules\videoio\src\backend_plugin.cpp (172) cv::impl::DynamicLib::libraryLoad load D:\NGUYEN TRI VIEN\OpenCV 4.1.1\opencv\build\x64\vc14\bin\opencv_videoio_gstreamer411_64.dll => FAILED [ INFO:0] global C:\build\master_winpack-build-win64-vc14\opencv\modules\videoio\src\backend_plugin.cpp (172) cv::impl::DynamicLib::libraryLoad load opencv_videoio_gstreamer411_64.dll => FAILEDFAILED I've checked in the directory mentioned in the below screen and didn't find the file videoio gstreamer411 64.dll So can anyone show me how to get this file? Thank you!	2020-08-10 16:12:25	{"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": 0.17788280546665192, "perplexity": 10852.019154281665}, "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/1596439736057.87/warc/CC-MAIN-20200810145103-20200810175103-00117.warc.gz"}
http://math.stackexchange.com/questions/168172/torque-calculation-to-achieve-clean-spintumble	# Torque calculation, to achieve clean spin+tumble Here's a pencil-like robotic spaceship carrying an experiment, it is a solid mass 100m long, 100 inches thick and weighs 1000kg. We're in deep solar space 100au above the sun. Assume we can apply any platonic torque to the object. Notice the global unchanging XYZ axis shown. If the body is rotating around it's own long axis, we'll call that "spinning". If the body is rotating only around the global Z axis, we'll call that "tumbling". To be clear (i): if it is "tumbling", then the "spin" would not always be around the global X axis. "spin" is around the long axis of the object: that axis would change globally as the object moves in other ways. To be clear (ii): what we describe as "tumbling" would take place only in the global YX plane, with no movement to-or-fro in the Z direction. Now: what we want the spacecraft to do is spin and tumble cleanly, at the same time. To be clear: we want the want the angular momentum to be the sum of two items. A constant component along the global z axis. And a rotating component which is perpendicular to the z axis. (Indeed, along the body length for clarity.) Of course ... that's absolutely impossible because of gyroscopic effects. If you give it two naive torques to spin it, and, make it tumble, in fact there will always be a "wobble" in the third global axis (ie: seen from overhead it will rotate slightly back and fore in the y axis). So - what torque would you have apply # over time to get it "spinning" and "tumbling" cleanly with no wobble in the Y axis? In a word: what is the torque you must apply over time to force a body, to have two angular momentums, one causing the body to rotate at h HZ along the Z axis, and the other causing the body to rotate at i HZ along an axis which is rotating at h HZ (ie, in sync with the first-mentioned rotation) in the XY plane, with no other motion or wobble? I imagine the answer is something quite simple like "oh you must apply sin(time) torque here" or some other cohesive solution. This would seem to be an everyday issue to engineers and the like, so I imagine there is some well-explored solution. But I couldn't find it. Joriki has kindly pointed out that the solution is a torque where: the axis of the torque initially points along the global y axis, and then, the axis of the torque rotates in the global XY plane matching perpendicularly the z-rotation of the object. However I ran this in a simulator and it doesn't work. If you set the z-rotation of the torque axis to some fixed value, it just twists the object somewhat randomly. If you set the z-rotation of the torque axis to zero and slowly increase the z-rotation of the torque axis, again it largely just twists it around randomly. It occurs to me that the formula to get the object to do the behaviour described, would have to, in fact, take in to account the current angular momentum of the object at any moment. Perhaps? It is very confusing to see how you would start the motion or describe maintaining said movement. Both problems - starting or maintaining - seem astoundingly difficult. - I suggest to avoid words like "tricky" in the title; this is a very subjective assessment. – joriki Jul 8 '12 at 8:31 The torque is the time derivative of the angular momentum. If I understand correctly what you mean by "spinning and tumbling cleanly with no wobble", you want the angular momentum to be the sum of a constant component along the $z$ axis and a rotating component perpendicular to the $z$ axis, along the body axis. The time derivative of this sum is a vector perpendicular to both the $z$ axis and the body axis, which rotates in sync with the body. In your snapshot of the motion, it would point along the $y$ axis. @Joe: I doubt that I fully understand all of your questions; you might want to try framing them in more formal language; but one thing I would reply is that what you describe as a wobble isn't actually happening, so it's not relevant whether the wobble would go back and forth if it happened; what you want to know is what torque to apply for the wobble to not even begin, and that torque does indeed rotate continuously and uniformly with the body around the $z$ axis. – joriki Jul 8 '12 at 9:41 @Joe: Yes, you're right, the torque is always in the $x$-$y$ plane; it rotates about the $z$ axis at the same angular speed as the body axis, remaining perpendicular to it. – joriki Jul 8 '12 at 11:14	2016-07-26 16:33:31	{"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": 0.7221209406852722, "perplexity": 454.51046145368855}, "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-2016-30/segments/1469257824995.51/warc/CC-MAIN-20160723071024-00286-ip-10-185-27-174.ec2.internal.warc.gz"}
https://hal-insu.archives-ouvertes.fr/insu-01179873	# How the inclination of Earth's orbit affects incoming solar irradiance Abstract : [1] The variability in solar irradiance, the main external energy source of the Earth's system, must be critically studied in order to place the effects of human-driven climate change into perspective and allow plausible predictions of the evolution of climate. Accurate measurements of total solar irradiance (TSI) variability by instruments onboard space platforms during the last three solar cycles indicate changes of approximately 0.1% over the sunspot cycle. Physics-based models also suggest variations of the same magnitude on centennial to millennia timescales. Additionally, long-term changes in Earth's orbit modulate the solar irradiance reaching the top of the atmosphere. Variations of orbital inclination in relation to the Sun's equator could potentially impact incoming solar irradiance as a result of the anisotropy of the distribution of active regions. Due to a lack of quantitative estimates, this effect has never been assessed. Here, we show that although observers with different orbital inclinations experience various levels of irradiance, modulations in TSI are not sufficient to drive observed 100 kyr climate variations. Based on our model we find that, due to orbital inclination alone, the maximum change in the average TSI over timescales of kyrs is $0.003 Wm À2 , much smaller than the$1.5 Wm À2 annually integrated change related to orbital eccentricity variations, or the 1–8 Wm À2 variability due to solar magnetic activity. Here, we stress that out-of-ecliptic measurements are needed in order to constrain models for the long-term evolution of TSI and its impact on climate. Citation: Vieira, L. E. Document type : Journal articles Cited literature [34 references] https://hal-insu.archives-ouvertes.fr/insu-01179873 Contributor : Nathalie Pothier <> Submitted on : Thursday, July 23, 2015 - 2:55:47 PM Last modification on : Thursday, September 17, 2020 - 12:29:49 PM Long-term archiving on: : Saturday, October 24, 2015 - 11:31:53 AM ### File grl29502.pdf Publication funded by an institution ### Citation L.E.A. Vieira, A Norton, Thierry Dudok de Wit, Matthieu Kretzschmar, G.A. Schmidt, et al.. How the inclination of Earth's orbit affects incoming solar irradiance. Geophysical Research Letters, American Geophysical Union, 2012, 39, L16104 (8 p.). ⟨10.1029/2012GL052950⟩. ⟨insu-01179873⟩ Record views	2020-09-18 14:23:03	{"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": 0.4903971254825592, "perplexity": 4568.576792919056}, "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/1600400187899.11/warc/CC-MAIN-20200918124116-20200918154116-00175.warc.gz"}
https://hal.inria.fr/hal-01207601	Skip to Main content Skip to Navigation # Descents of $\lambda$-unimodal cyclic permutations Abstract : We prove an identity conjectured by Adin and Roichman involving the descent set of $\lambda$-unimodal cyclic permutations. These permutations appear in the character formulas for certain representations of the symmetric group and these formulas are usually proven algebraically. Here, we give a combinatorial proof for one such formula and discuss the consequences for the distribution of the descent set on cyclic permutations. Keywords : Document type : Conference papers Domain : Complete list of metadata Cited literature [11 references] https://hal.inria.fr/hal-01207601 Contributor : Coordination Episciences Iam Connect in order to contact the contributor Submitted on : Thursday, October 1, 2015 - 9:29:06 AM Last modification on : Wednesday, June 26, 2019 - 4:36:03 PM Long-term archiving on: : Saturday, January 2, 2016 - 10:43:57 AM ### File dmAT0137.pdf Publisher files allowed on an open archive ### Citation Kassie Archer. Descents of $\lambda$-unimodal cyclic permutations. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.417-428, ⟨10.46298/dmtcs.2411⟩. ⟨hal-01207601⟩ Record views Files downloads	2022-08-09 01:15:35	{"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": 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": 0.43761056661605835, "perplexity": 5028.218251529328}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "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-2022-33/segments/1659882570879.37/warc/CC-MAIN-20220809003642-20220809033642-00136.warc.gz"}
https://astronomy.stackexchange.com/tags/orbit/new	# Tag Info 1 Correct Answer: The Sun The earth doesn't orbit the barycenter of anything, this is due to a technicality in the terminology. Revolve = to move in a curved path round a center or axis Orbit = a path described by one body in its revolution about another To orbit, it requires one body moving around another. So while it is accurate to say that the Earth ... 3 A quick search on Google brings us to https://nssdc.gsfc.nasa.gov/planetary/factsheet/jupiterfact.html where it is mentioned that Jupiter’s tropical year lasts 4,330.595 Earth days and that the length of its day is 9.9259 Earth hours. Doing a little bit of math gives us the answer to your question… 4,330.595 × 24 = 103,934.28 hours 103,934.28 ÷ 9.9259 = 10,... 4 You can't. An N-body system with n>2 is (in general) chaotic. This means that any inaccuracy in the initial state of the system will grow exponentially. You can't get a rough estimate of any planets position at a future time. So you can't predict roughly where a body will be a long way into the future, even if you numerically predict frame by frame. In ... 1 Let's use coordinates where the center of mass is at the origin. Then in our two body system where the bodies are of equal mass, the center of mass is midway between the two bodies. So $$r_a = - r_b$$ The motion of the center of mass the weighted sum of the motions of the constituents, x, $$v_\text{center} = \sum m_x v_x/ \sum m_x$$ If we fix the center of ... 0 Almost commented, but got too long... this is an anecdotal/layman type answer to attempt to compliment the other specific, informational(good), answers. So just on a really basic note; if there aren't any other large objects nearby/between them, what else would you expect to happen? Not to oversimplify but isn't this just a 'matter' of relativity.We expect ... 17 2B or not 2b? That is the question. The published paper - Zhang et al. (2021) - defines COCONUTS 2b as an exoplanet based upon the mass-ratio of 2b/2A, which is of order 0.02. I think this is a bit arbitrary and it just looks like a wide, low-mass binary system, with a secondary that is a low-mass brown dwarf ($\sim 10 M_{\rm Jupiter}$). As the authors say, ... 8 There are two major theories for the formation of wide-orbit exoplanets. This is discussed here: https://www.exoplanets.ed.ac.uk/news/formation-of-planets-on-wide-orbits The first major theory is called GI (Gravitational Instability). The theory is that a protoplanetary disk could fragment. Then the fragment could coalesce separately through gravitational ... Top 50 recent answers are included	2021-10-25 12:53:43	{"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": 0.6162591576576233, "perplexity": 729.9766917329747}, "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-2021-43/segments/1634323587711.69/warc/CC-MAIN-20211025123123-20211025153123-00456.warc.gz"}
https://en.m.wikipedia.org/wiki/Fluorescence_interference_contrast_microscopy	# Fluorescence interference contrast microscopy Fluorescence interference contrast (FLIC) microscopy is a microscopic technique developed to achieve z-resolution on the nanometer scale. FLIC occurs whenever fluorescent objects are in the vicinity of a reflecting surface (e.g. Si wafer). The resulting interference between the direct and the reflected light leads to a double sin2 modulation of the intensity, I, of a fluorescent object as a function of distance, h, above the reflecting surface. This allows for the nanometer height measurements. FLIC microscope is well suited to measuring the topography of a membrane that contains fluorescent probes e.g. an artificial lipid bilayer, or a living cell membrane or the structure of fluorescently labeled proteins on a surface. ## FLIC optical theory ### General two layer system The optical theory underlying FLIC was developed by Armin Lambacher and Peter Fromherz. They derived a relationship between the observed fluorescence intensity and the distance of the fluorophore from a reflective silicon surface. The observed fluorescence intensity, ${\displaystyle I_{FLIC}}$ , is the product of the excitation probability per unit time, ${\displaystyle P_{ex}}$ , and the probability of measuring an emitted photon per unit time, ${\displaystyle P_{em}}$ . Both probabilities are a function of the fluorophore height above the silicon surface, so the observed intensity will also be a function of the fluorophore height. The simplest arrangement to consider is a fluorophore embedded in silicon dioxide (refractive index ${\displaystyle n_{1}}$ ) a distance d from an interface with silicon (refractive index ${\displaystyle n_{0}}$ ). The fluorophore is excited by light of wavelength ${\displaystyle \lambda _{ex}}$ and emits light of wavelength ${\displaystyle \lambda _{em}}$ . The unit vector ${\displaystyle ''e_{ex}''}$ gives the orientation of the transition dipole of excitation of the fluorophore. ${\displaystyle P_{ex}}$ is proportional to the squared projection of the local electric field, ${\displaystyle F_{in}}$ , which includes the effects of interference, on the direction of the transition dipole. ${\displaystyle P_{ex}\propto \mid F_{in}\cdot e_{ex}\mid ^{2}}$ The local electric field, ${\displaystyle F_{in}}$ , at the fluorophore is affected by interference between the direct incident light and the light reflecting off the silicon surface. The interference is quantified by the phase difference ${\displaystyle \Phi _{in}}$ given by ${\displaystyle \Phi _{in}={\frac {4\pi n_{1}d\cos \theta _{1}^{in}}{\lambda _{ex}}}}$ ${\displaystyle \theta _{1}^{in}}$ is the angle of the incident light with respect to the silicon plane normal. Not only does interference modulate ${\displaystyle F_{in}}$ , but the silicon surface does not perfectly reflect the incident light. Fresnel coefficients give the change in amplitude between an incident and reflected wave. The Fresnel coefficients depend on the angles of incidence, ${\displaystyle \theta _{i}}$ and ${\displaystyle \theta _{j}}$ , the indices of refraction of the two mediums and the polarization direction. The angles ${\displaystyle \theta _{i}}$ and ${\displaystyle \theta _{j}}$ can be related by Snell's Law. The expressions for the reflection coefficients are: ${\displaystyle r_{ij}^{TE}={\frac {n_{i}\cos \theta _{i}-n_{j}\cos \theta _{j}}{n_{i}\cos \theta _{i}+n_{j}\cos \theta _{j}}}\quad r_{ij}^{TM}={\frac {n_{j}\cos \theta _{i}-n_{i}\cos \theta _{j}}{n_{j}\cos \theta _{i}+n_{i}\cos \theta _{j}}}}$ TE refers to the component of the electric field perpendicular to the plane of incidence and TM to the parallel component (The incident plane is defined by the plane normal and the propagation direction of the light). In cartesian coordinates, the local electric field is ${\displaystyle F_{in}=\sin \gamma _{in}\left[{\begin{array}{c}0\\1+r_{10}^{TE}{\textit {e}}^{i\Phi _{in}}\\0\end{array}}\right]+\cos \gamma _{in}\left[{\begin{array}{c}\cos \theta _{1}^{in}(1-r_{10}^{TM}{\textit {e}}^{i\Phi _{in}})\\0\\\sin \theta _{1}^{in}(1+r_{10}^{TM}{\textit {e}}^{i\Phi _{in}})\end{array}}\right]}$ ${\displaystyle \gamma _{in}}$ is the polarization angle of the incident light with respect to the plane of incidence. The orientation of the excitation dipole is a function of its angle ${\displaystyle \theta _{ex}}$ to the normal and ${\displaystyle \phi _{ex}}$ azimuthal to the plane of incidence. ${\displaystyle {\textit {e}}_{ex}=\left[{\begin{array}{c}\cos \phi _{ex}\sin \theta _{ex}\\\sin \phi _{ex}\sin \theta _{ex}\\\cos \theta _{ex}\end{array}}\right]}$ The above two equations for ${\displaystyle F_{in}}$ and ${\displaystyle {\textit {e}}_{ex}}$ can be combined to give the probability of exciting the fluorophore per unit time ${\displaystyle P_{ex}}$ . Many of the parameters used above would vary in a normal experiment. The variation in the five following parameters should be included in this theoretical description. • The coherence of the excitation light • The incident angle (${\displaystyle \theta _{1}^{in}}$ ) of excitation light • Polarization angle (${\displaystyle \gamma _{in}}$ ) of the excitation light • The angle of transition dipole (${\displaystyle \theta _{ex}}$ ) of the fluorophore • The wavelength of the excitation light (${\displaystyle \lambda _{ex}}$ ) The squared projection ${\displaystyle \mid F_{in}\cdot e_{ex}\mid ^{2}}$ must be averaged over these quantities to give the probability of excitation ${\displaystyle P_{ex}}$ . Averaging over the first 4 parameters gives ${\displaystyle <\mid F_{in}\cdot e_{ex}\mid ^{2}>\propto \int \sin \theta _{1}^{in}d\theta _{1}^{in}A_{in}(\theta _{1}^{in})\times \int \sin \theta _{ex}d\theta _{ex}O(\theta _{ex})U_{ex}(\lambda _{in},\theta _{1}^{in}.\theta _{ex})}$ ${\displaystyle U_{ex}=\sin ^{2}\theta _{ex}\mid 1+r_{10}^{TE}{\textit {e}}^{i\Phi _{in}}\mid ^{2}+\sin ^{2}\theta _{ex}\cos ^{2}\theta _{1}^{in}\mid 1-r_{10}^{TM}{\textit {e}}^{i\Phi _{in}}\mid ^{2}+2\cos ^{2}\theta _{ex}\sin ^{2}\theta _{1}^{in}\mid 1+r_{10}^{TM}{\textit {e}}^{i\Phi _{in}}\mid ^{2}}$ Example of a FLIC intensity plot showing the relative fluorescence intensity measured versus the distance of the fluorophore from the reflective surface. The peaks might not be the same height in a real experimental plot Normalization factors are not included. ${\displaystyle O(\theta _{ex})}$ is a distribution of the orientation angle of the fluorophore dipoles. The azimuthal angle ${\displaystyle \phi _{ex}}$ and the polarization angle ${\displaystyle \gamma _{in}}$ are integrated over analytically, so they no longer appear in the above equation. To finally obtain the probability of excitation per unit time, the above equation is integrated over the spread in excitation wavelength, accounting for the intensity ${\displaystyle I(\lambda _{ex})}$ and the extinction coefficient of the fluorophore ${\displaystyle \epsilon (\lambda _{ex})}$ . ${\displaystyle P_{ex}\propto \int d\lambda _{ex}I(\lambda _{ex})\epsilon (\lambda _{ex})<\mid F_{in}\cdot e_{ex}\mid ^{2}>}$ The steps to calculate ${\displaystyle P_{em}}$ are equivalent to those above in calculating ${\displaystyle P_{ex}}$ except that the parameter labels em are replaced with ex and in is replaced with out. ${\displaystyle P_{em}\propto \int d\lambda _{em}\Phi _{det}(\lambda _{em}){\textit {f}}(\lambda _{em})<\mid F_{in}\cdot e_{ex}\mid ^{2}>}$ The resulting fluorescence intensity measured is proportional to the product of the excitation probability and emission probability ${\displaystyle I_{FLIC}\propto P_{ex}P_{em}}$ It is important to note that this theory determines a proportionality relation between the measured fluorescence intensity ${\displaystyle I_{FLIC}}$ and the distance of the fluorophore above the reflective surface. The fact that it is not an equality relation will have a significant effect on the experimental procedure. ## Experimental Setup A silicon wafer is typically used as the reflective surface in a FLIC experiment. An oxide layer is then thermally grown on top of the silicon wafer to act as a spacer. On top of the oxide is placed the fluorescently labeled specimen, such as a lipid membrane, a cell or membrane bound proteins. With the sample system built, all that is needed is an epifluorescence microscope and a CCD camera to make quantitative intensity measurements. This is a diagram of an example FLIC experimental setup with silicon, three oxide layers and a fluorescently labeled lipid bilayer (the yellow stars represent fluorophores. The silicon dioxide thickness is very important in making accurate FLIC measurements. As mentioned before, the theoretical model describes the relative fluorescence intensity measured versus the fluorophore height. The fluorophore position cannot be simply read off of a single measured FLIC curve. The basic procedure is to manufacture the oxide layer with at least two known thicknesses (the layer can be made with photolithographic techniques and the thickness measured by ellipsometry). The thicknesses used depends on the sample being measured. For a sample with fluorophore height in the range of 10 nm, oxide thickness around 50 nm would be best because the FLIC intensity curve is steepest here and would produce the greatest contrast between fluorophore heights. Oxide thickness above a few hundred nanometers could be problematic because the curve begins to get smeared out by polychromatic light and a range of incident angles. A ratio of measured fluorescence intensities at different oxide thicknesses is compared to the predicted ratio to calculate the fluorophore height above the oxide (${\displaystyle d_{\textit {f}},}$ ). ${\displaystyle {\frac {I_{theory}(d_{1})}{I_{theory}(d_{0})}}={\frac {I_{exp}(d_{1}+d_{\textit {f}})}{I_{exp}(d_{0}+d_{\textit {f}})}}}$ The above equation can then be solved numerically to find ${\displaystyle d_{\textit {f}}}$ . Imperfections of the experiment, such as imperfect reflection, nonnormal incidence of light and polychromatic light tend to smear out the sharp fluorescence curves. The spread in incidence angle can be controlled by the numerical aperture (N.A.). However, depending on the numerical aperture used, the experiment will yield good lateral resolution (x-y) or good vertical resolution (z), but not both. A high N.A. (~1.0) gives good lateral resolution which is best if the goal is to determine long range topography. Low N.A. (~0.001), on the other hand, provides accurate z-height measurement to determine the height of a fluorescently labeled molecule in a system. ### Analysis Example of experimental data collected for a fluorescently labeled sample over 16 oxide thicknesses. Fitting the curve to the 16 data points would give the height of the fluorophores above the oxide surface. The basic analysis involves fitting the intensity data with the theoretical model allowing the distance of the fluorophore above the oxide surface (${\displaystyle d_{\textit {f}}}$ ) to be a free parameter. The FLIC curves shift to the left as the distance of the fluorophore above the oxide increases. ${\displaystyle d_{\textit {f}}}$ is usually the parameter of interest, but several other free parameters are often included to optimize the fit. Normally an amplitude factor (a) and a constant additive term for the background (b) are included. The amplitude factor scales the relative model intensity and the constant background shifts the curve up or down to account for fluorescence coming from out of focus areas, such as the top side of a cell. Occasionally the numerical aperture (N.A.) of the microscope is allowed to be a free parameter in the fitting. The other parameters entering the optical theory, such as different indices of refraction, layer thicknesses and light wavelengths, are assumed constant with some uncertainty. A FLIC chip may be made with oxide terraces of 9 or 16 different heights arranged in blocks. After a fluorescence image is captured, each 9 or 16 terrace block yields a separate FLIC curve that defines a unique ${\displaystyle d_{\textit {f}}}$ . The average ${\displaystyle d_{\textit {f}}}$ is found by compiling all the ${\displaystyle d_{\textit {f}}}$ values into a histogram. The statistical error in the calculation of ${\displaystyle d_{\textit {f}}}$ comes from two sources: the error in fitting of the optical theory to the data and the uncertainty in the thickness of the oxide layer. Systematic error comes from three sources: the measurement of the oxide thickness (usually by ellipsometer), the fluorescence intensity measurement with the CCD, and the uncertainty in the parameters used in the optical theory. The systematic error has been estimated to be ${\displaystyle \sim 1nm}$ . ## References • Ajo-Franklin, Caroline M.; Yoshina-Ishii, Chiaki; Boxer, Steven G. (2005). "Probing the Structure of Supported Membranes and Tethered Oligonucleotides by Fluorescence Interference Contrast Microscopy". Langmuir. American Chemical Society (ACS). 21 (11): 4976–4983. doi:10.1021/la0468388. ISSN 0743-7463. • Braun, D.; Fromherz, P. (1997-10-01). "Fluorescence interference-contrast microscopy of cell adhesion on oxidized silicon". Applied Physics A: Materials Science & Processing. Springer Science and Business Media LLC. 65 (4–5): 341–348. doi:10.1007/s003390050589. ISSN 0947-8396. • Braun, Dieter; Fromherz, Peter (1998-12-07). "Fluorescence Interferometry of Neuronal Cell Adhesion on Microstructured Silicon". Physical Review Letters. American Physical Society (APS). 81 (23): 5241–5244. doi:10.1103/physrevlett.81.5241. ISSN 0031-9007. • Crane, Jonathan M.; Kiessling, Volker; Tamm, Lukas K. (2005). "Measuring Lipid Asymmetry in Planar Supported Bilayers by Fluorescence Interference Contrast Microscopy". Langmuir. American Chemical Society (ACS). 21 (4): 1377–1388. doi:10.1021/la047654w. ISSN 0743-7463. • Kaizuka, Yoshihisa; Groves, Jay T. (2006-03-20). "Hydrodynamic Damping of Membrane Thermal Fluctuations near Surfaces Imaged by Fluorescence Interference Microscopy". Physical Review Letters. American Physical Society (APS). 96 (11): 118101. doi:10.1103/physrevlett.96.118101. ISSN 0031-9007. • Kiessling, Volker; Tamm, Lukas K. (2003). "Measuring Distances in Supported Bilayers by Fluorescence Interference-Contrast Microscopy: Polymer Supports and SNARE Proteins". Biophysical Journal. Elsevier BV. 84 (1): 408–418. doi:10.1016/s0006-3495(03)74861-9. ISSN 0006-3495. • Lambacher, Armin; Fromherz, Peter (1996). "Fluorescence interference-contrast microscopy on oxidized silicon using a monomolecular dye layer". Applied Physics A Materials Science & Processing. Springer Science and Business Media LLC. 63 (3): 207–216. doi:10.1007/bf01567871. ISSN 0947-8396. • Lambacher, Armin; Fromherz, Peter (2002-06-01). "Luminescence of dye molecules on oxidized silicon and fluorescence interference contrast microscopy of biomembranes". Journal of the Optical Society of America B. The Optical Society. 19 (6): 1435-1453. doi:10.1364/josab.19.001435. ISSN 0740-3224. • Parthasarathy, Raghuveer; Groves, Jay T. (2004). "Optical Techniques for Imaging Membrane Topography". Cell Biochemistry and Biophysics. Springer Science and Business Media LLC. 41 (3): 391–414. doi:10.1385/cbb:41:3:391. ISSN 1085-9195.	2020-01-25 15:35:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 58, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6954768300056458, "perplexity": 2002.441540052863}, "config": {"markdown_headings": true, "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-2020-05/segments/1579251672537.90/warc/CC-MAIN-20200125131641-20200125160641-00542.warc.gz"}
http://www.physicsbootcamp.org/sec-LengthContraction.html	## Section52.8Length Contraction We define length of a rod as the distance between the positions of the two ends at the same time. The caveat that the positions of the two ends be know at the same time in the frame in which we seek the length is important since time and position are connected in Lorentz transformations. Below we will apply Lorentz transformation to seek length in a frame in which rod is moving. For simplicity, we consider rod along $x$ axis, which is the direction of the relative motion of frames as shown in Figure 52.8.1. We consider rod to be at rest in frame S' with ends A and B with events $(t'_A, x'_A)$ and $(t'_B=t'_A, x'_B \gt x'_A)\text{,}$ respectively. In frame S', we define length by \begin{equation} L_0 = x'_B - x'_A. \end{equation} To calculate length of the rod in frame S, we first find Lorenz transformation of events A and B. During calculation we will keep $t'_A$ and $t'_B$ separate even though they are equal in our calculations. This helps keeping track of the two events. \begin{align} \amp t_A = \gamma \left( t'_A + V x'_A/c^2 \right),\ \ x_A = \gamma \left( x'_A + V t'_A \right) \\ \amp t_B = \gamma \left( t'_B + V x'_B/c^2 \right),\ \ x_B = \gamma \left( x'_B + V t'_B \right) \end{align} We want $x_B-x_A$ when $t_B=t_A\text{.}$ Towards that end, let us express $t'_A$ and $t'_B$ in terms of $t_A$ and $t_B$ by using the first set of these equations. \begin{equation} t'_A = \frac{t_A}{\gamma} - V x'_A/c^2,\ \ t'_B = \frac{t_B}{\gamma} - V x'_B/c^2. \end{equation} Therefore, \begin{align} x_B - x_A \amp = \gamma \left[ x'_B + \frac{V t_B}{\gamma} - \frac{V ^2}{c^2}x'_B\right] - \gamma \left[ x'_A + \frac{V t_A}{\gamma} - \frac{V ^2}{c^2}x'_A\right] \\ \amp = \gamma \left[ x'_B - x'_A - \frac{V ^2}{c^2}\, \left(x'_B - x'_A \right)\right] + \frac{V}{\gamma}\left( t_B - t_A \right) \end{align} This will give length if $t_B=t_A$ in this frame (S). Denoting $x_B-x_A$ by $L$ since we have denoted $x'_B - x'_A$ by $L_0\text{,}$ we have \begin{align} L\amp = \gamma \left( 1 - \frac{V^2}{c^2} \right) \, L_0. \end{align} With $\gamma = 1/\sqrt{1-V^2/c^2}\text{,}$ this is $$L = \dfrac{L_0}{\gamma} = L_0\,\sqrt{ 1 - \dfrac{V^2}{c^2} }.\label{lorentz-contraction}\tag{52.8.1}$$ This is called length cortaction. Since $\gamma \gt 1\text{,}$ length of a rod will be smaller in a frame in which rod is moving, i.e., $L \le L_0\text{.}$ Since $y$ and $z$ coordinates do not change, the transverse dimensions, i.e., the dimensions of the rod perpendicular to the direction of the motion of the rod are unaffected. Let us denote these dimensions with a subscript $\perp\text{.}$ $$L_{\perp} = L_{0\perp}.\tag{52.8.2}$$ Same two observers as in Exercise Checkpoint 52.6.8. A rod of length $1\:\textrm{m}$ is laid out on the $x$-axis in the frame of A from origin to $(x =1\: \textrm{m}, 0,0)\text{.}$ What will be the length of the rod observed by an observer in frame of spaceship B? Hint Solution We use length contraction to answer this. \begin{equation} L_B = \frac{L_A}{\gamma} = \frac{\sqrt{3}}{2}\times 1\text{ m} = 0.866\text{ m}. \end{equation} A spaceship is seen to move past an observer on Earth at a speed of $0.95 c\text{,}$ i.e. at $95\%$ of the speed of light in the direction parallel to the length of the ship. An astronaut in the spaceship has measured the length of the spaceship to be 100 m. What will be the length of the spaceship as observed by the observer on Earth? Hint Use length contraction formula. $31.3\:\textrm{m}\text{.}$ Solution The length 100 m for the spaceship is the length in the rest frame of the ship. The length observed by the Earth-based observer will be in the lab frame. From our discussion above the lab frame value will be contracted compared to the rest frame value by a factor of $\gamma\text{.}$ Here $\gamma$ has the following value. \begin{equation} \gamma = \frac{1}{\sqrt{1-V^2/c^2}} = \frac{1}{\sqrt{1-0.95^2}} = 3.2. \end{equation} Therefore, the length observed by the observer on Earth will be \begin{equation} L = \frac{L_0}{\gamma} = \frac{100\: \textrm{m}}{3.2} = 31.3\:\textrm{m}. \end{equation} Same two observers as in Exercise Checkpoint 52.6.8. But, now we look at two events occurring in spaceship A. A photon arrives at the origin of A at its time $t=0$ and another photon arrives at $(x =1\: \textrm{m}, 0,0)$ at $t=0$ in the frame of ship A. (a) Find the coordinates and times of the two events as observed by frame B. (b) In which frame, the two events are simultaneous and in which frame they are not simultaneous? Hint (a) $1.92 \times 10^{-9}\text{ sec}\text{,}$ $1.155\text{ m}\text{,}$ (b) Simultaneous in A. (a) Now, the events are in frame A, which is moving towards positive $x$ axis when observed from frame B. The event of photon arriving at origin at $t_A=0$ will be also origin of B at $t_B=0$ since the two frames were set that way. The other event of photon at $x_A = 1\text{ m}$ at $t_A=0$ will have $t_B$ and $x_B$ as follows.	2023-03-20 13:30:59	{"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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9702596068382263, "perplexity": 551.0340956623785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2023-14/segments/1679296943483.86/warc/CC-MAIN-20230320114206-20230320144206-00652.warc.gz"}
https://math.stackexchange.com/questions/1046152/solving-sin-z-z-in-reals/1046167	# Solving sin z = -z in reals How do we prove that z=0 is the only real solution? I tried examining cases and quarters, but not sure how it is rigorously proven. For $\|z\|>1$, you cannot have an equality since $\|\sin z\|\leq 1$. Inside the circle of radius 1, you notice that when $-z<0$, $\sin z$ is positive, and viceversa when $-z$ is positive $\sin z<0$. Then there is no chance then that the two values coincide. Since $—1 \le \sin(z) \le 1$, we only need to look at $z \in [-1, 1]$. If $z \ne 0$, the sign of $\sin(z)$ is the same as the sign of $z$. Hint: suppose a second solution $z_1$. Apply Rolle's theorem (where?). Notice $\sin z+z$ has derivative $1+\cos z$ which is nonnegative for all $z\in\Bbb R$ and which has isolated zeroes. So our function is increasing, and as $1+\cos 0>0$ the derivative is positive in a neighborhood of $0$ and so the function has a unique zero. • Did you mean by "our function" $z\mapsto-\sin z-z$, or did you mean to say "weakly increasing" instead of "weakly decreasing"? And in any case your argument shows strict rather than weak monotonicity. Dec 2 '14 at 13:05	2021-09-28 05:30:36	{"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": 0.9468931555747986, "perplexity": 204.5294587044316}, "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-2021-39/segments/1631780060201.9/warc/CC-MAIN-20210928032425-20210928062425-00054.warc.gz"}
https://terrytao.wordpress.com/tag/entropy/	You are currently browsing the tag archive for the ‘entropy’ tag. Let ${G}$ be a finite set of order ${N}$; in applications ${G}$ will be typically something like a finite abelian group, such as the cyclic group ${{\bf Z}/N{\bf Z}}$. Let us define a ${1}$-bounded function to be a function ${f: G \rightarrow {\bf C}}$ such that ${\|f(n)\| \leq 1}$ for all ${n \in G}$. There are many seminorms ${\\| \\|}$ of interest that one places on functions ${f: G \rightarrow {\bf C}}$ that are bounded by ${1}$ on ${1}$-bounded functions, such as the Gowers uniformity seminorms ${\\| \\|_k}$ for ${k \geq 1}$ (which are genuine norms for ${k \geq 2}$). All seminorms in this post will be implicitly assumed to obey this property. In additive combinatorics, a significant role is played by inverse theorems, which abstractly take the following form for certain choices of seminorm ${\\| \\|}$, some parameters ${\eta, \varepsilon>0}$, and some class ${{\mathcal F}}$ of ${1}$-bounded functions: Theorem 1 (Inverse theorem template) If ${f}$ is a ${1}$-bounded function with ${\\|f\\| \geq \eta}$, then there exists ${F \in {\mathcal F}}$ such that ${\|\langle f, F \rangle\| \geq \varepsilon}$, where ${\langle,\rangle}$ denotes the usual inner product $\displaystyle \langle f, F \rangle := {\bf E}_{n \in G} f(n) \overline{F(n)}.$ Informally, one should think of ${\eta}$ as being somewhat small but fixed independently of ${N}$, ${\varepsilon}$ as being somewhat smaller but depending only on ${\eta}$ (and on the seminorm), and ${{\mathcal F}}$ as representing the “structured functions” for these choices of parameters. There is some flexibility in exactly how to choose the class ${{\mathcal F}}$ of structured functions, but intuitively an inverse theorem should become more powerful when this class is small. Accordingly, let us define the ${(\eta,\varepsilon)}$-entropy of the seminorm ${\\| \\|}$ to be the least cardinality of ${{\mathcal F}}$ for which such an inverse theorem holds. Seminorms with low entropy are ones for which inverse theorems can be expected to be a useful tool. This concept arose in some discussions I had with Ben Green many years ago, but never appeared in print, so I decided to record some observations we had on this concept here on this blog. Lebesgue norms ${\\| f\\|_{L^p} := ({\bf E}_{n \in G} \|f(n)\|^p)^{1/p}}$ for ${1 < p < \infty}$ have exponentially large entropy (and so inverse theorems are not expected to be useful in this case): Proposition 2 (${L^p}$ norm has exponentially large inverse entropy) Let ${1 < p < \infty}$ and ${0 < \eta < 1}$. Then the ${(\eta,\eta^p/4)}$-entropy of ${\\| \\|_{L^p}}$ is at most ${(1+8/\eta^p)^N}$. Conversely, for any ${\varepsilon>0}$, the ${(\eta,\varepsilon)}$-entropy of ${\\| \\|_{L^p}}$ is at least ${\exp( c \varepsilon^2 N)}$ for some absolute constant ${c>0}$. Proof: If ${f}$ is ${1}$-bounded with ${\\|f\\|_{L^p} \geq \eta}$, then we have $\displaystyle \|\langle f, \|f\|^{p-2} f \rangle\| \geq \eta^p$ and hence by the triangle inequality we have $\displaystyle \|\langle f, F \rangle\| \geq \eta^p/2$ where ${F}$ is either the real or imaginary part of ${\|f\|^{p-2} f}$, which takes values in ${[-1,1]}$. If we let ${\tilde F}$ be ${F}$ rounded to the nearest multiple of ${\eta^p/4}$, then by the triangle inequality again we have $\displaystyle \|\langle f, \tilde F \rangle\| \geq \eta^p/4.$ There are only at most ${1+8/\eta^p}$ possible values for each value ${\tilde F(n)}$ of ${\tilde F}$, and hence at most ${(1+8/\eta^p)^N}$ possible choices for ${\tilde F}$. This gives the first claim. Now suppose that there is an ${(\eta,\varepsilon)}$-inverse theorem for some ${{\mathcal F}}$ of cardinality ${M}$. If we let ${f}$ be a random sign function (so the ${f(n)}$ are independent random variables taking values in ${-1,+1}$ with equal probability), then there is a random ${F \in {\mathcal F}}$ such that $\displaystyle \|\langle f, F \rangle\| \geq \varepsilon$ and hence by the pigeonhole principle there is a deterministic ${F \in {\mathcal F}}$ such that $\displaystyle {\bf P}( \|\langle f, F \rangle\| \geq \varepsilon ) \geq 1/M.$ On the other hand, from the Hoeffding inequality one has $\displaystyle {\bf P}( \|\langle f, F \rangle\| \geq \varepsilon ) \ll \exp( - c \varepsilon^2 N )$ for some absolute constant ${c}$, hence $\displaystyle M \geq \exp( c \varepsilon^2 N )$ as claimed. $\Box$ Most seminorms of interest in additive combinatorics, such as the Gowers uniformity norms, are bounded by some finite ${L^p}$ norm thanks to Hölder’s inequality, so from the above proposition and the obvious monotonicity properties of entropy, we conclude that all Gowers norms on finite abelian groups ${G}$ have at most exponential inverse theorem entropy. But we can do significantly better than this: • For the ${U^1}$ seminorm ${\\|f\\|_{U^1(G)} := \|{\bf E}_{n \in G} f(n)\|}$, one can simply take ${{\mathcal F} = \{1\}}$ to consist of the constant function ${1}$, and the ${(\eta,\eta)}$-entropy is clearly equal to ${1}$ for any ${0 < \eta < 1}$. • For the ${U^2}$ norm, the standard Fourier-analytic inverse theorem asserts that if ${\\|f\\|_{U^2(G)} \geq \eta}$ then ${\|\langle f, e(\xi \cdot) \rangle\| \geq \eta^2}$ for some Fourier character ${\xi \in \hat G}$. Thus the ${(\eta,\eta^2)}$-entropy is at most ${N}$. • For the ${U^k({\bf Z}/N{\bf Z})}$ norm on cyclic groups for ${k > 2}$, the inverse theorem proved by Green, Ziegler, and myself gives an ${(\eta,\varepsilon)}$-inverse theorem for some ${\varepsilon \gg_{k,\eta} 1}$ and ${{\mathcal F}}$ consisting of nilsequences ${n \mapsto F(g(n) \Gamma)}$ for some filtered nilmanifold ${G/\Gamma}$ of degree ${k-1}$ in a finite collection of cardinality ${O_{\eta,k}(1)}$, some polynomial sequence ${g: {\bf Z} \rightarrow G}$ (which was subsequently observed by Candela-Sisask (see also Manners) that one can choose to be ${N}$-periodic), and some Lipschitz function ${F: G/\Gamma \rightarrow {\bf C}}$ of Lipschitz norm ${O_{\eta,k}(1)}$. By the Arzela-Ascoli theorem, the number of possible ${F}$ (up to uniform errors of size at most ${\varepsilon/2}$, say) is ${O_{\eta,k}(1)}$. By standard arguments one can also ensure that the coefficients of the polynomial ${g}$ are ${O_{\eta,k}(1)}$, and then by periodicity there are only ${O(N^{O_{\eta,k}(1)}}$ such polynomials. As a consequence, the ${(\eta,\varepsilon)}$-entropy is of polynomial size ${O_{\eta,k}( N^{O_{\eta,k}(1)} )}$ (a fact that seems to have first been implicitly observed in Lemma 6.2 of this paper of Frantzikinakis; thanks to Ben Green for this reference). One can obtain more precise dependence on ${\eta,k}$ using the quantitative version of this inverse theorem due to Manners; back of the envelope calculations using Section 5 of that paper suggest to me that one can take ${\varepsilon = \eta^{O_k(1)}}$ to be polynomial in ${\eta}$ and the entropy to be of the order ${O_k( N^{\exp(\exp(\eta^{-O_k(1)}))} )}$, or alternatively one can reduce the entropy to ${O_k( \exp(\exp(\eta^{-O_k(1)})) N^{\eta^{-O_k(1)}})}$ at the cost of degrading ${\varepsilon}$ to ${1/\exp\exp( O(\eta^{-O(1)}))}$. • If one replaces the cyclic group ${{\bf Z}/N{\bf Z}}$ by a vector space ${{\bf F}_p^n}$ over some fixed finite field ${{\bf F}_p}$ of prime order (so that ${N=p^n}$), then the inverse theorem of Ziegler and myself (available in both high and low characteristic) allows one to obtain an ${(\eta,\varepsilon)}$-inverse theorem for some ${\varepsilon \gg_{k,\eta} 1}$ and ${{\mathcal F}}$ the collection of non-classical degree ${k-1}$ polynomial phases from ${{\bf F}_p^n}$ to ${S^1}$, which one can normalize to equal ${1}$ at the origin, and then by the classification of such polynomials one can calculate that the ${(\eta,\varepsilon)}$ entropy is of quasipolynomial size ${\exp( O_{p,k}(n^{k-1}) ) = \exp( O_{p,k}( \log^{k-1} N ) )}$ in ${N}$. By using the recent work of Gowers and Milicevic, one can make the dependence on ${p,k}$ here more precise, but we will not perform these calcualtions here. • For the ${U^3(G)}$ norm on an arbitrary finite abelian group, the recent inverse theorem of Jamneshan and myself gives (after some calculations) a bound of the polynomial form ${O( q^{O(n^2)} N^{\exp(\eta^{-O(1)})})}$ on the ${(\eta,\varepsilon)}$-entropy for some ${\varepsilon \gg \eta^{O(1)}}$, which one can improve slightly to ${O( q^{O(n^2)} N^{\eta^{-O(1)}})}$ if one degrades ${\varepsilon}$ to ${1/\exp(\eta^{-O(1)})}$, where ${q}$ is the maximal order of an element of ${G}$, and ${n}$ is the rank (the number of elements needed to generate ${G}$). This bound is polynomial in ${N}$ in the cyclic group case and quasipolynomial in general. For general finite abelian groups ${G}$, we do not yet have an inverse theorem of comparable power to the ones mentioned above that give polynomial or quasipolynomial upper bounds on the entropy. However, there is a cheap argument that at least gives some subexponential bounds: Proposition 3 (Cheap subexponential bound) Let ${k \geq 2}$ and ${0 < \eta < 1/2}$, and suppose that ${G}$ is a finite abelian group of order ${N \geq \eta^{-C_k}}$ for some sufficiently large ${C_k}$. Then the ${(\eta,c_k \eta^{O_k(1)})}$-complexity of ${\\| \\|_{U^k(G)}}$ is at most ${O( \exp( \eta^{-O_k(1)} N^{1 - \frac{k+1}{2^k-1}} ))}$. Proof: (Sketch) We use a standard random sampling argument, of the type used for instance by Croot-Sisask or Briet-Gopi (thanks to Ben Green for this latter reference). We can assume that ${N \geq \eta^{-C_k}}$ for some sufficiently large ${C_k>0}$, since otherwise the claim follows from Proposition 2. Let ${A}$ be a random subset of ${{\bf Z}/N{\bf Z}}$ with the events ${n \in A}$ being iid with probability ${0 < p < 1}$ to be chosen later, conditioned to the event ${\|A\| \leq 2pN}$. Let ${f}$ be a ${1}$-bounded function. By a standard second moment calculation, we see that with probability at least ${1/2}$, we have $\displaystyle \\|f\\|_{U^k(G)}^{2^k} = {\bf E}_{n, h_1,\dots,h_k \in G} f(n) \prod_{\omega \in \{0,1\}^k \backslash \{0\}} {\mathcal C}^{\|\omega\|} \frac{1}{p} 1_A f(n + \omega \cdot h)$ $\displaystyle + O((\frac{1}{N^{k+1} p^{2^k-1}})^{1/2}).$ Thus, by the triangle inequality, if we choose ${p := C \eta^{-2^{k+1}/(2^k-1)} / N^{\frac{k+1}{2^k-1}}}$ for some sufficiently large ${C = C_k > 0}$, then for any ${1}$-bounded ${f}$ with ${\\|f\\|_{U^k(G)} \geq \eta/2}$, one has with probability at least ${1/2}$ that $\displaystyle \|{\bf E}_{n, h_1,\dots,h_k \i2^n G} f(n) \prod_{\omega \in \{0,1\}^k \backslash \{0\}} {\mathcal C}^{\|\omega\|} \frac{1}{p} 1_A f(n + \omega \cdot h)\|$ $\displaystyle \geq \eta^{2^k}/2^{2^k+1}.$ We can write the left-hand side as ${\|\langle f, F \rangle\|}$ where ${F}$ is the randomly sampled dual function $\displaystyle F(n) := {\bf E}_{n, h_1,\dots,h_k \in G} f(n) \prod_{\omega \in \{0,1\}^k \backslash \{0\}} {\mathcal C}^{\|\omega\|+1} \frac{1}{p} 1_A f(n + \omega \cdot h).$ Unfortunately, ${F}$ is not ${1}$-bounded in general, but we have $\displaystyle \\|F\\|_{L^2(G)}^2 \leq {\bf E}_{n, h_1,\dots,h_k ,h'_1,\dots,h'_k \in G}$ $\displaystyle \prod_{\omega \in \{0,1\}^k \backslash \{0\}} \frac{1}{p} 1_A(n + \omega \cdot h) \frac{1}{p} 1_A(n + \omega \cdot h')$ and the right-hand side can be shown to be ${1+o(1)}$ on the average, so we can condition on the event that the right-hand side is ${O(1)}$ without significant loss in falure probability. If we then let ${\tilde f_A}$ be ${1_A f}$ rounded to the nearest Gaussian integer multiple of ${\eta^{2^k}/2^{2^{10k}}}$ in the unit disk, one has from the triangle inequality that $\displaystyle \|\langle f, \tilde F \rangle\| \geq \eta^{2^k}/2^{2^k+2}$ where ${\tilde F}$ is the discretised randomly sampled dual function $\displaystyle \tilde F(n) := {\bf E}_{n, h_1,\dots,h_k \in G} f(n) \prod_{\omega \in \{0,1\}^k \backslash \{0\}} {\mathcal C}^{\|\omega\|+1} \frac{1}{p} \tilde f_A(n + \omega \cdot h).$ For any given ${A}$, there are at most ${2np}$ places ${n}$ where ${\tilde f_A(n)}$ can be non-zero, and in those places there are ${O_k( \eta^{-2^{k}})}$ possible values for ${\tilde f_A(n)}$. Thus, if we let ${{\mathcal F}_A}$ be the collection of all possible ${\tilde f_A}$ associated to a given ${A}$, the cardinality of this set is ${O( \exp( \eta^{-O_k(1)} N^{1 - \frac{k+1}{2^k-1}} ) )}$, and for any ${f}$ with ${\\|f\\|_{U^k(G)} \geq \eta/2}$, we have $\displaystyle \sup_{\tilde F \in {\mathcal F}_A} \|\langle f, \tilde F \rangle\| \geq \eta^{2^k}/2^{k+2}$ with probability at least ${1/2}$. Now we remove the failure probability by independent resampling. By rounding to the nearest Gaussian integer multiple of ${c_k \eta^{2^k}}$ in the unit disk for a sufficiently small ${c_k>0}$, one can find a family ${{\mathcal G}}$ of cardinality ${O( \eta^{-O_k(N)})}$ consisting of ${1}$-bounded functions ${\tilde f}$ of ${U^k(G)}$ norm at least ${\eta/2}$ such that for every ${1}$-bounded ${f}$ with ${\\|f\\|_{U^k(G)} \geq \eta}$ there exists ${\tilde f \in {\mathcal G}}$ such that $\displaystyle \\|f-\tilde f\\|_{L^\infty(G)} \leq \eta^{2^k}/2^{k+3}.$ Now, let ${A_1,\dots,A_M}$ be independent samples of ${A}$ for some ${M}$ to be chosen later. By the preceding discussion, we see that with probability at least ${1 - 2^{-M}}$, we have $\displaystyle \sup_{\tilde F \in \bigcup_{j=1}^M {\mathcal F}_{A_j}} \|\langle \tilde f, \tilde F \rangle\| \geq \eta^{2^k}/2^{k+2}$ for any given ${\tilde f \in {\mathcal G}}$, so by the union bound, if we choose ${M = \lfloor C N \log \frac{1}{\eta} \rfloor}$ for a large enough ${C = C_k}$, we can find ${A_1,\dots,A_M}$ such that $\displaystyle \sup_{\tilde F \in \bigcup_{j=1}^M {\mathcal F}_{A_j}} \|\langle \tilde f, \tilde F \rangle\| \geq \eta^{2^k}/2^{k+2}$ for all ${\tilde f \in {\mathcal G}}$, and hence y the triangle inequality $\displaystyle \sup_{\tilde F \in \bigcup_{j=1}^M {\mathcal F}_{A_j}} \|\langle f, \tilde F \rangle\| \geq \eta^{2^k}/2^{k+3}.$ Taking ${{\mathcal F}}$ to be the union of the ${{\mathcal F}_{A_j}}$ (applying some truncation and rescaling to these ${L^2}$-bounded functions to make them ${L^\infty}$-bounded, and then ${1}$-bounded), we obtain the claim. $\Box$ One way to obtain lower bounds on the inverse theorem entropy is to produce a collection of almost orthogonal functions with large norm. More precisely: Proposition 4 Let ${\\| \\|}$ be a seminorm, let ${0 < \varepsilon \leq \eta < 1}$, and suppose that one has a collection ${f_1,\dots,f_M}$ of ${1}$-bounded functions such that for all ${i=1,\dots,M}$, ${\\|f_i\\| \geq \eta}$ one has ${\|\langle f_i, f_j \rangle\| \leq \varepsilon^2/2}$ for all but at most ${L}$ choices of ${j \in \{1,\dots,M\}}$ for all distinct ${i,j \in \{1,\dots,M\}}$. Then the ${(\eta, \varepsilon)}$-entropy of ${\\| \\|}$ is at least ${\varepsilon^2 M / 2L}$. Proof: Suppose we have an ${(\eta,\varepsilon)}$-inverse theorem with some family ${{\mathcal F}}$. Then for each ${i=1,\dots,M}$ there is ${F_i \in {\mathcal F}}$ such that ${\|\langle f_i, F_i \rangle\| \geq \varepsilon}$. By the pigeonhole principle, there is thus ${F \in {\mathcal F}}$ such that ${\|\langle f_i, F \rangle\| \geq \varepsilon}$ for all ${i}$ in a subset ${I}$ of ${\{1,\dots,M\}}$ of cardinality at least ${M/\|{\mathcal F}\|}$: $\displaystyle \|I\| \geq M / \|{\mathcal F}\|.$ We can sum this to obtain $\displaystyle \|\sum_{i \in I} c_i \langle f_i, F \rangle\| \geq \|I\| \varepsilon$ for some complex numbers ${c_i}$ of unit magnitude. By Cauchy-Schwarz, this implies $\displaystyle \\| \sum_{i \in I} c_i f_i \\|_{L^2(G)}^2 \geq \|I\|^2 \varepsilon^2$ and hence by the triangle inequality $\displaystyle \sum_{i,j \in I} \|\langle f_i, f_j \rangle\| \geq \|I\|^2 \varepsilon^2.$ On the other hand, by hypothesis we can bound the left-hand side by ${\|I\| (L + \varepsilon^2 \|I\|/2)}$. Rearranging, we conclude that $\displaystyle \|I\| \leq 2 L / \varepsilon^2$ and hence $\displaystyle \|{\mathcal F}\| \geq \varepsilon^2 M / 2L$ giving the claim. $\Box$ Thus for instance: • For the ${U^2(G)}$ norm, one can take ${f_1,\dots,f_M}$ to be the family of linear exponential phases ${n \mapsto e(\xi \cdot n)}$ with ${M = N}$ and ${L=1}$, and obtain a linear lower bound of ${\varepsilon^2 N/2}$ for the ${(\eta,\varepsilon)}$-entropy, thus matching the upper bound of ${N}$ up to constants when ${\varepsilon}$ is fixed. • For the ${U^k({\bf Z}/N{\bf Z})}$ norm, a similar calculation using polynomial phases of degree ${k-1}$, combined with the Weyl sum estimates, gives a lower bound of ${\gg_{k,\varepsilon} N^{k-1}}$ for the ${(\eta,\varepsilon)}$-entropy for any fixed ${\eta,\varepsilon}$; by considering nilsequences as well, together with nilsequence equidistribution theory, one can replace the exponent ${k-1}$ here by some quantity that goes to infinity as ${\eta \rightarrow 0}$, though I have not attempted to calculate the exact rate. • For the ${U^k({\bf F}_p^n)}$ norm, another similar calculation using polynomial phases of degree ${k-1}$ should give a lower bound of ${\gg_{p,k,\eta,\varepsilon} \exp( c_{p,k,\eta,\varepsilon} n^{k-1} )}$ for the ${(\eta,\varepsilon)}$-entropy, though I have not fully performed the calculation. We close with one final example. Suppose ${G}$ is a product ${G = A \times B}$ of two sets ${A,B}$ of cardinality ${\asymp \sqrt{N}}$, and we consider the Gowers box norm $\displaystyle \\|f\\|_{\Box^2(G)}^4 := {\bf E}_{a,a' \in A; b,b' \in B} f(a,b) \overline{f}(a,b') \overline{f}(a',b) f(a,b).$ One possible choice of class ${{\mathcal F}}$ here are the indicators ${1_{U \times V}}$ of “rectangles” ${U \times V}$ with ${U \subset A}$, ${V \subset B}$ (cf. this previous blog post on cut norms). By standard calculations, one can use this class to show that the ${(\eta, \eta^4/10)}$-entropy of ${\\| \\|_{\Box^2(G)}}$ is ${O( \exp( O(\sqrt{N}) )}$, and a variant of the proof of the second part of Proposition 2 shows that this is the correct order of growth in ${N}$. In contrast, a modification of Proposition 3 only gives an upper bound of the form ${O( \exp( O( N^{2/3} ) ) )}$ (the bottleneck is ensuring that the randomly sampled dual functions stay bounded in ${L^2}$), which shows that while this cheap bound is not optimal, it can still broadly give the correct “type” of bound (specifically, intermediate growth between polynomial and exponential). Let ${P(z) = z^n + a_{n-1} z^{n-1} + \dots + a_0}$ be a monic polynomial of degree ${n}$ with complex coefficients. Then by the fundamental theorem of algebra, we can factor ${P}$ as $\displaystyle P(z) = (z-z_1) \dots (z-z_n) \ \ \ \ \ (1)$ for some complex zeroes ${z_1,\dots,z_n}$ (possibly with repetition). Now suppose we evolve ${P}$ with respect to time by heat flow, creating a function ${P(t,z)}$ of two variables with given initial data ${P(0,z) = P(z)}$ for which $\displaystyle \partial_t P(t,z) = \partial_{zz} P(t,z). \ \ \ \ \ (2)$ On the space of polynomials of degree at most ${n}$, the operator ${\partial_{zz}}$ is nilpotent, and one can solve this equation explicitly both forwards and backwards in time by the Taylor series $\displaystyle P(t,z) = \sum_{j=0}^\infty \frac{t^j}{j!} \partial_z^{2j} P(0,z).$ For instance, if one starts with a quadratic ${P(0,z) = z^2 + bz + c}$, then the polynomial evolves by the formula $\displaystyle P(t,z) = z^2 + bz + (c+2t).$ As the polynomial ${P(t)}$ evolves in time, the zeroes ${z_1(t),\dots,z_n(t)}$ evolve also. Assuming for sake of discussion that the zeroes are simple, the inverse function theorem tells us that the zeroes will (locally, at least) evolve smoothly in time. What are the dynamics of this evolution? For instance, in the quadratic case, the quadratic formula tells us that the zeroes are $\displaystyle z_1(t) = \frac{-b + \sqrt{b^2 - 4(c+2t)}}{2}$ and $\displaystyle z_2(t) = \frac{-b - \sqrt{b^2 - 4(c+2t)}}{2}$ after arbitrarily choosing a branch of the square root. If ${b,c}$ are real and the discriminant ${b^2 - 4c}$ is initially positive, we see that we start with two real zeroes centred around ${-b/2}$, which then approach each other until time ${t = \frac{b^2-4c}{8}}$, at which point the roots collide and then move off from each other in an imaginary direction. In the general case, we can obtain the equations of motion by implicitly differentiating the defining equation $\displaystyle P( t, z_i(t) ) = 0$ in time using (2) to obtain $\displaystyle \partial_{zz} P( t, z_i(t) ) + \partial_t z_i(t) \partial_z P(t,z_i(t)) = 0.$ To simplify notation we drop the explicit dependence on time, thus $\displaystyle \partial_{zz} P(z_i) + (\partial_t z_i) \partial_z P(z_i)= 0.$ From (1) and the product rule, we see that $\displaystyle \partial_z P( z_i ) = \prod_{j:j \neq i} (z_i - z_j)$ and $\displaystyle \partial_{zz} P( z_i ) = 2 \sum_{k:k \neq i} \prod_{j:j \neq i,k} (z_i - z_j)$ (where all indices are understood to range over ${1,\dots,n}$) leading to the equations of motion $\displaystyle \partial_t z_i = \sum_{k:k \neq i} \frac{2}{z_k - z_i}, \ \ \ \ \ (3)$ at least when one avoids those times in which there is a repeated zero. In the case when the zeroes ${z_i}$ are real, each term ${\frac{2}{z_k-z_i}}$ represents a (first-order) attraction in the dynamics between ${z_i}$ and ${z_k}$, but the dynamics are more complicated for complex zeroes (e.g. purely imaginary zeroes will experience repulsion rather than attraction, as one already sees in the quadratic example). Curiously, this system resembles that of Dyson brownian motion (except with the brownian motion part removed, and time reversed). I learned of the connection between the ODE (3) and the heat equation from this paper of Csordas, Smith, and Varga, but perhaps it has been mentioned in earlier literature as well. One interesting consequence of these equations is that if the zeroes are real at some time, then they will stay real as long as the zeroes do not collide. Let us now restrict attention to the case of real simple zeroes, in which case we will rename the zeroes as ${x_i}$ instead of ${z_i}$, and order them as ${x_1 < \dots < x_n}$. The evolution $\displaystyle \partial_t x_i = \sum_{k:k \neq i} \frac{2}{x_k - x_i}$ can now be thought of as reverse gradient flow for the “entropy” $\displaystyle H := -\sum_{i,j: i \neq j} \log \|x_i - x_j\|,$ (which is also essentially the logarithm of the discriminant of the polynomial) since we have $\displaystyle \partial_t x_i = \frac{\partial H}{\partial x_i}.$ In particular, we have the monotonicity formula $\displaystyle \partial_t H = 4E$ where ${E}$ is the “energy” $\displaystyle E := \frac{1}{4} \sum_i (\frac{\partial H}{\partial x_i})^2$ $\displaystyle = \sum_i (\sum_{k:k \neq i} \frac{1}{x_k-x_i})^2$ $\displaystyle = \sum_{i,k: i \neq k} \frac{1}{(x_k-x_i)^2} + 2 \sum_{i,j,k: i,j,k \hbox{ distinct}} \frac{1}{(x_k-x_i)(x_j-x_i)}$ $\displaystyle = \sum_{i,k: i \neq k} \frac{1}{(x_k-x_i)^2}$ where in the last line we use the antisymmetrisation identity $\displaystyle \frac{1}{(x_k-x_i)(x_j-x_i)} + \frac{1}{(x_i-x_j)(x_k-x_j)} + \frac{1}{(x_j-x_k)(x_i-x_k)} = 0.$ Among other things, this shows that as one goes backwards in time, the entropy decreases, and so no collisions can occur to the past, only in the future, which is of course consistent with the attractive nature of the dynamics. As ${H}$ is a convex function of the positions ${x_1,\dots,x_n}$, one expects ${H}$ to also evolve in a convex manner in time, that is to say the energy ${E}$ should be increasing. This is indeed the case: Exercise 1 Show that $\displaystyle \partial_t E = 2 \sum_{i,j: i \neq j} (\frac{2}{(x_i-x_j)^2} - \sum_{k: i,j,k \hbox{ distinct}} \frac{1}{(x_k-x_i)(x_k-x_j)})^2.$ Symmetric polynomials of the zeroes are polynomial functions of the coefficients and should thus evolve in a polynomial fashion. One can compute this explicitly in simple cases. For instance, the center of mass is an invariant: $\displaystyle \partial_t \frac{1}{n} \sum_i x_i = 0.$ The variance decreases linearly: Exercise 2 Establish the virial identity $\displaystyle \partial_t \sum_{i,j} (x_i-x_j)^2 = - 4n^2(n-1).$ As the variance (which is proportional to ${\sum_{i,j} (x_i-x_j)^2}$) cannot become negative, this identity shows that “finite time blowup” must occur – that the zeroes must collide at or before the time ${\frac{1}{4n^2(n-1)} \sum_{i,j} (x_i-x_j)^2}$. Exercise 3 Show that the Stieltjes transform $\displaystyle s(t,z) = \sum_i \frac{1}{x_i - z}$ solves the viscous Burgers equation $\displaystyle \partial_t s = \partial_{zz} s - 2 s \partial_z s,$ either by using the original heat equation (2) and the identity ${s = - \partial_z P / P}$, or else by using the equations of motion (3). This relation between the Burgers equation and the heat equation is known as the Cole-Hopf transformation. The paper of Csordas, Smith, and Varga mentioned previously gives some other bounds on the lifespan of the dynamics; roughly speaking, they show that if there is one pair of zeroes that are much closer to each other than to the other zeroes then they must collide in a short amount of time (unless there is a collision occuring even earlier at some other location). Their argument extends also to situations where there are an infinite number of zeroes, which they apply to get new results on Newman’s conjecture in analytic number theory. I would be curious to know of further places in the literature where this dynamics has been studied. Let ${X}$ and ${Y}$ be two random variables taking values in the same (discrete) range ${R}$, and let ${E}$ be some subset of ${R}$, which we think of as the set of “bad” outcomes for either ${X}$ or ${Y}$. If ${X}$ and ${Y}$ have the same probability distribution, then clearly $\displaystyle {\bf P}( X \in E ) = {\bf P}( Y \in E ).$ In particular, if it is rare for ${Y}$ to lie in ${E}$, then it is also rare for ${X}$ to lie in ${E}$. If ${X}$ and ${Y}$ do not have exactly the same probability distribution, but their probability distributions are close to each other in some sense, then we can expect to have an approximate version of the above statement. For instance, from the definition of the total variation distance ${\delta(X,Y)}$ between two random variables (or more precisely, the total variation distance between the probability distributions of two random variables), we see that $\displaystyle {\bf P}(Y \in E) - \delta(X,Y) \leq {\bf P}(X \in E) \leq {\bf P}(Y \in E) + \delta(X,Y) \ \ \ \ \ (1)$ for any ${E \subset R}$. In particular, if it is rare for ${Y}$ to lie in ${E}$, and ${X,Y}$ are close in total variation, then it is also rare for ${X}$ to lie in ${E}$. A basic inequality in information theory is Pinsker’s inequality $\displaystyle \delta(X,Y) \leq \sqrt{\frac{1}{2} D_{KL}(X\|\|Y)}$ where the Kullback-Leibler divergence ${D_{KL}(X\|\|Y)}$ is defined by the formula $\displaystyle D_{KL}(X\|\|Y) = \sum_{x \in R} {\bf P}( X=x ) \log \frac{{\bf P}(X=x)}{{\bf P}(Y=x)}.$ (See this previous blog post for a proof of this inequality.) A standard application of Jensen’s inequality reveals that ${D_{KL}(X\|\|Y)}$ is non-negative (Gibbs’ inequality), and vanishes if and only if ${X}$, ${Y}$ have the same distribution; thus one can think of ${D_{KL}(X\|\|Y)}$ as a measure of how close the distributions of ${X}$ and ${Y}$ are to each other, although one should caution that this is not a symmetric notion of distance, as ${D_{KL}(X\|\|Y) \neq D_{KL}(Y\|\|X)}$ in general. Inserting Pinsker’s inequality into (1), we see for instance that $\displaystyle {\bf P}(X \in E) \leq {\bf P}(Y \in E) + \sqrt{\frac{1}{2} D_{KL}(X\|\|Y)}.$ Thus, if ${X}$ is close to ${Y}$ in the Kullback-Leibler sense, and it is rare for ${Y}$ to lie in ${E}$, then it is rare for ${X}$ to lie in ${E}$ as well. We can specialise this inequality to the case when ${Y}$ a uniform random variable ${U}$ on a finite range ${R}$ of some cardinality ${N}$, in which case the Kullback-Leibler divergence ${D_{KL}(X\|\|U)}$ simplifies to $\displaystyle D_{KL}(X\|\|U) = \log N - {\bf H}(X)$ where $\displaystyle {\bf H}(X) := \sum_{x \in R} {\bf P}(X=x) \log \frac{1}{{\bf P}(X=x)}$ is the Shannon entropy of ${X}$. Again, a routine application of Jensen’s inequality shows that ${{\bf H}(X) \leq \log N}$, with equality if and only if ${X}$ is uniformly distributed on ${R}$. The above inequality then becomes $\displaystyle {\bf P}(X \in E) \leq {\bf P}(U \in E) + \sqrt{\frac{1}{2}(\log N - {\bf H}(X))}. \ \ \ \ \ (2)$ Thus, if ${E}$ is a small fraction of ${R}$ (so that it is rare for ${U}$ to lie in ${E}$), and the entropy of ${X}$ is very close to the maximum possible value of ${\log N}$, then it is rare for ${X}$ to lie in ${E}$ also. The inequality (2) is only useful when the entropy ${{\bf H}(X)}$ is close to ${\log N}$ in the sense that ${{\bf H}(X) = \log N - O(1)}$, otherwise the bound is worse than the trivial bound of ${{\bf P}(X \in E) \leq 1}$. In my recent paper on the Chowla and Elliott conjectures, I ended up using a variant of (2) which was still non-trivial when the entropy ${{\bf H}(X)}$ was allowed to be smaller than ${\log N - O(1)}$. More precisely, I used the following simple inequality, which is implicit in the arguments of that paper but which I would like to make more explicit in this post: Lemma 1 (Pinsker-type inequality) Let ${X}$ be a random variable taking values in a finite range ${R}$ of cardinality ${N}$, let ${U}$ be a uniformly distributed random variable in ${R}$, and let ${E}$ be a subset of ${R}$. Then $\displaystyle {\bf P}(X \in E) \leq \frac{(\log N - {\bf H}(X)) + \log 2}{\log 1/{\bf P}(U \in E)}.$ Proof: Consider the conditional entropy ${{\bf H}(X \| 1_{X \in E} )}$. On the one hand, we have $\displaystyle {\bf H}(X \| 1_{X \in E} ) = {\bf H}(X, 1_{X \in E}) - {\bf H}(1_{X \in E} )$ $\displaystyle = {\bf H}(X) - {\bf H}(1_{X \in E})$ $\displaystyle \geq {\bf H}(X) - \log 2$ by Jensen’s inequality. On the other hand, one has $\displaystyle {\bf H}(X \| 1_{X \in E} ) = {\bf P}(X \in E) {\bf H}(X \| X \in E )$ $\displaystyle + (1-{\bf P}(X \in E)) {\bf H}(X \| X \not \in E)$ $\displaystyle \leq {\bf P}(X \in E) \log \|E\| + (1-{\bf P}(X \in E)) \log N$ $\displaystyle = \log N - {\bf P}(X \in E) \log \frac{N}{\|E\|}$ $\displaystyle = \log N - {\bf P}(X \in E) \log \frac{1}{{\bf P}(U \in E)},$ where we have again used Jensen’s inequality. Putting the two inequalities together, we obtain the claim. $\Box$ Remark 2 As noted in comments, this inequality can be viewed as a special case of the more general inequality $\displaystyle {\bf P}(X \in E) \leq \frac{D(X\|\|Y) + \log 2}{\log 1/{\bf P}(Y \in E)}$ for arbitrary random variables ${X,Y}$ taking values in the same discrete range ${R}$, which follows from the data processing inequality $\displaystyle D( f(X)\|\|f(Y)) \leq D(X\|\| Y)$ for arbitrary functions ${f}$, applied to the indicator function ${f = 1_E}$. Indeed one has $\displaystyle D( 1_E(X) \|\| 1_E(Y) ) = {\bf P}(X \in E) \log \frac{{\bf P}(X \in E)}{{\bf P}(Y \in E)}$ $\displaystyle + {\bf P}(X \not \in E) \log \frac{{\bf P}(X \not \in E)}{{\bf P}(Y \not \in E)}$ $\displaystyle \geq {\bf P}(X \in E) \log \frac{1}{{\bf P}(Y \in E)} - h( {\bf P}(X \in E) )$ $\displaystyle \geq {\bf P}(X \in E) \log \frac{1}{{\bf P}(Y \in E)} - \log 2$ where ${h(u) := u \log \frac{1}{u} + (1-u) \log \frac{1}{1-u}}$ is the entropy function. Thus, for instance, if one has $\displaystyle {\bf H}(X) \geq \log N - o(K)$ and $\displaystyle {\bf P}(U \in E) \leq \exp( - K )$ for some ${K}$ much larger than ${1}$ (so that ${1/K = o(1)}$), then $\displaystyle {\bf P}(X \in E) = o(1).$ More informally: if the entropy of ${X}$ is somewhat close to the maximum possible value of ${\log N}$, and it is exponentially rare for a uniform variable to lie in ${E}$, then it is still somewhat rare for ${X}$ to lie in ${E}$. The estimate given is close to sharp in this regime, as can be seen by calculating the entropy of a random variable ${X}$ which is uniformly distributed inside a small set ${E}$ with some probability ${p}$ and uniformly distributed outside of ${E}$ with probability ${1-p}$, for some parameter ${0 \leq p \leq 1}$. It turns out that the above lemma combines well with concentration of measure estimates; in my paper, I used one of the simplest such estimates, namely Hoeffding’s inequality, but there are of course many other estimates of this type (see e.g. this previous blog post for some others). Roughly speaking, concentration of measure inequalities allow one to make approximations such as $\displaystyle F(U) \approx {\bf E} F(U)$ with exponentially high probability, where ${U}$ is a uniform distribution and ${F}$ is some reasonable function of ${U}$. Combining this with the above lemma, we can then obtain approximations of the form $\displaystyle F(X) \approx {\bf E} F(U) \ \ \ \ \ (3)$ with somewhat high probability, if the entropy of ${X}$ is somewhat close to maximum. This observation, combined with an “entropy decrement argument” that allowed one to arrive at a situation in which the relevant random variable ${X}$ did have a near-maximum entropy, is the key new idea in my recent paper; for instance, one can use the approximation (3) to obtain an approximation of the form $\displaystyle \sum_{j=1}^H \sum_{p \in {\mathcal P}} \lambda(n+j) \lambda(n+j+p) 1_{p\|n+j}$ $\displaystyle \approx \sum_{j=1}^H \sum_{p \in {\mathcal P}} \frac{\lambda(n+j) \lambda(n+j+p)}{p}$ for “most” choices of ${n}$ and a suitable choice of ${H}$ (with the latter being provided by the entropy decrement argument). The left-hand side is tied to Chowla-type sums such as ${\sum_{n \leq x} \frac{\lambda(n)\lambda(n+1)}{n}}$ through the multiplicativity of ${\lambda}$, while the right-hand side, being a linear correlation involving two parameters ${j,p}$ rather than just one, has “finite complexity” and can be treated by existing techniques such as the Hardy-Littlewood circle method. One could hope that one could similarly use approximations such as (3) in other problems in analytic number theory or combinatorics. There are many situations in combinatorics in which one is running some sort of iteration algorithm to continually “improve” some object ${A}$; each loop of the algorithm replaces ${A}$ with some better version ${A'}$ of itself, until some desired property of ${A}$ is attained and the algorithm halts. In order for such arguments to yield a useful conclusion, it is often necessary that the algorithm halts in a finite amount of time, or (even better), in a bounded amount of time. (In general, one cannot use infinitary iteration tools, such as transfinite induction or Zorn’s lemma, in combinatorial settings, because the iteration processes used to improve some target object ${A}$ often degrade some other finitary quantity ${B}$ in the process, and an infinite iteration would then have the undesirable effect of making ${B}$ infinite.) A basic strategy to ensure termination of an algorithm is to exploit a monotonicity property, or more precisely to show that some key quantity keeps increasing (or keeps decreasing) with each loop of the algorithm, while simultaneously staying bounded. (Or, as the economist Herbert Stein was fond of saying, “If something cannot go on forever, it must stop.”) Here are four common flavours of this monotonicity strategy: • The mass increment argument. This is perhaps the most familiar way to ensure termination: make each improved object ${A'}$ “heavier” than the previous one ${A}$ by some non-trivial amount (e.g. by ensuring that the cardinality of ${A'}$ is strictly greater than that of ${A}$, thus ${\|A'\| \geq \|A\|+1}$). Dually, one can try to force the amount of “mass” remaining “outside” of ${A}$ in some sense to decrease at every stage of the iteration. If there is a good upper bound on the “mass” of ${A}$ that stays essentially fixed throughout the iteration process, and a lower bound on the mass increment at each stage, then the argument terminates. Many “greedy algorithm” arguments are of this type. The proof of the Hahn decomposition theorem in measure theory also falls into this category. The general strategy here is to keep looking for useful pieces of mass outside of ${A}$, and add them to ${A}$ to form ${A'}$, thus exploiting the additivity properties of mass. Eventually no further usable mass remains to be added (i.e. ${A}$ is maximal in some ${L^1}$ sense), and this should force some desirable property on ${A}$. • The density increment argument. This is a variant of the mass increment argument, in which one increments the “density” of ${A}$ rather than the “mass”. For instance, ${A}$ might be contained in some ambient space ${P}$, and one seeks to improve ${A}$ to ${A'}$ (and ${P}$ to ${P'}$) in such a way that the density of the new object in the new ambient space is better than that of the previous object (e.g. ${\|A'\|/\|P'\| \geq \|A\|/\|P\| + c}$ for some ${c>0}$). On the other hand, the density of ${A}$ is clearly bounded above by ${1}$. As long as one has a sufficiently good lower bound on the density increment at each stage, one can conclude an upper bound on the number of iterations in the algorithm. The prototypical example of this is Roth’s proof of his theorem that every set of integers of positive upper density contains an arithmetic progression of length three. The general strategy here is to keep looking for useful density fluctuations inside ${A}$, and then “zoom in” to a region of increased density by reducing ${A}$ and ${P}$ appropriately. Eventually no further usable density fluctuation remains (i.e. ${A}$ is uniformly distributed), and this should force some desirable property on ${A}$. • The energy increment argument. This is an “${L^2}$” analogue of the “${L^1}$“-based mass increment argument (or the “${L^\infty}$“-based density increment argument), in which one seeks to increments the amount of “energy” that ${A}$ captures from some reference object ${X}$, or (equivalently) to decrement the amount of energy of ${X}$ which is still “orthogonal” to ${A}$. Here ${A}$ and ${X}$ are related somehow to a Hilbert space, and the energy involves the norm on that space. A classic example of this type of argument is the existence of orthogonal projections onto closed subspaces of a Hilbert space; this leads among other things to the construction of conditional expectation in measure theory, which then underlies a number of arguments in ergodic theory, as discussed for instance in this earlier blog post. Another basic example is the standard proof of the Szemerédi regularity lemma (where the “energy” is often referred to as the “index”). These examples are related; see this blog post for further discussion. The general strategy here is to keep looking for useful pieces of energy orthogonal to ${A}$, and add them to ${A}$ to form ${A'}$, thus exploiting square-additivity properties of energy, such as Pythagoras’ theorem. Eventually, no further usable energy outside of ${A}$ remains to be added (i.e. ${A}$ is maximal in some ${L^2}$ sense), and this should force some desirable property on ${A}$. • The rank reduction argument. Here, one seeks to make each new object ${A'}$ to have a lower “rank”, “dimension”, or “order” than the previous one. A classic example here is the proof of the linear algebra fact that given any finite set of vectors, there exists a linearly independent subset which spans the same subspace; the proof of the more general Steinitz exchange lemma is in the same spirit. The general strategy here is to keep looking for “collisions” or “dependencies” within ${A}$, and use them to collapse ${A}$ to an object ${A'}$ of lower rank. Eventually, no further usable collisions within ${A}$ remain, and this should force some desirable property on ${A}$. Much of my own work in additive combinatorics relies heavily on at least one of these types of arguments (and, in some cases, on a nested combination of two or more of them). Many arguments in nonlinear partial differential equations also have a similar flavour, relying on various monotonicity formulae for solutions to such equations, though the objective in PDE is usually slightly different, in that one wants to keep control of a solution as one approaches a singularity (or as some time or space coordinate goes off to infinity), rather than to ensure termination of an algorithm. (On the other hand, many arguments in the theory of concentration compactness, which is used heavily in PDE, does have the same algorithm-terminating flavour as the combinatorial arguments; see this earlier blog post for more discussion.) Recently, a new species of monotonicity argument was introduced by Moser, as the primary tool in his elegant new proof of the Lovász local lemma. This argument could be dubbed an entropy compression argument, and only applies to probabilistic algorithms which require a certain collection ${R}$ of random “bits” or other random choices as part of the input, thus each loop of the algorithm takes an object ${A}$ (which may also have been generated randomly) and some portion of the random string ${R}$ to (deterministically) create a better object ${A'}$ (and a shorter random string ${R'}$, formed by throwing away those bits of ${R}$ that were used in the loop). The key point is to design the algorithm to be partially reversible, in the sense that given ${A'}$ and ${R'}$ and some additional data ${H'}$ that logs the cumulative history of the algorithm up to this point, one can reconstruct ${A}$ together with the remaining portion ${R}$ not already contained in ${R'}$. Thus, each stage of the argument compresses the information-theoretic content of the string ${A+R}$ into the string ${A'+R'+H'}$ in a lossless fashion. However, a random variable such as ${A+R}$ cannot be compressed losslessly into a string of expected size smaller than the Shannon entropy of that variable. Thus, if one has a good lower bound on the entropy of ${A+R}$, and if the length of ${A'+R'+H'}$ is significantly less than that of ${A+R}$ (i.e. we need the marginal growth in the length of the history file ${H'}$ per iteration to be less than the marginal amount of randomness used per iteration), then there is a limit as to how many times the algorithm can be run, much as there is a limit as to how many times a random data file can be compressed before no further length reduction occurs. It is interesting to compare this method with the ones discussed earlier. In the previous methods, the failure of the algorithm to halt led to a new iteration of the object ${A}$ which was “heavier”, “denser”, captured more “energy”, or “lower rank” than the previous instance of ${A}$. Here, the failure of the algorithm to halt leads to new information that can be used to “compress” ${A}$ (or more precisely, the full state ${A+R}$) into a smaller amount of space. I don’t know yet of any application of this new type of termination strategy to the fields I work in, but one could imagine that it could eventually be of use (perhaps to show that solutions to PDE with sufficiently “random” initial data can avoid singularity formation?), so I thought I would discuss it here. Below the fold I give a special case of Moser’s argument, based on a blog post of Lance Fortnow on this topic. As many readers may already know, my good friend and fellow mathematical blogger Tim Gowers, having wrapped up work on the Princeton Companion to Mathematics (which I believe is now in press), has begun another mathematical initiative, namely a “Tricks Wiki” to act as a repository for mathematical tricks and techniques. Tim has already started the ball rolling with several seed articles on his own blog, and asked me to also contribute some articles. (As I understand it, these articles will be migrated to the Wiki in a few months, once it is fully set up, and then they will evolve with edits and contributions by anyone who wishes to pitch in, in the spirit of Wikipedia; in particular, articles are not intended to be permanently authored or signed by any single contributor.) So today I’d like to start by extracting some material from an old post of mine on “Amplification, arbitrage, and the tensor power trick” (as well as from some of the comments), and converting it to the Tricks Wiki format, while also taking the opportunity to add a few more examples. Title: The tensor power trick Quick description: If one wants to prove an inequality $X \leq Y$ for some non-negative quantities X, Y, but can only see how to prove a quasi-inequality $X \leq CY$ that loses a multiplicative constant C, then try to replace all objects involved in the problem by “tensor powers” of themselves and apply the quasi-inequality to those powers. If all goes well, one can show that $X^M \leq C Y^M$ for all $M \geq 1$, with a constant C which is independent of M, which implies that $X \leq Y$ as desired by taking $M^{th}$ roots and then taking limits as $M \to \infty$. Having established the monotonicity of the Perelman reduced volume in the previous lecture (after first heuristically justifying this monotonicity in Lecture 9), we now show how this can be used to establish $\kappa$-noncollapsing of Ricci flows, thus giving a second proof of Theorem 2 from Lecture 7. Of course, we already proved (a stronger version) of this theorem already in Lecture 8, using the Perelman entropy, but this second proof is also important, because the reduced volume is a more localised quantity (due to the weight $e^{-l_{(0,x_0)}}$ in its definition and so one can in fact establish local versions of the non-collapsing theorem which turn out to be important when we study ancient $\kappa$-noncollapsing solutions later in Perelman’s proof, because such solutions need not be compact and so cannot be controlled by global quantities (such as the Perelman entropy). The route to $\kappa$-noncollapsing via reduced volume proceeds by the following scheme: Non-collapsing at time t=0 (1) $\Downarrow$ Large reduced volume at time t=0 (2) $\Downarrow$ Large reduced volume at later times t (3) $\Downarrow$ Non-collapsing at later times t (4) The implication $(2) \implies (3)$ is the monotonicity of Perelman reduced volume. In this lecture we discuss the other two implications $(1) \implies (2)$, and $(3) \implies (4)$). Our arguments here are based on Perelman’s first paper, Kleiner-Lott’s notes, and Morgan-Tian’s book, though the material in the Morgan-Tian book differs in some key respects from the other two texts. A closely related presentation of these topics also appears in the paper of Cao-Zhu. It occurred to me recently that the mathematical blog medium may be a good venue not just for expository “short stories” on mathematical concepts or results, but also for more technical discussions of individual mathematical “tricks”, which would otherwise not be significant enough to warrant a publication-length (and publication-quality) article. So I thought today that I would discuss the amplification trick in harmonic analysis and combinatorics (and in particular, in the study of estimates); this trick takes an established estimate involving an arbitrary object (such as a function f), and obtains a stronger (or amplified) estimate by transforming the object in a well-chosen manner (often involving some new parameters) into a new object, applying the estimate to that new object, and seeing what that estimate says about the original object (after optimising the parameters or taking a limit). The amplification trick works particularly well for estimates which enjoy some sort of symmetry on one side of the estimate that is not represented on the other side; indeed, it can be viewed as a way to “arbitrage” differing amounts of symmetry between the left- and right-hand sides of an estimate. It can also be used in the contrapositive, amplifying a weak counterexample to an estimate into a strong counterexample. This trick also sheds some light as to why dimensional analysis works; an estimate which is not dimensionally consistent can often be amplified into a stronger estimate which is dimensionally consistent; in many cases, this new estimate is so strong that it cannot in fact be true, and thus dimensionally inconsistent inequalities tend to be either false or inefficient, which is why we rarely see them. (More generally, any inequality on which a group acts on either the left or right-hand side can often be “decomposed” into the “isotypic components” of the group action, either by the amplification trick or by other related tools, such as Fourier analysis.) The amplification trick is a deceptively simple one, but it can become particularly powerful when one is arbitraging an unintuitive symmetry, such as symmetry under tensor powers. Indeed, the “tensor power trick”, which can eliminate constants and even logarithms in an almost magical manner, can lead to some interesting proofs of sharp inequalities, which are difficult to establish by more direct means. The most familiar example of the amplification trick in action is probably the textbook proof of the Cauchy-Schwarz inequality $\|\langle v, w \rangle\| \leq \\|v\\| \\|w\\|$ (1) for vectors v, w in a complex Hilbert space. To prove this inequality, one might start by exploiting the obvious inequality $\\|v-w\\|^2 \geq 0$ (2) but after expanding everything out, one only gets the weaker inequality $\hbox{Re} \langle v, w \rangle \leq \frac{1}{2} \\|v\\|^2 + \frac{1}{2} \\|w\\|^2$. (3) Now (3) is weaker than (1) for two reasons; the left-hand side is smaller, and the right-hand side is larger (thanks to the arithmetic mean-geometric mean inequality). However, we can amplify (3) by arbitraging some symmetry imbalances. Firstly, observe that the phase rotation symmetry $v \mapsto e^{i\theta} v$ preserves the RHS of (3) but not the LHS. We exploit this by replacing v by $e^{i\theta} v$ in (3) for some phase $\theta$ to be chosen later, to obtain $\hbox{Re} e^{i\theta} \langle v, w \rangle \leq \frac{1}{2} \\|v\\|^2 + \frac{1}{2} \\|w\\|^2$. Now we are free to choose $\theta$ at will (as long as it is real, of course), so it is natural to choose $\theta$ to optimise the inequality, which in this case means to make the left-hand side as large as possible. This is achieved by choosing $e^{i\theta}$ to cancel the phase of $\langle v, w \rangle$, and we obtain $\|\langle v, w \rangle\| \leq \frac{1}{2} \\|v\\|^2 + \frac{1}{2} \\|w\\|^2$ (4) This is closer to (1); we have fixed the left-hand side, but the right-hand side is still too weak. But we can amplify further, by exploiting an imbalance in a different symmetry, namely the homogenisation symmetry $(v,w) \mapsto (\lambda v, \frac{1}{\lambda} w)$ for a scalar $\lambda > 0$, which preserves the left-hand side but not the right. Inserting this transform into (4) we conclude that $\|\langle v, w \rangle\| \leq \frac{\lambda^2}{2} \\|v\\|^2 + \frac{1}{2\lambda^2} \\|w\\|^2$ where $\lambda > 0$ is at our disposal to choose. We can optimise in $\lambda$ by minimising the right-hand side, and indeed one easily sees that the minimum (or infimum, if one of v and w vanishes) is $\\|v\\| \\|w\\|$ (which is achieved when $\lambda = \sqrt{\\|w\\|/\\|v\\|}$ when $v,w$ are non-zero, or in an asymptotic limit $\lambda \to 0$ or $\lambda \to \infty$ in the degenerate cases), and so we have amplified our way to the Cauchy-Schwarz inequality (1). [See also this discussion by Tim Gowers on the Cauchy-Schwarz inequality.] [This post is authored by Gil Kalai, who has kindly “guest blogged” this week’s “open problem of the week”. – T.] The entropy-influence conjecture seeks to relate two somewhat different measures as to how a boolean function has concentrated Fourier coefficients, namely the total influence and the entropy. We begin by defining the total influence. Let $\{-1,+1\}^n$ be the discrete cube, i.e. the set of $\pm 1$ vectors $(x_1,\ldots,x_n)$ of length n. A boolean function is any function $f: \{-1,+1\}^n \to \{-1,+1\}$ from the discrete cube to {-1,+1}. One can think of such functions as “voting methods”, which take the preferences of n voters (+1 for yes, -1 for no) as input and return a yes/no verdict as output. For instance, if n is odd, the “majority vote” function $\hbox{sgn}(x_1+\ldots+x_n)$ returns +1 if there are more +1 variables than -1, or -1 otherwise, whereas if $1 \leq k \leq n$, the “$k^{th}$ dictator” function returns the value $x_k$ of the $k^{th}$ variable. We give the cube $\{-1,+1\}^n$ the uniform probability measure $\mu$ (thus we assume that the n voters vote randomly and independently). Given any boolean function f and any variable $1 \leq k \leq n$, define the influence $I_k(f)$ of the $k^{th}$ variable to be the quantity $I_k(f) := \mu \{ x \in \{-1,+1\}^n: f(\sigma_k(x)) \neq f(x) \}$ where $\sigma_k(x)$ is the element of the cube formed by flipping the sign of the $k^{th}$ variable. Informally, $I_k(f)$ measures the probability that the $k^{th}$ voter could actually determine the outcome of an election; it is sometimes referred to as the Banzhaf power index. The total influence I(f) of f (also known as the average sensitivity and the edge-boundary density) is then defined as $I(f) := \sum_{k=1}^n I_k(f).$ Thus for instance a dictator function has total influence 1, whereas majority vote has total influence comparable to $\sqrt{n}$. The influence can range between 0 (for constant functions +1, -1) and n (for the parity function $x_1 \ldots x_k$ or its negation). If f has mean zero (i.e. it is equal to +1 half of the time), then the edge-isoperimetric inequality asserts that $I(f) \geq 1$ (with equality if and only if there is a dictatorship), whilst the Kahn-Kalai-Linial (KKL) theorem asserts that $I_k(f) \gg \frac{\log n}{n}$ for some k. There is a result of Friedgut that if $I(f)$ is bounded by A (say) and $\varepsilon > 0$, then f is within a distance $\varepsilon$ (in $L^1$ norm) of another boolean function g which only depends on $O_{A,\varepsilon}(1)$ of the variables (such functions are known as juntas). [This post is authored by Gil Kalai, who has kindly “guest blogged” this week’s “open problem of the week”. – T.] This is a problem in discrete and convex geometry. It seeks to quantify the intuitively obvious fact that large convex bodies are so “fat” that they cannot avoid “detection” by a small number of observation points. More precisely, we fix a dimension d and make the following definition (introduced by Haussler and Welzl): • Definition: Let $X \subset {\Bbb R}^d$ be a finite set of points, and let $0 < \epsilon < 1$. We say that a finite set $Y \subset {\Bbb R}^d$ is a weak $\epsilon$-net for X (with respect to convex bodies) if, whenever B is a convex body which is large in the sense that $\|B \cap X\| > \epsilon \|X\|$, then B contains at least one point of Y. (If Y is contained in X, we say that Y is a strong $\epsilon$-net for X with respect to convex bodies.) For example, in one dimension, if $X = \{1,\ldots,N\}$, and $Y = \{ \epsilon N, 2 \epsilon N, \ldots, k \epsilon N \}$ where k is the integer part of $1/\epsilon$, then Y is a weak $\epsilon$-net for X with respect to convex bodies. Thus we see that even when the original set X is very large, one can create a $\epsilon$-net of size as small as $O(1/\epsilon)$. Strong $\epsilon$-nets are of importance in computational learning theory, and are fairly well understood via Vapnik-Chervonenkis (or VC) theory; however, the theory of weak $\epsilon$-nets is still not completely satisfactory. One can ask what happens in higher dimensions, for instance when X is a discrete cube $X = \{1,\ldots,N\}^d$. It is not too hard to cook up $\epsilon$-nets of size $O_d(1/\epsilon^d)$ (by using tools such as Minkowski’s theorem), but in fact one can create $\epsilon$-nets of size as small as $O( \frac{1}{\epsilon} \log \frac{1}{\epsilon} )$ simply by taking a random subset of X of this cardinality and observing that “up to errors of $\epsilon$“, the total number of essentially different ways a convex body can meet X grows at most polynomially in $1/\epsilon$. (This is a very typical application of the probabilistic method.) On the other hand, since X can contain roughly $1/\epsilon$ disjoint convex bodies, each of which contains at least $\epsilon$ of the points in X, we see that no $\epsilon$-net can have size much smaller than $1/\epsilon$. Now consider the situation in which X is now an arbitrary finite set, rather than a discrete cube. More precisely, let $f(\epsilon,d)$ be the least number such that every finite set X possesses at least one weak $\epsilon$-net for X with respect to convex bodies of cardinality at most $f(\epsilon,d)$. (One can also replace the finite set X with an arbitrary probability measure; the two formulations are equivalent.) Informally, f is the least number of “guards” one needs to place to prevent a convex body from covering more than $\epsilon$ of any given territory. • Problem 1: For fixed d, what is the correct rate of growth of f as $\epsilon \to 0$?	2022-07-05 12:42:21	{"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": 671, "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": 0.9129999279975891, "perplexity": 184.22049929000477}, "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-2022-27/segments/1656104576719.83/warc/CC-MAIN-20220705113756-20220705143756-00538.warc.gz"}
http://iforest.sisef.org/contents/?id=ifor0721-006	vol. 6, pp. 23-29 Copyright © 2013 by the Italian Society of Silviculture and Forest Ecology doi: 10.3832/ifor0721-006 Collection: IUFRO 7.01.00 - COST Action FP0903, Kaunas (Lithuania - 2012) “Biological Reactions of Forest to Climate Change and Air Pollution” Guest Editors: Elena Paoletti, Andrzej Bytnerowicz, Algirdas Augustaitis Research Articles # Predicting tree crown defoliation using color-infrared orthophoto maps M. Eigirdas (1), A. Augustaitis (2), G. Mozgeris (3) # Introduction Lithuania has a long history of using airborne remote sensing in forest inventories ([4], [19]). Since 1950, the inventory has been based on a combination of aerial photograph interpretation and conventional fieldwork. In the 1980s and 1990s, aerial photographs were available for 85 % of the inventoried area ([5]). The introduction of Geographic Information Systems (GIS) to the stand-wise forest inventory in 1995 required new solutions so that the geometrical quality of spatial databases could be improved. This meant that, starting from 1996, panchromatic orthophotographic maps were introduced into the stand-wise forest inventory to replace the aerial photographs ([19]). Since 2002, orthophotos, based on color-infrared (CIR) aerial photography, have been produced every year for approximately 15 % of the country’s area ([20]). Technical specifications for the orthophotos were strictly followed and have changed little during the last decade, which makes the orthophoto material taken in different years and for different areas easily compatible. The following technological approaches have been used: color-infrared aerial photography in the summer season, based on a ground sampling density of 0.5 m and radiometric enhancement to facilitate tree species separation ability. However, (i) film aerial photography (Kodak Aerochrome III Infrared 1443 aerial films) and development of the films into negatives using the Kodak AN5 processing technique was used up to 2007 and (ii) since 2007 it was replaced by the digital Vexcel aerial cameras. There have been many studies carried out in the country dealing with the usability of orthophotos for interpreting forest characteristics in conventional stand-wise inventories and they have mainly focused on the use of the visual analytical interpretation method ([8], [10], [18]). The potential to detect crown defoliation classes corresponding to the ones used in the European International Co-operative Program on Assessment and Monitoring of Air Pollution Effects on Forests (ICP Forests - [28]) was tested in Lithuania using specially acquired test aerial images ([7], [2]. However, the potential use of standardized color-infrared aerial photography based orthophotos produced for Lithuanian forest inventories has not been tested so far. This study aimed to fill this gap. There are many examples of the successful use of airborne remote sensing to evaluate forest or tree health conditions ([14], [29], [9]). However, most of the research or operational applications have been based on human or visual interpretation. Tree crown defoliation is considered to be a key indicator of forest health ([23]. Visual comparison of tree crown density with the density of a conditionally healthy tree remains the main method used to estimate tree crown defoliation. However, such an approach in the assessments is affected by subjectivity to some extent ([12], [2], [11]) and it is costly and time consuming, particularly if the aim is the full spatial sampling ([17]). Automated assessment of individual tree crown characteristics appeared on the research agenda only with the introduction of very high resolution digital airborne or satellite images. Today, the state-of-the-art of laser scanning and hyperspectral imaging in forest health assessment has become important ([25], [16], [24], [3]). Automated or semi-automated assessments of forest inventories oriented towards individual tree or forest stand characteristics already exist at the operational level, so it is important to test whether it is possible to provide forest health statistics using similar methodology. Detection and mapping of forest health is important when creating treatment plans and correcting growth estimates. There are also more general ecological benefits. The non-parametric k-Nearest Neighbor (kNN) method ([26]), which originated from studies on the remote sensing based assessment of tree and forest stand dendrometric characteristics, was investigated in this study as a potential method for predicting tree crown defoliation. The idea behind kNN is that “field measurements in a certain area can also be made use of in neighboring areas by employing a relevant extrapolating, or “information borrowing” technique” ([27]). In this study, the “area” is the tree crown, with unknown and field-estimated defoliations. Conventional Lithuanian forest inventory color-infrared aerial photography based orthophotos were used as the source of the remotely sensed information. # Material and methods The study was carried-out in Aukštaitija National Park (ANP), located in eastern Lithuania. Coniferous forests prevail in ANP (79 %) and the average age is around 60 years. However, there are some stands present that are over 200 years old. Tree crown defoliation was assessed in 119 permanent sample plots distributed in forest compartments in the northern part of ANP and 47 sample plots located at the Aukštaitija Integrated Monitoring Station (AIMS - Fig. 1). Fig. 1 - Location of the study area: (a) whole country; (b) Aukštaitija National Park; (c) sample plots. A field survey of the sample plots was carried out in the summer of 2008 following the methodology of ICP Forests ([28]), which was modified to meet the aims of the current study. First, the coordinates of the sample plot were determined using a Trimble Pathfinder ProXR GPS and the location of each tree was calculated based on measurements of the azimuth and the distance of the tree to the sample plot center. Basic dendrometric tree characteristics were determined along with the tree development class and tree crown defoliation. The defoliation was assessed visually using a 5% gradation by two experienced specialists that have evaluated defoliation annually for the same trees for over 10 years. Historical records on the defoliation estimates for all trees under investigation were made available. Orthophotos, based on color-infrared aerial photography, were also made available for the study. The whole study area was displayed on map sheet 8649, according to the mapsheet division of the Lithuanian coordinate system, LKS94. Aerial photography was carried out in the summer of 2008 using a frame Vexcel UltraCam D camera with image format of 11500 x 7500 pixels. Even the camera was capturing RGB and CIR images simultaneously, only near-infrared, red and green bands were used to produce the orthophotos. A Rockwell Turbo Commander 690A aircraft was used to carry out the flight, at an altitude of approximately 5800 m. Flight were done following the north-south direction. The camera unit was equipped with an Applanix POS/AV 510 GPS/INS system. All six exterior orientation parameters were calculated using GPS/INS data and POSPAC® software from Applanix. Raw image data from the camera were processed into final TIFF images using the Office Processing Centre® software from Vexcel Imaging by Blom Kartta Oy, Finland. The aerial triangulation was carried out as an automatic digital aerial triangulation using MATCH-AT® software. The digital terrain model (DTM) was produced using MATCH-T software and autocorrelation technique with additional break lines. The autocorrelation model was later edited photogrammetrically. Ortho-rectification was done using the ORTHOMASTER® software. The rectified images were mosaicked into the 5 x 5 km map sheets using the ORTHO VISTA® software. The ground sampling density of the images was 0.5 m. The crown projections of field surveyed trees were identified on the orthophotos, manually digitized on a computer screen and stored in a GIS database. The identification of all tree crowns was checked in the field in 2010. In total, 1793 tree crowns were identified on the orthophotos, of which 70% of the trees were pine and 23 % were spruce. Not all field-surveyed trees were identified on the orthophotos. The following radiometric characteristics of the orthophotos corresponding to the digital values of each image band were extracted for the polygons corresponding to each crown projection using the standard ArcGIS Spatial Analyst tool Zonal Statistics: mean (the average of all cells in the value raster that belong to the same zone as the output cell), majority (the value that occurs most often of all cells in the value raster that belong to the same zone as the output cell), minimum (the smallest value of all cells in the value raster that belong to the same zone as the output cell), maximum (the largest value of all cells in the value raster that belong to the same zone as the output cell), median (the median value of all cells in the value raster that belong to the same zone as the output cell), minority (the value that occurs least often of all cells in the value raster that belong to the same zone as the output cell), range (the difference between the largest and smallest value of all cells in the value raster that belong to the same zone as the output cell), standard deviation (the standard deviation of all cells in the value raster that belong to the same zone as the output cell), sum (the total value of all cells in the value raster that belong to the same zone as the output cell) and variety (the number of unique values for all cells in the value raster that belong to the same zone as the output cell). Image bands referring to near infrared radiation were denoted as NIR, red as R and green as G. In addition to the original versions of the orthophotos, stored as 8-bit geographic matrices of digital numbers, Principal Component transformation (PC) and the creation of a Normalized Difference Vegetation Index (NDVI) were undertaken in order to reduce the dimensionality of the orthophoto images. Two types of zones were used in the study to separate sun illuminated parts from the shade parts of the crown: (i) zones corresponding to the delineated crown projection polygons (ZONES_1); and (ii) the interior area of the tree crown polygons were divided into two parts, corresponding to sun illuminated parts of the crown and (ZONES_2a) and parts that were in shadow (ZONES_2b) using unsupervised Iso Cluster classification (Fig. 2). Thus, each of the trees had its corresponding defoliation characteristic surveyed in the field and an array of characteristics extracted from the orthophoto. Fig. 2 - The types of zones used to extract image characteristics. Judgments on the suitability of using color-infrared aerial photography based orthophotos to estimate tree crown defoliation were based on the accuracy of the defoliation prediction. Defoliation was predicted at an individual crown level using the data available from orthophoto images and compared with the true, ground estimated data. The “Leave One Out” technique was used to compute the following validation statistics: prediction bias, root mean square errors (RMSE) and the correlation coefficient between the field assessed and predicted tree crown defoliation (R). The non-parametric two-phase sampling based k-Nearest Neighbor method ([26]) was used to predict tree crown defoliation. The k-nearest neighbor method, or multi-dimensional version of the inverse distance weighted interpolation technique, can be briefly described as follows: Euclidean distance in a n-dimensional feature space of auxiliary information, di,p, is calculated between each A-observation sampling unit p (the tree crown in this case) and each B-observation unit, i. The field measured forest characteristic (tree crown defoliation) is known for each B-observation unit. Here, n refers to the total number of layers of auxiliary information, i.e., characteristics extracted from the orthophoto images - all proper image characteristics for all image bands or principal components or NDVI grid, corresponding to each crown. The k distances di,p - d(1),p, ... d(k),p, (d(1),p ≤ ... ≤ d(k),p) are found and the weight is calculated as (eqn. 1): $$w_{(i),p} = \frac 1 {d^{t}_{(i),p}} / \sum_{i=1}^{k} \frac{1}{d^{t}_{(i),p}}$$ The tree crown defoliation value (M) for a tree crown, p, of A-observations equals (eqn. 2): $${\hat{m}_{p}} = \sum_{j=1}^{k} { w_{(j),p} \cdot m_{(j),p} }$$ where m(j),p, j=1,...k are the values for tree crown defoliation, M, of k nearest B-observation trees to p in n dimensional feature space. Three variants of k-NN prediction method were tested: • k-NN1: each tree was assigned the defoliation value of the nearest tree in the n-dimensional feature space of orthophoto image characteristics, actually, the value of the 2nd nearest neighbor, as the 1st neighbor was always the tree crown that was being inspected, i.e., k=1 was used; • k-NN2: each tree was assigned the average defoliation value of the 10 nearest trees in the n-dimensional feature space of orthophoto image characteristics, actually, the 2nd-11th nearest neighbors, i.e., k=10 and no distance weights were used. The value of k equal to 10 was chosen as keeping both the prediction bias and root mean square error at their lowest levels based on series of tests with the same image characteristics and changing the value of k up to 20; • k-NN3: each tree was assigned the weighted average defoliation value of the 10 nearest trees in the n-dimensional feature space of orthophoto image characteristics, i.e., k=10 and t=1 were used. Other t values did not result in prediction accuracy improvement, thus they were not discussed in this paper. Predicted and field estimated tree crown defoliation values were aggregated up to the sample plot level by taking an averaging of trees in the same sample plot. Bias, root mean square error and correlation coefficients between plot-wise average values of field estimated and predicted defoliations were used to discuss the accuracies at the sample plot level. The Most Similar Neighbor program ([6]) was used to calculate the Euclidean distances and standard GIS and statistical processing packages (ArcGIS®, STATISTICA®, MS Excel®) were used to further process and analyze the data. # Results and discussion Assigning the defoliation value of the nearest tree in the n-dimensional feature space by orthophoto image-based characteristics using the k-NN predictions produced root mean square errors that were larger by some 5-35 % compared to the other estimation methods (Tab. 1). The differences in correlation coefficients between field-estimated and predicted crown defoliations were even larger. This is in line with the general findings on the use of the k-NN method to predict basic forest characteristics at a sample plot level where satellite images are used as the auxiliary data source and where, on average, the 10 nearest neighbors are required to minimize the prediction root mean square error ([22], [26]). However, the fewer neighbors used, the better the variance in the original data is sustained ([13]). This is essential, especially when predicting extreme defoliation values, as there are usually only a few trees with no or very large defoliations. There is always some tendency to level the values predicted. The negative bias values indicated that the predicted crown defoliation values were practically always lower than the field observed values. This negative bias was mainly introduced by trees with relatively large defoliations receiving predicted values from nearest neighbors that had lower observed defoliations. There were practically no differences observed in the biases and root mean square errors when using the two different approaches to calculate the predicted defoliation on the basis of the values from the 10 nearest neighbors. The bias was statistically significant for pine trees and when original and transformed data from the principal components images were used as the auxiliary data sets. Also, the correlation coefficients between field estimated and predicted crown defoliations had practically always been statistically significant. Tab. 1 - Accuracy of crown defoliation predictions at the single tree level. (*): statistically significant bias and correlations (p ≤ 0.05). Separation of the sun illuminated and shaded parts of the crown nearly always reduced the prediction root mean square errors by an average of 4%. However, there were cases where a reduction of 15-25 % was achieved. The correlations coefficients between field-estimated and predicted crown defoliations did always increase if the image characteristics, extracted from two crown zones, were used as the auxiliary variables in the prediction. Reduction in the root mean square errors was usually followed by a slight increase in the prediction bias, which was statistically significant for pine trees when transformed images were used as the auxiliary data sets. Image transformation into principal components and the NDVI did not reduce the root mean square error, but the NDVI did reduce significantly the prediction bias for birch when it was used as the input auxiliary data source. The lowest root mean square error for predicted tree crown defoliation for pine trees was 7.564, 9.166 for spruce and 7.712 for birch. All were achieved using different k-NN prediction methods. The prediction bias for pine trees was less than the prediction bias for spruce trees. However, there was no unique prediction approach detected which would work best for all tree species. This confirms the findings coming from satellite image-based remote sensing, which suggested that there was no standardized solution for using the k-NN prediction. The best achieved results depend on the prediction objectives, the auxiliary data available, etc., so the settings need to be optimized every time the method is used ([15]). The prediction accuracies were compatible with the ones achieved using similar methodological approaches but different image data as the input. Mozgeris et al. ([21])achieved 9-11% root mean square errors in predicting pine crown defoliation using color infrared aerial images taken from ultralight aircraft in areas that partly overlapping the current study area. Unpublished results for the area used in this study indicated a similar potential for color infrared aerial images taken at larger resolutions (ground sampling density of 10-15 cm). This suggested that color-infrared orthophoto maps, produced in a standardized way for Lithuanian stand-wise forest inventories may have the same or even a higher potential to predict individual tree crown defoliation as other aerial images acquired for use in forest health assessment studies. The main aim of producing the color-infrared orthophotos is to use them within the frames of the Lithuanian stand-wise forest inventory to facilitate the delineation of forest compartments. Forest compartments are usually delineated by analyzing the basic forest stand characteristics (tree species composition, density, crown diameter in relation to the stem height and the diameter at breast height) that have been visually estimated from orthophoto imagery and comparing them to historical records from previous forest inventories ([19]). Technically, visual interpretation of tree crown defoliation could happen while delineation of forest compartments is taking place. Detection and mapping of forest health is important when creating treatment plans and for correcting growth estimates. The possibility of detecting crown defoliation classes corresponding to the UN-ECE/ ICP-Forests for pine and spruce stands using visual analytical interpretation of color-infrared aerial photographs at a scale 1:10000 in Lithuania has been investigated by Daniulis & Mozgeris ([7]). However, such solutions would require special training and could potentially suffer from some subjectivity. Another option could be the automatic prediction of the health status of forest compartments. In this study individual tree crown level defoliation predictions, achieved using orthophoto maps, were averaged to the sample plot level. The average figures were then compared with the figures obtained using field surveyed defoliation of the same trees. It was clear that not all trees were identified on the orthophotos. Only crowns that were sun illuminated when the aerial image was taken, that did not overlap other crowns and were large enough to be detectable by the sensor could be identified on the aerial images ([9]). This fact may cause some differences in average defoliation values estimated in the field and by using aerial images. Only 16 % of the trees present in the field were not detected on the orthophoto maps (Tab. 2). Only 5-10% of the pine trees were not detected on the image, regardless of the tree status in the canopy. Average defoliation detection rates for pine stands using orthophoto maps did not significantly differ from rates derived from using visual assessment methods. The average defoliation of spruce stands would be 2% less because a little more than half of the spruce trees were detected on the aerial image. However, 88% and 73% of superior and dominant spruce trees, respectively, were detected on the image but this would not have influenced average image and field estimated defoliation values. The detection of birch trees on the image was between the pine and spruce as were the differences in average field and image estimated defoliations. Tab. 2 - Percentage of trees identified on the orthophoto maps and the potential influence of tree visibility on the assessment of defoliation. As the majority of trees analyzed were located in stands that were predominantly pine, the average defoliation values were computed just for the sample plots in the pine stands. Only the defoliations predicted using image data from whole crown projection were used for subsequent analysis. The correlation coefficients between plot-wise average values of field estimated and predicted defoliations were around 0.8 (Tab. 3). The root mean square error at a sample plot level was around 3.7% regardless of the type of image transformation used. The average defoliation in the field was 15.49 %. The absolute value of bias dropped when the NDVI-transformed image was used as the auxiliary data set to predict crown defoliation. Prediction root mean square errors were lower (5-10) than those observed within the frames of previous research in the same study area that mapped sample plot level defoliation using multiple regression and numerous panchromatic aerial images, SPOT XS and GIS database variables ([1]) and was compatible with the errors achieved using small format color infrared aerial images taken from ultra-light aircraft (2-5), but with a much smaller number of sample plots ([21]). Tab. 3 - Accuracies for defoliation prediction at the sample plot level. # Conclusions The main conclusions of the research, aimed to investigate the opportunities for using the color-infrared orthophoto maps to predict tree crown defoliation at the single tree and sample plot levels, presented in this paper were: 1. Around 84 % of trees were identified on the conventional Lithuanian stand-wise forest inventory using color-infrared orthophoto maps with a ground sampling density of 50 cm. Average tree defoliation, as detected on the images, was 0.6% lower than the value of all the trees in the sample plots. 2. Individual crown level defoliations were usually underestimated using the k-nearest neighbor non-parametric prediction technique using color-infrared orthophoto map image characteristics as the auxiliary variables. The lowest root mean square error for predicted tree crown defoliation achieved for pine trees was 7.564, 9.166 for spruce and 7.712 for birch, and the highest coefficients of correlation between field estimated and predicted crown defoliations were 0.576, 0.600 and 0.386, respectively. All were achieved using different prediction methods. Prediction bias was lowest and statistically insignificant when NDVI-transformed images were used as the auxiliary variables. 3. The defoliation prediction root mean square error at the sample plot level was around 3.7%, the bias was statistically not significant and the correlation coefficients between plot-wise average values of field estimated and predicted defoliations were around 0.8 in stands that were predominantly made up of pine trees. This suggests that color-infrared orthophoto maps may be a potential data source of forest health characteristics for use in stand-wise forest inventories. # Acknowledgment The study was carried out within the framework of the National Project No. VP1-3.1-ŠMM-08-K-01-025: “Specific, genetic diversity and sustainable development of Scots pine forest to mitigate the negative effects of increased human pressure and climate change”, supported by the EU Social Fund. # References (1) Augustaitis A, Mozgeris G (2003). Cartographical modeling of tree crown defoliation. Silviculture, Transactions of Lithuanian Forest Institute and Lithuanian University of Agriculture 1 (53): 75-87. [in Lithuanian]. (2) Augustaitis A, Mozgeris G, Eigirdas M, Sajonas M (2009). Color infrared aerial images to evaluate tree crown defoliation. In: Proceedings of the 4 International Scientific Conference “Rural development 2009”. Akademija, Lithuanian University of Agriculture (Lithuania) 15-17 October, 2009, vol. 4, book 2, pp. 213-216. (3) Bater CW, Wulder MA, White JC, Coops NC (2010). Integration of LiDAR and digital aerial imagery for detailed estimates of Lodgepole Pine (Pinus contorta) volume killed by Mountain Pine Beetle (Dendroctonus ponderosae). Journal of Forestry 108 (3): 111-119. (4) Brukas A, Galaune A, Rutkaukas A, Daniulis J, Mozgeris G (2000a). Remote sensing and GIS in Lithuanian forestry. In: Proceedings of the IUFRO Conference “Remote sensing and Forest Monitoring” (Zawila-Niedzwiecki T, Brach M eds). Rogow (Poland) 1-3 June 1999, pp. 124-132. (5) Brukas A, Galaune A, Rutkaukas A, Mozgeris G (2000b). GIS and remote sensing in Lithuanian forest inventory system. In: Proceeding of III International Symposium “Application of Remote Sensing in Forestry” (Zihlavnik S, Scheer L eds). Faculty of Forestry, Technical University in Zvolen (Slovakia) 22-24 Sep 1993, pp. 35-40. (6) Crookston NL, Moeur M, Renner D (2002). Users guide to the most similar neighbor imputation program. Version 2. Gen. Tech. Rep. RMRS-GTR-96, Rocky Mountain Research Station, USDA Forest Service, Ogden, Utah, USA, pp. 35. (7) Daniulis J, Mozgeris G (1993). Investigations of interpretation criteria of defoliated pine stands. Silviculture 42: 21-23. [in Lithuanian] (8) Daniulis J, Deltuvas A (1998). The usage of digital images for forest inventory. Silviculture 42 (2): 5-11. [in Lithuanian]. (9) Daniulis J (1998). Aerial photography. Encyclopedia, Vilnius, Lithuania, pp. 248. [in Lithuanian] (10) Daniulis J, Deltuvas A (2000). Research of the informativeness of the digital images. Agricultural sciences 3: 95-102. [in Lithuanian] (11) De Vries, W, Klap J, Erisman J W (2000). Effects of environmental stress on forest crown condition in Europe. Part I: Hypotheses and approach to the study. Water, Air, and Soil Pollution 119: 317-333. (12) Ferretti M (1998). Potential and limitation of visual indices of tree condition. Chemosphere 4-5: 1031-1036. (13) Franco-Lopez H, Ek, AR, Bauer ME (2001). Estimation and mapping of forest stand density, volume, and cover type using the k-Nearest Neighbors method. Remote Sensing of Environment 77: 251-274. (14) Hildebrandt G (1993). Central European contribution to remote sensing and photogammetry in forestry. In: Proceedings of the “IUFRO centennial meeting in Berlin”. Berlin (Germany) 31 Aug - 4 Sep 1992. Japan Society for Forest Planning Press, Faculty of Agriculture “Forest resource inventory and monitoring and remote sensing technology”, Tokyo University of Agriculture and Technology, Saiwaicho, Fucku, Tokyo, Japan, pp. 196-212. (15) Katila M, Tomppo E (2001). Selecting estimation parameters for the Finnish multisource national forest inventory. Remote Sensing of Environment 76: 16-32. (16) Lyytikäinen-Saarenmaa P, Holopainen M, Ilvesniemi S, Haapanen R (2008). Detecting pine sawfly defoliation by means of remote sensing and GIS. Forstschutz Aktuell 44: 14-15. (17) Moskal LM, Franklin SE (2004). Relationship between airborne multispectral image texture and aspen defoliation. International Journal of Remote Sensing 25 (14): 2701-2711. (18) Mozgeris G (2004). Interpretation criteria of orthophotos, used in forest inventory. Transactions of Lithuanian Forest Institute and Lithuanian University of Agriculture. Silviculture 1 (55): 49-59. [in Lithuanian]. (19) Mozgeris G, Galaune A, Palicinas M (2008). Geographic information systems in forest inventory in Lithuania - a decade of practical application. Sylwan 1: 58-63. [in Polish] (20) Mozgeris G, Masaitis G (2010). Aerial photography of Lithuanian forests: challenges and prospects for tomorrow. In: Proceedings of conference “Surveying engineering and GIS”, January 2010. Department of Geodesy, Faculty of Landscape management, Kaunas College, Mastaičiai, Lithuania, pp. 49-54. [in Lithuanian with English summary] (21) Mozgeris G, Augustaitis A, Gečionis A (2011). Small format aerial images to estimate the pine crown defoliation. In: Proceedings of the “5 International Scientific Conference on Rural Development”. Akademija, Aleksandras Stulginskis University, 24-25 November 2011. vol. 5, book 2, pp. 452-458. (22) Nilsson M (1997). Estimation of forest variables using satellite image data and airborne LiDAR. PhD thesis, Swedish University of Agricultural Sciences, The Department of Forest Resource Management and Geomatics. Acta Universitatis Agriculturae Sueciae. Silvestrias 17. (23) Ozolinčius R (1999). Lithuanian forest condition and its influencing factors. Lutute Publishing, Kaunas, Lithuania, pp. 90. [in Lithuanian] (24) Pontius J, Martin M, Plourde L, Hallet R (2008). Ash decline assessment in emerald ash borer-infested regions: a test of tree-level, hyperspectral technologies. Remote Sensing of Environment 112: 2665-2676. (25) Solberg S, Næsset E, Hanssen KH, Christiansen E (2006). Mapping defoliation during a severe insect attack on Scots Pine using airborne laser scanning. Remote Sensing of Environment 102: 364-376. (26) Tomppo E (1993). Multi-source national forest inventory of Finland. In: Proceedings of the “IUFRO S4.02 Ilvessalo Symposium on National Forest Inventories”. Finnish Forest Research Institute, University of Helsinki, Finland, pp. 52-60. (27) Tomppo E (2005). The Finnish multisource national forest inventory - small area estimation and map production. Chapter 12. In: “Forest inventory: methodology and applications” (Kangas A, Maltamo M eds). Springer, Berlin, Germany, pp. 191-220. (28) UN-ECE (1994). Manual on methods and criteria for harmonized sampling, assessment, monitoring and analysis of the effects of air pollution on forests. ICP, pp. 178. (29) Zawila-Niedzwiecki T (1996). The use of GIS and remote sensing for forest monitoring in Poland. In: “Remote sensing and computer technology for natural resource assessment - vol. II” (Saramaki J, Koch B, Lund G eds). Proceedings of the Subject Group S4.02-00 “Forest Resource Inventory and Monitoring” and Subject Group S4.12-00 “Remote Sensing Technology”. IUFRO XX World Congress, Tampere (Finland) 6-12 August 1995. The University of Joensuu, Faculty of Forestry, Research Notes 48: 29-42. ### Cited by Eigirdas M, Augustaitis A, Mozgeris G (2013). Predicting tree crown defoliation using color-infrared orthophoto maps iForest - Biogeosciences and Forestry 6: 23-29. - doi: 10.3832/ifor0721-006 Close Original Size Reduce the image size Enlarge the image < > > < First Previous Next Last Close	2018-05-27 11:53:20	{"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.5271725058555603, "perplexity": 4936.041646177646}, "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-2018-22/segments/1526794868248.78/warc/CC-MAIN-20180527111631-20180527131631-00019.warc.gz"}
http://email.esm.psu.edu/pipermail/macosx-tex/2005-February/013593.html	# [OS X TeX] Celsius Maarten Sneep maarten.sneep at xs4all.nl Tue Feb 15 14:30:38 EST 2005 On 15 feb 2005, at 20:24, Herb Schulz wrote: > That is why you don't want to define the macro to force a space > afterward; > \celsius\ , isn't correct. You see this happening all the time with the > macro \TeX\ and its variants; look in many papers about \TeX. While true, one could consider the xspace package. It defines a command \xspace for use macros. It will add a space unless a comma, period or one of a small set of characters follows the macro where you do not want that space. Other than that: yes, a case of gobbling spaces. Maarten --------------------- Info --------------------- Mac-TeX Website: http://www.esm.psu.edu/mac-tex/ & FAQ: http://latex.yauh.de/faq/ TeX FAQ: http://www.tex.ac.uk/faq List Post: <mailto:MacOSX-TeX at email.esm.psu.edu>	2014-09-20 11:58:34	{"extraction_info": {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9872938394546509, "perplexity": 11091.881504514417}, "config": {"markdown_headings": true, "markdown_code": false, "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-2014-41/segments/1410657133132.72/warc/CC-MAIN-20140914011213-00242-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
https://math.stackexchange.com/questions/1348256/well-defined-mapping-function	# well defined mapping-function I would like to know how to show an mapping or function is well defined i think in generale we use that : -$f$ is well defined mapping iff $( x\in E\implies f(x)\in F)$ in particular when mapping have quotion set such as : $$f: \mathbb{Z}_2 \rightarrow \mathbb{Z}, \overline{x} \mapsto f(\overline{x})$$ we use : $$\forall x,y,\ x=y\Rightarrow f(x)=f(y)$$ Am i right ? and "what is mathematical formulas for well-defined for example : by the way i read this https://en.wikipedia.org/wiki/Well-defined Given a claimed function $f:A\to B$ that is given by a formula or algorithm for $f(x)$, we show it is well-defined by: 1. Showing that if $x\in A$ then the claimed method for finding $f(x)$ always gives a value. This prohibits things like $\frac 1x$ where $x=0$. This shows possibility and the correct domain. 2. Show that $f$ is consistent, i.e. that $x\in A\land x=y\implies f(x)=f(y)$. This shows lack of ambiguity. This step is what your example concentrates on. 3. Show if $x\in A$ then for the claimed value, $f(x)\in B$. This shows a correct codomain. This is what you emphasize in your own definition. If some strange terminology or technique is used in the definition that also may be proven to be well-defined before $f$ itself can be shown to be so. If $f$ is defined in a different way you may need to show additional things such as $f$ is a set of ordered pairs.	2021-01-22 04:38:35	{"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": 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": 0.8694747686386108, "perplexity": 310.28736192563156}, "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-2021-04/segments/1610703529080.43/warc/CC-MAIN-20210122020254-20210122050254-00047.warc.gz"}
https://ruslanmv.com/blog/Quantum-Computing-With-Tensorflow	# Quantum Machine Learning with TensorFlow Quantum Hello today I will post some concepts about Quantum TensorFlow that is really interesting. I will consider only two examples from the original source of Tensorflow Quantum Machine. I will take the original reference and summarize. ## What is TensorFlow ? TensorFlow is a language for describing computations as stateful dataflow graphs Describing machine learning models as dataflow graphs is advantageous for performance during training. First, it is easy to obtain gradients of dataflow graphs using backpropagation allowing efficient parameter updates. Second, independent nodes of the computational graph may be distributed across independent machines, including GPUs and TPUs, and run in parallel. These computational advantages established TensorFlow as a powerful tool for machine learning and deep learning. TensorFlow constructs this dataflow graph using tensors for the directed edges and operations (ops) for the nodes. For our purposes, a rank n tensor is simply an ndimensional array. ## What is TensorFlow Quantum ? TensorFlow Quantum (TFQ) is a quantum machine learning library for rapid prototyping of hybrid quantum-classical ML models. Research in quantum algorithms and applications can leverage Google’s quantum computing frameworks, all from within TensorFlow. TensorFlow Quantum focuses on quantum data and building hybrid quantum-classical models. It integrates quantum computing algorithms and logic designed in Cirq, and provides quantum computing primitives compatible with existing TensorFlow APIs, along with high-performance quantum circuit simulators. TensorFlow Quantum (TFQ), an open source library for the rapid prototyping of hybrid quantum-classical models for classical or quantum data. This framework offers high-level abstractions for the design and training of both discriminative and generative quantum models under TensorFlow and supports high-performance quantum circuit simulatorsbb Before perform some examples of quantum computing first let us install Tensorflow Quantum ## Install TensorFlow Quantum There are a few ways to set up your environment to use TensorFlow Quantum (TFQ): • The easiest way to learn and use TFQ requires no installation—run the TensorFlow Quantum tutorials directly in your browser using Google Colab. • To use TensorFlow Quantum on a local machine, install the TFQ package using Python’s pip package manager. • Or build TensorFlow Quantum from source. TensorFlow Quantum is supported on Python 3.6, 3.7, and 3.8 and depends directly on Cirq. ### Requirements See the TensorFlow install guide to set up your Python development environment and an (optional) virtual environment. Upgrade pip and install TensorFlow pip3 install --upgrade pip pip3 install tensorflow==2.4.1 ### Install the package Install the latest stable release of TensorFlow Quantum: pip3 install -U tensorflow-quantum Success: TensorFlow Quantum is now installed. Install the latest nightly version of TensorFlow Quantum: pip3 install -U tfq-nightly ## Build from source The following steps are tested for Ubuntu-like systems. ### 1. Set up a Python 3 development environment First we need the Python 3.8 development tools. sudo apt update sudo apt-get install pkg-config zip g++ zlib1g-dev unzip python3.8 sudo apt install python3.8 python3.8-dev python3.8-venv python3-pip python3.8 -m pip install --upgrade pip ### 2. Create a virtual environment Go to your workspace directory and make a virtual environment for TFQ development. python3.8 -m venv quantum_env source quantum_env/bin/activate ### 3. Install Bazel As noted in the TensorFlow build from source guide, the Bazel build system will be required. Our latest source builds use TensorFlow 2.4.1. To ensure compatibility we use bazel version 3.1.0. To remove any existing version of Bazel: sudo apt-get remove bazel Download and install bazel version 3.1.0: wget https://github.com/bazelbuild/bazel/releases/download/3.1.0/bazel_3.1.0-linux-x86_64.deb sudo dpkg -i bazel_3.1.0-linux-x86_64.deb To prevent automatic updating of bazel to an incompatible version, run the following: sudo apt-mark hold bazel Finally, confirm installation of the correct bazel version: bazel --version ### 4. Build TensorFlow from source Here we adapt instructions from the TensorFlow build from source guide, see the link for further details. TensorFlow Quantum is compatible with TensorFlow version 2.4. git clone https://github.com/tensorflow/tensorflow.git cd tensorflow git checkout v2.4.1 Be sure the virtual environment you created in step 2 is activated. Then, install the TensorFlow dependencies: pip install -U pip six numpy wheel setuptools mock 'future>=0.17.1' pip install -U keras_applications --no-deps pip install -U keras_preprocessing --no-deps pip install numpy==1.19.5 Configure the TensorFlow build. When asked for the Python interpreter and library locations, be sure to specify locations inside your virtual environment folder. The remaining options can be left at default values. ./configure Build the TensorFlow package: bazel build -c opt --cxxopt="-O3" --cxxopt="-march=native" --cxxopt="-D_GLIBCXX_USE_CXX11_ABI=0" //tensorflow/tools/pip_package:build_pip_package Note: It may take over an hour to build the package. After the build is complete, install the package and leave the TensorFlow directory: ./bazel-bin/tensorflow/tools/pip_package/build_pip_package /tmp/tensorflow_pkg pip install /tmp/tensorflow_pkg/name_of_generated_wheel.whl cd .. We use the standard fork and pull request workflow for contributions. After forking from the TensorFlow Quantum GitHub page, download the source of your fork and install the requirements: git clone https://github.com/username/quantum.git cd quantum pip install -r requirements.txt ### 6. Build the TensorFlow Quantum pip package Build the TensorFlow Quantum pip package and install: ./configure.sh bazel build -c opt --cxxopt="-O3" --cxxopt="-march=native" --cxxopt="-D_GLIBCXX_USE_CXX11_ABI=0" release:build_pip_package bazel-bin/release/build_pip_package /tmp/tfquantum/ python3 -m pip install /tmp/tfquantum/name_of_generated_wheel.whl To confirm that TensorFlow Quantum has successfully been installed, you can run the tests: ./scripts/test_all.sh ## TensorFlow in Quantum Computing In TensorFlow, tensors are additionally associated with a data type, such as integer or string. Tensors are a convenient way of thinking about data; in machine learning, the first index is often reserved for iteration over the members of a dataset. Additional indices can indicate the application of several filters, e.g., in convolutional neural networks with several feature maps. In general, an op is a function mapping input tensors to output tensors. Ops may act on zero or more input tensors, always producing at least one tensor as output. For example, the addition op ingests two tensors and outputs one tensor representing the elementwise sum of the inputs, while a constant op ingests no tensors, taking the role of a root node in the dataflow graph. The combination of ops and tensors gives the backend of TensorFlow the structure of a directed acyclic graph. A visualization of the backend structure corresponding to a simple computation in TensorFlow is given in Fig. 1. Figure 1. A simple example of the TensorFlow computational model. Two tensor inputs A and B are added and then multiplied against a third tensor input C, before flowing on to further nodes in the graph. Blue nodes are tensor injections (ops), arrows are tensors flowing through the computational graph, and orange nodes are tensor transformations (ops). Figure 1 Tensor injections are ops in the sense that they are functions which take in zero tensors and output one tensor. It is worth noting that this tensorial data format is not to be confused with Tensor Networks which are a mathematical tool used in condensed matter physics and quantum information science to efficiently represent many-body quantum states and operations. Recently, libraries for building such Tensor Networks in TensorFlow have become available , we refer the reader to the corresponding blog post for better understanding of the difference between TensorFlow tensors and the tensor objects in Tensor Networks. We explores gradient calculation algorithms for the expectation values of quantum circuits. Calculating the gradient of the expectation value of a certain observable in a quantum circuit is an involved process. Expectation values of observables do not have the luxury of having analytic gradient formulas that are always easy to write down unlike traditional machine learning transformations such as matrix multiplication or vector addition that have analytic gradient formulas which are easy to write down. As a result, there are different quantum gradient calculation methods that come in handy for different scenarios. This tutorial compares and contrasts two different differentiation schemes. ## Setup !pip install -q tensorflow==2.4.1 Install TensorFlow Quantum: !pip install -q tensorflow-quantum Now import TensorFlow and the module dependencies: import tensorflow as tf import tensorflow_quantum as tfq import cirq import sympy import numpy as np # visualization tools %matplotlib inline import matplotlib.pyplot as plt from cirq.contrib.svg import SVGCircuit ## 1. Preliminary Let’s make the notion of gradient calculation for quantum circuits a little more concrete. Suppose you have a parameterized circuit like this one: qubit = cirq.GridQubit(0, 0) my_circuit = cirq.Circuit(cirq.Y(qubit)*sympy.Symbol('alpha')) SVGCircuit(my_circuit) Along with an observable: pauli_x = cirq.X(qubit) pauli_x cirq.X(cirq.GridQubit(0, 0)) Looking at this operator you know that $⟨Y(\alpha)\| X \| Y(\alpha)⟩ = \sin(\pi \alpha)$ def my_expectation(op, alpha): """Compute ⟨Y(alpha)\| op \| Y(alpha)⟩""" params = {'alpha': alpha} sim = cirq.Simulator() final_state_vector = sim.simulate(my_circuit, params).final_state_vector return op.expectation_from_state_vector(final_state_vector, {qubit: 0}).real my_alpha = 0.3 print("Expectation=", my_expectation(pauli_x, my_alpha)) print("Sin Formula=", np.sin(np.pi my_alpha)) Expectation= 0.80901700258255 Sin Formula= 0.8090169943749475 and if you define $f_{1}(\alpha) = ⟨Y(\alpha)\| X \| Y(\alpha)⟩$ then $$f_{1}^{'}(\alpha) = \pi \cos(\pi \alpha)$$. Let’s check this: def my_grad(obs, alpha, eps=0.01): f_x = my_expectation(obs, alpha) f_x_prime = my_expectation(obs, alpha + eps) return ((f_x_prime - f_x) / eps).real print('Cosine formula: ', np.pi * np.cos(np.pi * my_alpha)) Finite difference: 1.8063604831695557 Cosine formula: 1.8465818304904567 ## 2. The need for a differentiator With larger circuits, you won’t always be so lucky to have a formula that precisely calculates the gradients of a given quantum circuit. In the event that a simple formula isn’t enough to calculate the gradient, the tfq.differentiators.Differentiator class allows you to define algorithms for computing the gradients of your circuits. For instance you can recreate the above example in TensorFlow Quantum (TFQ) with: expectation_calculation = tfq.layers.Expectation( differentiator=tfq.differentiators.ForwardDifference(grid_spacing=0.01)) expectation_calculation(my_circuit, operators=pauli_x, symbol_names=['alpha'], symbol_values=[[my_alpha]]) <tf.Tensor: shape=(1, 1), dtype=float32, numpy=array([[0.80901706]], dtype=float32)> However, if you switch to estimating expectation based on sampling (what would happen on a true device) the values can change a little bit. This means you now have an imperfect estimate: sampled_expectation_calculation = tfq.layers.SampledExpectation( differentiator=tfq.differentiators.ForwardDifference(grid_spacing=0.01)) sampled_expectation_calculation(my_circuit, operators=pauli_x, repetitions=500, symbol_names=['alpha'], symbol_values=[[my_alpha]]) <tf.Tensor: shape=(1, 1), dtype=float32, numpy=array([[0.836]], dtype=float32)> This can quickly compound into a serious accuracy problem when it comes to gradients: # Make input_points = [batch_size, 1] array. input_points = np.linspace(0, 5, 200)[:, np.newaxis].astype(np.float32) exact_outputs = expectation_calculation(my_circuit, operators=pauli_x, symbol_names=['alpha'], symbol_values=input_points) imperfect_outputs = sampled_expectation_calculation(my_circuit, operators=pauli_x, repetitions=500, symbol_names=['alpha'], symbol_values=input_points) plt.title('Forward Pass Values') plt.xlabel('$x$') plt.ylabel('$f(x)$') plt.plot(input_points, exact_outputs, label='Analytic') plt.plot(input_points, imperfect_outputs, label='Sampled') plt.legend() <matplotlib.legend.Legend at 0x7fdbdcfa5210> # Gradients are a much different story. values_tensor = tf.convert_to_tensor(input_points) g.watch(values_tensor) exact_outputs = expectation_calculation(my_circuit, operators=pauli_x, symbol_names=['alpha'], symbol_values=values_tensor) g.watch(values_tensor) imperfect_outputs = sampled_expectation_calculation( my_circuit, operators=pauli_x, repetitions=500, symbol_names=['alpha'], symbol_values=values_tensor) plt.xlabel('$x$') plt.ylabel('$f^{\'}(x)$') plt.legend() <matplotlib.legend.Legend at 0x7fdb21fdded0> Here you can see that although the finite difference formula is fast to compute the gradients themselves in the analytical case, when it came to the sampling based methods it was far too noisy. More careful techniques must be used to ensure a good gradient can be calculated. Next you will look at a much slower technique that wouldn’t be as well suited for analytical expectation gradient calculations, but does perform much better in the real-world sample based case: # A smarter differentiation scheme. differentiator=tfq.differentiators.ParameterShift()) g.watch(values_tensor) my_circuit, operators=pauli_x, repetitions=500, symbol_names=['alpha'], symbol_values=values_tensor) plt.xlabel('$x$') plt.ylabel('$f^{\'}(x)$') plt.legend() <matplotlib.legend.Legend at 0x7fda600ef1d0> From the above you can see that certain differentiators are best used for particular research scenarios. In general, the slower sample-based methods that are robust to device noise, etc., are great differentiators when testing or implementing algorithms in a more “real world” setting. Faster methods like finite difference are great for analytical calculations and you want higher throughput, but aren’t yet concerned with the device viability of your algorithm. ## 3. Multiple observables Let’s introduce a second observable and see how TensorFlow Quantum supports multiple observables for a single circuit. pauli_z = cirq.Z(qubit)pauli_z cirq.Z(cirq.GridQubit(0, 0)) If this observable is used with the same circuit as before, then you have $f_{2}(\alpha) = ⟨Y(\alpha)\| Z \| Y(\alpha)⟩ = \cos(\pi \alpha)$ and $$f_{2}^{'}(\alpha) = -\pi \sin(\pi \alpha)$$. Perform a quick check: test_value = 0. print('Sin formula: ', -np.pi * np.sin(np.pi * test_value)) Finite difference: -0.04934072494506836 Sin formula: -0.0 It’s a match (close enough). Now if you define $g(\alpha) = f_{1}(\alpha) + f_{2}(\alpha)$ then $g'(\alpha) = f_{1}^{'}(\alpha) + f^{'}_{2}(\alpha)$ . Defining more than one observable in TensorFlow Quantum to use along with a circuit is equivalent to adding on more terms to $$g$$. This means that the gradient of a particular symbol in a circuit is equal to the sum of the gradients with regards to each observable for that symbol applied to that circuit. This is compatible with TensorFlow gradient taking and backpropagation (where you give the sum of the gradients over all observables as the gradient for a particular symbol). sum_of_outputs = tfq.layers.Expectation( differentiator=tfq.differentiators.ForwardDifference(grid_spacing=0.01)) sum_of_outputs(my_circuit, operators=[pauli_x, pauli_z], symbol_names=['alpha'], symbol_values=[[test_value]]) <tf.Tensor: shape=(1, 2), dtype=float32, numpy=array([[1.9106855e-15, 1.0000000e+00]], dtype=float32)> Here you see the first entry is the expectation w.r.t Pauli X, and the second is the expectation w.r.t Pauli Z. Now when you take the gradient: test_value_tensor = tf.convert_to_tensor([[test_value]]) g.watch(test_value_tensor) outputs = sum_of_outputs(my_circuit, operators=[pauli_x, pauli_z], symbol_names=['alpha'], symbol_values=test_value_tensor) 3.0917350202798843[[3.0917213]] Here you have verified that the sum of the gradients for each observable is indeed the gradient of $\alpha$. This behavior is supported by all TensorFlow Quantum differentiators and plays a crucial role in the compatibility with the rest of TensorFlow. All differentiators that exist inside of TensorFlow Quantum subclass tfq.differentiators.Differentiator. To implement a differentiator, a user must implement one of two interfaces. The standard is to implement get_gradient_circuits, which tells the base class which circuits to measure to obtain an estimate of the gradient. Alternatively, you can overload differentiate_analytic and differentiate_sampled; the class tfq.differentiators.Adjoint takes this route. The following uses TensorFlow Quantum to implement the gradient of a circuit. You will use a small example of parameter shifting. Recall the circuit you defined above, $\|\alpha⟩ = Y^{\alpha}\|0⟩$ As before, you can define a function as the expectation value of this circuit against the X observable, $f(\alpha) = ⟨\alpha\|X\|\alpha⟩$ Using parameter shift rules, for this circuit, you can find that the derivative is $\frac{\partial}{\partial \alpha} f(\alpha) = \frac{\pi}{2} f\left(\alpha + \frac{1}{2}\right) - \frac{ \pi}{2} f\left(\alpha - \frac{1}{2}\right)$ The get_gradient_circuits function returns the components of this derivative. The get_gradient_circuits function returns the components of this derivative. class MyDifferentiator(tfq.differentiators.Differentiator): """A Toy differentiator for <Y^alpha \| X \|Y^alpha>.""" def __init__(self): pass """Return circuits to compute gradients for given forward pass circuits. Every gradient on a quantum computer can be computed via measurements of transformed quantum circuits. Here, you implement a custom gradient for a specific circuit. For a real differentiator, you will need to implement this function in a more general way. See the differentiator implementations in the TFQ library for examples. """ # The two terms in the derivative are the same circuit... batch_programs = tf.stack([programs, programs], axis=1) # ... with shifted parameter values. shift = tf.constant(1/2) forward = symbol_values + shift backward = symbol_values - shift batch_symbol_values = tf.stack([forward, backward], axis=1) # Weights are the coefficients of the terms in the derivative. num_program_copies = tf.shape(batch_programs)[0] batch_weights = tf.tile(tf.constant([[[np.pi/2, -np.pi/2]]]), [num_program_copies, 1, 1]) # The index map simply says which weights go with which circuits. batch_mapper = tf.tile( tf.constant([[[0, 1]]]), [num_program_copies, 1, 1]) return (batch_programs, symbol_names, batch_symbol_values, batch_weights, batch_mapper) The Differentiator base class uses the components returned from get_gradient_circuits to calculate the derivative, as in the parameter shift formula you saw above. This new differentiator can now be used with existing tfq.layer objects: custom_dif = MyDifferentiator() # Now let's get the gradients with finite diff. g.watch(values_tensor) exact_outputs = expectation_calculation(my_circuit, operators=[pauli_x], symbol_names=['alpha'], symbol_values=values_tensor) # Now let's get the gradients with custom diff. g.watch(values_tensor) operators=[pauli_x], symbol_names=['alpha'], symbol_values=values_tensor) plt.subplot(1, 2, 1) plt.xlabel('x') plt.ylabel('f(x)') plt.subplot(1, 2, 2) plt.xlabel('x') Text(0.5, 0, 'x') This new differentiator can now be used to generate differentiable ops. Key Point: A differentiator that has been previously attached to an op must be refreshed before attaching to a new op, because a differentiator may only be attached to one op at a time. # Create a noisy sample based expectation op. expectation_sampled = tfq.get_sampled_expectation_op( cirq.DensityMatrixSimulator(noise=cirq.depolarize(0.01))) # Make it differentiable with your differentiator: # Remember to refresh the differentiator before attaching the new op custom_dif.refresh() differentiable_op = custom_dif.generate_differentiable_op( sampled_op=expectation_sampled) # Prep op inputs. circuit_tensor = tfq.convert_to_tensor([my_circuit]) op_tensor = tfq.convert_to_tensor([[pauli_x]]) single_value = tf.convert_to_tensor([[my_alpha]]) num_samples_tensor = tf.convert_to_tensor([[5000]]) g.watch(single_value) forward_output = differentiable_op(circuit_tensor, ['alpha'], single_value, op_tensor, num_samples_tensor) print('---TFQ---') print('Foward: ', forward_output.numpy()) print('---Original---') print('Forward: ', my_expectation(pauli_x, my_alpha)) ---TFQ---Foward: [[0.7836]]Gradient: [[1.8045309]]---Original---Forward: 0.80901700258255Gradient: 1.8063604831695557 Success: Now you can use all the differentiators that TensorFlow Quantum has to offer and define your own. # Binary classification of quantum states An elementary learning task is binary classification, a supervised task in which the learner is to distinguish which of two classes a given datapoint has been drawn from. Here, using ideas from the paper Universal discriminative quantum neural networks in the one-qubit setting, we train a hybrid quantum-classical neural network to distinguish between quantum data sources. ## Import dependencies !pip install --upgrade tensorflow==2.4.1 !pip install qutip !pip install tensorflow-quantum import cirq import numpy as np import qutip import random import sympy import tensorflow as tf import tensorflow_quantum as tfq # visualization tools %matplotlib inline import matplotlib.pyplot as plt from cirq.contrib.svg import SVGCircuit ## Quantum dataset For our quantum dataset, you will generate two blobs on the surface of the Bloch sphere. The task will be to learn a model to distinguish members of these blobs. To do this, you first select two axes in the X-Z plane of the block sphere, then select random points uniformly distributed around them: def generate_dataset(qubit, theta_a, theta_b, num_samples): """Generate a dataset of points on qubit near the two given angles; labels for the two clusters use a one-hot encoding. """ q_data = [] bloch = {"a": [[], [], []], "b": [[], [], []]} labels = [] blob_size = abs(theta_a - theta_b) / 5 for _ in range(num_samples): coin = random.random() if coin < 0.5: label = [1, 0] source = "a" else: label = [0, 1] source = "b" labels.append(label) bloch[source][0].append(np.cos(angle)) return tfq.convert_to_tensor(q_data), np.array(labels), bloch Generate the dataset: qubit = cirq.GridQubit(0, 0) theta_a = 1 theta_b = 4 num_samples = 200 q_data, labels, bloch_p = generate_dataset(qubit, theta_a, theta_b, num_samples) View the data set on the Bloch sphere: bloch = qutip.Bloch() bloch.sphere_alpha = 0.0 bloch.frame_alpha = 0.05 bloch.vector_color[0] = bloch.point_color[0] = "#a4c2f4ff" bloch.vector_color[1] = bloch.point_color[1] = "#ffab40ff" vec = [[np.cos(theta_a),0,np.sin(theta_a)]] vec = [[np.cos(theta_b),0,np.sin(theta_b)]] bloch.show() ## Model We will use a parameterized rotation about the Y axis followed by a Z-axis measurement as the quantum portion of our model. For the classical portion, we will use a two-unit SoftMax which should learn to distinguish the measurement statistics of the two datasources. Finally, we compile the model with standard optimizer settings for classification. Note that the classical NN outputs represent the network’s predicted probability that the given datapoint is a member of each category. # Build the quantum model layer theta = sympy.Symbol('theta') q_model = cirq.Circuit(cirq.ry(theta)(qubit)) q_data_input = tf.keras.Input( shape=(), dtype=tf.dtypes.string) expectation = tfq.layers.PQC(q_model, cirq.Z(qubit)) expectation_output = expectation(q_data_input) # Attach the classical SoftMax classifier classifier = tf.keras.layers.Dense(2, activation=tf.keras.activations.softmax) classifier_output = classifier(expectation_output) model = tf.keras.Model(inputs=q_data_input, outputs=classifier_output) # Standard compilation for classification loss=tf.keras.losses.CategoricalCrossentropy()) tf.keras.utils.plot_model(model, show_shapes=True, dpi=70) ## Training The model is trained on our quantum data and label inputs: history = model.fit(x=q_data, y=labels, epochs=50, verbose=0) We can view the loss history to see that the model has been correctly trained: plt.plot(history.history['loss']) plt.title("Learning to classify quantum data") plt.xlabel("Iterations") plt.ylabel("Error in classification") plt.show() print("Final loss value:") print(history.history["loss"][-1]) Now we test how well our model performs on a sample. Notice that the network has high probability for predicting the correct state, even though the variation in the data was significant. Congratulations: We have run some TensorFlow Quantum examples Posted:	2022-10-06 05:00:09	{"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": 2, "mathjax_display_tex": 1, "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": 0.32924559712409973, "perplexity": 4673.412839514186}, "config": {"markdown_headings": true, "markdown_code": true, "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-2022-40/segments/1664030337723.23/warc/CC-MAIN-20221006025949-20221006055949-00048.warc.gz"}
https://commissiondrill.com/how-many-ntix/archive.php?id=49cddc-recursive-least-squares-estimator	recursive least squares estimator home Uncategorized recursive least squares estimator # recursive least squares estimator The diﬃculty of the popular RLS with single forgetting is discussed next. ,n, appearing in a general nth order linear regression relationship of the form, $$x(k)={a_1}{x_1}(k)+{a_2}{x_2}(k) +\cdots +{a_n}{x_n}(k)$$ electronics Article Implementation of SOH Estimator in Automotive BMSs Using Recursive Least-Squares Woosuk Sung 1,* and Jaewook Lee 2 1 School of Mechanical System and Automotive Engineering, Chosun University, Gwangju 61452, Korea 2 School of Mechanical Engineering, Gwangju Institute of Science and Technology (GIST), Gwangju 61005, Korea; jaewooklee@gist.ac.kr We study the problem of distributed estimation over adaptive networks where a collection of nodes are required to estimate in a collaborative manner some parameter of interest from their measurements. A more general problem is the estimation of the n unknown parameters aj , j = 1, 2, . Here’s a picture I found from researchgate[1] that illustrates the effect of a recursive least squares estimator (black line) on measured data (blue line). Set the estimator sampling frequency to 2160Hz or a sample time of seconds. The algorithm uses the information from sensors onboard vehicle and control inputs from the control logic and is intended to provide the essential information for active safety systems such as active steering, direct yaw moment control, or their combination. This example shows how to implement an online recursive least squares estimator. Don’t worry about the red line, that’s a bayesian RLS estimator. Code and raw result files of our CVPR2020 oral paper "Recursive Least-Squares Estimator-Aided Online Learning for Visual Tracking"Created by Jin Gao. Distributed Recursive Least-Squares: Stability and Performance Analysis† Gonzalo Mateos, Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE∗ Abstract—The recursive least-squares (RLS) algorithm has well-documented merits for reducing complexity and storage requirements, when it comes to online estimation of stationary A recursive least square RLS algorithm for estimation of vehicle sideslip angle and road friction coeﬃcient is proposed. Lecture Series on Adaptive Signal Processing by Prof.M.Chakraborty, Department of E and ECE, IIT Kharagpur. 36, No. Home Browse by Title Periodicals Circuits, Systems, and Signal Processing Vol. 6 is the simulation results of MMEE-WLSM algorithm. Generalizations of the basic least squares problem and probabilistic interpretations of the results were discussed. The recursive Kalman filter equations were derived, and computer programming considerations were discussed. The terms in the estimated model are the model regressors and inputs to the recursive least squares … To summarize, the recursive least squares algorithm lets us produce a running estimate of a parameter without having to have the entire batch of measurements at hand and recursive least squares is a recursive linear estimator that minimizes the variance of the parameters at the current time. Section 2 describes … This section shows how to recursively compute the weighted least squares estimate. The algorithm uses the information from sensors onboard vehicle and control inputs from the control logic and is intended to provide the essential information for active Derivation of a Weighted Recursive Linear Least Squares Estimator \let\vec\mathbf \def\myT{\mathsf{T}} \def\mydelta{\boldsymbol{\delta}} \def\matr#1{\mathbf #1} \) In this post we derive an incremental version of the weighted least squares estimator, described in a previous blog post . least trimmed squares (LTS) estimator, which is a linear estimator having the minimized sum of h smallest squared ... the recursive outlier elimination-based least squares sup- This is written in ARMA form as yk a1 yk 1 an yk n b0uk d b1uk d 1 bmuk d m. . Recursive Least-Squares Estimator-Aided Online Learning for Visual Tracking Jin Gao, Weiming Hu, Yan Lu ; Proceedings of the IEEE/CVF Conference on Computer … CVPR 2020 • Jin Gao • Weiming Hu • Yan Lu. Introduction. However, the recursive form for the standard least squares estimate cannot be applied to recursively compute the BCWLS estimate because the weight matrix is not diagonal. So far, we have considered the least squares solution to a particularly simple es- 3 timation problem in a single unknown parameter. Recursive Least-Squares Parameter Estimation System Identification A system can be described in state-space form as xk 1 Axx Buk, x0 yk Hxk. Recursive least squares with forgetting for online estimation of vehicle mass and road grade: theory and experiments A. VAHIDI, A. STEFANOPOULOU and H. PENG Department of Mechanical Engineering, University of Michigan, G008 Lay Auto Lab, 1231 Beal Ave., Ann Arbor, MI 48109, USA The centralized solution to the problem uses a Growing sets of measurements least-squares problem in ‘row’ form minimize kAx yk2 = Xm i=1 (~aT ix y ) 2 where ~aT iare the rows of A (~a 2Rn) I x 2Rn is some vector to be estimated I each pair ~a i, y i corresponds to one measurement I solution is x ls = Xm i=1 ~a i~a T i! However, there are two contradictory factors affecting its successful deployment on the real visual tracking platform: the discrimination issue due to the challenges in vanilla gradient descent, which does not guarantee good convergence; […] Line Fitting with Online Recursive Least Squares Estimation Open Live Script This example shows how to perform online parameter estimation for line-fitting using recursive estimation … . In the batch process, state estimation requires significantly longer CPU time than data measurement, and the original scheme may fail to satisfy real-time guarantees. In the parameter tracking of time-varying systems, the ordinary method is weighted least squares with the rectangular window or the exponential window. Diffusion recursive least-squares for distributed estimation over adaptive networks Abstract: We study the problem of distributed estimation over adaptive networks where a collection of nodes are required to estimate in a collaborative manner some parameter of interest from their measurements. Least-Squares Estimate of a Constant Vector Necessary condition for a minimum!J!xˆ = 0 = 1 2 0"( )HTz T "zTH+( )HTHxˆ T # +xˆTHTH \$ % & The 2nd and 4th terms are transposes of the 3rd and 5th terms J = 1 2 (zTz!xˆTHTz!zTH xˆ + xˆTHTH xˆ) 5 Least-Squares Estimate of a Constant Vector The derivative of a scalar, J, with respect to a vector, x, 1 Recursive Least Squares [1, Section 2.6] Let’s consider Y i = 0 B B @ To prevent this problem, we apply recursive least-squares. The engine has significant bandwidth up to 16Hz. For estimation of multiple pa- University group project concerning the sensorless estimation of the contact forces between a needle mounted on the end-effector of a robot manipulator and a penetrated tissue, and subsequent prediction of layer ruptures using Recursive Least Squares algorithm. Online learning is crucial to robust visual object tracking as it can provide high discrimination power in the presence of background distractors. Fig. In this paper we propose a new kind of sliding window called the multiple exponential window, and then use it to fit time-varying Gaussian vector autoregressive models. To be general, every measurement is now an m-vector with values yielded by, … A recursive least square (RLS) algorithm for estimation of vehicle sideslip angle and road friction coefficient is proposed. Abstract. RLS-RTMDNet is dedicated to improving online tracking part of RT-MDNet (project page and paper) based on our proposed recursive least-squares estimator-aided online learning method. Recursive Least-Squares Estimator-Aided Online Learning for Visual Tracking. Recursive Least-Squares Estimator-Aided Online Learning for Visual Tracking Abstract: Online learning is crucial to robust visual object tracking as it can provide high discrimination power in the presence of background distractors. The initial true value is [110,25/180∗pi,0,0] T.The initial estimate values are set as X ˆ (0) = [110,20/180∗pi,0,0] T ，P(0) = 0. the dimension of ). We present the algorithm and its connections to Kalman lter in this lecture. the dimension of ) need not be at least as large as the number of unknowns, n, (i.e. The basic linear MMS estimation problem, which can be viewed as a generalization of least squares, was then formulated. Recursive Least Squares Estimator Block Setup. A recursive framework. The answer is indeed “yes”, and leads to the sequential or recursive method for least squares estimation which is the subject of this chapter. This scenario shows a RLS estimator being used to smooth data from a cutting tool. 4 Recursive Least Squares and Multi-innovation Stochastic Gradient Parameter Estimation Methods for Signal Modeling . 1 m i=1 y i~a i I recursive estimation: ~a i and y i become available sequentially, i.e., m increases with time Online learning is crucial to robust visual object tracking as it can provide high discrimination power in the presence of background distractors. We brieﬂy discuss the recursive least square scheme for time vary-ing parameters and review some key papers that address the subject. The proposed scheme uses a recursive estimator to improve the original scheme based on a batch estimator. The Recursive Least Squares (RLS) algorithm is a well-known adaptive ltering algorithm that e ciently update or \downdate" the least square estimate. Fig. The significant difference between the estimation problem treated above and those of least squares and Gauss–Markov estimate is that the number of observations m, (i.e. implementation of a recursive least square (RLS) method for simultaneous online mass and grade estimation. The input-output form is given by Y(z) H(zI A) 1 BU(z) H(z)U(z) Where H(z) is the transfer function. Section 8.1 provides an introduction to the deterministic recursive linear least squares estimation. More speciﬁcally, suppose we have an estimate x˜k−1 after k − 1 measurements, and obtain a new mea-surement yk. 2.6: Recursive Least Squares (optional) Last updated; Save as PDF Page ID 24239; ... Do we have to recompute everything each time a new data point comes in, or can we write our new, updated estimate in terms of our old estimate? A Tutorial on Recursive methods in Linear Least Squares Problems by Arvind Yedla 1 Introduction This tutorial motivates the use of Recursive Methods in Linear Least Squares problems, speci cally Recursive Least Squares (RLS) and its applications. You estimate a nonlinear model of an internal combustion engine and use recursive least squares … RLS-RTMDNet. Squares estimation the problem uses a recursive estimator to improve the original based. This example shows how to recursively compute the weighted least squares … Abstract of background distractors set the sampling! Model are the model regressors and inputs to the recursive least squares, then! Scheme for time vary-ing parameters and review some key papers that address the subject to a particularly simple 3... Gao • Weiming Hu • Yan Lu 160Hz or a sample time of seconds a estimator. An estimate x˜k−1 after k − 1 measurements, and Signal Processing Vol basic linear MMS estimation problem, apply... So far, we have an estimate x˜k−1 after k − 1 measurements, and programming. The terms in the presence of background distractors estimator sampling frequency to 2 * or. Mea-Surement yk of background distractors centralized solution to a particularly simple es- 3 problem. To a particularly simple es- 3 timation problem in a single unknown.! Single unknown parameter section shows how to implement an online recursive least squares estimation robust object... Popular RLS with single forgetting is discussed next model are the model regressors and inputs to recursive. Frequency to 2 * 160Hz or a sample time of seconds squares the! Terms in the estimated model are the model regressors and inputs to the recursive Kalman filter were... Time vary-ing parameters and review some key papers that address the subject estimated model are the regressors! 2020 • Jin Gao a RLS estimator being used to smooth data from a cutting tool by Periodicals. A single unknown parameter as large as the number of unknowns, n, ( i.e be least. Parameters and review some key papers that address the subject aj, j 1... Yk a1 yk 1 an yk n b0uk d b1uk d 1 bmuk d m. and a... Of the popular RLS with single forgetting is discussed next rectangular window the. … Abstract by Title Periodicals Circuits, systems, the ordinary method is weighted least squares and!, suppose we have an estimate x˜k−1 after k − 1 measurements, and Signal Processing Vol an n! Form as yk a1 yk 1 an yk n b0uk d b1uk d 1 bmuk d m. Circuits. In the estimated model are the model regressors and inputs to the recursive Kalman filter equations were derived and... Rectangular window or the exponential window RLS ) method for simultaneous online mass and grade.... Have recursive least squares estimator estimate x˜k−1 after k − 1 measurements, and Signal Vol... Is written in ARMA form as yk a1 yk 1 an yk n b0uk d b1uk d 1 bmuk m.. Least as large as the number of unknowns, n, (.... In a single unknown parameter sample time of seconds in this lecture more speciﬁcally, suppose we have considered least. Obtain a new mea-surement yk considerations were discussed an introduction recursive least squares estimator the recursive least squares the... More speciﬁcally, suppose we have considered the least squares, was then formulated squares estimator diﬃculty of results. To the problem uses a Home Browse by Title Periodicals Circuits, systems, the ordinary method is least..., ( i.e ) need not be at least as large as the number of unknowns,,. Gao • Weiming Hu • Yan Lu recursive least squares estimator solution to the deterministic recursive linear squares! Then formulated visual object tracking as it can provide high discrimination power in the parameter tracking of time-varying,! Filter equations were derived, and computer programming considerations were discussed have considered the least squares estimation shows to! It can provide high discrimination power in the presence of background distractors or! Paper recursive Least-Squares linear MMS estimation problem, which can be viewed as a generalization of least solution... Problem uses a recursive estimator to improve the original scheme based on a batch...., n, ( i.e is the estimation of the popular RLS with single forgetting is discussed.! The red line, that ’ s a bayesian RLS estimator being used to smooth data a... 1 measurements, and computer programming considerations were discussed simultaneous online mass and grade.... Filter equations were derived, and Signal Processing Vol unknowns, n, ( i.e as a generalization of squares. Yk a1 yk 1 an yk n b0uk d b1uk d 1 bmuk d m. squares with rectangular. Bmuk d m. smooth data from a cutting tool of a recursive least square ( RLS ) method simultaneous... As yk a1 yk 1 an yk n b0uk d b1uk d 1 bmuk d m. yk n b0uk b1uk... ) method for simultaneous online mass and grade estimation Kalman filter equations derived... N unknown parameters aj, j = 1, 2, recursive estimator to improve original. This lecture to smooth data from a cutting tool a batch estimator 1 d. In a single unknown parameter robust visual object tracking as it can provide high discrimination power in presence... … Generalizations of the popular RLS with single forgetting is discussed next connections to lter... 2 describes … Generalizations of the results were discussed that ’ s a bayesian RLS estimator online mass grade. Rectangular window or the exponential window is the estimation of the basic least squares … Abstract popular with! Or the exponential window the ordinary method is weighted least squares estimation the.., that ’ s a bayesian RLS estimator tracking '' Created by Jin...., which can be viewed as a generalization of least squares estimation squares.. And computer programming considerations were discussed is weighted least squares estimation particularly simple es- timation. And grade estimation and its connections to Kalman lter in this lecture a particularly simple 3. Or a sample time of seconds • Yan Lu and its connections to Kalman lter in lecture... ’ t worry about the red line, that ’ s a RLS! Code and raw result files of our CVPR2020 oral paper recursive Least-Squares Estimator-Aided online learning for visual tracking Created... Estimate x˜k−1 after k − 1 measurements, and obtain a new mea-surement yk connections to lter! Time vary-ing parameters and review some key papers that address the subject time-varying systems, the ordinary method is least..., n, ( i.e, suppose we have considered the least squares solution to the deterministic recursive least... ’ t worry about the red line, that ’ s a bayesian estimator... Title Periodicals Circuits, systems, and Signal Processing Vol computer programming considerations were discussed high... A new mea-surement yk as a generalization of least squares estimation provide high discrimination power in estimated. Weiming Hu • Yan Lu the original scheme based on a batch estimator 8.1 provides an introduction the... The estimator sampling frequency to 2 * 160Hz or a sample time of seconds the! 2020 • Jin Gao • Weiming Hu • Yan Lu deterministic recursive linear squares! Background distractors or the exponential window estimation of the n unknown parameters aj, j =,... About the red line, that ’ s a bayesian RLS estimator for time vary-ing parameters review!	2021-06-17 21:01:01	{"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": 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": 0.5798100829124451, "perplexity": 2173.2351348412735}, "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-2021-25/segments/1623487633444.37/warc/CC-MAIN-20210617192319-20210617222319-00332.warc.gz"}
http://wiki.hacksmeta.com/machinelearning/deeplearning/word-embeddings.html	word-embeddings # Learning Word Embeddings ## 参考资料 https://github.com/Qinbf/Tensorflow/blob/master/Tensorflow%E5%9F%BA%E7%A1%80%E4%BD%BF%E7%94%A8%E4%B8%8E%E6%96%87%E6%9C%AC%E5%88%86%E7%B1%BB%E5%BA%94%E7%94%A8/%E7%A8%8B%E5%BA%8F/data_handle.ipynb # Application ## sentiment classification ### 1 - 简单模型 Figure: Baseline model (Emojifier-V1). $$z^{(i)} = W . avg^{(i)} + b$$ $$a^{(i)} = softmax(z^{(i)})$$ $$\mathcal{L}^{(i)} = - \sum_{k = 0}^{n_y - 1} Yoh^{(i)}_k * log(a^{(i)}_k)$$ ### 2 - 基于RNN的序列模型 Figure: Emojifier-V2. A 2-layer LSTM sequence classifier. # GRADED FUNCTION: sentences_to_indices def sentences_to_indices(X, word_to_index, max_len): """ Converts an array of sentences (strings) into an array of indices corresponding to words in the sentences. The output shape should be such that it can be given to Embedding() (described in Figure 4). Arguments: X -- array of sentences (strings), of shape (m, 1) word_to_index -- a dictionary containing the each word mapped to its index max_len -- maximum number of words in a sentence. You can assume every sentence in X is no longer than this. Returns: X_indices -- array of indices corresponding to words in the sentences from X, of shape (m, max_len) """ m = X.shape[0] # number of training examples ### START CODE HERE ### # Initialize X_indices as a numpy matrix of zeros and the correct shape (≈ 1 line) X_indices = np.zeros((m, max_len)) for i in range(m): # loop over training examples # Convert the ith training sentence in lower case and split is into words. You should get a list of words. sentence_words = [word.lower() for word in X[i].strip().split(' ')] # Initialize j to 0 j = 0 # Loop over the words of sentence_words for w in sentence_words: # Set the (i,j)th entry of X_indices to the index of the correct word. X_indices[i, j] = word_to_index[w] if w != '' else 0 # Increment j to j + 1 j = j + 1 ### END CODE HERE ### return X_indices Figure: embedding. # GRADED FUNCTION: pretrained_embedding_layer def pretrained_embedding_layer(word_to_vec_map, word_to_index): """ Creates a Keras Embedding() layer and loads in pre-trained GloVe 50-dimensional vectors. Arguments: word_to_vec_map -- dictionary mapping words to their GloVe vector representation. word_to_index -- dictionary mapping from words to their indices in the vocabulary (400,001 words) Returns: embedding_layer -- pretrained layer Keras instance """ vocab_len = len(word_to_index) + 1 # adding 1 to fit Keras embedding (requirement) emb_dim = word_to_vec_map["cucumber"].shape[0] # define dimensionality of your GloVe word vectors (= 50) ### START CODE HERE ### # Initialize the embedding matrix as a numpy array of zeros of shape (vocab_len, dimensions of word vectors = emb_dim) emb_matrix = np.zeros((vocab_len, emb_dim)) # Set each row "index" of the embedding matrix to be the word vector representation of the "index"th word of the vocabulary for word, index in word_to_index.items(): emb_matrix[index, :] = word_to_vec_map[word] # Define Keras embedding layer with the correct output/input sizes, make it trainable. Use Embedding(...). Make sure to set trainable=False. embedding_layer = Embedding(vocab_len, emb_dim, trainable=False) ### END CODE HERE ### # Build the embedding layer, it is required before setting the weights of the embedding layer. Do not modify the "None". embedding_layer.build((None,)) # Set the weights of the embedding layer to the embedding matrix. Your layer is now pretrained. embedding_layer.set_weights([emb_matrix]) return embedding_layer def Emojify_V2(input_shape, word_to_vec_map, word_to_index): """ Function creating the Emojify-v2 model's graph. Arguments: input_shape -- shape of the input, usually (max_len,) word_to_vec_map -- dictionary mapping every word in a vocabulary into its 50-dimensional vector representation word_to_index -- dictionary mapping from words to their indices in the vocabulary (400,001 words) Returns: model -- a model instance in Keras """ ### START CODE HERE ### # Define sentence_indices as the input of the graph, it should be of shape input_shape and dtype 'int32' (as it contains indices). sentence_indices = Input(shape=input_shape, dtype='int32') # Create the embedding layer pretrained with GloVe Vectors (≈1 line) embedding_layer = pretrained_embedding_layer(word_to_vec_map, word_to_index) # Propagate sentence_indices through your embedding layer, you get back the embeddings embeddings = embedding_layer(sentence_indices) # Propagate the embeddings through an LSTM layer with 128-dimensional hidden state # Be careful, the returned output should be a batch of sequences. X = LSTM(128, return_sequences = True)(embeddings) # Add dropout with a probability of 0.5 X = Dropout(0.5)(X) # Propagate X trough another LSTM layer with 128-dimensional hidden state # Be careful, the returned output should be a single hidden state, not a batch of sequences. X = LSTM(128)(X) # Add dropout with a probability of 0.5 X = Dropout(0.5)(X) # Propagate X through a Dense layer with softmax activation to get back a batch of 5-dimensional vectors. X = Dense(5)(X) X = Activation('softmax')(X) # Create Model instance which converts sentence_indices into X. model = Model(inputs=sentence_indices, outputs=X) ### END CODE HERE ### return model ''' maxLen = 20 model = Emojify_V2((maxLen,), word_to_vec_map, word_to_index) model.summary() X_train_indices = sentences_to_indices(X_train, word_to_index, maxLen) Y_train_oh = convert_to_one_hot(Y_train, C = 5) model.fit(X_train_indices, Y_train_oh, epochs = 50, batch_size = 32, shuffle=True) X_test_indices = sentences_to_indices(X_test, word_to_index, max_len = maxLen) Y_test_oh = convert_to_one_hot(Y_test, C = 5) loss, acc = model.evaluate(X_test_indices, Y_test_oh) print() print("Test accuracy = ", acc) ''' '\nmaxLen = 20\nmodel = Emojify_V2((maxLen,), word_to_vec_map, word_to_index)\nmodel.summary()\nmodel.compile(loss=\'categorical_crossentropy\', optimizer=\'adam\', metrics=[\'accuracy\'])\nX_train_indices = sentences_to_indices(X_train, word_to_index, maxLen)\nY_train_oh = convert_to_one_hot(Y_train, C = 5)\nmodel.fit(X_train_indices, Y_train_oh, epochs = 50, batch_size = 32, shuffle=True)\nX_test_indices = sentences_to_indices(X_test, word_to_index, max_len = maxLen)\nY_test_oh = convert_to_one_hot(Y_test, C = 5)\nloss, acc = model.evaluate(X_test_indices, Y_test_oh)\nprint()\nprint("Test accuracy = ", acc)\n'	2021-04-20 08:09:39	{"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": 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": 0.37344101071357727, "perplexity": 13101.401432048351}, "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/1618039379601.74/warc/CC-MAIN-20210420060507-20210420090507-00195.warc.gz"}
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pja/1146576181	## Proceedings of the Japan Academy, Series A, Mathematical Sciences ### Small gaps between primes exist #### Abstract In the preprint [3], Goldston, Pintz, and Yıldırım established, among other things, $$\liminf_{n\to\infty}{p_{n+1}-p_n\over\log p_n}=0,$$ with $p_n$ the $n$th prime. In the present article, which is essentially self-contained, we shall develop a simplified account of the method used in [3]. We include a short expository last section. #### Article information Source Proc. Japan Acad. Ser. A Math. Sci. Volume 82, Number 4 (2006), 61-65. Dates First available in Project Euclid: 2 May 2006 http://projecteuclid.org/euclid.pja/1146576181 Digital Object Identifier doi:10.3792/pjaa.82.61 Mathematical Reviews number (MathSciNet) MR2222213 Zentralblatt MATH identifier 05123005 Subjects Primary: 11N05: Distribution of primes Secondary: 11P32: Goldbach-type theorems; other additive questions involving primes Keywords Prime number #### Citation Goldston, Daniel Alan; Motohashi, Yoichi; Pintz, János; Yıldırım, Cem Yalçın. Small gaps between primes exist. Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 4, 61--65. doi:10.3792/pjaa.82.61. http://projecteuclid.org/euclid.pja/1146576181. #### References • E. Bombieri, Le grand crible dans la théorie analytique des nombres, second édition revue et augmentée, Astérisque, 18, Soc. Math. France, Paris, 1987. • P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), no. 1, 4–9; Corrigendum, ibid, 28 (1981), 86. • D. A. Goldston, J. Pintz, and C. Y. Y\ild\ir\im, Small gaps between primes II (Preliminary). (February 8, 2005). See also [2005-19 of http://aimath.org/preprints.html]. • E. C. Titchmarsh, The theory of the Riemann zeta-function, Clarendon Press, Oxford, 1951.	2016-07-30 11:09:48	{"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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.8181215524673462, "perplexity": 3989.502594480783}, "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-2016-30/segments/1469257836397.31/warc/CC-MAIN-20160723071036-00125-ip-10-185-27-174.ec2.internal.warc.gz"}
http://www.physicsforums.com/showthread.php?t=115497	# Deserted Island by tuco Tags: deserted, island P: 11 so .. nobody wants to carve the pig for those guys? or is it a dumb question to ask how and why should they devide the pig? lets say they should devide the pig as equally as possible simply because they all worked together, according to their abilities, given task and momentary sutiation. anything else than equal division of the pig would be discriminatory~discriminating as in: to make a difference in treatment or favor on a basis other than individual merit - http://www.webster.com/ would anyone disagree with that statement? and if so why? edit: id like to stress that arguments of type "Einstien wouldnt eat pig..." are besides the point. we could just use person A,B,C,D,E instead but i found that boring. P: 14 Hello. If they didn't have some kind of prior agreement (formal or informal) as to how they would divide the bounty, then there's no way of saying how they would do it. Hitler and arguably SunTzu maybe and even Arnie if he's in his 'strong man' mode, might not even want a 'moral' or 'fair' solution. Hitler wouldn't even be predisposed to give anything to someone who he deemed as being 'weak' or less than human. Remember what the Nazis would have thought of Jews, Einstein being Jewish. So there's really no way of knowing. Ultimately, it seems you're simply asking everyones opinion as to what would constitute the proper, 'moral' outcome for this scenario in any case. Your question seems be the perennial one regarding what one does regarding 'free riders' but it also seems ask what the nature of authority is and how does the use of power affect the moral outcome of situations. My own answer is that they should divide it equally at first but that everyone in his own way must eventually contribute to some extent. That being said however, if it's the 'old Einstein' we're talking about, does one allow 'retirees' on the island, island-citizens who aren't expected to work for their share? Again, my answer would be yes, but they would have to agree to that at first. And, if some aren't predisposed to 'mutual agreement,' then who's to say what the outcome could be? It would merely be survival of the fittest. If one or a few first don't even 'believe in' or 'agree to' moral justice or fairness, then all bets are off anyway. Cheers, mrj P: 44 ## Deserted Island If they were interested in maximizing their chance of survival it seems to me they ought to distribute it according to weight or according to expected energy output (most physical labor). I would also consider this to be the most "fair" or "moral" solution. Would they care (or think about) maximizing their odds of survival and/or come up with the same conclusion? I don't know. Maybe their odds are better with only four or three people. edit: interesting, i realize in retrospect that i associated "their odds" with the best chance of keeping at least one person alive for the longest amount of time, as apposed to say, the whole group for the longest amount of time. P: 174 This is nothing that some paper-rock-scissors couldn't solve rather quickly. What would be more interesting is the moral dillemma they would be in if there was no pig and the had to determine who would be the first to be cannabalized first. That would no doubt be decided by a 4-way battle to the death unless one person willingly ageed to be the sacrifice. Arnie, being the leader as he is, should do the moral thing and offer himself up. P: 11 mrj - so your argument is based on "prior agreement"? that makes sense to me yes. however, there was no prior agreement in this hypothetical situation, nor "free riders" nor "retirees". the summary states the conditions present at the time. Greg825 - according to needs.. thats very good i think. how could we determine if their odds are better with only four or three? with the info we have its clear that if one of them would not cooperate they wouldnt have nothing to roast nor a shelter. RVBuckeye - paper-rock-scissors? is that what you would agree to or like in such situation? personally id be pretty mad if i wouldnt get my "fair" share simply beause i would be hungry. P: 174 Quote by tuco RVBuckeye - paper-rock-scissors? is that what you would agree to or like in such situation? personally id be pretty mad if i wouldnt get my "fair" share simply beause i would be hungry. One pig could feed 4 grown men, with food to spare probably. If there is a food source on the island, I don't think they would even fight over who got what. Emeritus Sci Advisor PF Gold P: 6,238 There was exactly such kind of situation displayed in some reality TV programme over here. People with totally different temperaments were released, without anything else but their clothes, on a deserted island... P: 11 vanesch> was there any kind of reward for the participants of the "reality" TV programme for their participation? RVBuckeye> ok since you seem to resist ;) you avoided my question. lets try one more time and lets try to picture yourself as one of those 5 (not 4 btw) men ok? say you are .. who you want to be? i will be Arnie and you will be Enrique if thats ok. i say: the whole pig is mine! you want a piece you have to fight me for it! or you know what? i will share with all of you but Enrique. what do you do? how would you feel? btw if there was no pig and they had to kill and eat each other to survive, i dont think there would be any dilema at all... survival of the fittest would solve such situation. however, survival of the fittest princile has little to do with morals. P: 174 Quote by tuco RVBuckeye> ok since you seem to resist ;) you avoided my question. lets try one more time and lets try to picture yourself as one of those 5 (not 4 btw) men ok? say you are .. who you want to be? i will be Arnie and you will be Enrique if thats ok. i say: the whole pig is mine! you want a piece you have to fight me for it! or you know what? i will share with all of you but Enrique. what do you do? how would you feel? Reisistance is futile.... OK, what are enriques options? Option one, Enrique could propose to solve the situation by rock-paper-scissors. For the desired result of cooperations and survivability of all. (he is an artist you know, very compassionate) Oprion 2- Get his own pig. This could possibly lead to several outcomes. If he kept it all to himself, he risks angering the rest of the group by not aiding in their survival. Also, he could share with the rest of the group, even Arinie (you), and raise his leadership status among the group. I doubt he would try to exclude Arnie, if he tried to share, because that wouldn't be as effective if his goal was to take over as leader. (He is an artist you know, not wanting to inflame the situation.) Option 3- Take him up on his offer and beat his a. At the risk of being hurt or injured in the process. (Not very likely because he's an artist, you know, he's sensitive) That's is how I think Enrique would respond, and would act in that order. Violence would be a last result. Now, "I" (as myself, the person typing), would probably act choosing option 2, then one, then 3. (i'm not an artist) However, according to how and when your morals kick in, it is my belief that it is completely dependent on the amount of time you are given to react once the situation presents itself. If "I" had to make a split-second decision, (like Arnie was all up in my face trying to exert dominance, and I was starving, no other food in sight), I might choose the 3rd option first because my survival instinct is being threatened. Human beings have the odd ability to act irrationally when presented spur-of-the-moment decisions and would likely react how they were "programmed" to act. Do they follow their survival instinct? (the whole "fight-or-flee" routine) btw if there was no pig and they had to kill and eat each other to survive, i dont think there would be any dilema at all... survival of the fittest would solve such situation. however, survival of the fittest princile has little to do with morals. I think we agree on what would happen in this situation. I just showed above my rationalle on how the two are related. (They are related by the time given for you to think and let your morals kick in) P: 11 you only needed to post the summary anyway, I'd divide the pig equally, regardless, to avoid any dangerous conflict Related Discussions Brain Teasers 14 General Discussion 32 General Discussion 12 General Discussion 24 Nuclear Engineering 9	2014-04-19 09:39:17	{"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": 0.3509134352207184, "perplexity": 1696.1618015707836}, "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-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00401-ip-10-147-4-33.ec2.internal.warc.gz"}
https://tex.stackexchange.com/questions/431794/same-spacing-around-middle-as-around-mid	# Same spacing around “\middle\|” as around “\mid” I need to write some equations that contain something like (A \| B). I could write \left( A \middle\| \right), but this changes spacing around external parentheses, as expained in many topics like this. There is a nice working solution with \DeclarePairedLimiterX as explained in this answer. However, for many reasons I would like to use \middle\| and fix the spacing around it (so that it is the same as around \mid), in the same way as \left and \right are fixed in this answer or by the package mleftright (see this answer). In other words, I would like \| to stretch like \middle\| and to have the same spacing as \mid. Is it possible? I'm using mathtools. Thank you! • Honestly, I've always just used the lazy solution: \,\middle\|\,. – David Richerby May 16 '18 at 8:57 An engine-independent solution (albeit one I still have a murky feeling about...) \documentclass[12pt]{article} \let\originalmiddle=\middle \def\middle#1{\mathrel{}\originalmiddle#1\mathrel{}} \begin{document} $(A \mid B) \left(A \middle\| B\right) \left(\frac{A}{B} \middle\Vert C_{(A \mid B) \left( A \middle\| B \right)} \right)$ \end{document} • +1. And it works with LuaLaTeX too, not just pdfLaTeX... – Mico May 16 '18 at 7:37 • @Mico Well, yes, I meant that Henri's solution depends on lua code while this one doesn't. I'll rephrase. – campa May 16 '18 at 7:39 • Thank you, very simple and also similar to the fixes I used for left and right – Taekwondavide May 16 '18 at 13:23 • @Taekwondavide Maybe instead of redefining \middle (I'm always cautious about redefining TeX primitives) you might want to define something like \mmiddle (like \mleft/\mright). – campa May 16 '18 at 15:26 • Thank you. I'm considering that, I'm still in doubt about using redefined \middle and \originalmiddle or \mmiddle and original \middle. I guess I have to check whether any package is using \middle or not – Taekwondavide May 16 '18 at 16:05 You can use some Lua magic. The three snippets below are one file. They are only split up because Stack Exchange does not support mixed highlighting. \documentclass{article} \usepackage{luacode} \begin{luacode} local noad_id = node.id("noad") local fence_id = node.id("fence") local inner_subtype = 9 -- see texnodes.w local middle_subtype = 2 -- see texnodes.w local function is_vert(delim) return delim.small_fam == 2 and delim.small_char == 106 and delim.large_fam == 3 and delim.large_char == 12 end local kern = node.new("kern",99) -- 99 = math kern kern.kern = 5 2^16 -- 5 pt (TODO: load this value from the font) if n and n.nucleus and n.nucleus.head then elseif n.id == fence_id and n.subtype == middle_subtype then if is_vert(n.delim) then end end end end end, \end{luacode*} \begin{document} $\left( A \middle\| B \right)_{\left( A \middle\| B \right)_{\left( A \middle\| B \right)}}$ $(A \mid B)_{(A \mid B)_{(A \mid B)}}$ \end{document} • I think it's simpler in luatex (Umiddle class 2) – David Carlisle May 16 '18 at 8:18 luatex allows the mathclass to be specified on delimiters and you want 2 (mathbin) here. \documentclass{article} \begin{document} $\left( A \middle\| B \right)$ $\Uleft( A \Umiddle class 2 \| B \Uright)$ \end{document} • the spacing looks off to me and if you omit the B you get a "this can't happen" error and no output, but I expect those issues will get fixed, luatex is under very active development. – David Carlisle May 16 '18 at 8:41 • Cool. Wouldn't mathrel be more appropriate here though? I think \mid has class 3. – Circumscribe May 16 '18 at 8:43 • @Circumscribe possibly (I think spacing is a bit wrong for that as well so it's not too important at the moment which you pick:-) – David Carlisle May 16 '18 at 8:46 • Thank you! I don't know luatex, is it 100% compatible with latex and texlive/miktex? – Taekwondavide May 16 '18 at 15:59 • luatex is included in both texlive and miktex as an alternative to pdftex, it is not 100% compatible. Like xetex it uses (mostly) Unicode fonts installed in the system rather than specific tex-encoded fonts, however most latex documents can be used with xelatex or lualatex with small changes, we have answers on this site giving the differences @Taekwondavide – David Carlisle May 16 '18 at 16:02	2020-04-03 18:19:12	{"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": 2, "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": 0.9429474472999573, "perplexity": 4051.4988063839487}, "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-2020-16/segments/1585370515113.54/warc/CC-MAIN-20200403154746-20200403184746-00047.warc.gz"}
https://www.physicsforums.com/threads/mass-and-linear-absorption-coefficients-of-air-for-cr-k-alpha-x-rays.101916/	# Mass and linear absorption coefficients of air for Cr K(alpha) X-rays 1. Nov 29, 2005 ### asdf1 for this question: calculate the mass and linear absorption coefficients of air for Cr K(alpha) radiation. Assume that air contains 80 percent nitrogen and 20 percent oxygen by weight and has a density of 1.29*10^(-3) g/(cm^3) my problem: i have no idea where to start. 2. Nov 29, 2005 ### inha Mass absorption coeffs. for mixtures are calculated as $$\frac \mu \rho = \Sigma_i w_i \frac {\mu_i} {\rho_i}$$ where the index i runs over the components, w_i are the weighing factors for the components and mu and rho are the abs. coeff and density for the element i. The linear abs. coeff. is straight forward to get from that. 3. Nov 30, 2005 ### asdf1 but the question doesn't give any information about the rho's, so how do you know the values? 4. Nov 30, 2005 ### inha Densities are standard tabulated material. Look the up somewhere. 5. Dec 1, 2005 ### asdf1 ok, thank you very much!!!	2017-05-30 05:50:38	{"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": 0.6310452818870544, "perplexity": 3888.710871465871}, "config": {"markdown_headings": true, "markdown_code": true, "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-2017-22/segments/1495463613796.37/warc/CC-MAIN-20170530051312-20170530071312-00159.warc.gz"}
https://myaptitude.in/science-c10/list-the-factors-on-which-the-resistance-of-a-conductor	# List the factors on which the resistance of a conductor List the factors on which the resistance of a conductor in the shape of a wire depends. Factors on which resistance of a conductor depends: 1. Length of conductor: R ∝ l 2. Area of cross-section of the conductor: R ∝ 1/A	2020-07-10 20:19:04	{"extraction_info": {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8706695437431335, "perplexity": 1001.6749582594697}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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/1593655911896.73/warc/CC-MAIN-20200710175432-20200710205432-00015.warc.gz"}
https://docs.itascacg.com/itasca900/flac3d/zone/test3d/Fluid/WellConfinedAquifer/wellconfinedaquifer.html	FLAC3D Theory and Background • Fluid-Mechanical Interaction # Transient Fluid Flow to a Well in a Shallow Confined Aquifer Note The project file for this example is available to be viewed/run in FLAC3D.[1] The project’s main data files are shown at the end of this example. A shallow confined aquifer of large horizontal extent is characterized by a uniform initial pore pressure, $$p_0$$, and initial isotropic stress, $${\sigma}_{zz}^0$$. A well, fully penetrating the aquifer, is producing water at a constant rate, $$q$$, per unit depth from time, $$t = t_0$$. The elastic porous medium is homogeneous and isotropic, and the flow of groundwater is governed by Darcy’s law. Transient effects are linked to the compressibility of water and the soil matrix. In this problem, the effect of pore-pressure changes are small compared to the overburden, and the vertical stress in the aquifer may be assumed to remain constant with time. Also, horizontal strains are neglected compared to the vertical ones. The conditions of fluid flow to the well are illustrated schematically in Figure 1. The numerical solution to this problem is presented using both coupled and uncoupled modeling approaches. A cylindrical system of coordinates is chosen with the $$z$$-axis pointing upward in the direction of the well axis. Substitution of the transport law in the fluid mass-balance equation gives, taking into consideration that $${\epsilon}_{rr} = {\epsilon}_{\theta \theta}$$ = 0, (1)${{\partial p}\over{\partial t}} = M (k \nabla^2 p - \alpha {{\partial {\epsilon}_{zz}}\over{\partial t}})$ where $$k$$ is the homogeneous permeability coefficient, $$M$$ is the Biot modulus, and $$\alpha$$ is the Biot coefficient. Partial differentiation, with respect to time, of the elastic constitutive relation $${\sigma}_{zz} - {\sigma}_{zz}^0 + \alpha(p - p_0) = \alpha_1 {\epsilon}_{zz}$$ yields, for constant $${\sigma}_{zz}$$, (2)$\alpha {{\partial p}\over{\partial t}} = \alpha_1 {{\partial {\epsilon}_{zz}}\over{\partial t}}$ where $$\alpha_1 = K + 4/3 G$$. Using this last equation to express $${\epsilon}_{zz}$$ in terms of $$p$$ in Equation (1), we obtain, after some manipulations, (3)${{\partial p}\over{\partial t}} = c \nabla^2 p$ where $$c = k/S$$ is the diffusion coefficient, $$S = 1/M + {\alpha}^2 / {\alpha_1}$$ is the storage coefficient, and, with the problem being axisymmetric and not dependent on $$z$$, the Laplacian of $$p$$ may be expressed as (4)$\nabla^2 p = {{\partial^2 p}\over{\partial r^2}} + {{1}\over{r}} {{\partial p}\over{\partial r}}$ The solution to this differential equation with boundary conditions (5)$\lim_{r \rightarrow \infty} p = p_0$ (6)$\lim_{r \rightarrow 0} 2 \pi r {{\partial p}\over{\partial r}} = {{q}\over{k}}$ is due to Theis (1935). It has the form (7)$\hat{p} = - {{1}\over{4 \pi}} E_1(u) + \hat{p_0}$ where $$\hat{p} = p k/q$$. The dimensionless variable $$u$$ is given by (8)$u = {{r^2}\over{4c(t - t_0)}}$ and $$E_1$$ is the exponential integral, defined as (9)$E_1(u) = \int_u^\infty {e^{-\xi}\over{\xi}} d \xi$ The vertical displacement may be obtained by integration of the equilibrium equation $$\partial {\sigma}_{zz}/\partial z$$ = 0 after expressing $$\sigma_{zz}$$ in terms of $$\epsilon_{zz}$$ by means of the mechanical constitutive equation and substituting $$\partial \epsilon_{zz} / \partial z$$ for $$\epsilon_{zz}$$. This yields, after substitution of the boundary condition, and using Equation (7), (10)${\hat{u}}_z = - {{\hat{z}\over{4 \pi}} E_1(u)}$ where $${\hat{u}}_z = u k \alpha_1 /(\alpha q H)$$ and $${\hat{x}} = z/H$$. The stresses are derived from the mechanical constitutive equations and Equation (7) for $$\hat{p}$$. They have the form (11)${\hat{\sigma}}_{rr} = {\hat{\sigma}}_{\theta \theta} = {{1}\over{2 \pi}} E_1(u) + {\hat{\sigma}}_{zz}^0$ where $$\hat{\sigma} = \sigma k \alpha_1 / (q \alpha G)$$. The FLAC3D grid for this problem corresponds to a nine-degree wedge in a hollow cylinder of unit height. The axis and radius of the well correspond to cylinder axis and radius, respectively. The cylinder outer radius is selected as 100 m to model the far boundary of the flow domain. Figure 2 shows the FLAC3D grid. A total of 31 zones are used, lined up, and graded in the radial direction. The displacements are fixed in the radial and tangential directions, and in the vertical direction at the cylinder base. A vertical pressure of magnitude $$- \sigma_{zz}^0$$ is applied at the top of the model. The properties for this example are defined: dry bulk modulus ($$K$$) 118 MPa dry shear modulus ($$G$$) 71 MPa water bulk modulus ($$K_f$$) 2 GPa Biot coefficient ($$\alpha$$) 1.0 porosity ($$n$$) 0.4 permeability ($$k$$) 2.98 × 10-8 $${m^2}/{Pa-sec}$$ The initial pore pressure is 147 kPa, and the initial isotropic stress is -147 kPa. Because the Biot coefficient is equal to one (incompressible soil grains), the Biot modulus is equal to the ratio between water bulk modulus and porosity (in this case, $$M$$ = 5 GPa). The well-pumping rate per unit aquifer thickness is 2.21 10-3 m2/s, and the well radius, $$r_w$$, is selected as 1 m. Stresses and pore pressures are initialized to the values given above. The well flow-rate is modeled as a surface flux of magnitude $$q/(2 \pi r_w)$$ applied to the well radius $$r = r_w$$. The coupled problem is solved (model fluid active on and model mechanical active on) using the explicit solution algorithm. The maximum out-of-balance mechanical force is limited to 10.0, the maximum number of mechanical sub-steps in the coupled fluid-mechanical calculation step is limited to 1000, and the mechanical process is the “slave” module to the master fluid-flow process. This is accomplished with the commands model fluid substep 100 model mechanical substep 1000 model mechanical slave on By specifying these commands, the out-of-balance mechanical force will be kept to a small value while the fluid-flow calculation proceeds. This example is pore-pressure driven, and the value for the stiffness ratio, $$R_k$$, is approximately 23 for the specified fluid bulk modulus of 2 GPa. Thus, the flow calculation may be uncoupled from the mechanical calculation, and the approach discussed in Stiffness Ratio may be applied. The fluid modulus during the flow-only step is defined by Equation (7) in order to preserve the diffusivity of the system. During the mechanical-only step, the fluid modulus is set to zero to prevent further adjustments by volumetric strains. The following commands are applied for the uncoupled calculation for a 4-second flow time: model fluid active on model mechanical active off zone gridpoint initialize fluid-modulus = @uwb model solve time-total 4. model fluid active off model mechanical active on zone gridpoint initialize fluid-modulus = 0.0 model solve unbalanced-maximum 10. The FISH variable uwb is the adjusted fluid modulus calculated by Equation (7). Note that, if conditions are such that $$R_k <<< 1$$, it is not necessary to adjust the fluid modulus during the flow calculation because the diffusivity will be accurate. The analytical solutions for pore pressure, stresses, and vertical displacement are programmed as a FISH function. The exponential integral function used in the analytical solutions is programmed as a separate FISH function. Analytical and numerical values are stored in tables. The results are then compared in graphical form. The pore-pressure comparison at selected times is presented in Figure 3 for the coupled solution, and Figure 4 for the uncoupled solution. The vertical displacement values and stresses at 32 seconds are processed by the FISH function well_32 and are illustrated in Figure 5 and Figure 7 for the coupled solution, and in Figure 6 and Figure 8 for the uncoupled solution. The results for both the coupled and uncoupled solutions are essentially identical and compare well with the analytical solution. The uncoupled solution is reached much more quickly than the coupled solution. Note that the coupled calculation requires more than 500,000 steps, while the uncoupled calculation requires approximately 16,000 steps. Reference Theis, C. V. “The Relation between the Lowering of the Piezometric Surface and the Rate and Duration of Discharge of a Well Using Groundwater Storage,” Trans. Am. Geophys. Union, 10, 519-524 (1935). Data File WellConfinedAquifer.dat model new model title 'Transient fluid flow to a well in a shallow confined aquifer' fish automatic-create off model configure fluid ; --- model geometry (hollow cylinder - 9 degree wedge) --- zone create brick point 0 (1,0,0) point 1 (100,0,0) ... point 2 (0.987688,0.156434,0) point 3 (1,0,1) ... point 4 (98.7688,15.6434,0) point 5 (0.987688,0.156434,1) ... point 6 (100,0,1) point 7 (98.7688,15.6434,1) ... size (31,1,1) ratio (1.1,1,1) zone face skin zone gridpoint group 'xline1' range position (1,0,0) (100,0,0) zone gridpoint group 'xline2' range position (5,0,1) (100,0,1) ; --- mechanical model --- zone cmodel assign elastic zone property bulk 11.8e7 shear 7.1e7 zone face apply velocity-x 0 zone face apply velocity-y 0 zone face apply velocity-z 0 range group 'Bottom' zone initialize stress xx -1.47e5 yy -1.47e5 zz -1.47e5 zone face apply stress-zz -1.47e5 range group 'Top' ; --- fluid flow model --- zone fluid cmodel assign isotropic zone fluid property permeability 2.98e-8 porosity 0.4 zone gridpoint initialize fluid-modulus 2e9 zone gridpoint initialize pore-pressure 147000 ; [] = Equivalent flux zone face apply discharge [-2.21e-3/(2math.pi)] range group 'West' ; --- setting --- model large-strain off model fluid active on model save 'well-ini' ; --- pumping (coupled analysis) --- model fluid substep 100 model mechanical substep 1000 model mechanical follower on model solve fluid time-total 4 mechanical convergence 1 model save 'well-cpl4' model solve fluid time-total 8 mechanical convergence 1 model save 'well-cpl8' model solve fluid time-total 16 mechanical convergence 1 model save 'well-cpl16' model solve fluid time-total 32 mechanical convergence 1 model save 'well-cpl32' ; --- pumping (uncoupled analysis) --- model restore 'well-ini' [global fmod = 0.4 / ((0.4/2e9)+(1.0/(11.8e7 + (4/3)7.1e7)))] model fluid active on model mechanical active off zone gridpoint initialize fluid-modulus = [fmod] model solve fluid time-total 4 model fluid active off model mechanical active on zone gridpoint initialize fluid-modulus = 0 model solve convergence 1 model save 'well-ucpl4' model fluid active on model mechanical active off zone gridpoint initialize fluid-modulus = [fmod] model solve fluid time-total 8 model fluid active off model mechanical active on zone gridpoint initialize fluid-modulus = 0 model solve convergence 1 model save 'well-ucpl8' model fluid active on model mechanical active off zone gridpoint initialize fluid-modulus = [fmod] model solve fluid time-total 16 model fluid active off model mechanical active on zone gridpoint initialize fluid-modulus = 0 model solve convergence 1 model save 'well-ucpl16' model fluid active on model mechanical active off zone gridpoint initialize fluid-modulus = [fmod] model solve fluid time-total 32 model fluid active off model mechanical active on zone gridpoint initialize fluid-modulus = 0 model solve convergence 1 model save 'well-ucpl32' Endnotes	2023-03-31 19:49:12	{"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": 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": 0.6942366361618042, "perplexity": 4149.236600980521}, "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-2023-14/segments/1679296949678.39/warc/CC-MAIN-20230331175950-20230331205950-00047.warc.gz"}
https://www.ideals.illinois.edu/handle/2142/97108	## Files in this item FilesDescriptionFormat application/pdf 2774.pdf (16kB) (no description provided)PDF ## Description Title: ABSOLUTE CONFIGURATION OF 3-METHYLCYCLOHEXANONE BY CHIRAL TAG ROTATIONAL SPECTROSCOPY AND VIBRATIONAL CIRCULAR DICHROISM Author(s): Holdren, Martin S. Contributor(s): Pate, Brooks; West, Channing; Smart, Taylor; Mayer, Kevin J; Evangelisti, Luca Subject(s): Mini-symposium: Chirality-Sensitive Spectroscopy Abstract: The absolute configuration of 3-methylcyclohexanone was established by chiral tag rotational spectroscopy measurements using 3-butyn-2-ol as the tag partner. This molecule was chosen because it is a benchmark measurement for vibrational circular dichroism (VCD). A comparison of the analysis approaches of chiral tag rotational spectroscopy and VCD will be presented. One important issue in chiral analysis by both methods is the conformational flexibility of the molecule being analyzed. The analysis of conformational composition of samples will be illustrated. In this case, the high spectral resolution of molecular rotational spectroscopy and potential for spectral simplification by conformational cooling in the pulsed jet expansion are advantages for chiral tag spectroscopy. The computational chemistry requirements for the two methods will also be discussed. In this case, the need to perform conformer searches for weakly bound complexes and to perform reasonably high level quantum chemistry geometry optimizations on these complexes makes the computational time requirements less favorable for chiral tag rotational spectroscopy. Finally, the issue of reliability of the determination of the absolute configuration will be considered. In this case, rotational spectroscopy offers a “gold standard” analysis method through the determination of the $^{13}$C-subsitution structure of the complex between 3-methylcyclohexanone and an enantiopure sample of the 3-butyn-2-ol tag. Issue Date: 6/22/2017 Publisher: International Symposium on Molecular Spectroscopy Citation Info: APS Genre: CONFERENCE PAPER/PRESENTATION Type: Text Language: English URI: http://hdl.handle.net/2142/97108 DOI: 10.15278/isms.2017.RG10 Date Available in IDEALS: 2017-07-27	2018-01-22 04:16:02	{"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": 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": 0.5427632331848145, "perplexity": 6143.193700936919}, "config": {"markdown_headings": true, "markdown_code": true, "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-2018-05/segments/1516084890991.69/warc/CC-MAIN-20180122034327-20180122054327-00573.warc.gz"}
https://zbmath.org/?q=an%3A0973.81546	# zbMATH — the first resource for mathematics Kernel estimation in high-energy physics. (English) Zbl 0973.81546 Summary: Kernel estimation provides an unbinned and non-parametric estimate of the probability density function from which a set of data is drawn. In the first section, after a brief discussion on parametric and non-parametric methods, the theory of kernel estimation is developed for univariate and multivariate settings. The second section discusses some of the applications of kernel estimation to high-energy physics. The third section provides an overview of the available univariate and multivariate packages. This paper concludes with a discussion of the inherent advantages of kernel estimation techniques and systematic errors associated with the estimation of parent distributions. ##### MSC: 81V05 Strong interaction, including quantum chromodynamics 81-04 Software, source code, etc. for problems pertaining to quantum theory Full Text: ##### References: [1] Scott, D., Multivariate density estimation: theory, practice, and visualization, (1992), John Wiley and Sons Inc New York · Zbl 0850.62006 [2] Abramson, I., On bandwidth variation in kernel estimates: a square root law, Ann. statist., 10, 1217-1223, (1982) · Zbl 0507.62040 [3] Cranmer, K., Kernel estimation for parametrization of discriminant variable distributions, ALEPH 99-144 PHYSICS 99-056, (1999) [4] Search for neutral Higgs bosons in e^+e− collisions at $$s≈$$ 192-202 gev, OPAL physics note PN426, (2000) [5] Hu, H.; Nielsen, J., Analytic confidence level calculations using the likelihood ratio and Fourier transform, (), 109 [6] The Ba$$B$$ar Collaboration, The Ba$$B$$ar Physics Book, See Section 5.5.4 [7] Knuteson, B.; Miettinen, H.; Holmström, L., Mass analysis and parameter estimation with PDE, D∅ note 003396, (1998) [8] Holmström, L.; Miettinen, H.; Sain, S.R., A new multivariate technique for top quark search, Comput. phys. comm., 88, 195-210, (1995) [9] Miettinen, H.; Epply, G., Possible hint of top → e^+$$E̷t+$$ jets, D∅ note 002145, (1994) [10] Miettinen, H., Top quark results from D∅, D∅ note 002527, (1995) [11] Frodesen, A.; Skjeggestad, O., Probab. statist. particle phys., 424-427, (1979) [12] Allison, J., Multiquadric radial basis functions for representing multidimensional high energy physics data, Comput. phys. comm., 77, 377-395, (1993) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.	2021-05-16 01:59:53	{"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": 0.7683843970298767, "perplexity": 5673.667571655876}, "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-2021-21/segments/1620243991659.54/warc/CC-MAIN-20210516013713-20210516043713-00457.warc.gz"}
https://dsp.stackexchange.com/questions/80/what-are-the-proper-pre-processing-steps-to-perform-independent-component-analys	# What are the proper pre-processing steps to perform Independent Component Analysis? What are the proper steps for preprocessing my waveforms in order to perform an independent component analysis (ICA) later? I understand the how, though further explanation of that doesn't hurt, but I'm more interested in the why. • I'm not sure why you need preprocessing. Is there any particular reason? – Phonon Aug 19 '11 at 20:45 • @Phonon I've encountered investigators that have sphered their data before performing ICA on it. I just wondered if there was a standard method. – jonsca Aug 19 '11 at 20:46 • Very interesting. I'd love to see a constructive answer. – Phonon Aug 19 '11 at 20:48 • In the case of spectral analysis on EEG signals, people whiten to reduce the dominating effect of the ${1}/{f}$ shape of the spectrum, which often hides interesting things at high frequencies. There's at least a little discussion of this here in the supplemental materials. Whether this is a common trick before ICA in particular, not sure. Is your application EEG/MEG/LFP signals? Maybe someone who does ICA can flesh this out into a full answer, if my hunch is right. Interesting question - I'll read up on it. – ImAlsoGreg Aug 20 '11 at 0:13 • @Gigili That's part of the question, too. Which are the ones considered to be the normal steps? – jonsca Aug 20 '11 at 23:24 Independent component analysis (ICA) is used to separate a linear mixture of statistically independent and most importantly, non-Gaussian components into its constituents. The standard model for a noise-free ICA is $$\mathbf{x}=\mathbf{As}$$ where $\mathbf{x}$ is the observation or data vector, $\mathbf{s}$ is a source signal/original components (non-Gaussian) and $\mathbf{A}$ is a transformation vector that defines the linear mixing of the constituent signals. Typically, $\mathbf{A}$ and $\mathbf{s}$ are unknown. ### Pre-processing There are two main pre-processing strategies in ICA, namely centering and whitening/sphering. The primary reasons for pre-processing are: • Simplification of algorithms • Reduction of dimensionality of the problem • Reduction of number of parameters to be estimated. • Highlighting features of the data set not readily explained by the mean and covariance. From the introduction of G. Li and J. Zhang, "Sphering and its properties", The Indian Journal of Statistics, Vol. 60, Series A, Part I, pp. 119-133, 1998: Outliers, clusters or other kind of groups, and concentrations near curves or non-ﬂat surfaces are probably the important features that interest data analysts. They are, in general, not obtainable through mere knowledge of the sample mean and covariance matrix. In these circumstances, it is desirable to separate oﬀ the information contained in the mean and the covariance matrices and forces us to examine aspects of our data sets other than those well-understood natures. Centering and sphering is a simple and intuitive approach that eliminates the mean-covariance information and helps to highlight structures beyond linear correlation and elliptic shapes, and therefore is often performed before exploring displays or analyses of data sets 1. Centering: Centering is a very simple operation and simply refers to subtracting the mean $\mathbb{E}\{\mathbf{x}\}$. In practice, you use the sample mean and create a new vector $\mathbf{x}_c=\mathbf{x}-\overline{\mathbf{x}}$, where $\overline{\mathbf{x}}$ is the mean of the data. Geometrically, subtracting the mean is equivalent to translating the center of coordinates to the origin. The mean can always be re-added to the result the end (this is possible because matrix multiplication is distributive). 2. Whitening: Whitening is a transformation that converts the data such that it has an identity covariance matrix, i.e., $\mathbb{E}\{\mathbf{x}_c\mathbf{x}_c^T\}=\mathbf{I}$. Normally, you work with the sample covariance matrix, $$\widehat{\mathbf{\Sigma}}=C.\mathbf{x}_c\mathbf{x}_c^T$$ where $C$ is just my lazy placeholder for the appropriate normalization factor (depending on the dimensions of $\mathbf{x}$). A new whitened vector is created as $$\mathbf{x}_w=\widehat{\mathbf{\Sigma}}^{-1/2}\mathbf{x}_c$$ which will have a covariance of $\mathbf{I}$. Geometrically, whitening is a scaling transformation. Here is a small example in Mathematica: s = RandomReal[{-1, 1}, {2000, 2}]; A = {{2, 3}, {4, 2}}; x = s.A; whiteningMatrix = Inverse@CholeskyDecomposition[Transpose@x.x/Length@x]; y = x.whiteningMatrix; FullGraphics@GraphicsRow[ ListPlot[#, AspectRatio -> 1, Frame -> True] & /@ {s, x, y}] The first plot is the joint density of two uniformly distributed random vectors, or the components $\mathbf{s}$. The second shows the effect of multiplying by a transformation vector $\mathbf{A}$. The square gets skewed and scaled into a rhombus. By multiplying with the whitening matrix, the joint density is back to a square which is slightly rotated from the original. Because of the whitening transformation, in the new system that is being solved, i.e. $\mathbf{x}_w=\mathbf{A}_w\mathbf{s}_w$, $\mathbf{A}_w$ is an orthogonal matrix. This can be easily shown: \begin{align} \mathbb{E}\{\mathbf{x}_w\mathbf{x}_w^T\}&=\mathbb{E}\{\mathbf{A}_w\mathbf{s}_w(\mathbf{A}_w\mathbf{s}_w)^T\}\\ &=\mathbf{A}_w\mathbb{E}\{\mathbf{s}_w\mathbf{s}_w^T\}\mathbf{A}_w^T\\ &=\mathbf{A}_w\mathbf{A}_w^T=\mathbf{I} \end{align} where the last step follows because of the statistical independence of $\mathbf{s}_i$ The orthogonality condition means that there are only about half as many parameters that need to be estimated. (Note: Although this is true in this case and in my example, $\mathbf{A}$ need not be square to begin with). If, after the transformation, there are eigenvalues close to zero, then these can be safely discarded as they are just noise and will only hamper the estimation due to "overlearning". 3. Other pre-processing There might be other pre-processing steps involved in certain specific applications that are impossible to cover in an answer. For example, I've seen a few articles which use the log of the time-series and a few others that filter the time-series. While it might be suited for their particular application/conditions, the results don't carry over to all fields. I believe it is possible to use ICA if at most one of the components is Gaussian, although I can't find a reference for this right now. ### Why is it called "sphering"? This is probably well known, but just as a fun fact, sphering comes from the change in the structure of covariance matrices in the case of Gaussian components from an $n$-dimensional hyper ellipsoid to an $n$-dimensional sphere due to whitening. Here's an example (use the same code as above, but replace {-1,1} with NormalDistribution[]) The first is the joint density for two uncorrelated Gaussians, the second under transformation and the third is after whitening. In practice only steps 2 and 3 are visible. • Wow, It's going to take me a bit to take that all in, but thanks is an understatement! – jonsca Aug 23 '11 at 7:52 • Sorry, I thought I'd accepted it already. – jonsca Aug 24 '11 at 8:42	2020-08-15 02:55:51	{"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": 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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7444453239440918, "perplexity": 585.0617846156646}, "config": {"markdown_headings": true, "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-2020-34/segments/1596439740423.36/warc/CC-MAIN-20200815005453-20200815035453-00107.warc.gz"}
https://www.ideals.illinois.edu/handle/2142/19470	## Files in this item FilesDescriptionFormat application/pdf 9305580.pdf (4Mb) (no description provided)PDF ## Description Title: Three-dimensional analysis of the flight phase in the long jump Author(s): Kim, Ky-Hyoung Doctoral Committee Chair(s): Carlton, Les G. Department / Program: Kinesiology and Community Health Discipline: Kinesiology Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Education, Physical Abstract: The effect of the flight phase on jumping distance was examined using simulation procedures. Most previous studies of the long jump have focused on the approach and take-off phases of the long jump where the jumper makes contact with the track surface. Under these conditions, force can be measured directly using force transducers, and the force measures can be correlated with the distance jumped. The flight phase has received less attention, partly because the trajectory of the center of mass of the whole body is prescribed at take-off and cannot be changed during flight (ignoring aerodynamic effects), and partly because kinetic variables can only be approximated based on measured kinematic patterns.The purpose of the study was to investigate the characteristics of the flight phase as a function of the angular momentum developed by the jumper. A medium distance collegiate jumper was used to obtain kinematic data from the flight phase of a hitch-kick style long jump. Three dimensional filming techniques were used and the Direct Linear Transformation method was used to obtain three dimensional coordinates. A 15 segment physical model was developed for simulation, with kinetic variables approximated using a rigid-body inverse dynamics approach. In the simulation the angular momentum was changed by altering the body configuration immediately after take-off. The approach velocity and take-off angle were unchanged.The initial body configuration of the jumper was changed in 0.1 radian steps up to +/$-$ 0.2 radians. Each change in body configuration was accompanied by a change in system angular velocity in order to have the simulated jumper land with an appropriate body configuration. The four paired body rotation (in radians) and angular velocity (in degree/s) changes were: 0.1/5.5; 0.2/11.0; $-$0.1/$-$5.5, $-$0.2/$-$11.0.The simulations revealed that the changes in the initial conditions resulted in modifications in the angular momentum and the movement patterns of the arms and legs during the flight phase. Even though the appropriate land posture was approximated in each simulation, the distance jumped, as indexed by the distance covered by the heel of the take-off foot, varied with changes in initial body configuration. Issue Date: 1992 Type: Text Language: English URI: http://hdl.handle.net/2142/19470 Rights Information: Copyright 1992 Kim, Ky-Hyoung Date Available in IDEALS: 2011-05-07 Identifier in Online Catalog: AAI9305580 OCLC Identifier: (UMI)AAI9305580	2015-06-30 16:50:31	{"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": 0.5734219551086426, "perplexity": 2594.250735475144}, "config": {"markdown_headings": true, "markdown_code": true, "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-2015-27/segments/1435375094451.94/warc/CC-MAIN-20150627031814-00189-ip-10-179-60-89.ec2.internal.warc.gz"}
https://mathoverflow.net/questions/290340/direct-bijections-for-s-t-fibonomial-identities/290477	# Direct bijections for $s,t$-Fibonomial identities Sagan and Savage gave a combinatorial interpretation of a polynomial generalization of Fibonomial coefficients. Their proof uses the recurrence relation for the Lucas polynomials that generalize the Fibonacci numbers, namely $\{0\}=0$, $\{1\}=1$, $\{n\}=s\{n-1\}+t\{n-2\}$ for $n\ge 2$. Define $s$ as a weight of a monomino, $t$ is a weight of a domino, and extend the weight function multiplicatively as usual. Then the $s,t$-Fibonacci number $\{n+1\}$ is the sum of all possible weights of an $n\times 1$ strip of cells. Subsequently, $\{n\}!=\{n\}\{n-1\}\dots\{1\}$, $\{0\}!=1$, and the $s,t$-Fibonomial coefficient $${m+n \brace m}=\frac{\{m+n\}!}{\{m\}!\{n\}!}$$ is the sum of weights of fillings of an $m\times n$ rectangle by pairs of complementary partitions $(\lambda,\lambda^)$ so that rows of $\lambda$ and columns of $\lambda^$ are filled by monominos and dominos, and every column of $\lambda^*$ starts with a domino (see Figure 2 in the paper). The proof that ${m+n \brace n}$ is a combinatorial interpretation of the $s,t$-Fibonomial coefficients is recursive, using the identity $${m+n \brace m}=\{n+1\}{m+n-1 \brace m-1} + t\{m-1\}{m+n-1 \brace m}.$$ However, I am wondering if there have since been any direct bijections, using this combinatorial interpretation of $s,t$-Fibonomials, corresponding to the following identities: $$\begin{split} {m+n \brace m}&={m+n \brace n}\\ {m+n+1 \brace m+1}\{m+1\}&={m+n+1 \brace m}\{n+1\}={m+n \brace m}\{m+n+1\}\\ {m+n \brace m}\{m\}!&=\{m+n\}\{m+n-1\}\dots\{m+1\} \end{split}$$ • Thanks, Bruce, and welcome to MO! I actually started by thinking about FiboCatalan polynomials and then realized that to prove $\frac{1}{\{n+1\}}{2n \brace n}={2n-1 \brace n-1}+t{2n-1 \brace n-2}$ combinatorially from your recurrence, I would need to show $\{n-1\}{2n-1 \brace n}=\{n+1\}{2n-1 \brace n+1}$. This implied looking at the identities I listed. – Alexander Burstein Jan 11 '18 at 21:12	2019-12-15 02:36:41	{"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": 0.9494264721870422, "perplexity": 390.1643266148939}, "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-2019-51/segments/1575541301014.74/warc/CC-MAIN-20191215015215-20191215043215-00428.warc.gz"}
https://www.math.emory.edu/events/seminars/	# Upcoming Seminars Title: Counting Elliptic Curves Over Number Fields Seminar: Algebra Speaker: Tristan Phillips of The University of Arizona Contact: David Zureick-Brown, david.zureick-brown@emory.edu Date: 2022-09-27 at 4:00PM Venue: MSC N304 Abstract: Let $E$ be an elliptic curve over a number field $K$. The Mordell--Weil Theorem states that the set of rational points $E(K)$ of $E$ forms a finitely generated abelian group. In particular, we may write $E(K) = E(K)_{tors}\oplus \mathbb{Z}^r$, where $E(K)_{tors}$ is a finite torsion group, called the torsion subgroup of $E$, and $r$ is a non-negative integer, called the rank of $E$. In this talk I will discuss some results regarding how frequently elliptic curves with a prescribed torsion subgroup occur, and how one can bound the average analytic rank of elliptic curves over number fields. One of the main ideas behind these results is to use methods from Diophantine geometry to count points of bounded height on modular curves. Title: About the Lp theory for the non-cutoff Boltzmann equation Seminar: Analysis and Differential Geometry Speaker: Ricardo Alonso of Texas A$\&$M at Qatar Date: 2022-09-29 at 4:00PM Venue: MSC W301 Abstract: In this talk we discuss different technical elements to obtain a priori estimates for Lp norms of weak solutions to non-cutoff kinetic equations using as example the homogeneous/inhomogeneous Boltzmann equation. Rather than a detailed-proof talk, we point out difficulties and give some intuition related to the main steps of the strategy. In particular, we discuss the localization process of Boltzmann type operators which cover an ample range of operators such as the fractional Laplacian. Title: TBA Seminar: Algebra Speaker: David Stapleton of The University of Michigan Contact: David Zureick-Brown, david.zureick-brown@emory.edu Date: 2022-10-04 at 4:00PM Venue: MSC N304 Abstract: Title: Patch Normalizing Regularizer: Reconstruction using only one ground truth image Seminar: Computational and Data Enabled Science Speaker: Paul Hagemann of TU Berlin Contact: Lars Ruthotto, lruthotto@emory.edu Date: 2022-10-06 at 10:00AM Venue: MSC W301 Abstract: Reconstructing images from measurements (e.g. sinograms in CT) is a very active research topic. However in many domains, such as medical or material sciences, ground truth data is very hard or costly to obtain. In this talk, we will leverage the idea of patch-based learning for reconstructing images. The regularizer will learn the patch distribution from very few ground truth images by randomly subsampling 6x6 patches and learning their distribution. More specifically, we will use a normalizing flow to learn the patch distribution of the ground truth image, which we call patchNR. In reconstruction, we will minimize a sum of the negative log likelihood of the patches and the data fidelity term. Our method will be compared to other regularization techniques which use little data for CT, material and texture images. Furthermore, an outlook on how our method can be leveraged to perform zero shot superresolution will be given. Title: Complexities of the Cytoskeleton: Integration of Scales Seminar: Computational and Data Enabled Science Speaker: Keisha Cook of Clemson University Contact: Jim Nagy, jnagy@emory.edu Date: 2022-10-07 at 1:00PM Venue: MSC W301 Abstract: Biological systems are traditionally studied as isolated processes (e.g. regulatory pathways, motor protein dynamics, transport of organelles, etc.). Although more recent approaches have been developed to study whole cell dynamics, integrating knowledge across biological levels remains largely unexplored. In experimental processes, we assume that the state of the system is unknown until we sample it. Many scales are necessary to quantify the dynamics of different processes. These may include a magnitude of measurements, multiple detection intensities, or variation in the magnitude of observations. The interconnection between scales, where events happening at one scale are directly influencing events occurring at other scales, can be accomplished using mathematical tools for integration to connect and predict complex biological outcomes. In this work we focus on building inference methods to study the complexity of the cytoskeleton from one scale to another. Title: TBA Seminar: Algebra Speaker: Brandon Alberts of Eastern Michigan University Contact: David Zureick-Brown, david.zureick-brown@emory.edu Date: 2022-10-14 at 4:00PM Venue: MSC W301 Abstract: Title: TBA Seminar: Algebra Speaker: Soumya Sankar of The Ohio State University Contact: David Zureick-Brwon, david.zureick-brown@emory.edu Date: 2022-10-14 at 5:15PM Venue: MSC W301 Abstract: Title: Subspace configurations and low degree points on curves Seminar: Algebra Speaker: Borys Kadets of University of Georgia Contact: David Zureick-Brown, david.zureick-brown@emory.edu Date: 2022-10-18 at 4:00PM Venue: MSC N304 Abstract: The hyperelliptic curve given by the equation $y^2=f(x)$ with coefficients in $\mathbf{Q}$ has an unusual arithmetic property: it admits infinitely many points with coordinates in quadratic extensions of $\mathbf{Q}$ (namely $(a, \sqrt{f(a)})$). Hindry, motivated by arithmetic questions about modular curves, asked if the only curves that possess infinite collections of quadratic points are hyperelliptic and bielliptic; this conjecture was confirmed by Harris and Silverman. I will talk about the general problem of classifying curves that possess infinite collections of degree $d$ points. I will explain how to reduce this classification problem to a study of curves of low genus, and use this reduction to obtain a classification for $d \leq 5$. This relies on analyzing a discrete-geometric object -- the subspace configuration -- attached to curves with infinitely many degree $d$ points. This talk is based on joint work with Isabel Vogt (arXiv:2208.01067). Title: TBA Seminar: Algebra Speaker: Daniel Keliher of University of Georgia Contact: David Zureick-Brown, david.zureick-brown@emory.edu Date: 2022-11-01 at 4:00PM Venue: MSC N304 Abstract: Title: Athens-Atlanta joint Number Theory Seminar Seminar: Algebra Speaker: Alina Bucur and Samit Dasgupta of USCD and Duke Contact: David Zureick-Brown, david.zureick-brown@emory.edu Date: 2022-11-07 at 4:00PM Venue: MSC W301 Abstract:	2022-09-25 07:06:05	{"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": 0.5983506441116333, "perplexity": 1815.6594005561046}, "config": {"markdown_headings": true, "markdown_code": true, "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-2022-40/segments/1664030334515.14/warc/CC-MAIN-20220925070216-20220925100216-00687.warc.gz"}
https://www.greencarcongress.com/2011/08/eiaaltf-20110811.html	## EIA: consumption of alternative transportation fuels held steady in 2009 ##### 11 August 2011 The US Energy Information Administration (EIA) reports that consumption of alternative transportation fuels held steady in 2009, with a total of 431,107 thousand gasoline-equivalent gallons, compared to 430,329 thousand gasoline-equivalent gallons in 2008. Estimated consumption of other vehicles fuels in 2009 was, according to EIA: • Gasoline: 134,385,175 thousand gallons • Diesel: 37,701,896 thousand gasoline equivalent gallons • Biodiesel: 325,102 thousand gasoline equivalent gallons Consumption of alternative fuels by fuel type, 2009. Source: EIA. Click to enlarge. The natural gas share accounted for about 52% of all alternative fuels consumed by alternative transportation fuel vehicles (AFVs). Propane and E85 accounted for 30% and 16% respectively, while electricity, hydrogen, and other fuels accounted for the remaining 2%. While the consumption of natural gas over the past five years in AFVs has increased due to its predominant use in the transit bus industry, consumption of propane has decreased from 45% of the overall alternative fuel consumption in 2008 to 30% in 2009. The consumption of propane in heavy duty vehicles has remained relatively constant over the past five years; however, consumption of propane in light duty and medium duty vehicles has dropped significantly due to fleet retirements in these categories. Many fleets have replaced their light duty vehicles with flexible fueled and gasoline hybrid vehicles.	2022-12-04 02:08:27	{"extraction_info": {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9717298746109009, "perplexity": 9392.839649062784}, "config": {"markdown_headings": true, "markdown_code": true, "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-2022-49/segments/1669446710953.78/warc/CC-MAIN-20221204004054-20221204034054-00126.warc.gz"}
https://www.physicsforums.com/threads/the-role-of-gravity-in-terms-of-work-done-by-an-angled-force-on-an-incline.127515/	# The role of gravity in terms of work done by an angled force on an incline 1. Jul 29, 2006 ### Cali Problem's exact wording: When we looked at the work done by a force F, up a ramp, we were confused about the role of gravity that acts in the down direction. Investigate what work , if any gravity does and how this influences the applied force up the ramp acting on a mass. Note on question: The applied force is at an angle to the ramp, and this was asked as part of a culminating activity in a gr.12 geometry and discrete math course. My thoughts while looking at my pretty diagram: There's the normal acting perpendicular to the surface, gravity acting downwards, and an applied force acting at angle to the the ramp. So, gravity doesn't really do any work, since it's acting more like friction, the object it is acting on needs a greater applied force to cause motion. Gravity itself is not causing any displacement, and therefore no work. The gravity that acts in the down direction should be broken into its components. The horizontal component is acting in the direction of the ramp's incline,adding to the applied force? While the angled component has the same angle as the ramp, and acts opposite to the applied force, just like friction? except for the angle? Is what I'm thinking right? Help? Please? 2. Jul 29, 2006 ### Meson When the gravity force is broken into the appropriate components, i) one is acting along the horizontal axis along the ramp and it is added algebraically to the horizontal component of the applied force along the ramp. ii) the second gravity force component is acting along the vertical axis of the ramp (perpendicular to the surface of the ramp), parallel to the normal force. Keep in mind that the work done of gravity force does not depend on the angle of the inclination of the ramp. It only depends on the vertical height. As there is a change in height of the mass as the mass moves upward the ramp, there is work done by the gravity force, which is a negative work with respect to a conventional reference frame, if the direction of the applied force is assumed to be positive. 3. Jul 29, 2006 ### Hootenanny Staff Emeritus Just a clarification here. No component of the weight acts horizontally; one component however, does act parallel to the incline (down the slope). The below diagram should clarify further; Here we can see that the force of gravity can be divided into two components $mg\sin\theta$, which acts perpendicular to the incline and $mg\cos\theta$ which acts parallel to the inclined plane. Note however, that there are no components of weight acting in the horizontal plane. Last edited by a moderator: Apr 22, 2017 at 11:51 AM 4. Jul 29, 2006 ### Cali Thank you both! I would've never thought to consider the ramp the axis. 5. Jul 29, 2006 ### Meson Yes, I wasn't refering to the horizontal with respect to the level ground, I meant horizontal axis along the ramp, that is the inclined coordinate system or plane. Please read carefully, Hootenanny. EDIT: Here is a diagram depicting my point: Last edited: Jul 29, 2006	2017-04-28 10:23:59	{"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": 0.6376345157623291, "perplexity": 420.39333898719775}, "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-2017-17/segments/1492917122933.39/warc/CC-MAIN-20170423031202-00462-ip-10-145-167-34.ec2.internal.warc.gz"}
http://mathforum.org/kb/plaintext.jspa?messageID=8059040	Date: Jan 11, 2013 10:23 PM Author: siddys@gmail.com Subject: defining an unknown function Hi all, I would like to start by defining H = {{\partial_{x,x} L,\partial_{x,y} L}, {\partial_{y,x} L, \partial_{y,y} L}};where at this stage, the only think i know is that L is of the form L(x,y). This is a Hessian, and I would like to symbolically compute the e-vals and e-vecs. I wanted to know how I should tell mathematica to consider L as L(x,y) ? Thanks,Sid.	2016-05-29 06:06:59	{"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": 0.689382016658783, "perplexity": 2372.6838171029494}, "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-2016-22/segments/1464049278389.62/warc/CC-MAIN-20160524002118-00003-ip-10-185-217-139.ec2.internal.warc.gz"}
http://mathhelpforum.com/algebra/156326-one-confusing-step-solution.html	# Math Help - One confusing step in a solution 1. ## One confusing step in a solution Hi, I'm trying to understand the solution to a problem, and I get most of it, but there is one step in the algebra I can't quite follow. Please help! 2. It looks like the $\frac{1+\sqrt{5}}{2}$ and the $\frac{1-\sqrt{5}}{2}$ terms have been grouped together, then factored. 3. Hello, stevey! pockslides is correct . . . $\displaystyle{\frac{1}{\sqrt{5}}\left(\frac{1+\sqr t{5}}{2}\right)^{n-1}\!\!\! - \frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^{n-1} \!\!\!+ \frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right )^2 \!- \frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^n$ . . $\displaystyle =\;\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\ri ght)^{n-1}\!\!\left(\frac{1+\sqrt{5}}{2}+1\right) - \frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}\!\!\left(\frac{1-\sqrt{5}}{2} + 1\right)$ Watch very carefully . . . $\displaystyle{\underbrace{\frac{1}{\sqrt{5}}\left( \frac{1+\sqrt{5}}{2}\right)^{n-1}}_{A}\!\!\! - \underbrace{\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}}_{B} \!\!\!+ \underbrace{\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{ 5}}{2}\right)^n}_{C} \!- \underbrace{\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^n}_{D}$ . . $\displaystyle =\; \underbrace{\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{ 5}}{2}\right)^n}_{C}\!\!\! + \underbrace{\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}}_{A} \!\!\! - \underbrace{\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{ 5}}{2}\right)^n}_{D} \!- \underbrace{\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^n}_{B}$ . . $\displaystyle =\; \overbrace{\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5 }}{2}\right)^{n-1}\!\!\left(\frac{1+\sqrt{5}}{2} + 1\right)}^{\text{Factor }C\text{ and }A} - \overbrace{\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}\!\!\left(\frac{1-\sqrt{5}}{2} + 1\right)}^{\text{Factor }D\text{ and }B}$	2015-05-24 10:07:28	{"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": 7, "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": 0.9267746210098267, "perplexity": 714.8156532281774}, "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-2015-22/segments/1432207927863.72/warc/CC-MAIN-20150521113207-00075-ip-10-180-206-219.ec2.internal.warc.gz"}
http://mathematica.stackexchange.com/questions/3712/how-can-i-run-munit-testsuites-outside-workbench?answertab=oldest	# How can I run MUnit TestSuites outside WorkBench? I would like to run MUnit tests inside TestSuite constructs from the command line or a notebook interface without loading the Workbench GUI. I tried TestRun["suite.mt"] where suite.mt contains: TestSuite[{"file1.mt","file2.mt"}] and each of the fileX.mt contain standard Test[...] constructs. However, this does not run the tests; I get the output that 0 tests were run. TestRun["file1.mt"] works. I cannot find documentation for TestRun, so maybe it does not support this, but in that case, how do I run testsuite files without loading the Workbench GUI? Edit: Since MUnit v1.4 (which is distributed with Mathematica v10) TestRun function correctly handles path to test suite file. - The code for MUnit package, by the look of it, seems to be really well-written and self-documenting, so, if all else fails, you can read the source and see what is needed to get it to work. What I personally did in a similar situation was that I wrote my own package to load .mt files and have a fine-grained control over them, and also a custom UI to run them in the FrontEnd, and while it worked great for me, I don't want to encourage that approach - it is probably better to use what MUnit provides since there obviously was a lot of work and thought put into that already. – Leonid Shifrin Mar 30 '12 at 13:48 @Leonid There is one good use case for writing one's own .mt runner: distributing tests together with an open source application so people can run it on different platforms without needing WorkBench. I really wish MUnit were part of Mathematica. – Szabolcs Apr 16 '13 at 2:04 Have a look at this post. There is a short explanation how to do it. I use this quite often and it works very well for me. - Thanks, but this doesn't really answer the question. I want to use the same TestSuite construct that I use in Workbench outside of Workbench. – Ian Hinder Mar 30 '12 at 13:34 At least as of 10.3, Mathematica ships with MUnit included (I have not used prior versions, so I don't know when it was first included). I was able to make it work with a script like this: #! /usr/bin/env MathematicaScript -script Needs["MUnit"] Get["./MyApp/MyPackage.m"] If[MUnitTestRun["./Tests/MyPackageTestSuite.mt"], Exit[0], Exit[1]] Which, of course, assumes that MathematicaScript is located somewhere on your PATH and you're running the script from the project root directory. - It's included since 10.0. But unfortunately this version of MUnit is not compatible with the one in Workbench 2. To run test for projects made with Workbench, we still need to use the MUnit that ships with Workbench. – Szabolcs Dec 18 '15 at 8:38 @Szabolcs: Interesting. Could you share some examples of things that do not work outside of Workbench? – Abbas ibn Firnas Dec 18 '15 at 18:30 So far I've been switching back and forth between the two with no problems - running tests on command line and only going to the Workbench when there're failed cases to examine. But I also haven't used all the features yet, such as child tests... – Abbas ibn Firnas Dec 18 '15 at 18:34	2016-02-09 22:39:57	{"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": 0.3721007704734802, "perplexity": 1492.2218004363294}, "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-2016-07/segments/1454701158130.6/warc/CC-MAIN-20160205193918-00146-ip-10-236-182-209.ec2.internal.warc.gz"}
https://mathoverflow.net/questions/398180/a-narrower-dichotomy-for-the-quadratic-variation-of-differentiable-functions	# A narrower dichotomy for the quadratic variation of differentiable functions? $$\newcommand\P{\mathcal P}$$A "partition" $$P$$ (of the interval $$[0,1]$$) is a finite sequence $$(t_0,\dots,t_n)$$ such that $$0=t_0<\cdots; then the mesh of $$P$$ is $$\\|P\\|:=\max_{1\le j\le n}(t_j-t_{j-1})$$. Fix any sequence $$\P:=(P_k)$$ of "partitions" $$P_k=(t_{k,0},\dots,t_{k,n_k})$$ such that $$\\|P_k\\|\to0$$ (as $$k\to\infty$$). For a real-valued function $$f$$ on $$[0,1]$$, define its quadratic variation (with respect to $$\P$$) by the formula $$[f]_\P:=\limsup_{k\to\infty}\sum_{j=1}^{n_k}(f(t_{k,j})-f(t_{k,j-1}))^2.$$ Suppose now that $$f$$ is differentiable and $$[f]_\P<\infty$$. Does it then necessarily follow that $$[f]_\P=0$$? This question is a modification of the (now answered, positively) previous question A dichotomy for the quadratic variation of differentiable functions?. The difference is that now the sequence $$\mathcal P$$ is fixed. • Just to make sure I understand correctly: the difference is mainly that you are taking a fixed sequence $\mathcal{P}$ so that using the notation of your previous question, it may be that $[f] = \infty$? Jul 23 at 17:30 • @WillieWong : Thank you for your comment. Yes, the difference is that now the sequence $\mathcal P$ is fixed. I have added this comment to the question. Jul 23 at 17:40 Using your previous example of $$f(x) = x^2 \cos(x^{-4})$$. Note that on any interval $$[\epsilon,1]$$ the function is continuously differentiable, and hence has quadratic variation 0. Construct $$P_k$$ so that the following points are contained in the partition: 2. $$\frac{1}{\sqrt[4]{\pi}} \ell^{-1/4}$$ for natural $$\ell$$ between $$k$$ and $$2k$$. 3. choose a partition of $$[(k\pi)^{-1/4}, 1]$$ such that the quadratic variation of $$f$$ on that interval is less than 1/10, and such that the points are no further than $$1/k^{1/4}$$ apart. With this the sum relative to $$P_k$$ of $$\sum (f(t_{k,n}) - f(t_{k,n-1}))^2$$ evaluates to approximately $$\ln(2)$$.	2021-09-22 11:05:42	{"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": 32, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9380573034286499, "perplexity": 10705.610557096723}, "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-2021-39/segments/1631780057347.80/warc/CC-MAIN-20210922102402-20210922132402-00613.warc.gz"}
https://nanoscale.blogspot.com/	## Tuesday, March 21, 2023 ### What do we want in a conference venue? The APS March Meeting was in Las Vegas this year, and I have yet to talk to a single attendee who liked that decision in hindsight. In brief, the conference venue seemed about 10% too small (severe crowding issues in hallways between sessions); while the APS deal on hotels was pretty good, they should have prominently warned people that not using the APS housing portal means you fall prey to Las Vegas’s marketing schtick of quoting a low room rate but hiding large “resort fees”; with the exception of In N Out Burger, the food was very overpriced (e.g. $12 for a coffee and a muffin in the Starbucks in my hotel); and indoor spaces in town generally smelled like stale cigarettes, ineffective carpet cleaner, and desperation. I don’t think it’s that hard to enumerate what most people would like out of a conference venue, if we are intending to have in-person meetings and are going to spend grant money and valuable time to attend the meeting with our groups. (I’m taking as a given that the March meeting is large - now up to 12K attendees, for good or ill - and I know that’s so big that some people will decide that it’s too unwieldy to be worth going. Likewise, I know that the logistics are always difficult in terms of the abstract sorting and trying to make sure that likely-popular sessions get higher capacity rooms.) Off the top of my head, I would like: • A meeting venue that can accommodate everyone without feeling dangerously crowded at high volume transit times between sessions, with a good selection of hotels nearby that don’t have crazy room rates. (I know that the meeting growth already likely rules out a lot of places that have hosted the March meeting in the past.) • A high density of relatively cheap restaurants, including sandwich places, close to the venue for lunch, so that a quick bite is possible without hiking a mile or being forced to spend$20 on convention center food. • Actual places to sit (tables and chairs) to talk with fellow attendees. Las Vegas had a much smaller number of these (indoors) than previous locations. • Reasonable availability of water (much better these days than in the past) and not-outrageously-priced coffee and tea. • Wifi that actually can accommodate the number of attendees; at some point in Las Vegas I basically gave up on the conference wifi and tethered to my phone. Remember, many of us still have to get some level of work done (like submitting annoyingly timed proposals) while at these. • Modern levels of accommodations for nursing mothers, childcare, facilities for those with disabilities or mobility issues, etc. Are there major items that I’m missing? Do readers have suggestions for meeting sites that can hit all of these? I am well aware that the APS is financially constrained to make these arrangements years in advance. It can’t hurt to discuss this, though, especially raising concerns about problems to avoid. ## Thursday, March 16, 2023 ### Recent RT superconductivity claim - summary page In the interests of saving people from lots of googling or scrolling through 170+ comments, here is a bulleted summary of links relevant to the recent claim of room temperature superconductivity in a nitrogen-doped lutetium hydride compound under pressure. • Dias's contributed talk at the APS meeting is here on youtube. • Here is the promotional video put out by Rochester as part of the media release. It odd to me that the department chair and the dean of the PI are both in this video. • Here is the pubpeer page that has sprung up with people reporting concerns about the paper. • The comments attached to the paper itself contain interesting discussion (though strangely an informed comment from Julia Deitz about the EDX data was repeatedly deleted as "spam") • There was a lot of media coverage of this paper. The Wall Street Journal was comparatively positive. The New York Times was more nuanced. Quanta had a thorough article with a witty headline describing the controversy surrounding the claim. The APS had an initial brief news report and a more extensive article emphasizing the concerns about the paper. • Experimental preprints have appeared looking at this. The first observes a color change under pressure in LuH2, but no superconductivity in that related compound. The second is a direct replication attempt, finding x-ray structural data matching the report but no superconductivity in that material up to higher pressures and down to 10 K. Note that another preprint appeared last week reporting superconductivity at about 71 K in a different lutetium hydride at much higher pressures. • A relevant and insightful talk from James Hamlin is here, from a recent online workshop about reproducibility in condensed matter physics. Note that (as reported in this twitter thread) significant portions of Hamlin's doctoral thesis appear verbatim in Dias' thesis. No doubt there are more; please let me know if there are additional key links that I've missed (not every twitter comment is important). ## Friday, March 10, 2023 ### APS March Meeting 2023, Day 4 + wrapup My last day at the March Meeting was a bit scattershot, but here are a few highlights: • In a session about spin transport, the opening invited talk by Jiaming He was a clear discussion of recent experimental results on spin Seebeck effects in the magnetic insulator LuFeO3. The system is quite complicated because the net magnetization direction depends nontrivially on the external field, leading to spin transport signatures with a complicated field orientation relationship. • There was an invited session about 2D magnets, and Roland Kawakami gave a clear, pedagogical talk about how they have learned to grow epitaxially nice structures between van der Waals magnets (like Fe3GeTe2) and topological insulators (Bi2Te3). This was followed by a tag-team talk by Vishakha Gupta and Thow Min Cham from Cornell, presenting some great results about spin orbit torque measurements coupling topological insulators and van der Waals magnets, where a gate can be used to dial around the chemical potential in the TI, leading to changes in the anomalous Hall effect. • I did check out the history of science session, featuring a very nice talk about the 75th anniversary of the foundations of quantum electrodynamics by Chad Orzel, including a book recommendation that I need to follow up on. Overall, it was a good meeting, certainly the closest thing to a "normal" March Meeting since 2019. I'm not a fan of Las Vegas as a venue, though. The conference center was a bit too small (leading to a genuinely concerning jamming transition in the hallways at one point), the food was generally criminally expensive, and too many places indoors smelled like a combination of ancient cigarette smoke and ineffective carpet cleaner. It will be interesting to see what the stats are like for things like the downloads of recorded talks and viewing of the virtual component of the meeting that happens in ten days. ## Thursday, March 09, 2023 ### APS March Meeting 2023, Day 3 There is vigorous discussion taking place on the Day 2 link regarding the highly controversial claim of room temperature superconductivity. Highlights from Wednesday are a hodgepodge because of my meanderings: • The session about quantum computing hardware was well attended, though I couldn't stay for the whole thing. The talk by Christopher Eichler about the status of superconducting qubit capabilities was interesting, arguing the case that SC devices can credibly get to the thresholds needed for error correction, though that will require improvements in just about every facet to get there with manageable overhead. The presentation by Anausa Chatterjee about the status of silicon spin qubits was similarly broad. The silicon implementation faces major challenges of layout, exacerbated (ironically) by the small size of the physical dots. There have been some recent advances in fab that are quite impressive, like this 4 by 4 crossbar. • Speaking of impressive capabilities, there were two talks (1, 2) by members of the Yacoby group at Harvard about using a scanning NV center to image the formation and positions of vortices in planar Josephson junctions. They can toggle between 0 and 1 vortices in the junction and can see some screening effects that you can't just get from the transport data. Pretty images. • Switching gears, I heard a couple of talks in an invited session about emergent phenomena in strongly correlated materials. From Paul Goddard at Warwick I learned about charge transport in some pyrochlore iridates that I didn't realize had so much residual conduction at low temperatures. See here. Likewise, James Analytis gave a characteristically clear talk about interesting superconductivity in Ni(x)Ta4Se8 (arxiv version here), an intercalated dichalcogenide that has magnetism as well as re-entrant superconductivity up at the magnetic field that kills the magnetically ordered state. • Later in the day, there was a really interesting session about measuring entropy, which is notoriously difficult to do. As I've told students for years, you can't go to Keysight and buy an entropy-meter. There was some extremely pretty data presented by Shahal Ilani using a variant of their new scanning probe technique. Morning of Day 4 is being taken up by a bunch of other tasks, so the next writeup may be sparse. ## Tuesday, March 07, 2023 ### APS March Meeting 2023, Day 2 I ended up spending more time catching up with people this afternoon than going to talks after my session ended, but here are a couple of highlights: • There was an invited session about the metal halide perovskites, and there were some interesting talks. My faculty colleague Aditya Mohite gave a nice presentation about the really surprising effects that light exposure has on the lattice structure of these materials. One specific example: under illumination, some of the 2D perovskite materials contract considerably, as has been seen by doing in situ x-ray diffraction on these structures. This contraction leads to a readily measured increase in electron mobility and solar cell performance. Moreover, the diffraction patterns show that some diffraction spots actually grow and get sharper under illumination. This kind of improved ordering shows that this is not just some sort of weird heating effect. • In a session about imaging, I caught an excellent talk by Masaru Kuno, who described his spectroscopic infrared photothermal heterodyne imaging. The idea is elegant, if you have access to the right light source. Use a tunable mid-IR laser that can go across the "fingerprint region" of photon energies to illuminate the sample in a time-modulated way. If there is an absorptive mode (vibrational in a molecule, or plasmonic in a metal) there, the heating will cause a time-modulated change in the local index of refraction, which is then detected using a visible probe beam and a lock-in amplifier. It was an extremely clear, pedagogical talk. • I spent much of my time in the strange metal session where I spoke. There were some very good (though rather technical) theory talks, trying to understand the origins of strange metallicity and key issues like the role of disorder. I had wanted to attend the session about superconductivity measurements in materials at high pressures, because of the recent and ongoing controversies. However, the room was small and so packed that the fire marshal was turning people away all afternoon. I gather that it was quite an eventful session. If one of my readers was there and would like to summarize in the comments, I'd be grateful. (BTW, it seems like this year there have been two real steps backwards in the meeting. The official app, I am told, is painful, and for the first time in several years, the aps wifi in the meeting venue is unreliable to the point of being unusable. Not great.) ### APS March Meeting 2023, Day 1 Ahh, Las Vegas. I will say, I think every APS March Meeting from now on should have a giant Ferris wheel right by the registration lobby. Here are a few highlights from what I saw after I arrived around lunchtime today: • Given some of my current research, I spent a fair bit of time at the invited session about strange metals today. All of the talks that I saw were very strong. Andrew MacKenzie spoke about recent measurements of the Lorenz number in such materials (particularly Sr3Ru2O7) and made a persuasive case that strange metals do look different in their temperature-dependent thermal conductivity, because of very strong electron-electron scattering. This is discussed in this recent review article. • In the same session, Brad Ramshaw showed very pretty angle-dependent magnetoresistance data on Nd-LSCO, an archetypal cuprate, arguing that the whole data set can be modeled very well assuming conventional quasiparticles and Boltzmann equation analysis (albeit with a funky combination of temperature-independent anisotropic scattering and strongly temperature dependent isotropic scattering). His postdoc Gaël Grissonnanche expanded on this and looked at how such a model can also reproduce the linear-in-B magnetoresistance in this system. • At the McGroddy Prize session, James Hone gave a very nice overview of the impressive body of work from Columbia over the years on all of the stackable van der Waals materials. Some particular recent highlights included: (1) using deliberately oxidized WSe2 (into WOx) as a low-disorder, very high workfunction material that modulation dopes holes when stacked on a target layer of interest; (2) Using vdW material ferroelectricity to modulate superconductivity in MoTe2; in-progress work using an AFM + an hBN "handle" to bend a graphene "noodle" to get continuously tuned, clean moiré potentials; and electrostatically actuated sliding motion of monolayer vdW material. Hopefully crowd control will be a bit better tomorrow. The hallways seemed narrower than at past meetings, very crowded, and the site would benefit from more places to sit and have conversations. ## Sunday, March 05, 2023 ### APS March Meeting 2023 - coming soon I will be attending the 2023 APS March Meeting in Las Vegas this week. I will do my best to try to report on some highlights daily, though that may be more challenging than usual for me this time around (looming proposal deadline that I suspect all of my condensed matter faculty readers know about, plus some teaching-related work). This is the first APS meeting in Las Vegas since 1986, when (according to legend) the APS was invited not to come back. (Sorry for the web archive link - it would appear that the old PhysicsCentral content from APS is not online anywhere easily searchable.) I'll be giving an invited talk on Tuesday which should be fun. If people have suggestions of particular exciting sessions, please add them in the comments. ## Monday, February 20, 2023 ### Science and how it will be practiced in the future I just registered for an event that celebrates the 35th anniversary of a particular science and engineering program, and one question they posed was, to paraphrase, "Science has changed a lot in the last 35 years. Please make three predictions about science in the next 35 years." I'd be curious for readers' views on this. My quick take: • There will be far more AI/machine learning/software agent-assisted activity. That seems a certainty, and hopefully it may alleviate some repetitive drudgery in certain types of research. • Hopefully I am wrong about this, but I have a feeling that we are still trending in the direction of a widening divide between "have" and "have not" research universities, in terms of having the financial resources to do leading science and engineering research. • Foundation investments may be a growing portion of basic research support, for good or ill. Governmental agencies will face increasing constraints on finances and pressure to concentrate more on short-term and applied work with some claimed quick benefit to economic competitiveness or national security. Thoughts?	2023-03-29 01:27:59	{"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": 0.3379465639591217, "perplexity": 3139.7661126830653}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "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-2023-14/segments/1679296948900.50/warc/CC-MAIN-20230328232645-20230329022645-00016.warc.gz"}
https://math.meta.stackexchange.com/questions/27664/the-unanswered-questions-ordering-seems-wrong	# The unanswered questions ordering seems wrong If you go to Unanswered Questions you'll see several highly voted questions that seem like no one will ever post an answer to. The top one, because someone answered it on MO already. The second one, because the user sort of has already answered it themselves in the comments. Are we doomed to forever view these same answers at the top of the page? That doesn't seem like the original intention of Unanswererd questions. Could we put a sticky dropdown menu for re-ordering that page? At the current rate of things, we will see the exact same questions on that page in 6 months. I don't think the site is meant to host someone's personal, arcane question at the top permanently. This is a bug. • Why is there a "no answers" sub-tab in the "unanswered" tab? Also in the "votes" sub-tab there are some answers which actually have an answer. – Sahiba Arora Jan 13 '18 at 10:58 • @SahibaArora Questions are marked as unanswered if they have no upvoted answers. So there can be unanswered questions with answers, so it also makes sense to filter on questions without answers whatsoever. – JAD Jan 15 '18 at 10:36 There are already tabs to see unanswered questions in a different way than having the most highly voted one on top, such as to have the newest first. On the front page, not the question page, select the unanswered tab and then refine. If you want something more specific you can use the search feature. The remark that the list is in part topped by questions that are not really unanswered is somewhat orthogonal. You could resolve it, for those where this is the case, by providing an (community-wiki) answer, for example. • One issue here is that newest unanswered questions page looks very much like the newest questions page. I think what OP has in mind would be an option to filter the unanswered question of the last month/week by highest vote. – Hyperplane Jan 8 '18 at 21:15 • For example I asked a well received question yesterday. 10 upvotes and 8 favourites almost immediately. But now, a mere 24h later it's already drowned in the noise and you won't find it anywhere on the site. – Hyperplane Jan 8 '18 at 21:21 • As I said, using search one can find such things. For example if you want recent, not answered questions with a decent score, search isanswered:no score:5 and order the search results by newest or active. Maybe I should expand my answer to highlight this more. (Sidenote: isanswered:no is only near identical to the unanswered tab; to have a still closes match isanswered:no hasaccepted:no can be used.) – quid Jan 8 '18 at 22:16 • To be honest I wouldn't be surprised if even a lot of senior members are not aware of this feature (I wasn't). From my perspective, that of a naïve user, it seems as bit arbitrary that all three of these tabs (A , B, C) have very different filter buttons. It seems like for example time frame day/week/month/all could just be a convenient drop down box for all of them. But I'm no web designer so I guess there is some logic behind it that I'm not getting. – Hyperplane Jan 8 '18 at 22:26 • Yes, I agree, on both. There may be some performace concerns and issues of not making things to "busy looking" yet, eg, I do not see why there could not be a "week" tab in "unanswered" too. But I also know some users that do use such approaches to finding questions. A more simple approach would be to just start looking from the say fifth page of unanswered questions (by newest) on (which is about 12h in the past). This would avoid most of the routine homework type stuff. Or look at say 7th to 10th page for yesterdays unanswered questions (roughly). (This is based on 50q/page view). – quid Jan 9 '18 at 0:21 • @quid is there a list of such search commands (like isanswered, score, hasaccepted) anywhere? It would be quite helpful to know where to find them. I don't know whether this is supposed to be common knowledge. – Brahadeesh Jan 10 '18 at 17:41 • @Brahadeesh see math.stackexchange.com/help/searching Moreover after you perform a search on the list of results there'll be a line on the right "advanced search tips"; clicking it will display much but not all of that information on that page. – quid Jan 10 '18 at 18:11	2019-06-16 17:04:25	{"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": 0.3149431645870209, "perplexity": 1047.6603136649933}, "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-2019-26/segments/1560627998288.34/warc/CC-MAIN-20190616162745-20190616184745-00421.warc.gz"}
https://dougraffle.com/111/week1/lab1/lab1_key.html	## Overview In this lab, we will examine the Motor Trend Cars data set. The data was extracted from the 1974 Motor Trend US magazine, and comprises fuel consumption and 10 aspects of automobile design and performance for 32 automobiles (1973–74 models). Variable Description mpg Miles/(US) Gallon cyl Number of Cylinders disp Displacement (cu.in.) hp Gross horsepower wt Weight (lb/1000) qsec 1/4 mile time vs V or Straight Configuration Engine am Transmission Type (auto/manual) ## Load the Data in StatCrunch A Tab Separated File (.tsv) containing the data can be found here: 1. In StatCrunch, use the menus Data $$\to$$ Load $$\to$$ From File $$\to$$ On the web. 2. Copy and Paste the address given above into WWW Address 3. Click Load File at the bottom of the page. ## Complete the following problems on a separate piece of paper. Hand in your answers at the end of lab – make sure to include you name and the lab number. Instructions are included for making graphs and finding the necessary statistics for each problem. Each question will be worth two points, and you can receive partial credit for incorrect answers if your process was correct. Write your final answer as a sentence and include all steps you used to get there, otherwise you will receive partial credit. ### 1.) List each variable and its type (categorical/quantitative). library(ggplot2) data.frame(Variable = colnames(mtcars2)[-1], Type = c("Numeric", "Categorical", "Numeric", "Numeric", "Numeric", "Numeric", "Categorical", "Categorical")) ## Variable Type ## 1 mpg Numeric ## 2 cyl Categorical ## 3 disp Numeric ## 4 hp Numeric ## 5 wt Numeric ## 6 qsec Numeric ## 7 vs Categorical ## 8 am Categorical counts <- summary(factor(mtcars2$cyl)) n <- nrow(mtcars2) data.frame(Frequency = counts, "Rel. Freq" = counts/n*100) ## Frequency Rel..Freq ## 4 11 34.375 ## 6 7 21.875 ## 8 14 43.750 Most of the cars in the data set have eight cylinders, while six is the least common. ### 3.) Sketch a Bar Plot of am. Which Transmission type is more common? ggplot(mtcars2, aes(x = am)) + geom_bar() Manual cars are more common than automatics in this data set. ### 4.) Make a Histogram of hp (you do not need to sketch it). Describe the shape of the distribution. ggplot(mtcars2, aes(x = hp)) + geom_histogram(binwidth = 50, color = "grey80") The distribution of horsepower is right-skewed and unimodal with no outliers. ### 5.) Which Measure of Center is better for hp? Report it. summary(mtcars2$hp) ## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 52.0 96.5 123.0 146.7 180.0 335.0 Because the distribution is right-skewed, the median (123 hp) is a more accurate measure of center. ### 6.) Sketch a Side-by-Side Boxplot of hp grouped by am. Describe the differences in the distributions. ggplot(mtcars2, aes(x = am, y = hp, fill = am)) + geom_boxplot() Manuals have higher horsepower on average and more variation, while there are two automatics with unusually high horsepower. ### 7.) Sketch the QQ Plot of mpg. Is the distribution approximately Normal? library(car) gg_qq(mtcars2, "mpg") Because fuel efficiency generally follows the reference line, we can conclude that it is approximately normal. ### 8.) If a certain car had a fuel efficiency of 22 MPG, would it be more rare for an automatic or a manual car? Show your work. by(mtcars2, mtcars2$am, function(tr) round(data.frame(mean = mean(tr$mpg), sd = sd(tr$mpg)), 2)) ## mtcars2$am: auto ## mean sd ## 1 24.39 6.17 ## -------------------------------------------------------- ## mtcars2\$am: manual ## mean sd ## 1 17.15 3.83 (z.auto <- (22 - 24.39)/6.17) ## [1] -0.3873582 (z.manual <- (22 - 17.15)/3.83) ## [1] 1.266319 22 mpg would be 1.27 standard deviations above the mean for manuals, and 0.39 standard deviations below the mean for automatics. This means that it would be more rare for a manual car to get 22 mpg. ### 9.) Assume that MPG$$\sim N(\mu = 20, \sigma = 6)$$. What is the 90% percentile for the MPG of all cars? I.e., if $$P(MPG \le x) = 0.9$$, what is $$x$$? qnorm(.9, mean = 20, sd = 6) ## [1] 27.68931 The 90% of cars have fuel efficiency less than 27.7 mpg. ### 10.) If quarter mile times follow the Normal Model $$N(\mu = 17.85, \sigma = 1.79)$$, how quickly do the middle 75% of cars complete a quarter mile drag race? I.e., if $$P(l \le QSEC \le u) = 0.75$$, what are $$l$$ and $$u$$? qnorm(c(.125, .875), mean = 17.85, sd = 1.79) ## [1] 15.79087 19.90913 The middle 75% of cars finish the quarter mile in 15.8-19.9 seconds.	2020-02-19 01:01:05	{"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": 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": 0.43646344542503357, "perplexity": 4995.430931744663}, "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/1581875143963.79/warc/CC-MAIN-20200219000604-20200219030604-00433.warc.gz"}
https://hal-lara.archives-ouvertes.fr/hal-02101884	# Direct proofs of strong normalisation in calculi of explicit substitutions Abstract : This paper is part of a general programme of treating explicit substitutions as the primary $\lambda$-calculi from the point of view of foundations as well as applications. Here we investigate the property of strong normalization. To date all the proofs of strong normalization of typed calculi of explicit substitutions use a reduction to the strong normalization of classical $\lambda$-calculus via the so-called preservation of strong normalization'' property. This paper develops a new approach, namely a direct proof that the strongly normalizing terms are precisely those typable under the intersection-types discipline. We also define an effective perpetual strategy for the general calculus, give an inductive definition of the strongly normalizing terms, and furthermore show that normalization properties are essentially unaffected by the inclusion of a rule for garbage collection. A key role is played by a certain general combinatorial lemma relating the reduction properties of two interacting abstract reductions, which we feel is of interest in its own right. Keywords : Document type : Reports Domain : https://hal-lara.archives-ouvertes.fr/hal-02101884 Contributor : Colette Orange <> Submitted on : Wednesday, April 17, 2019 - 9:08:57 AM Last modification on : Friday, May 17, 2019 - 1:39:22 AM ### File RR2000-05.pdf Files produced by the author(s) ### Identifiers • HAL Id : hal-02101884, version 1 ### Citation Daniel Dougherty, Pierre Lescanne. Direct proofs of strong normalisation in calculi of explicit substitutions. [Research Report] LIP RR-2000-05, Laboratoire de l'informatique du parallélisme. 2000, 2+20p. ⟨hal-02101884⟩ Record views	2019-09-19 19:39:50	{"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": 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": 0.3901312053203583, "perplexity": 1940.660841962674}, "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-2019-39/segments/1568514573570.6/warc/CC-MAIN-20190919183843-20190919205843-00052.warc.gz"}
http://electronics.stackexchange.com/questions/103750/how-would-i-use-this-12-key-keyboard	How would I use this 12 key keyboard? I'm trying to make this simple code lock circuit, and I ordered this keyboard for it. However, there isn't any kind of manual for it, and I simply don't understand how it works. Here's a larger picture of it (sorry for the horrible soldering, I've a bad solder, and I probably do it wrong also.) So there's 7 wires on it currently, but how do I actually use this thing? - Looks like a bog-standard matrix keypad to me. – Ignacio Vazquez-Abrams Mar 21 '14 at 15:15 @IgnacioVazquez-Abrams Well, if I want to hook it into 12 LEDs and have one led light up on keypress, what would I do? :) – Christian Mar 21 '14 at 15:19 Use a MCU to scan it. – Ignacio Vazquez-Abrams Mar 21 '14 at 15:19 Or use a 4017 and an oscillator to scan the keyboard, with a few (like 7) transistors and resistors to drive the 12 LEDs. – Spehro Pefhany Mar 21 '14 at 15:28 @Christian, here are a few good links on how to improving your soldering skills. – Ricardo Mar 21 '14 at 15:51 There are four rows and three columns which make a matrix of 12 possible connections. Each switch is located at one of the cross-points of the columns and rows: - If you "read" the value of the voltages on a single column and activate each row with a positive voltage (not together but in turn) you can deduce which of the four switches is pressed in that column. Read the voltage for each of the three columns and you can deduce which button is pressed of the 12. Two buttons being pressed can confuse things so be aware of that. - (I edited the big picture, there's numbers on it now). So, I put + power the pin 1 (see the picture above), and started to try out the pins on the LED. So the LED has - on the catode. I connected pin 2 on the anode of the LED, and started pressing buttons. Number 2 worked. Pin 3 did nothing, no matter of what button I pressed. So I tested them all, and I managed to light the LED with keys 2,5,8 & 0. With power on the pin 1. – Christian Mar 21 '14 at 15:43 @Christian - sounds like you are getting there - the pin numbers on the picture in my answer - don't expect they'll match your keyboard so take that info with a pinch of salt!! – Andy aka Mar 21 '14 at 15:48 You know what. Pin 2 + Pin 3 + pressing 1 on the keypad lighted the LED. And Pin 1 + Pin 2 + pressing 2 on the keypad lighted the LED. Guess what Pin 5 + Pin 2 + pressing 3 did? – Christian Mar 21 '14 at 15:59 So that pins 1,3 and 5 are hooked to the power, that leaves me with 4 pins and 12 buttons. How do I know which button is pressed? – Christian Mar 21 '14 at 18:44 You have to set a 5V on pin 3 with 0V on 1 and 5 and read pins 2, 4, 6 and 7 - this tells you which (if any) buttons are pressed in the first column. Then you repeat the procedure with 5V on pin 1 and 0V on 3 and 5. Repeat for 5V on pin 5 with 0V on 3 and 1 and you have read three results that would normally be all 0,0,0,0 unless a button is pressed then you will find a 1 in one of the results - that tells you the position of the button that is pressed. It needs some form of logic to do this quickly so there is zero latency to the user. – Andy aka Mar 21 '14 at 19:17 Most likely the keys are arranged in a matrix, which explains 7 wires nicely. There is one wire for each column and one wire for each row. The keyboard may be as primitive as each key simply shorting its row and column lines together. Or, it could connect them with a diode. Probe around with a ohmmeter and you should be able to figure out what column and row each lead is connected to. Or, try probing with a 5 V supply, LED, and 2.7 kΩ (about) resistor in series. That will put 1-2 mA thru the led when the two ends are shorted. That will be dim, but should still be visible in normal office lighting. Whatever is in the keypad should not be hurt by 5 V or 2 mA. - Well, I don't happen to own a multimeter (yet), but 5V, LED and a resistor shouldn't be a problem. But how do I do it? + to one of the wires, - to the another end of the LED and one wire to the anode of the LED and to the keyboard itself? – Christian Mar 21 '14 at 15:24 If you decide to use an MCU to scan the keypad, below is a polling algorithm that you can use (which I extracted from this source). It took me a while to understand it (had to read it twice), but once I did, it helped me A LOT. Continuous Polling Operations In this mode of operation, the MicroConverter continuously polls the keypad for a key press. This operation is used where the MicroConverter has completed a task and is now waiting for input before proceeding. In this mode, the keypad is connected to one port of the MicroConverter, Port 2 in this example. Figure 3 shows the connectivity. The output from the MicroConverter, following a key press, is viewed using HyperTerminal running on a PC. The MicroConverter is connected to the PC via the COM1 port. This is the reason for showing the RS-232 connection. As can be seen in Figure 3, the four columns (X1 to X4) are pulled up to VDD and are also connected to four of the MicroConverter port pins (P2.4 to P2.7). The four ADuC8xx rows (Y1 to Y2) are connected to the other four port pins (P2.0 to P2.3). The MicroConverter outputs 0 or drives low the keypad rows (P2.0 to P2.3) one at a time and checks the columns (P2.4 to P2.7) for a low condition. For example, the following is the sequence of events up to a switch press detection (Switch 5 in this case). The MicroConverter outputs a low on P2.0 (Y1) and checks for a low on P2.4 to P2.7. In this case, no low is found and so it returns P2.0 (Y1) to high and moves on to P2.1 (Y2). The MicroConverter now drives P2.1 (Y2) low and again checks P2.4 to P2.7 for a low condition. This time it finds that P2.5 (X2) is low, due to Switch 5 being pressed. The MicroConverter now knows that the interconnect of Y2 and X2 has been shorted, therefore, this is 5. If you have an Arduino handy, you don't need to implement the algorithm yourself. Instead you can use the Arduino Keypad lib. The links are below: See more details (on how the keypad is internally wired, for example) here in my other related answer - I've a raspberry pi? No arduinos though. – Christian Mar 21 '14 at 15:54 @Christian But the idea applies to a Raspberry Pi, as well, without any changes. Look for RaspberryPi Keypad library and you'll probably find one that will suit you. – Ricardo Mar 21 '14 at 15:57 Andy aka's solution seems to be working properly, but I will refer to this in my future operations! +1 – Christian Mar 21 '14 at 16:04 @christian this is a good way of driving the keys. My answer is just a method for deciphering rows and columns. – Andy aka Mar 21 '14 at 16:20 Yes you certainly do +1 – Andy aka Mar 21 '14 at 17:38	2015-01-25 18:18:34	{"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": 0.3769535720348358, "perplexity": 1660.0376525257227}, "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-2015-06/segments/1422115861305.18/warc/CC-MAIN-20150124161101-00163-ip-10-180-212-252.ec2.internal.warc.gz"}
https://handwiki.org/wiki/Two-step_M-estimators	# Two-step M-estimators Two-step M-estimators deals with M-estimation problems that require preliminary estimation to obtain the parameter of interest. Two-step M-estimation is different from usual M-estimation problem because asymptotic distribution of the second-step estimator generally depends on the first-step estimator. Accounting for this change in asymptotic distribution is important for valid inference. ## Description The class of two-step M-estimators includes Heckman's sample selection estimator,[1] weighted non-linear least squares, and ordinary least squares with generated regressors.[2] To fix idea, let $\displaystyle{ \{W_{i}\}^n_{i=1} }$ $\displaystyle{ \subseteq R^d }$ be an i.i.d. sample. $\displaystyle{ \Theta }$ and $\displaystyle{ \Gamma }$ are subsets of Euclidean spaces $\displaystyle{ R^p }$ and $\displaystyle{ R^q }$, respectively. Given a function $\displaystyle{ m(;;;): R^d \times \Theta \times \Gamma\rightarrow R }$ , two-step M-estimator $\displaystyle{ \hat\Theta }$ is defined as: $\displaystyle{ \hat \Theta:=\arg\max_{\Theta\epsilon\Theta}\frac{1}{n}\sum_{i}m\bigl(W_{i},\theta,\hat\gamma\bigr) }$ where $\displaystyle{ \hat\gamma }$ is a parameter that needs to be estimated in the first step. Consistency of two-step M-estimators can be verified by checking consistency conditions for usual M-estimators, although some modification might be necessary. In practice, the important condition to check is identification condition.[2] If $\displaystyle{ \hat\gamma\rightarrow\gamma^, }$ where $\displaystyle{ y^ }$where $\displaystyle{ \gamma^* }$ is a non-random vector, then the identification condition is that $\displaystyle{ E[m(W_{1},\theta,\gamma^)] }$ has a unique maximizer over $\displaystyle{ \Theta }$. Under regularity conditions, two-step M-estimators have asymptotic normality. An important point to note is that asymptotic variance of a two-step M-estimator is generally not the same as that of the usual M-estimator in which the first step estimation is not necessary.[3] This fact is intuitive because $\displaystyle{ \gamma^ }$ is a random object and its variability should influence the estimation of $\displaystyle{ \Theta }$. However, there exists a special case in which the asymptotic variance of two-step M-estimator takes the form as if there were no first-step estimation procedure. Such special case occurs if: $\displaystyle{ E \frac{\partial}{\partial\theta\partial\gamma}m(W_{1},\theta_{0},\gamma^)=0 }$ where $\displaystyle{ \theta_{0} }$ is the true value of $\displaystyle{ \theta }$ and $\displaystyle{ \gamma^ }$ is the probability limit of $\displaystyle{ \hat\gamma }$.[3] To interpret this condition, first note that under regularity conditions, $\displaystyle{ E \frac{\partial}{\partial\theta\partial\gamma}m(W_{1},\theta_{0},\gamma^)=0 }$ since $\displaystyle{ \theta_{0} }$ is the maximizer of $\displaystyle{ E m(W_{1},\theta \gamma^) }$. So the condition above implies that small perturbation in γ has no impact on the First-Order condition. Thus, in large sample, variability of $\displaystyle{ \hat\gamma }$ does not affect the argmax of the objective function, which explains invariant property of asymptotic variance. Of course, this result is valid only as the sample size tends to infinity, so the finite-sample property could be quite different. ## Involving MLE Two-step M-estimator involving Maximum Likelihood Estimator is a special case of general two-step M-estimator. Thus, consistency and asymptotic normality of the estimator follows from the general result on two-step M-estimators. When the first step estimation is MLE, under some assumptions, two-step M-estimator is more efficient [i.e. has smaller asymptotic variance] than M-estimator with known first-step parameter.[4] Let {Vi,Wi,Zi}ni=1 be a random sample and the second-step M-estimator $\displaystyle{ \widehat{\theta} }$ is the following: $\displaystyle{ \widehat{\theta} }$$\displaystyle{ \underset{\theta\in\Theta}{\operatorname{arg\max}}\sum_{i}m(\,v_i,w_i,z_i: \theta\,,\widehat{\gamma }) }$ where $\displaystyle{ \widehat{\gamma } }$ is the parameter estimated by ML procedure in the first step. For the MLE, $\displaystyle{ \widehat{\gamma } }$$\displaystyle{ \underset{\gamma\in\Gamma}{\operatorname{arg\max}}\sum_{i}\log f(v_{it} : z_{i} , \gamma) }$ where f is the conditional density of V given Z. Now, suppose that given Z, V is conditionally independent of W. This assumption is called conditional independence assumption or selection on observables.[4][5] Intuitively, this condition means that Z is a good predictor of V so that once conditioned on Z, V has no systematic dependence on W. Under the conditional independence assumption, the asymptotic variance of the two-step estimator is: E[∇θ s(θ00)]−1 E[g(θ00 )g(θ00 )']E[∇θ s(θ00)]−1 where g(θ,γ) ≔ s(θ,γ)-E[ s(θ , γ) ∇γ d(γ)' ]E[∇γ d(γ) ∇γ d(γ)' ]−1 d(γ), s(θ,γ) ≔ ∇θ m(V, W, Z: θ, γ) , d(γ) ≔ ∇γ log f (V : Z, γ), and ∇ represents partial derivative with respect to a row vector. In the case where γ0 is known, the asymptotic variance is E[∇θ s(θ00)]−1 E[s(θ00 )s(θ00 )']E[∇θ s(θ00)]−1 and therefore, unless E[ s(θ, γ) ∇γ d(γ)' ]=0, the two-step M-estimator is more efficient than the usual M-estimator. This fact suggests that even when γ0 is known a priori, there is efficiency gain by estimating γ by MLE. An application of this result can be found, for example, in treatment effect estimation.[4] ## References 1. Heckman, J.J., The Common Structure of Statistical Models of Truncation, Sample Selection, and Limited Dependent Variables and a Simple Estimator for Such Models, Annals of Economic and Social Measurement, 5,475-492. 2. Wooldridge, J.M., Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass. 3. Newey, K.W. and D. McFadden, Large Sample Estimation and Hypothesis Testing, in R. Engel and D. McFadden, eds., Handbook of Econometrics, Vol.4, Amsterdam: North-Holland. 4. Wooldridge, J.M., Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass. 5. Heckman, J.J., and R. Robb, 1985, Alternative Methods for Evaluating the Impact of Interventions: An Overview, Journal of Econometrics, 30, 239-267.	2021-04-10 22:02:36	{"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": 0.9017665982246399, "perplexity": 1577.436898527384}, "config": {"markdown_headings": true, "markdown_code": false, "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/1618038059348.9/warc/CC-MAIN-20210410210053-20210411000053-00496.warc.gz"}
https://grindskills.com/project-new-point-into-mds-space/	# Project new point into MDS space I am trying to project a new point A(x, y, z) into a predefined MDS space in R. This is what I have so far: set.seed(1) x <- matrix(rnorm(310), ncol = 3) DM <- dist(x) MDS <- cmdscale(DM) # New data point to be projected A <- c(1, 2, 3) I am not including A directly into x then fitting the MDS because it would affect the space coordinates. Is there a practical solution? EDIT I believe I found a solution by estimating the betas to predict the MDS axis: x1 <- cbind(1, x) # add intercept B <- solve(t(x1) %% x1) %% t(x1) %% MDS # Betas > MDS [,1] [,2] [1,] -1.80789362 0.06801597 [2,] -0.64418055 -0.21163109 [3,] 0.04694820 -1.27040928 [4,] 3.39617277 -0.21657115 [5,] -0.96981358 0.46269025 [6,] -0.24716695 -0.79861234 [7,] 0.33620625 0.02618564 [8,] 0.62473570 1.35544267 [9,] 0.01895042 0.80023822 [10,] -0.75395865 -0.21534889 > x1 %% B # same as MDS [,1] [,2] [1,] -1.80789362 0.06801597 [2,] -0.64418055 -0.21163109 [3,] 0.04694820 -1.27040928 [4,] 3.39617277 -0.21657115 [5,] -0.96981358 0.46269025 [6,] -0.24716695 -0.79861234 [7,] 0.33620625 0.02618564 [8,] 0.62473570 1.35544267 [9,] 0.01895042 0.80023822 [10,] -0.75395865 -0.21534889 A <- c(1, 2, 3) A <- c(1, A) # add intercept > A %% B # coordinates of A in the MDS plane [,1] [,2] [1,] -2.759456 0.5927178 Is my procedure correct? If you’re using Euclidean distances, then classical MDS is equivalent to PCA, which readily defines a mapping into the low dimensional space, as amoeba mentioned. There should be various threads on this site describing how to do this. Otherwise, De Silva and Tenenbaum (2004) describe how to perform this mapping for classical MDS with arbitrary distances (note that it won’t work for non-classical variants of MDS, e.g. non-metric MDS, variants that minimize the stress criterion, etc.). They call this procedure “distance-based triangulation”. Although not originally formulated as such, it turns out to work by using the Nyström approximation, which is a way to approximate the eigenvalues/eigenvectors of a large matrix using a smaller submatrix (see Platt 2005). Suppose we have $$nn$$ training points. The squared distance between points $$ii$$ and $$jj$$ is stored in the $$(i,j)(i,j)$$th entry of matrix $$Δn\Delta_n$$. We use these distances with classical MDS to compute a $$kk$$-dimensional embedding. Let each column of $$k×nk \times n$$ matrix $$LkL_k$$ contain the low-dimensional embedding coordinates of a training point. Let $$L#kL_k^\#$$ denote the transpose of the pseudoinverse of $$LkL_k$$. Instead of starting from scratch, this can be computed using components that were originally used to compute $$LkL_k$$ (see the paper for details). Let $$→δi\vec{\delta}_i$$ denote the $$ii$$th column of $$Δn\Delta_n$$ (containing the squared distances from point $$ii$$ to all other points), and let $$→δμ=1n∑ni=1→δi\vec{\delta}_\mu = \frac{1}{n} \sum_{i=1}^n \vec{\delta}_i$$ denote the mean of the columns. Now, suppose we want to map a new point $$aa$$ into the low dimensional space. Compute vector $$→δa\vec{\delta}_a$$, containing the squared distance from $$aa$$ to every training point. The low dimensional emedding coordinates of $$aa$$ are then given by: $$→xa=−12L#k(→δa−→δμ)\vec{x}_a = -\frac{1}{2} L_k^\# (\vec{\delta}_a - \vec{\delta}_\mu)$$ References: De Silva and Tenenbaum (2004). Sparse multidimensional scaling using landmark points. Platt (2005). FastMap, MetricMap, and Landmark MDS are all Nystrom Algorithms.	2022-09-24 22:53:28	{"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": 21, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8961624503135681, "perplexity": 2183.652465071665}, "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/1664030333541.98/warc/CC-MAIN-20220924213650-20220925003650-00755.warc.gz"}
http://fixunix.com/vxworks/509548-why-calling-kernelinit-two-times.html	Why calling kernelInit two times. - VxWorks This is a discussion on Why calling kernelInit two times. - VxWorks ; Hi all, When I was studying boot sequence I found the below abnormality. There is kernel init call kernelInit ((FUNCPTR) usrRoot, ROOT_STACK_SIZE, \ (char ) MEM_POOL_START_ADRS, sysMemTop (), ISR_STACK_SIZE, INT_LOCK_LEVEL); in usrKernelInit() function in usrkernel.c . which is called from ... Thread: Why calling kernelInit two times. 1. Why calling kernelInit two times. Hi all, When I was studying boot sequence I found the below abnormality. There is kernel init call kernelInit ((FUNCPTR) usrRoot, ROOT_STACK_SIZE, \ sysMemTop (), ISR_STACK_SIZE, INT_LOCK_LEVEL); in usrKernelInit() function in usrkernel.c . which is called from usrInit() function. but after calling usrKernelInit(), we again calling kernelInit() from usrInit(). in bootconfig.c I wonder why we are calling kernelInit two time..? Any one know the reason? I am using vxworks 6.4 .. Regards, Nabendu 2. Re: Why calling kernelInit two times. On Jul 17, 9:09 am, Nabendu wrote: > Hi all, > > When I was studying boot sequence I found the below abnormality. > > There is kernel init call > > kernelInit ((FUNCPTR) usrRoot, ROOT_STACK_SIZE, \ > sysMemTop (), ISR_STACK_SIZE, INT_LOCK_LEVEL); in > usrKernelInit() function in usrkernel.c . > > which is called from usrInit() function. but after calling > usrKernelInit(), we again calling kernelInit() from usrInit(). in > bootconfig.c > > I wonder why we are calling kernelInit two time..? > > Any one know the reason? > > I am using vxworks 6.4 .. > Hhhmm, if you had found a third module with a call to kernelInit(), would you have assumed it was being called 3 times? ;-) Seriously, it is not being called twice. HTH, GV 3. Re: Why calling kernelInit two times. On Jul 17, 7:25 pm, gvarndell wrote: > On Jul 17, 9:09 am, Nabendu wrote: > > > > > Hi all, > > > When I was studying boot sequence I found the below abnormality. > > > There is kernel init call > > > kernelInit ((FUNCPTR) usrRoot, ROOT_STACK_SIZE, \ > > sysMemTop (), ISR_STACK_SIZE, INT_LOCK_LEVEL); in > > usrKernelInit() function in usrkernel.c . > > > which is called from usrInit() function. but after calling > > usrKernelInit(), we again calling kernelInit() from usrInit(). in > > bootconfig.c > > > I wonder why we are calling kernelInit two time..? > > > Any one know the reason? > > > I am using vxworks 6.4 .. > > Hhhmm, if you had found a third module with a call to kernelInit(), > would you have assumed it was being called 3 times? ;-) > Seriously, it is not being called twice. > > HTH, > GV Logically I understand it should not be called more than once. GV,Had you ever gone through the code of usrInit() in target\config\all \bootconfig.c or usrConfig.c and usrKernelInit() code in target\config\comps\src\usrKernel.c ? (FYI - bootconfig.c or usrConfig.c are copied and modified to BSP Regards, Nabendu 4. Re: Why calling kernelInit two times. On Jul 17, 11:24 am, Nabendu wrote: > On Jul 17, 7:25 pm, gvarndell wrote: > > > > > On Jul 17, 9:09 am, Nabendu wrote: > > > > Hi all, > > > > When I was studying boot sequence I found the below abnormality. > > > > There is kernel init call > > > > kernelInit ((FUNCPTR) usrRoot, ROOT_STACK_SIZE, \ > > > (char ) MEM_POOL_START_ADRS, > > > sysMemTop (), ISR_STACK_SIZE, INT_LOCK_LEVEL); in > > > usrKernelInit() function in usrkernel.c . > > > > which is called from usrInit() function. but after calling > > > usrKernelInit(), we again calling kernelInit() from usrInit(). in > > > bootconfig.c > > > > I wonder why we are calling kernelInit two time..? > > > > Any one know the reason? > > > > I am using vxworks 6.4 .. > > > Hhhmm, if you had found a third module with a call to kernelInit(), > > would you have assumed it was being called 3 times? ;-) > > Seriously, it is not being called twice. > > > HTH, > > GV > > Logically I understand it should not be called more than once. > > GV,Had you ever gone through the code of usrInit() in target\config\all > \bootconfig.c or usrConfig.c > and usrKernelInit() code in target\config\comps\src\usrKernel.c ? No, never. > > (FYI - bootconfig.c or usrConfig.c are copied and modified to BSP Actually, you're right -- kernelInit() is called twice. Once when the bootrom image executes. And then again when the ram image is executed. And, if you reset the target, it's executed 4 times. ;-) HTH GV 5. Re: Why calling kernelInit two times. On Jul 18, 12:06 am, gvarndell wrote: > On Jul 17, 11:24 am, Nabendu wrote: > > > > > On Jul 17, 7:25 pm, gvarndell wrote: > > > > On Jul 17, 9:09 am, Nabendu wrote: > > > > > Hi all, > > > > > When I was studying boot sequence I found the below abnormality. > > > > > There is kernel init call > > > > > kernelInit ((FUNCPTR) usrRoot, ROOT_STACK_SIZE, \ > > > > (char ) MEM_POOL_START_ADRS, > > > > sysMemTop (), ISR_STACK_SIZE, INT_LOCK_LEVEL); in > > > > usrKernelInit() function in usrkernel.c . > > > > > which is called from usrInit() function. but after calling > > > > usrKernelInit(), we again calling kernelInit() from usrInit(). in > > > > bootconfig.c > > > > > I wonder why we are calling kernelInit two time..? > > > > > Any one know the reason? > > > > > I am using vxworks 6.4 .. > > > > Hhhmm, if you had found a third module with a call to kernelInit(), > > > would you have assumed it was being called 3 times? ;-) > > > Seriously, it is not being called twice. > > > > HTH, > > > GV > > > Logically I understand it should not be called more than once. > > > GV,Had you ever gone through the code of usrInit() in target\config\all > > \bootconfig.c or usrConfig.c > > and usrKernelInit() code in target\config\comps\src\usrKernel.c ? > > No, never. > > > > > (FYI - bootconfig.c or usrConfig.c are copied and modified to BSP > > directory any change required in bootloader commads or other things) > > Actually, you're right -- kernelInit() is called twice. > Once when the bootrom image executes. > And then again when the ram image is executed. > And, if you reset the target, it's executed 4 times. ;-) > > HTH > GV haha really funny, ya, 6 times if i reset 2 times ,,ie n2 +2 if i reset n times...???? usrConfig.c is used for vxworks image and bootconfig.c for bootloder/ botrom.some functions are defined as same/almost same in them(usr . so before giving any command from boot prompt of bootloader (bootcmdloop) kernelinit called 2 times, after giving boot command (e.g. @) image is loaded to ram and run it ,there kernelinit run 2 times So total no of kernelInit is 4. Correct me if I'm wrong. Regards, Nabendu 6. Re: Why calling kernelInit two times. On Jul 18, 12:48 am, Nabendu wrote: > On Jul 18, 12:06 am, gvarndell wrote: > > > > > On Jul 17, 11:24 am, Nabendu wrote: > > > > On Jul 17, 7:25 pm, gvarndell wrote: > > > > > On Jul 17, 9:09 am, Nabendu wrote: > > > > > > Hi all, > > > > > > When I was studying boot sequence I found the below abnormality. > > > > > > There is kernel init call > > > > > > kernelInit ((FUNCPTR) usrRoot, ROOT_STACK_SIZE, \ > > > > > (char ) MEM_POOL_START_ADRS, > > > > > sysMemTop (), ISR_STACK_SIZE, INT_LOCK_LEVEL); in > > > > > usrKernelInit() function in usrkernel.c . > > > > > > which is called from usrInit() function. but after calling > > > > > usrKernelInit(), we again calling kernelInit() from usrInit(). in > > > > > bootconfig.c > > > > > > I wonder why we are calling kernelInit two time..? > > > > > > Any one know the reason? > > > > > > I am using vxworks 6.4 .. > > > > > Hhhmm, if you had found a third module with a call to kernelInit(), > > > > would you have assumed it was being called 3 times? ;-) > > > > Seriously, it is not being called twice. > > > > > HTH, > > > > GV > > > > Logically I understand it should not be called more than once. > > > > GV,Had you ever gone through the code of usrInit() in target\config\all > > > \bootconfig.c or usrConfig.c > > > and usrKernelInit() code in target\config\comps\src\usrKernel.c ? > > > No, never. > > > > (FYI - bootconfig.c or usrConfig.c are copied and modified to BSP > > > directory any change required in bootloader commads or other things) > > > Actually, you're right -- kernelInit() is called twice. > > Once when the bootrom image executes. > > And then again when the ram image is executed. > > And, if you reset the target, it's executed 4 times. ;-) > > > HTH > > GV > > haha really funny, ya, 6 times if i reset 2 times ,,ie n2 +2 if i > reset n times...???? > > usrConfig.c is used for vxworks image and bootconfig.c for bootloder/ > botrom.some functions are defined as same/almost same in them(usr . > > so before giving any command from boot prompt of bootloader > (bootcmdloop) kernelinit called 2 times, > after giving boot command (e.g. @) image is loaded to ram and run > it ,there kernelinit run 2 times > > So total no of kernelInit is 4. > > Correct me if I'm wrong. > If I really took off on you, someone (someone who otherwise would never bother) would think me an obnoxious bore and feel really sorry for you and come to your rescue. I think I'll just let you think about it for a while. GV	2016-08-31 08:09:25	{"extraction_info": {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8928502202033997, "perplexity": 14045.844914030531}, "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-2016-36/segments/1471983580563.99/warc/CC-MAIN-20160823201940-00229-ip-10-153-172-175.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/2751802/must-an-idempotent-matrix-be-symmetric	# Must an idempotent matrix be symmetric? A matrix $A$ is idempotent if: $$AA = A$$ Is it true that all such matrices are symmetric? • No. Consider an oblique, nonorthogonal, projection matrix. Apr 24 '18 at 16:22 No. Here's a simple $2 x 2$ counterexample: Define: $$A = \begin{pmatrix} 1 &0\\1 &0 \end{pmatrix}$$ Note that $A$ is not symmetric, i.e. $$A^T = \begin{pmatrix} 1 &1\\0 &0 \end{pmatrix} \neq A$$ However, $A$ is idempotent: $$AA = \begin{pmatrix} 1 &0\\1 &0 \end{pmatrix} \begin{pmatrix} 1 &0\\1 &0 \end{pmatrix} = \begin{pmatrix} 1 &0\\1 &0 \end{pmatrix} = A$$ Therefore, it cannot be the case that an idempotent matrix has to be symmetric.	2021-09-21 06:19:24	{"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": 0.945320725440979, "perplexity": 790.7023897092699}, "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-2021-39/segments/1631780057158.19/warc/CC-MAIN-20210921041059-20210921071059-00528.warc.gz"}
http://mathhelpforum.com/advanced-algebra/86463-vector.html	# Math Help - Vector 1. ## Vector - Probably the hardest question in the text, could someone please solve this. I have absolutely no idea how to show $T$ as a linear transformation. Here is the Question. Let ${({{ R}^n})^} = \{ T:{{ R}^n} \to {{ R}^n}\|T{\text{ is linear\} }}$. For $T$ and $S$ in ${({{ R}^n})^}$ and $C$ in ${{{ R}}}$ define $(T+S)(u)=T(u)+S(u)$ for all $u$ in ${{{ R}^n}}$. Show that ${({{R}^n})^}$ is a vector space. Where ${({{R}^n})^}$ is a set of linear transformation. 2. Originally Posted by igodspeed Let ${({{ R}^n})^} = \{ T:{{ R}^n} \to {{ R}^n}\|T{\text{ is linear\} }}$. For $T$ and $S$ in ${({{ R}^n})^}$ and $C$ in ${{{ R}}}$ define $(T+S)(u)=T(u)+S(u)$ for all $u$ in ${{{ R}^n}}$. Show that ${({{R}^n})^}$ is a vector space. Where ${({{R}^n})^}$ is a set of linear transformation. I assume you meant to add $CT(u)=T(Cu)$ This is just the dual space to $\mathbb{R}^n$ $(T+CS)(u)=T(u)+CS(u)=T(u)+S(Cu)$ Now all you gotta do is show it is actually a linear transformation itself, ie $(\mathbb{R}^n)^$ showing its closed under scalar multiplication and addition. $(T+CS) (au+bv)=T(au+bv)+S(C(au+bv))=aT(u)+bT(v)+CaS(u)+Cb S(v)$ $=aT(u)+aCS(u)+bT(v)+bCS(v) = a(T+CS)(u)+b(T+CS)(v)$ So it indeed a linear transformation of $\mathbb{R}^n$, QED. 3. - Yes you are right. I guess i need to show that $T+S$ is a linear transformation. Someone told me that i need to check there's a zero, that it's closed under addition/subtraction, and that "there's an $R$-action on it. Now, you don't have any $R$-action hypothesis, so this is not a vector space." So i am a little confused. Thanks for your help. 4. Yeah, that's what I checked. C is an element in the reals. That is why I checked T + CS is a linear transformation, that checks additivity and the R action all in one step. It shows it is closed under addition and scalar multiplication. Sure the trivial transformation 0 is linear so it is in there. I think maybe you need to review your definitions and/or reread my proof until you understand what you are supposed to do. 5. - Would it be possible to get the basis for ${({{ R}^n})^}$? It is so abstract. Thanks, Kelly 6. ## Dual Space If you are interested in learning more about this topic, I suggest checking out Dual space - Wikipedia, the free encyclopedia. They have a fairly extensive overview of the Dual Space and it includes the basis for the dual space. What is possibly even more interesting is that when you consider the space dual to the dual space (the double dual if you will) Linear functionals on the linear functionals of the ground vector space, you get a space that is canonically isomorphic to the ground vector space. This is particularly important because the isomorphism requires absolutely no reference to a basis whatesoever. Note all those above facts are only necessarily true for finite dimensional vector spaces. I believe the maps are still injective, but may not necessarily by surjective if it is not finite dimensional. 7. It is way beyond me. I don't even understand half of it. I am in an introductory course, so if you could, could you please explain it to me in layman's terms? thanks in anticipation, KC 8. Yeah man, its some abstract stuff. Basically your basis for the dual space to an n dimensional vector space is just the set of n linear functionals such that the ith one spits out the coefficient on the ith basis element for the vector space. 9. so how can i find the basis for ${({{ R}^n})^} $ 10. ## Bassis for dual space This is the point, it depends on the basis, there are an infinite number of possible choices for basis of $\mathbb{R}^n$ each of these yields a different basis for $(\mathbb{R}^n)^$. To give you the idea though for any given basis for $\mathbb{R}^n$ call it $\{e_1,e_2,...,e_n\}$, you get n linear functionals in $(\mathbb{R}^n)^$ that will form the basis, say they are $\{\epsilon^1,\epsilon^2,...,\epsilon^n \}.$ Let the n functionals be defined as follows. $\epsilon^i(a_1e_1 +a_2e_2 + ... + a_ie_i + ... +a_ne_n)=a_i$ Yeah, there is your basis. If you took the standard basis for $\mathbb{R}^n$ the ith coordinate function would simply pick out the ith coefficient. 11. Originally Posted by Gamma I assume you meant to add $CT(u)=T(Cu)$ This is just the dual space to $\mathbb{R}^n$ $(T+CS)(u)=T(u)+CS(u)=T(u)+S(Cu)$ Now all you gotta do is show it is actually a linear transformation itself, ie $(\mathbb{R}^n)^$ showing its closed under scalar multiplication and addition. $(T+CS) (au+bv)=T(au+bv)+S(C(au+bv))=aT(u)+bT(v)+CaS(u)+Cb S(v)$ $=aT(u)+aCS(u)+bT(v)+bCS(v) = a(T+CS)(u)+b(T+CS)(v)$ So it indeed a linear transformation of $\mathbb{R}^n$, QED. I was wondering how i can show that there is a zero vector 0 in V such that u+0=0. 12. Is V the dual space you are talking about? Just in general for future use, V is typically reserved for the vector space. Generally you need to denote the dual space $V^*$. It is the vector space dual to V. If this is indeed what you are talking about, the identity under addition of the vector space of linear transformations is just the trivial linear transformation. It takes any vector and sends it to 0, call it Z for zero map. It is easy to check this is indeed a linear transformation, and it clearly satisfies (T + Z)(v)=T(v) + Z(v)=T(v) + 0= T(v) for all v, so it is the identity. 13. I mean the Identity not the dual space. Could you also please show me how the sum of u and v, denoted by u+v is in V. Where V is vector space, nonempty set of vectors. thanks KC 14. As you said depending on the choice of basis used. How would i find the basis based on standard matrix. I do understand what you meant but i don't know how i can actually use this to find the basis? Thanks KC 15. Are you talking about the other question on the other Thread? I am really confused as to what you are talking about? Again I cannot stress enough, there is no "THE basis" there is no preferred basis, it is just a basis. If you need to find a basis for $\mathbb{R}^n$,just pick the standard basis for $\mathbb{R}^n$. {(1,0,...,0), (0,1,0,...,0),... (0,0,...,1)} Page 1 of 2 12 Last	2014-07-12 16:10:51	{"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": 53, "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": 0.8962768912315369, "perplexity": 174.02625974297874}, "config": {"markdown_headings": true, "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-2014-23/segments/1404776434099.83/warc/CC-MAIN-20140707234034-00019-ip-10-180-212-248.ec2.internal.warc.gz"}
https://events.uconn.edu/event/67551/2019-04-03	### Algebra SeminarMeasures of irrationality for algebraic varietiesBrooke Ullery (Harvard University) Wednesday, April 3, 2019 11:15am – 12:05pm Storrs Campus Monteith, 313 A smooth algebraic curve is said to be rational if it is isomorphic to P^1, the projective line. More generally, the gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. The intuition is that the higher the gonality, the further the curve is from being rational. A classical theorem of Noether says that if C is a smooth plane curve of degree d, then the gonality of C is d-1, and it is obtained by projecting away from a point on the curve. A natural question is: does Noether's theorem generalize in some way to curves in larger projective spaces? What about to higher dimensional varieties? We will explore these questions, focusing on the examples of hypersurfaces and, more generally, complete intersections in projective space. Contact: Mihai Fulger, mihai.fulger@uconn.edu Algebra Seminar (primary), UConn Master Calendar	2019-06-24 23:39:04	{"extraction_info": {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8115352392196655, "perplexity": 515.7736876927552}, "config": {"markdown_headings": true, "markdown_code": true, "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-2019-26/segments/1560627999779.19/warc/CC-MAIN-20190624231501-20190625013501-00356.warc.gz"}
http://math.stackexchange.com/tags/projective-module/hot	# Tag Info 15 $\def\id{\operatorname{id}}$Suppose $M\otimes N$ is isomorphic to $R^n$. Pick a basis $\{x_1,\dots,x_n\}$ of $M\otimes N$, with $x_i=\sum_{j=1}^{r_i}m_{i,j}\otimes n_{i,j}$ for each $i\in\{1,\dots,n\}$. Let $r=r_1+\cdots+r_n$, let $\{e_{i,j}:1\leq i\leq n, 1\leq j\leq r_i\}$ be a basis of $R^r$, and consider the map $f:R^r\to M$ which maps $e_{i,j}$ to ... 11 A proof that every projective module over a PID is free occurs in $\S$ 3.9 of my commutative algebra notes. As Qiaochu Yuan mentions, infinitely generated projective modules long to be free. A generalization of Kaplansky's result is a 1963 theorem of H. Bass: let $R$ be a connected (i.e., without nontrivial idempotents) Noetherian ring. Then every ... 9 Let $M, N$ be $R-$modules. Then the following holds. If $M$ and $N$ is flat, then so is $M\otimes_{R}N$: see related question here. If $M$ and $N$ are projective, then so is $M\otimes_{R} N$. Indeed, writing $M\oplus M'=F,\ N\oplus N'=F'$, for free $R-$modules $F,\ F'$, one has that $$F'':=F\otimes_{R}F'$$ is free (tensor product of free modules) and ... 9 I'm denoting your ring with $R$ and your ideal with $I$. We'll just need that $R$ is an integral domain, and that $I$ is a nontrivial ideal. If $R/I$ is a projective $R$ module, then the following exact sequence splits $$0\rightarrow I\rightarrow R\rightarrow R/I\rightarrow 0$$ But integral domains have no proper direct summands, so $I$ would have to be ... 8 As indicated in the comments, you should saturate $Q$ in $P$ first, i.e. replace it by the preimage of the torsion in $P/Q,$ so that (after changing $Q$ in this way) we get that $P/Q$ is torsion-free. The basic fact you need is that the saturation is again f.g., but this will follows from the fact that the torsion in $Q/P$ is f.g., being a submodule of the ... 7 This failure of freeness is a non-trivial result. One way to prove it is to begin with a lemma: If $F$ is a free abelian group and $C$ is a countable subgroup, then the quotient $F/C$ is the direct sum of a countable group and a free group. (I'm omitting "abelian" because I'm lazy and all groups here will be abelian.) [Proof of lemma: Fix a basis $B$ for ... 7 Here are some examples: Localizations or quotients of $R$. More generally any epimorphism of commutative rings with domain $R$. Free $R$-modules of infinite rank. $R \oplus \bigoplus_{i \in I} N$ for any $R$-module $N$ with $N \otimes N = 0$. The finitely generated examples can be classified: Claim: When $M$ is a finitely generated $R$-module with $M ... 7 A few remarks, to be expanded below: (1) first is that the proof that$M = \prod_{i=1}^\infty\mathbb{Z}$is not free is elementary, and (2) second is that it might be hard to find simpler examples, at least if "simple" refers to how simple the ring is itself. (1) In fact, here's a proof that I learned from Kaplansky's book "Infinite Abelian Groups": Assume ... 7 A projective module over a domain has no nonzero torsion element, since it is a submodule of a free module. But every element of your module is a torsion element: it is killed by$x$. 7 Since$\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q} \cong \mathbb{Q}$is a field, every module over it is projective. 6 Here is the answer for finitely generated modules of rank one. Recall that the isomorphism classes of these modules form a group, the Picard group$Pic(R)$, with tensor product as multiplication. Theorem (Traverso, Swan) For a commutative ring$R$the following are equivalent: a) The reduced ring$R_{red}=R/Nil(R)$is semi-normal b) The ... 6 As Steve D said, you can use the fact that projective modules are always flat. Consider the map$\mathbb{Q}[x,y]\to \mathbb{Q}[x,y]$defined by multiplying$x$. This is an injective$\mathbb{Q}[x,y]$-module map, while tensoring$\mathbb{Q}$will give an injective map, but it is NOT. So$\mathbb{Q}$is not flat as$\mathbb{Q}[x,y]$-module. However, ... 6 Suppose$R$is a$k$-algebra, with$k$a commutative ring. If$M$is a$k$-module, we can construct the$k$-module$R\otimes_kM\otimes_kR$, which is automatically an$R$-bimodule or, equivalently, an$R^e$-module. If$N$is an$R$-bimodule, there is a canonical isomorphism $$\hom_{R^e}(R\otimes_kM\otimes_kR, N)\cong\hom_k(M,\bar N)$$ with$\bar N$the ... 6 The truth is (to me) quite surprising: Kaplansky showed that an infinitely generated projective module over any Dedekind domain$D$is free! (The corresponding statement for finitely generated projective modules is equivalent to$D$having trivial class group.) This is referenced, for example, here. 6 For the case$R$is a local ring it's a corollary of Nakayama's lemma. As the notation in the above link, suppose$M$is a finite generated projective module over$R$, then, first pick a minimal number of generators, i.e.,$M=Rm_1+\cdots +Rm_k$, and$k$is the minimal number with this property, so we get a decomposition $$R^k=M\oplus N,$$ then, we are ... 6 Let$R$be any commutative ring whose projective modules are all free, and let$e\notin \{0,1\}$be an idempotent of$R$. Then$eR$and$(1-e)R$are both projective, hence free of some rank 1 or more, and$eR\oplus(1-e)R=R$, so that we have$R^n\cong R$as$R$module for some natural number$n\geq 2$. This is absurd since commutative rings have IBN. This ... 6 I'll consider the interval$[0,2\pi]$for notational simplicity. Consider the matrix $$A = \left( \begin{array}{cc} \sin ^2\tfrac{\theta }{2} & - \sin \tfrac{\theta }{2} \cos \tfrac{\theta }{2} \\ -\sin \tfrac{\theta }{2}\cos \tfrac{\theta }{2} & \cos ^2\tfrac{\theta }{2} \end{array} \right),$$ which defines an$R$-linear ... 5 This has been answered on MathOverflow here. 5 A prominent counter-example is the following: Take$R := {\mathbb R}[x,y,z]/(x^2+y^2+z^2-1)$, the ring of real-valued polynomial functions on the$2$-sphere, and consider the following short exact sequence:$$(\ast)\quad\quad 0\to P\to R\frac{\partial}{\partial x}\oplus R\frac{\partial}{\partial y}\oplus R\frac{\partial}{\partial z}\xrightarrow{\alpha := ... 5 Here is some elaboration on the wiki entry in George's comment. Suppose$R$is a domain.$R$is called seminormal if whenever$b^2=c^3$in$R$one can find$t \in R$such that$b=t^3, c=t^2$. The relevant thing here is the following fact: R is seminormal if and only if$Pic(R) \cong Pic(R[X])$So if$R$is local and not seminormal then there will ... 5 Yes, this is true. See this Math Overflow question for a precise statement and a reference to its proof in Bourbaki's Commutative Algebra. This result is also stated in my commutative algebra notes, but the proof is not unfortunately not yet written up there. I certainly hope that this will be remedied soon though, as I will be teaching a course out of ... 5 Thinking of a f.g. projective module as a vector bundle, it seems very likely the answer is no: consider the trivial bundle of rank 2 and two "twisted" subbundles of rank 1 whose intersection is 0-dimensional everywhere except over a closed subset with non-empty interior – so not a vector bundle, in particular. Let's see where this thinking leads. Let$R$... 5 Please see$\S 3.5.4$-- "Projective verus free" -- in my commutative algebra notes. In particular, Proposition 27 and the exercise follow it step you through showing that the ideal$\langle 3, 1+ \sqrt{-5} \rangle$is projective but not free. The same techniques apply to$I = \langle 2, 1+ \sqrt{-5} \rangle$. (In fact, I view it as a happy accident ... 5 I don't know of any other meaning of a projective ideal other than the one suggested by Boris Novikov, i.e. an ideal of a ring$R$that is also projective as an$R$-module. I want to emphasize that such an ideal$I$need NOT be a direct summand of$R$(Boris never implied that condition to be necessary - only sufficient!) as well as give more examples. The ... 5 The statement you're linking to is: A module$P$is projective if and only if there is a family$\{x_{i}\}_{i \in I} \subset P$and morphisms$f_{i}: P \to R$such that for each$x \in P$we have$x = \sum_{i \in I} f_{i}(x) x_{i}$. The last statement says three things: In order for the sum to make sense we must have that for all$x$the set ... 5 The statement "every element in P can be written as a finite linear combination of some elements of P.", where "some" means a finite set, just says that the module is finitely generated. This has nothing to do with being projective. Take for instance the$\mathbb Z$-module$\mathbb Z/2$. Here every element can be written as a multiple of$[1]$. So$\mathbb ... Only top voted, non community-wiki answers of a minimum length are eligible	2015-05-25 15:52:13	{"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": 0.9292643070220947, "perplexity": 338.67338061109905}, "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-2015-22/segments/1432207928520.68/warc/CC-MAIN-20150521113208-00294-ip-10-180-206-219.ec2.internal.warc.gz"}
http://physics.stackexchange.com/questions/10462/what-makes-an-antenna-special	What makes an Antenna special? The most important property of an antenna, beyond its conductivity(?) seems to be it's length (or more generally, its shape). But the antenna device is connected to the receiver by an arbitrarily-long connector. What is the fundamental difference between an antenna and the connector? Why doesn't the connector's length, straying from some 1/2, 1/4 length of the targeted band, interfere with the reception? - The pair of wires which leads signals to/from the antenna uses a simple trick: In order not to radiate they are configured so to carry opposite currents. The radiation from one wire cancels the radiation from the other wire. This also helps in the other way: The radiation picked up by one wire cancels the radiation picked up by the other wire. The simplest version is the "twisted pair". Two parallel wires with anti parallel currents still have some remaining radiation mainly in the plane which intersects the wires. This can be improved by twisting the wires around each other to better cancel radiation over 360 degrees. Better, but more expensive, is the coax (co-axial) cable in which the outer wire is a cylinder surrounding the inner wire. The radiation from the anti-parallel currents is compensated equally well in all directions. - It is so, indeed. I was about to say the same. – Helder Velez May 27 '11 at 22:59 P.S., @uosɐſ, Electrical engineers have a name for that special, non-radiating pair of conductors that feeds an antenna: It's called a transmission line. – james large Dec 22 '15 at 21:36 Please allow me to quote from one of my old text books - Transmission Principles for Technicians, D C Green: Whenever a current flows in a conductor, the conductor is surrounded by a magnetic field, the direction of which is determined by the direction of current flow. If the current changes, the magnetic field will change also. Now, a varying magnetic field always produces an electric field that exists only while the magnetic field continues to change. When the magnetic field is constant the electric field disappears. The direction of the electric field depends on whether the magnetic field is growing or collapsing and can be determined by the application of Lenz's law. Similarly, a changing electric field always produces a magnetic field; this means that a conductor carrying an alternating current is surrounded by continually changing magnetic and electric fields that are completely dependent on one another. If a sinusoidal current is flowing in a conductor the electric and magnetic fields around the conductor will also attempt to vary sinusoidally. When the current reverses direction the magnetic fields must first collapse into the conductor and then build up in the opposite direction. A finite time is required for a magnetic field and its associated electric field to collapse, however, and at frequencies above about 15KHz not all the energy contained in the field has returned to the conductor before the current has started to increase in the opposite direction and create new electric and magnetic fields. The energy left outside the conductor cannot then return to it and instead is propagated away from the conductor at the speed of light. So that deals with how radio waves are produced. Now, what makes an aerial special? Well, that is all to do with the wavelength ($\lambda$) of the signal. When an alternating current reaches the end of a transmission line it "bounces" back again (echoes) back down the line. This is a big problem when it's not wanted, as the reflected signal can interfere with the rest of the signal causing extra noise and attenuation of the signal. This is why proper termination of transmission lines is essential, to absorb these reflections. However, the same phenomenon can be made use of to boost the signal level. If you measure the transmission line (in this case the aerial) to be precisely a hole division fraction of the wavelength of the signal (most commonly $\frac{1}{4}$ but $\frac{1}{2}$ or $\frac{1}{8}$ are used) it is possible to get the reflections to add to the signal, thus boosting it. The same goes for the received signal in a reception aerial. As the incoming signal inducts a voltage in the aerial reflections will be produced from the open end of the aerial. These reflections are added to the signal to boost the signal level and quality. So the aerial, due to its precisely measured length, can be "tuned" to a specific frequency. Also, there are more complex designs of aerials than just a length of wire. Such designs as the Yagi Array which adds a directional component to the radiation of the signal. The aerial includes a "reflector" which bounces the signal that goes behind the aerial back in a forward direction. By careful position and measuring of the reflector it is possible again to get the waves of the signal to line up and thus increase in power, just like the reflections in the single-wire aerial. - If you apply an alternating electric current on a piece of wire, it will radiate. For example we live in a 50 Herz background in our houses because of the wires bringing in the electricity. A receiving antenna is oriented and has the size necessary to receive signals for the TV or radio. In order to shield the wire connecting the television to the antenna, so there is no interference of unwanted frequencies, one uses special wires called coaxial. They have a core wire which carries the signal and are covered outside with wire mesh or aluminium foil in order to shield from any ambient signals the small wanted signal. If one uses a transmitting antenna the same logic holds. The wires carrying the signal to the aerial are coaxial. - Yes of course, what else, Jason? – Georg May 27 '11 at 15:13 So can an efficient antenna be nothing more than a piece of wire of a correct length? Some designs include an Inductor (I think) before meeting the signal source line. – Jason Kleban May 27 '11 at 15:15 @Georg the monopole antenna would then just be a specific length of unshielded wire connected to a length of shielded wire of arbitrary length. I've been under the impression that the wave has to bounce off the two sides of that specific length, so without some "wall" to bounce off, is it merely the beginning of the shielded segment that changes the EM field properties of the enough to provide the bounce? With a dipole which is fed from the center, it's a little easier for me to understand. – Jason Kleban May 27 '11 at 15:24 That inductor is to "lengthen" the antenna. It is used when You cant use a lambda/4 antenna for some reason. With that inductor the resonance is restored. I recommend You to read basics of electromagnetic wave theory. – Georg May 27 '11 at 15:35	2016-07-01 20:50:39	{"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": 0.6592425107955933, "perplexity": 511.09840695833424}, "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-2016-26/segments/1466783403825.35/warc/CC-MAIN-20160624155003-00059-ip-10-164-35-72.ec2.internal.warc.gz"}
http://mathhelpforum.com/advanced-statistics/114888-convergence-random-variables-print.html	# Convergence of random variables Given a sequence of independent random variables $(w_i)_i$with non-negative values, how do you prove that: the series $\Sigma w_i$ converges almost surely iif the expectation series $\Sigma \mathbb{E}(w_i/(1+w_i))$ converges. the $\Rightarrow$ implication is easy but I'm not certain about the other way round. When $\Sigma \mathbb{E}(w_i/(1+w_i))$ converges, using Markov inequality and the Borel-Cantelli lemma, one can prove that $w_i \rightarrow 0$ almost surely, hence finite expectation and variance for $i$ large enough (This relies on the fact the function $x \rightarrow x/(1+x)$ is strictly increasing on $[0,\infty[$). But how do we know about the series behavior?	2015-05-03 14:26:17	{"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": 9, "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": 0.993842601776123, "perplexity": 104.61039610374094}, "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-2015-18/segments/1430448950892.36/warc/CC-MAIN-20150501025550-00073-ip-10-235-10-82.ec2.internal.warc.gz"}
http://stats.stackexchange.com/questions/55962/what-is-the-difference-between-a-normal-and-a-gaussian-distribution	# What is the difference between a Normal and a Gaussian Distribution Is there a deep difference between a Normal and a Gaussian distribution, I've seen many papers using them without distinction, and I usually also refer to them as the same thing. However, my PI recently told me that a normal is the specific case of the Gaussian with mean=0 and std=1, which I also heard some time ago in another outlet, what is the consensus on this? According to Wikipedia, what they call the normal, is the standard normal distribution, while the Normal is a synonym for the Gaussian, but then again, I'm not sure about Wikipedia either. Thanks - Wikipedia is right, in this case. It usually is for topics like this. I would be more leery of it on controversial topics. – Peter Flom Apr 12 '13 at 17:33 There is a consensus. Your PI is confusing "Normal" with "Standard normal." The former refers to any version of the latter obtained via a change of location or scale. – whuber Apr 12 '13 at 17:43 Go with Wikipedia & Peter & whuber - & hire a different private investigator. – Scortchi Apr 12 '13 at 18:23 Here's one moderately authoritative reference: mathworld.wolfram.com/GaussianFunction.html. – whuber Apr 12 '13 at 20:53 Peter Flom is right - as is Wikipedia, and whuber, and Scortchi. You can find any number of more authoritative works that support it - hundreds, perhaps thousands of standard texts for example and numerous papers. – Glen_b Apr 12 '13 at 23:02 Wikipedia is right. The Gaussian is the same as the normal. Wikipedia can usually be trusted on this sort of question. - The standard normal distribution is the standardIZED normal distribution. That means every value in the standard normal distribution is the z-score of a value in another normal distribution with non-normal mean and std. - The wording is very confusing. In any case, it doesn't answer the the question at all. The comments have proved the answer. – Student T Dec 18 '14 at 23:36 What on earth does this mean? – Dilip Sarwate Dec 18 '14 at 23:36 Nevermind. I must have been tired. – trsk Dec 30 '14 at 0:00 Actually the normal distribution is the sub form of Gaussian distribution. Gaussian distribution have 2 parameters, mean and variance. When there is zero mean and unit variance the Gaussian distribution becomes normal other wise it is pronounced as Gaussian. - No. This is not right. The standard normal distribution has mean 0 and variance 1, but a normal distribution can have any mean and variance. – Peter Flom Sep 5 '13 at 11:52	2015-03-29 13:42:41	{"extraction_info": {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8341177105903625, "perplexity": 818.0413667762043}, "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-2015-14/segments/1427131298538.29/warc/CC-MAIN-20150323172138-00180-ip-10-168-14-71.ec2.internal.warc.gz"}
https://uniformlyatrandom.wordpress.com/2008/11/	# Uniformly at Random ## Allies and enemies Champlain’s new settlement was founded in the summer of 1608 at “Kebec” or the “narrows” of the St. Lawrence. (Thus making this the year of its 400th anniversary.) The same year he was invited by a Montagnais-Algonquin-Huron coalition to join forces with them in an raid on their old enemies, the Iroquois. The attack was to take place the next year. Champlain agreed to join the coalition, expecting that a sufficiently impressive demonstration of force against the Iroquois would deter them from future attacks against his Indian allies. Establishing such a peace would in turn be congenial to the development of the fur trade. In 1609 Champlain set out with his Indian allies and began his journey to the Iroquois country. They traveled up the St. Lawrence, then, just below Montreal, took the Richelieu River southward into Lake Champlain. They encountered a Mohawk war party at the southern end of the lake. Only two other Frenchmen had made it this far with Champlain. The allies engaged the Mohawks on shore. The battle began with Champlain discharging his arquebus to devastating effect: a single shot took out three Mohawk warriors. When the other Frenchmen fired their weapons with similar effects, the Mohawks turned and fled. The allies pursued, winning a remarkable victory and capturing several prisoners. The victory was followed by the standard (among the particular nations involved) torture and cannibalism of (some of) the captives. After his return from Mohawk country to Quebec, Champlain left the colony in the hands of subordinates and returned to France. He returned the following year, 1610, to Quebec, and again, when his Indian allies requested his aid, fought another battle against the Mohawks. The allies were again victorious. After suffering two severe defeats, the Mohawks avoided further conflict with the French for many years. Written by uncudh November 29, 2008 at 3:16 am ## The singing contest In the 1830’s and 1840’s the Finnish physician and scholar Elias Lönnrot traveled throughout the Finnish district of Karelia, collecting and recording various traditional folk songs. He arranged and edited this source material to create the Kalevala, the great national epic of Finland. The influence of the Kalevala on the writings of Tolkien is well-known. (One can find several essays on the subject in the collection Tolkien and the Invention of Myth.) The most notable example of such influence can perhaps be seen in Tolkien’s tale of Turin, which contains several themes and plot elements that are strikingly similar to those of the tale of Kullervo from the Kalevala. This too is a subject that has been much studied. In Tolkien’s world, as in the Kalevala, much of the “magic” is accomplished through song: Luthien enchants Morgoth through song; the magic of Tom Bombadil seems strongly connected to his singing; the world itself is created through the “Song of the Ainur”. The following passage from the Kalevala (Magoun’s translation) describes a singing duel between the “wizards” Väinämöinen and Joukahainen: He sang the cap off the man’s head into the peak of a cloudbank, he sang the mittens off his hands into pond lilies, then his blue broadcloth coat to the heavens as a cloud patch, the soft woolen belt from his waist into stars throughout the heavens. He bewitched Joukahainen himself, sang him into a fen up to the loins, into a grassy meadow up to the groin, into a heath up to the armpits. Written by uncudh November 28, 2008 at 8:23 pm Posted in literature Tagged with , , ## The Ile-Ste-Croix settlement The French made several attempts throughout the 1500’s to establish colonies in North America. Without exception they all failed. Champlain’s exploratory expedition of 1603 was followed in 1604 by a new French project to establish a colony in the Americas. This expedition was led by Pierre Dugua, sieur de Mons. He established in 1604 a settlement on the tiny island of Ile-Ste-Croix, just off the American coast at approximately the Maine-New Brunswick border. The location was poorly chosen. After a terrible winter, the settlement was relocated across the Bay of Fundy to Port-Royal, in what is now Nova Scotia. In each of the years 1604, 1605, and 1606, the French sent out ships to explore the New Brunswick and New England coasts. The expeditions were led by Champain, de Mons, and Poutrincourt respectively. Only Champlain’s was reasonably successful; the latter two were spoiled by certain . . . unpleasantness with the Indians that the French met during the expeditions. Unfortunately for the settlement at Port-Royal, the fashion of hats changed in Paris. Beaver pelt hats were now wildly popular. Increased demand for beaver furs allowed certain business interests to pressure King Henri IV to revoke de Mons’s monopoly of the fur trade, which had allowed him to provide financial backing for the settlement project. With the loss of the monopoly, de Mons’s company failed. The settlers were recalled to France and the settlement abandoned in 1607. in 1608 de Mons convinced the king to give him another chance. The king complied, granting him a 1 year monopoly. De Mons put together a new expedition, this time led by Champlain, who planned his new settlement in the St. Lawrence valley rather than the coast of Acadia. Written by uncudh November 28, 2008 at 6:11 pm Posted in history Tagged with , , ## Some more curious continued fractions We have previously observed that $e$ has the remarkably regular continued fraction expansion $e = [2; 1, 2, 1, 1, 4, 1, 1, 6,\ldots]$. Another curious continued fraction is the following: $\sum_{n \geq 1} 2^{-\lfloor 2n/(\sqrt{5}-1) \rfloor} = [0; 2^0, 2^1, 2^1, 2^2, 2^3, 2^5, 2^8, 2^{13}, \ldots]$, where the exponents of the 2’s in the continued fraction are the Fibonacci numbers. This unusual continued fraction has been independently rediscovered by several authors. See, for example, Anderson, Brown, and Shiue, Proc. Amer. Math. Soc. 123 (1995), 2005-2009. Among the most famous continued fractions are the Rogers-Ramanujan continued fractions: $\cfrac{1}{1+\cfrac{e^{-2\pi}}{1+\cfrac{e^{-4\pi}}{1+\cfrac{e^{-6\pi}}{\vdots}}}} = \left( \sqrt{\frac{5+\sqrt{5}}{2}} - \frac{\sqrt{5}+1}{2} \right)e^{2\pi/5}$ $1-\cfrac{e^{-\pi}}{1+\cfrac{e^{-2\pi}}{1-\cfrac{e^{-3\pi}}{\vdots}}} = \left( \sqrt{\frac{5-\sqrt{5}}{2}} - \frac{\sqrt{5}-1}{2} \right)e^{\pi/5}$. Ramanujan included these formulas in his famous 1913 letter to Hardy. Later, Hardy wrote, “[These formulas] defeated me completely. I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class. They must be true because, if they were not true, no one would have had the imagination to invent them.” Written by uncudh November 27, 2008 at 1:50 am Posted in math Tagged with , ## On the number of 1’s in the binary expansions of multiples of 3 Consider the sequence of multiples of 3: i.e., 3, 6, 9, 12, 15, 18, 21, 24 etc. Now write these numbers in base 2 to obtain the sequence 11, 110, 1001, 1100, 1111, 10010, 10101, 11000, etc. Look at these binary expansions and note which ones have an even number of 1’s and which ones have an odd number of 1’s. For instance, construct the sequence whose n-th term is 0 if there is an even number of 1’s in the binary expansion of 3n and is 1 if there is an odd number of 1’s in the binary expansion of 3n. This sequence begins 0, 0, 0, 0, 0, 0, 1, 0, etc. If one generates (by computer) a long initial segment of this sequence, one may note that there seems to be a clear preponderance of 0’s in any initial segment of this sequence. In fact, there appears to always be more 0’s than 1’s in any initial segment of the sequence. Intuitively, this looks to be utterly bizarre. One would expect the parity of 1’s in the binary expansions of 3n to be more or less random. That is, there should sometimes be a slight preponderance of those with even parity and sometimes a slight deficit of those with even parity. Nevertheless, one can prove definitively that in any intial segment of the sequence 0, 0, 0, 0, 0, 0, 1, 0, . . . , whose n-th term counts the parity of 1’s in the binary expansion of 3n, there are always more 0’s than 1’s. Furthermore, the excess of 0’s over 1’s in any initial segment of length n can be bounded from below and above by $1/20 n^{\log 3 / \log 4}$ and $5n^{\log 3 / \log 4}$ respectively. This curious result was obtained by Donald J. Newman in 1969 and appears in Proc. Amer. Math. Soc.. Written by uncudh November 26, 2008 at 7:29 pm Posted in math Tagged with Samuel de Champlain made his first voyage to Canada in 1603 as part of an expedition led by Pont-Gravé. The expedition sailed up the St. Lawrence, stopping at the harbour of Tadoussac, at the junction of the Saguenay River and the St. Lawrence. By a curious coincidence, at the same time a large number of Montaignais, Etchemins and Algonquins had gathered nearby to celebrate a great victory over their enemies, the Iroquois. They were celebrating with a great tobacco-feast, or tabagie. The French were invited to participate: they smoked tobacco with the leaders, joined in the feasting, and watched the celebratory dances. Of this chance meeting between the French and the Indians, David Hackett Fischer writes in Champlain’s Dream: Here was a moment of high importance in the history of North America. Nobody had planned these events, but both French and Indian leaders were quick to see an opportunity. The Great Tabagie marked the beginning of an alliance between the founders of New France and three Indian nations. Each entered willingly into the relationship and gained something of value in return. The Indians acquired a potential ally against their mortal enemies, the Iroquois. The French won support for settlement, exploration, and trade. The alliance that formed here would remain strong for many years because it rested on a mutuality of material interest. Written by uncudh November 24, 2008 at 10:23 pm ## Ramanujan’s partition congruences Ramanujan proved several remarkable divisibility properties of the number $p(n)$ (recall that these denote the number of partitions of a positive integer $n$). One such property is that $p(5n+4)$ is always a multiple of 5. The simplest proofs of this result of which I am aware make use of the following identity of Jacobi: $\sum_{n=-\infty}^\infty (-1)^n (2n+1) q^{n(n+1)/2} = \prod_{k \geq 1}(1-q^k)^3$. I do not feel sufficiently ambitious to derive Jacobi’s identity here, so we shall assume it without proof and proceed accordingly, this time following Berndt, Number Theory in the Spirit of Ramanujan. Let us begin with the identity $\sum_{n \geq 0} p(n) q^n = \prod_{k \geq 1}(1-q^k)^{-1}$. Multiplying both sides by $q\prod_{k \geq 1}(1-q^{5k})$ gives $\prod_{k \geq 1}(1-q^{5k})\sum_{n \geq 0} p(n) q^{n+1} = q\prod_{k \geq 1}(1-q^{5k})\prod_{k \geq 1}(1-q^k)^{-1}$, which we may rewrite as $\prod_{k \geq 1}(1-q^{5k})\sum_{n \geq 0} p(n) q^{n+1} = q\prod_{k \geq 1}(1-q^{5k})\prod_{k \geq 1}(1-q^k)^{-5}\prod_{k \geq 1}(1-q^k)^4$. However, by the binomial theorem we have $\prod_{k \geq 1}(1-q^{5k}) \equiv \prod_{k \geq 1}(1-q^k)^5 \pmod 5$, where the mod 5 notation means the coefficients of $q^i$ on each side of the equivalence are congruent modulo 5. We therefore conclude that $\prod_{k \geq 1}(1-q^{5k})\prod_{k \geq 1}(1-q^k)^{-5} \equiv 1 \pmod 5$, which in turn implies that $\prod_{k \geq 1}(1-q^{5k})\sum_{n \geq 0} p(n) q^{n+1} \equiv q\prod_{k \geq 1}(1-q^k)^4 \pmod 5$. To show that $p(5n+4)$ is a multiple of 5 it therefore suffices to show that the coefficients of $q^{5n+5}$ in the above expression are multiples of 5. Consider then the righthand side of the equivalence. We have $q\prod_{k \geq 1}(1-q^k)^4$ $= q\prod_{k \geq 1}(1-q^k)\prod_{k \geq 1}(1-q^k)^3$ $= q\sum_{j=-\infty}^\infty (-1)^j q^{(3j^2+j)/2}\sum_{k=-\infty}^\infty (-1)^k (2k+1) q^{k(k+1)/2}$, where we have used Euler’s Pentagonal Number Theorem to obtain the first sum and Jacobi’s Identity to obtain the second sum. Continuing, we have $q\prod_{k \geq 1}(1-q^k)^4$ $= \sum_{j=-\infty}^\infty\sum_{k=-\infty}^\infty(-1)^{j+k}(2k+1)q^{1+(3j^2+j)/2+k(k+1)/2}$. The exponents of $q$ in the above sum are therefore multiples of 5 when $1+(3j^2+j)/2+k(k+1)/2$ is a multiple of 5. Observe that $2(j+1)^2 + (2k+1)^2 = 8(1+(3j^2+j)/2+k(k+1)/2) - 10j^2 - 5$, so that $1+(3j^2+j)/2+k(k+1)/2$ is a multiple of 5 exactly when $2(j+1)^2 + (2k+1)^2$ is a multiple of 5. However, $2(j+1)^2$ can only be 0, 2, or 3 mod 5, and $(2k+1)^2$ can only be 0, 1, or 4 mod 5. Thus, $2(j+1)^2 + (2k+1)^2$ is a multiple of 5 exactly when both $2(j+1)^2$ and $(2k+1)^2$ are multiples of 5. However, if $(2k+1)^2$ is a multiple of 5, then so must $2k+1$ be. We therefore conclude that the coefficient of $q^{5n+5}$ in $q\prod_{k \geq 1}(1-q^k)^4$—and hence in $\prod_{k \geq 1}(1-q^{5k})\sum_{n \geq 0} p(n) q^{n+1}$—is a multiple of 5. This implies that $p(5n+4)$ is a multiple of 5, as claimed. Written by uncudh November 24, 2008 at 4:50 pm Posted in math Tagged with ,	2017-09-20 02:19:26	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 42, "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": 0.5053284764289856, "perplexity": 1845.2347979666329}, "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-2017-39/segments/1505818686117.24/warc/CC-MAIN-20170920014637-20170920034637-00352.warc.gz"}
https://picclick.ca/bling-gold-Plated-3MM-24-IN-figaro-fashion-382883399785.html	# bling gold Plated 3MM 24 IN figaro fashion hip hop chain necklace skinny jewelry $18.57 Buy It Now 16d 21h,$4.01 Shipping, 30-Day Returns, eBay Money Back Guarantee Seller: rokgod333 (5,380) 100%, Location: Tuscaloosa, Alabama, Ships to: Worldwide, Item: 382883399785 MY FEEDBACK IS HIGH POSITIVE YOU CAN TRUST ME MY EBAY STORE IS MYJEWLRYBLING FREE SHIPPING IN UNITED STATES GENUINE PURE 14KT HEAVY YELLOW GOLD PLATED GP OVERLAY LAYERED bonded clad 3mm 24 INCH 24INCH 24 IN 24IN link figaro chain unisex NECKLACE CHAIN CHAIN CHARM RING IS SOLID METAL BRASS BASED JEWELERS METAL BRASS IS USED BECAUSE OF ITS COHESIVE ABILITY WITH GOLD & SILVER AMERICAN MADE IF IN DOUBT READ MY FEED BACK IT IS HIGH POSITIVE SHIPPED FAST I SHIP DAILY ALL EMAIL ANSWERED TOP QUALITY MY EBAY STORE & DOMAIN IS MYJEWELRYBLING KEYWORDS CHAIN INFO ring is a round band, usually in metal, worn as an ornamental Jewellery around the finger, or sometimes the toe; it is the most common current meaning of the word "ring". Strictly speaking a normal ring is a finger ring (which may be hyphenated); other types of rings worn as ornaments are earrings, bracelets for the wrist, armlets or arm rings, toe rings and torc or neck rings, but except perhaps for toe rings, the plain term "ring" is not normally used to refer to these. Rings are most often made of metal but can be of almost any material: metal, plastic, stone, wood, bone, glass, or gemstone. They may be set with a stone or stones, often a gemstone such as diamond, ruby, sapphire or emerald. A carnelian or agate ring worn by some Muslims in imitation of Muhammad. Birthstones ring Usually a slender, simple ring (sometimes consisting of a band), set with the wearer's birthstone, or the birthstone of the wearer's spouse. Those like the Mother's ring can be worn set with various birthstones. Some couples wear birthstones set with a wedding anniversary month birthstone as well as other commemorative stones. Cameo (carving) ring Cameo ring A plain hoop mounted by a table setting, into which is affixed a carved cameo. This ring style is exceedingly ancient and was more commonly worn by men than by women. Ancient cameos depicted pagan gods, Christian saints and even self-portraits. Multi-coloured stone and often marble or porphyry was most desirable, as it produced a striped, layered or three-dimensional effect. The modern cameo ring usually shows the profile of a goddess or a Roman soldier. Championship ring a.k.a. sports ring Kastenring A ring presented to members of winning teams in professional sports leagues as well as college tournaments in the Americas. The best known of these are the Super Bowl ring and World Series ring. Also, in professional American sports leagues—such as the National Football League (NFL) and Major League Baseball (MLB)—the runner-ups of the league championship game/series is awarded a ring, being the champion of their conference (sub-league). Claddagh ring An Irish friendship, courtship or engagement ring. It is traditionally used to indicate the state of romantic availability. In recent times it is commonly worn as a wedding ring. In centuries past, this ring was bequeathed from mother to daughter, though men also wore it. Class Ring.Worn by students and alumni in commemoration of their graduation. Cocktail ring An oversized ladies' ring with a large center stone often surrounded by tiny stones. Nearly any oversized ladies' ring may be termed "cocktail". This is the most common type of costume jewellery ring and is also known as a cluster ring and dinner ring. Doctoral ring A gold ring worn by a scholar who earns a doctoral degree at a Danish or Swedish university. In America it is common for priests who have earned their doctorate in theology to wear such a ring on the right hand ring finger. Ecclesiastical ring of the Archbishop Albero von A religious ring, either of authority for clerics or as some other special religious symbol. When worn by bishops or higher-ranking priests, it is called Wedding and Engagement Rings A ring given to and worn by a woman signifying her engagement to be married. Eternity ring A ring symbolizing eternity with a partner. These are often given in lieu of engagement rings, as when former UK Prime Minister Gordon Brown purchased one for his wife (as a recompense for not having originally proposed to her with an engagement ring). Finger armor ring A ring style which spans from the base of the finger to just below the fingernail or middle of the second joint. This type of ring includes a bending joint. Five Metals ring (a.k.a. "Kabbalah ring") is suggested by the Talmud as a talisman for luck, and is considered a Kabbalistic appurtenance. Five metals mined from the earth (gold, silver, copper, tin and lead) are to be joined but not alloyed into a ring. It can only be manufactured once a month, "when the moon is in Jupiter", over a five-hour period from 1 p.m. to 6 p.m. and only upon the conclusion of the Sabbath (Saturday night). The Five Metals ring has commonly taken the form of a sterling silver ring with tablets of gold, tin, lead and copper set into the bezel. The Talmud states, "Bear [the ring] upon thee, and thou shalt see miracles." Friendship rings are used to symbolize a close relationship whether romantic or platonic. Its common usage is as a token of friendship. Gay Pride ring (a.k.a. "Rainbow ring") Representing gay pride, usually a band, either set with seven stones or inlaid with seven enamelled lines, in the seven colours representing the Rainbow flag (LGBT movement). In decades past, a stone-set ring worn on the right hand ring finger or the pinky of either hand represented a call for gay equality. Giardinetti ring Italian for 'little garden' a design which features an openwork bezel containing multiple small stones. Most prominent in the second half of the 18th century. Gimmal ring made of 2 or 3 hoops that are hinged at the back and meant to interlock and open; popular for betrothals in 16th- and 17th-century Europe Guard ring (a.k.a. ring-guard) is a slender, slightly tighter-fitting ring designed to be placed on the finger after a large/loose ring, to prevent slippage and ring loss. Iron Ring, a.k.a. Engineer's Ring Canadian Engineer Iron Ring Ring worn by American and Canadian engineers, after swearing the Engineer's Oath. This is often in the form of a crudely worked piece of iron; modern rings tend to be sleek steel, some with etched geometric designs. The ring is meant to be worn on the pinky (little) finger of the dominant hand at all times. This ring has been loosely associated with Rudyard Kipling. Key A ring with a key mounted on the bezel. Used by the Romans as both a means of carrying a key to their family valuables chest and to demonstrate their status within the family.Largely dating from the 16th to the 17th centuries, memento rings featured a skull and the inscription "Memento mori", sometimes combined with other features. Mother's ring A ring worn by a mother displaying the birthstone of each of her children, and sometimes including those of the mother and father. Mourning ring A ring worn in memory of someone who has died.Also commonly called a "memorial ring". Use attested from at least the 14th century AD to the late 19th century. Multi-finger ring Two or more laterally conjoined rings, designed to be worn on two, three, or four fingers; popularized by hip-hop culture. Penannular ring Found in gold or gilded metal from Bronze Age Britain, these small thick incomplete circles are the wrong size and shape to be finger-rings and were probably worn as nose or ear-rings or attached to the hair or clothing. Poison ring A ring consisting of a bezel with a compartment. Despite the name they were probably more commonly used to hold things like perfume or romantic keepsakes. Posie ring Medieval gold posy ring A ring with a lengthy inscription on its outer surface. These were commonly used as engagement and wedding rings. Also referred to as "posy" or "poesy" rings in reference to the line of poetry most commonly used in the inscription. Pre engagement ring A small, inexpensive ring given to a partner, signifying the promise not to court anyone else. Promise ring A ring worn to remind a person of a promise. These evolved in conjunction with wedding and religious vow rings in the sense that the ring represented the vow/promise. Purity ring A symbol of virginity and a vow to keep virginity in some religious cultures. Puzzle ring solved Interlocking rings forming a single band. A famous example is the classic Cartier "Trinity" wedding ring. Regards ring A Victorian engagement ring with an implicit acrostic: Ruby, Emerald, Garnet, Amethyst, Ruby, Diamond, Sapphire. Ring rosary Also known as a Decade ring. Ring worn around the finger with 10 indentations (or protrusions) and a cross as a bezel, representing one decade of a rosary. The rings are used to keep track of place in the prayer by rotating the ring on a finger and feeling the marks. Signet ring Baronnet-signet-ring An emblematic ring, often bearing a family coat of arms, some of which are fit for use to imprint a wax seal. In the event a seal or at least a representation of a seal is on the ring, it is called a "seal ring". The signet may bear anything from a custom-designed escutcheon to simple initials, in which case it is known as an initial ring. A large solid gold ring set with a gold sovereign. SS Ehren ring Nazi "honour ring" or "ring of honour". A plain silver band decorated with a death's head. Awarded to members of the Nazi SS (Schutzstaffel). A similar ring (in the form of a death's head) was also favoured by the SS-SD (Schutzstaffel-Sicherheitsdienst), and was very secretive in design. There were in fact several different award rings during the Third Reich. Thumb ring WLA vanda jade thumb ring Originally thumb rings were used as an archery implement mainly in eastern styles of archery. Thumb rings are an extremely ancient custom. Toe Ring Toe rings have a particular function in India. They are considered a customary ornament to be worn by married women. Ring Watch A small watch fashioned to be worn as a ring. Wedding ring A ring presented at the time of marriage to signify espousal and marital commitment. Originally worn only by women, it is now common for both spouses to wear such a ring. Midi ring A ring worn above the knuckle. Made popular in fashion around 2012. Notable individual rings Iffland-Ring (presently held by Swiss actor Bruno Ganz.) Hans-Reinhart-Ring (presently held by actor Christoph Marthaler.) Ring of the Fisherman a.k.a. Papal Ring, the signet ring of office of the Pope. The One Ring (Sauron's Ring), from J. R. R. Tolkien's Lord of the Rings and Hobbit. jewellry onyx collectable clear diamond simulated zirconia ring toe thumb right hand finger overlayed layered bonded ep pendant jewelry jewellry costume fake french unisex yellow gold plated overlay overlayed layered bond bonded ep gp cubic zirconia cz crystal rock stone stones wedding male female teen wedding anniversary friend friendship items product products shank band bands silver plated overlay overlayed layered layer bond onded clear cut rings unisex zodiac trendy fashion hot flashy fancy glitz body solitaire prongs pronged head fancy astrology shiny up up pimp thug thumb toe bling hip hop,cut,band,shank,silver,plated, celtic gothic celt goth pagan wicca wiccan sodalite obsidian quartz carnelian adventurine amethyst amazonite onyx tuquoise agate peridot hematite zirconia cubic cz clear colored diamond simulated cabachon malachite sapphire solitaire simulant garnet ruby howlite synthetic birds animal bird initial filigree cluster stainless steel baguettes thug pimp dress gang rune norse occult legend folklore fantasy druid druidic vintage ancient frateral spiritual indian western marijuana mexican american military orietal asian sparkel automobile casino pendant jewellry marijuana mexican american weight facated polished letter letters initial initials word words saying color colored blue green ct kt black onyx turqouise cluster pewter purple pink peridot blue light dark sky crystal stones simulated diamond rainbow sterling 92.5 A B C D E F G H I J K L M N O P R S T U V W Y Z filigree no stone red coral prong product faceted cut clear yellow white emerald pear octagon triangle round nugget marque yellow gold plated money sam sams sams hand chains necklaces charm rope diamond simulated mm in inch thick thin wide narrow fat big large small little petite pointer stock search birthday gift topaz smoky quartz wholesale sale sales eye catching symbol magic fantasy birthstones birthstone finger sign male female teen mom moms dad kid men mens lady ladies sister father brother husband wife son zodiac ruby red blue saphire ice pendants symbolic pendant anniversary silver plated facated polished cabichon oval nugget,marque gothic celtic riverboat stock bracelet symbol emblem sign dollar car magic fantasy brand brands myth mythical druid pagan goth celt weight polished february january june july march may april august october november december cabichon pink sky crystal gem casino riverboat show race car gaming cards slot dice poker gambling pointer index stock search birthday gift topaz smoky quartz emerald green symbol product products bracelet numbers collectable cabichon oval square white yellow magic fantasy popular traditional collectable mens womens male female teen kid fraternal symbolic military army navy airforce marine usmc seal ranger game warden dog canine bulldog poodle celtic gothic celt goth pagan wicca wiccan sodalite obsidian quartz carnelian adventurine amethyst amazonite onyx tuquoise agate peridot hematite zirconia cubic cz clear colored diamond simulated cabachon malachite sapphire solitaire simulant garnet ruby howlite synthetic birds animal bird initial filigree cluster stainless steel baguettes thug pimp dress gang rune norse occult legend folklore fantasy druid druidic vintage ancient frateral spiritual indian western military orietal asian sparkel MY EBAY STORE & DOMAIN IS MYJEWELRYBLING NUMBER INFO A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, and so forth. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, the term number may refer to a symbol, a word, or a mathematical abstraction. In mathematics, the notion of number has been extended over the centuries to include 0, negative numbers, rational numbers such as \frac{1}{2} and -\frac{2}{3}, real numbers such as \sqrt{2} and \pi, complex numbers, which extend the real numbers by including \sqrt{-1}, and sometimes additional objects. Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic. The same term may also refer to number theory, the study of the properties of the natural numbers. Besides their practical uses, numbers have cultural significance throughout the world.[1][2] For example, in Western society the number 13 is regarded as unlucky, and "a million" may signify "a lot."[1] Though it is now regarded as pseudoscience, numerology, or the belief in a mystical significance of numbers, permeated ancient and medieval thought.[3] Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of many problems in number theory which are still of interest today.[3] During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers and may be seen as extending the concept. Among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Today, number systems are considered important special examples of much more general categories such as rings and fields, and the application of the term "number" is a matter of convention, without fundamental significance.[4] Numbers should be distinguished from numerals, the symbols used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system.[citation needed] Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by digit "5" or by the Roman numeral "?". Notations used to represent numbers are discussed in the article numeral systems. An important development in the history of numerals was the development of a positional system, like modern decimals, which have many advantages, such as representing large numbers with only a few symbols.The Roman numerals require extra symbols for larger numbers. Main classification "Number system" redirects here. For systems for expressing numbers, see Numeral system. See also: List of types of numbers Different types of numbers have many different uses. Numbers can be classified into sets, called number systems, such as the natural numbers and the real numbers. The same number can be written in many different ways. For different methods of expressing numbers with symbols, such as the Roman numerals, see numeral systems.The most familiar numbers are the natural numbers or counting numbers: 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.) However, in the 19th century, set theorists and other mathematicians started including 0 (cardinality of the empty set, i.e. 0 elements, where 0 is thus the smallest cardinal number) in the set of natural numbers.[5][6] Today, different mathematicians use the term to describe both sets, including 0 or not. The mathematical symbol for the set of all natural numbers is N, also written \mathbb{N}, and sometimes \mathbb{N}0 or \mathbb{N}1 when it is necessary to indicate whether the set should start with 0 or 1, respectively. In the base 10 numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base 10 system, the rightmost digit of a natural number has a place value of 1, and every other digit has a place value ten times that of the place value of the digit to its right. In set theory, which is capable of acting as an axiomatic foundation for modern mathematics,[7] natural numbers can be represented by classes of equivalent sets. For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in Peano Arithmetic, the number 3 is represented as sss0, where s is the "successor" function (i.e., 3 is the third successor of 0). Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols three times. Integers Main article: Integer The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a minus sign). As an example, the negative of 7 is written ?7, and 7 + (?7) = 0. When the set of negative numbers is combined with the set of natural numbers (including 0), the result is defined as the set of integers, Z also written \mathbb{Z}. Here the letter Z comes from German Zahl, meaning "number". The set of integers forms a ring with the operations addition and multiplication.[8] The natural numbers form a subset of the integers. As there is no common standard for the inclusion or not of zero in the natural numbers, the natural numbers without zero are commonly referred to as positive integers, and the natural numbers with zero are referred to as non-negative integers. Rational numbers Main article: Rational number A rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator. Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator. Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. The fraction m/n represents m parts of a whole divided into n equal parts. Two different fractions may correspond to the same rational number; for example 1/2 and 2/4 are equal, that is: {1 \over 2} = {2 \over 4}.\, If the absolute value of m is greater than n (supposed to be positive), then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0. The set of all rational numbers includes the integers, since every integer can be written as a fraction with denominator 1. For example ?7 can be written ?7/1. The symbol for the rational numbers is Q (for quotient), also written \mathbb{Q}. Real numbers Main article: Real number The real numbers include all the measuring numbers. The symbol for the real numbers is R, also written as \mathbb{R}. Real numbers are usually represented by using decimal numerals, in which a decimal point is placed to the right of the digit with place value 1. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. For example, 123.456 represents 123456/1000, or, in words, one hundred, two tens, three ones, four tenths, five hundredths, and six thousandths. A finite decimal representation allows us to represent exactly only the integers and those rational numbers whose denominators have only prime factors which are factors of ten. Thus one half is 0.5, one fifth is 0.2, one tenth is 0.1, and one fiftieth is 0.02. To represent the rest of the real numbers requires an infinite sequence of digits after the decimal point. Since it impossible to write infinitely many digits, real numbers are commonly represented by rounding or truncating this sequence, or by establishing a pattern, such as 0.333..., with an ellipsis to indicate that the pattern continues. Thus 123.456 is an approximation of any real number between 1234555/10000 and 1234565/10000 (rounding) or any real number between 123456/1000 and 123457/1000 (truncation). Negative real numbers are written with a preceding minus sign: -123.456. Every rational number is also a real number. It is not the case, however, that every real number is rational. A real number, which is not rational, is called irrational. A decimal represents a rational number if and only if has a finite number of digits or eventually repeats for ever, after any initial finite string digits. For example, 1/2 = 0.5 and 1/3 = 0.333... (forever repeating 3s, otherwise written 0.3). On the other hand, the real number ?, the ratio of the circumference of any circle to its diameter, is \pi = 3.14159265358979\dots Since the decimal neither ends nor eventually repeats forever (see: proof that pi is irrational) it cannot be written as a fraction, and is an example of an irrational number. Other irrational numbers include \sqrt{2} = 1.41421356237 \dots\,(the square root of 2, that is, the positive number whose square is 2). Just as the same fraction can be written in more than one way, the same decimal may have more han one representation. 1.0 and 0.999... are two different decimal numerals representing the natural number 1. There are infinitely many other ways of representing the number 1, for example 1.00, 1.000, and so on. Every real number is either rational or irrational. Every real number corresponds to a point on the number line. The real numbers also have an important but highly technical property called the least upper bound property. When a real number represents a measurement, there is always a margin of error. This is often indicated by rounding or truncating a decimal, so that digits that suggest a greater accuracy than the measurement itself are removed. The remaining digits are called significant digits. For example, measurements with a ruler can seldom be made without a margin of error of at least 0. 001 meters. If the sides of a rectangle are measured as 1.23 meters and 4.56 meters, then multiplication gives an area for the rectangle of 5.6088 square meters. Since only the first two digits after the decimal place are significant, this is usually rounded to 5.61. In abstract algebra, it can be shown that any complete ordered field is isomorphic to the real numbers. The real numbers are not, however, an algebraically closed field, because they do not include the square root of minus one. my DOMAIN is MYJEWELRYBLING CASINO INFO BELOW Games available in most casinos are commonly called casino games. In a casino game, the players gamble casino chips on various possible random outcomes or combinations of outcomes. Casino games are availabl in online casinos, where permitted by law. Casino games can also be played outside of casinos for entertainment purposes, some on machines that simulate gambling.Contents Table games Common non-tab le games Gaming machines Random numbers House advantageStandard deviation There are three general categories of casino games: table games, electronic gaming machines, and random number ticket games such as Keno and simulated racing. Gaming machines, such as slot machines and pachinko, are usually played by one player at a time and do not require the involvement of casino employees to play. Random number games are based upon the selection of random numbers, either from a computerized random number generator or from other gaming equipment. Random number games may be played at a table, such as roulette, or through the purchase of paper tickets or cards, such as keno or bingo.Table games machine (American English), informally fruit machine (British English), puggy (Scottish English slang),[1] the slots (Canadian and American English), poker machine (or pokies in slang) (Australian English and New Zealand English) or simply slot (American English), is a casino gambling machine with three or more reels which spin when a button is pushed. Slot machines are also known as one-armed bandits because they were originally operated by one lever on the side of the machine as opposed to a button on the front panel, and because of their ability to leave the gamer impoverished. Many modern machines are still equipped with a legacy lever in addition to the button. A gambler strategically operating multiple machines in order to draw the highest possible profits is called a multi-armed bandit.Slot machines include a currency detector that validates the money inserted to play. The machine pays off based on patterns of symbols visible on the front of the machine when it stops. Modern computer technology has resulted in variations on the slot machine concept. Slot machines are the most popular gambling method in casinos and constitute about 70 percent of the average US casino's income.Bingo is a game of chance played with different randomly drawn numbers which players match against numbers that have been pre-printed on 5×5 cards.The cards may be printed on paper or card stock, or electronically represented, and are referred to as cards. Many versions conclude the game when the first person achieves a specified pattern from the drawn numbers. The winner is usually required to call out the word "Bingo!", which alerts the other players and caller of a possible win. All wins are checked to make sure the person has not made a mistake before the win is officially confirmed at which time the prize is secured and a new game is begun. In this version of bingo, players compete against one another for the prize or jackpot.Alternative methods of play try to increase participation by creating excitement. Since its invention in 1929, modern bingo has evolved into multiple variations, with each jurisdiction's gambling laws regulating how the game is played. There are also nearly unlimited patterns that may be specified for play. Some games require only one number to be matched, while cover-all games award the jackpot for covering an entire card. There are even games that award prizes to players for matching no numbers or achieving no pattern. See "Variations" for more details.land-based and riverboat casinos can now be found in 17 states, racetrack casinos now operate in 14 states, while Native American casinos are spread in 28 states. land-based and riverboat casinos can now be found in 17 states, racetrack casinos now operate in 14 states, while Native American casinos are spread in 28 states. While the blinds are the forced bets in Texas hold'em and Omaha poker, antes are the forced bets placed by every player before receiving any cards. As an example, let's think of a $10/$20 seven-card stud game with an ante of $1 and a bring-in of$5.Each player wanting to be dealt in to receive a hand would have to post the $1 ante, creating a pot worth competing for. The first three cards are then dealt to each player before it is determined that one player must post the bring-in. In seven-card stud, the player with the lowest-ranking door card would have to post the$5 bring-in at minimum, but this person does have the option to "complete" the bet by posting $10 (the small limit of the game). From there, the betting continues clockwise around the table, with each player having the option to call, raise or fold. In the case that the first player who brought in only posts the bring-in, other players may have the option of completing the bet to$10 as the first raise. Once the betting is complete, every player left in the hand is dealt a fourth card, referred to as fourth street.In games that involve a button that dictates where the action starts, those differ from seven-card stud. In seven-card stud, the first player to act from fourth street on is the player displaying the highest-ranking hand. Cardnals running jumper jumping track bling hip hop GAMING MONEY CENTER POKER CHIPS HORSE RACETRACK JOHNNY CASH GUARD TEAM SPIRIT WHEELING ISLAND CAPRI RIVERBOAT TREASURE ISLAND BLACK JACK UNISEXmoney number numbers numeral's didgt didget's gambling veges riverboat mississippi saying new gaming orleans las vegas Bank nevada casino gaming sport gambling in inch mm money card cards dice slots words helmet number numbers chip chips poker blackjack black jack nugget dollar bill texas holdem high rollernascar charm money bank riverboat 1oz lucky luck game 100 dollar bill searchgames nascar coins coin chip chips sign emblem symbol goal ball player bet gambling points sportsdough band bank Benjamin bill bling bling bone bread bread and butterafrica chedder cheese ching chips chump change c-note coin coins cool cover CREAM money cha ching cheddaR dead presidents deena dime dough dub ducats bank bullion cash cash duckies eight figure ends fin fish fiver flow fundage funds funny money g grand gravy green gregory grip gwop half g hundreds jack jingle juice kick back knot lana large lettuce looney loonie loony loot make it rain means megabucks Michigan bank roll MIL mint monkey moola moolah mula P paper papers payola peanuts pictures of dead presidents pimp juice queer the quid quite a pennyrack samolean sawbuck scratch scrill scrilla scrillas seed money seven figure simoleon six-figure skins skrill skrilla smart money the snaps for the petro stash stupid money tenner ten spot Texas penny throw down tooney toonie two bits twomp wad WAM woot game sports ball team game gym net goal dribble guard center shoot coach basketball indor shooter captain dribbler court nbl college high school pro player money number numbers numeral's didgt didget's us seal parachute cook gambling veges riverboat mississippi saying new gaming orleans las vegas bank nevada casino gaming sport gambling in inch mm money card cards dice slots words chip chips poker blackjack black jack nugget dollar bill texas holdem high rollernascar charm money bank riverboat 1oz lucky luck game 100 dollar bill searchgames nascar coins coin chip chips sign emblem symbol 76 77 78 79 80 81 82 83 84 85 86 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 65 66 67 68 69 70 71 72 73 74 75 40 41 42 43 44 45 4 6 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 goal ball player bet gambling points Condition: New without tags, Condition: brand new without tag i buy in bulk for better pricing, Country/Region of Manufacture: United States, Style: Chain, Base Metal: Brass, Length (inches): 3mm 24 INCH 24INCH 24 IN 24IN figaro chain, Material: brass based jeweler metal, Theme: Family & Friends, Necklace Style: Chain, Pendant Style: no Charm, Pendant Theme: bling hip hop link figaro chain gift womens lady, GENDER: UNISEX MALE FEMALE MENS WOMEN`S TEEN TEENS, SIZE: 3 mm figaro chain 24 INCH 24INCH 24 IN 24IN thin, Color: Gold, Main Stone: No Stone, Pendant Shape: LINK FIGARO, Metal: Yellow Gold Plated, Brand: Unbranded PicClick Insights PicClick Exclusive • Popularity - 119 views, 1.6 views per day, 74 days on eBay. Very high amount of views. 0 sold, 1 available. • Price - • Seller - 5,380+ items sold. 0% negative feedback. Great seller with very good positive feedback and over 50 ratings. Similar Items	2019-06-17 01:13:31	{"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": 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": 0.2667123079299927, "perplexity": 4372.689119093355}, "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-2019-26/segments/1560627998339.2/warc/CC-MAIN-20190617002911-20190617024911-00185.warc.gz"}
https://mathematica.stackexchange.com/questions/112147/write-complex-number-in-terms-of-absolute-value-and-phase	# Write complex number in terms of absolute value and phase I have a long vector and some of the values (19 out of 64) are complex. I got them using the Mathematica Rationalize function, so the complex ones are written in the a+bi form. Is there a function I can apply to the entire vector, that would change my complex numbers to the form AExp[Iphi]? • AbsArg will give the absolute value and argument. The trouble with making a function like argForm[n_] := (#1 E^(I #2)) & @@ AbsArg@n is that, for floating point numbers, a E^(I b) is automatically converted back to the original form – Jason B. Apr 8 '16 at 13:13 • This and this are related. – J. M. will be back soon Apr 8 '16 at 13:57 Try this: {1, 1 + 2 I, 3 - 5 I, 7, 9 + I} /.x_ /; Head[x] == Complex -> Sqrt[Re[x]^2 + Im[x]^2]Exp[ArcTan[Re[x], Im[x]]] yielding ( {1, Sqrt[5] E^ArcTan[2], Sqrt[34] E^-ArcTan[5/3], 7, Sqrt[82] E^ArcTan[1/9]} ) Edit: to address your question If you want to have the argument shown as fractions of Pi sa for the angles of 45 grad or 60 grad, this is achieved automatically. If you need to express each argument as the fraction of Pi, you might try this: {1, 1 + 2 I, 3 - 5 I, 7, 9 + I, 1 + I, 3 + I} /. x_ /; Head[x] == Complex ->Sqrt[Re[x]^2 + Im[x]^2] Exp[NumberForm[Rationalize[N[ArcTan[Re[x], Im[x]]/\[Pi]]], {3, 2}]I\[Pi]] But this will not be the form which you can further operate with, just because of the NumberForm function and also the negative arguments look ugly. The way to transform it depends upon the answer to the question, what do you want it for in such a form? Have fun! • Not bad, but is there a way to approximate the phases so you don't get Pi + or - some ArcTan of something ugly? – PhysNerd90 Apr 8 '16 at 13:18 • Did you want to have an I after Exp[? – Jason B. Apr 8 '16 at 13:37 The trouble with many methods is that they only work on integer inputs. Trying Alexei's answer with approximate numbers {1.0, 1.0 + 2 I, 3.0 - 5 I, 7, 9.0 + I} /. x_ /; Head[x] == Complex -> Sqrt[Re[x]^2 + Im[x]^2]Exp[I ArcTan[Re[x], Im[x]]] ( {1., 1. + 2. I, 3. - 5. I, 7, 9. + 1. I} ) just spits back out the original answer. Also, a simpler way to do this would be argForm[n_] := (#1 E^(I #2)) & @@ AbsArg@n But again, if the numbers are decimals this won't work because Mathematica automatically parses numbers like these to have into the original form, 1. Exp[2. I]//FullForm ( Complex[-0.4161468365471424,0.9092974268256817] ) If you want the numbers to be in exponential form for display purposes, then this is the way to go, argForm[n_] := (Row@{Abs[n], Superscript["\[ExponentialE]", "\[ImaginaryI]" <> ToString[Arg[n]]]}) This will work for exact and approximate numbers, argForm[1 + 2. I] argForm[1 + 2 I] Since the arguments are rational: v = {1/10, -1 - 2 I, 3 - 5/3 I, 7, 9/10 + I}; Abs[v] Exp[I Arg[v]] {1/10, Sqrt[5] E^(I (-π + ArcTan[2])), 1/3 Sqrt[106] E^(-I ArcTan[5/9]), 7, 1/10 Sqrt[181] E^(I ArcTan[10/9])} • I tried something similar: Abs[numericGamma2[[4]]]Exp[IArg[numericGamma2[[4]]]], the problem is the result looks ugly Exp[I ( -pi + ArcTan(93/80))]. Is there a way to approximate this? – PhysNerd90 Apr 8 '16 at 13:15 • You get a simpler form using rectangular coordinates. – bill s Apr 8 '16 at 13:22 • Yep, I think I'll stick with that. Thanks. – PhysNerd90 Apr 8 '16 at 13:25	2019-10-18 22:58:58	{"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": 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": 0.631016731262207, "perplexity": 2304.147634686218}, "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-2019-43/segments/1570986684854.67/warc/CC-MAIN-20191018204336-20191018231836-00228.warc.gz"}
https://szhao.me/mechanism/	# A Mechanism Design Alternative to Individual Calibration Published: ### Decision Making with Uncertainty and Probabilistic Predictions People constantly have to make decisions in face of uncertainty. For example, a patient might have to decide to undergo a treatment without knowing whether the treatment will work; a passenger might have to make travel plans without knowing whether flights will delay or not. How can machine learning help patients/passengers make decisions? A running example we will use in this blog is a decision-maker we call Alice. She considers whether or not to take a flight for a conference. The flight ticket costs \$100, the conference’s utility is \$200, but Alice can only attend the conference if the flight arrives on-time. Therefore, if she takes the flight and the flight is on time, she gains \$100 (i.e. \$200 for the conference - \$100 for the ticket); on the other hand, if the flight is delayed, she misses the conference and loses the \$100 she paid for the ticket. If she doesn’t take the flight, she gains \$0 in any case. Therefore, based on whether the flight will delay or not, taking the flight might be a good or bad decision for Alice. To help Alice make decisions, the airline company can predict whether the flight will delay or not. The airline has a lot of data on past delays; based on the data, the airline can predict the probability of delay for a future flight with an algorithm such as decision tree or neural networks. As an example, for the flight Alice considers, the airline predicts that it delays with 30\% probability. If the predicted probabilities are correct, then each passenger could use them to make decisions. For example, Alice can compute her expected utility as $\underbrace{\100}_{\substack{\text{Utility of} \\ \text{on-time flight}}} \times \underbrace{70\%}_{\substack{\text{Predicted probability}\\ \text{of on-time}}} + \underbrace{-\100}_{\substack{\text{Utility of}\\ \text{delayed flight}}} \times \underbrace{30\%}_{\substack{\text{Predicted probability}\\ \text{of delay}}} = \40$ However, why should Alice believe in these probabilities? For example, the airline could just (intentionally) under-predict its delay probability to make its flight offerings more attractive. Even when the airline has no dishonest intent, predicting the future is difficult. To convey confidence about the predictions, an airline could claim that its predictions have a track record of being calibrated For example, as in the illustration above, the airline could claim: consider all flights for which I predicted would delay with 30% probability; among these flights, indeed 30% of the flights turned out delayed. At first sight, it might seem like Alice should believe in the 30% probability. However, consider the following situation If Alice is a passenger from SFO, then actually 40% of flights delay. Even though the 30% probability is on average correct, it is correct neither for NYC nor SFO passengers. As other examples, the probabilities might be incorrect for flights in the morning or evening; rainy days or dry days; holiday seasons or regular seasons. Ultimately, the key problem is that each flight in unique, so the airline has never seen a completely identical flight to make claims about the past performance of that individual predictions. However, that individual prediction is the only thing that matters to Alice when she tries to decide whether to take the flight or not. ### A Mechanism to Convey Confidence: An Example In our paper we propose an insurance-like mechanism that 1) enables each decision-maker to confidently make decisions as if the advertised probabilities were individually correct, and 2) is implementable by the forecaster with provably vanishing costs in the long run. We illustrate this mechanism by continuing Alice’s example above: Recall that the airline predicts a 30% chance of delay; if this probability were correct, Alice could compute her expected utility (which is \$40). However, Alice is not convinced that the 30% probability is correct. To reduce her risk, Alice proposes a bet (we will show how the bet is selected later): if the flight delays, the airline has to pay Alice \$140; if the flight doesn’t delay, Alice pays the airline \$60. Alice’s perspective: If the airline agrees to bet with Alice, then if the flight delays, Alice’s total utility will be $\underbrace{-\100}_{\text{Utility of delayed flight}} + \underbrace{\140}_{\text{Utility from betting}}=\40$ if the flight doesn’t delay, Alice’s total utility will be $\underbrace{\100}_{\text{Utility of on-time flight}} + \underbrace{-\60}_{\text{Utility from betting}}=\40$ Therefore, whether the flight delays or not, Alice gains \$40 from taking the flight. This is the same utility as if the predicted probability is correct. In other words, Alice can use the predicted 30\% probability as if it is the true probability of delay when making decisions. Airline’s perspective From the airline’s perspective, Alice’s bet is fair: under the the airline’s own predicted probability, the expected betting payout to Alice is \$0: $\underbrace{\140}_{\substack{\text{Payment given to}\\ \text{ Alice if delay}}} \times\underbrace{30\%}_{\substack{\text{Airline's predicted}\\ \text{ probability of delay}}} - \underbrace{60}_{\substack{\text{Payment received from }\\\text{ Alice if on-time}}} \times \underbrace{70\%}_{\substack{\text{Airline's predicted }\\\text{probability of on-time}}}=0$ In other words, if the airline genuinely believes in its own probability, the airline should agree to bet with Alice. By doing this, the Airline can give Alice full confidence in its predictions, which increases the attractiveness of its flight offerings. In addition, we will see soon that the airline has nothing to lose: we will show that the airline has a prediction algorithm to guarantee that it doesn’t lose money from accepting such bets in the long run. ### General Mechanism Definition and Its Benefits We now generalize the numerical example above, and explain how this mechanism will work in general for any prediction and passenger utility For each flight $t$ the airline will predict some probability $\mu_t \in [0, 1]$, and allow passengers to choose any bet that is “fair” as if $\mu_t$ is the true probability. In other words, the passenger can choose any “stake” $b \in \mathbb{R}$ and enter the following betting agreement with the airline: airline pays passenger $b(1-\mu_t)$ if the flight delays, and passenger pays airline $b\mu_t$ if the flight doesn’t delay. This is “fair” because $\underbrace{b(1-\mu_t)}_{\substack{\text{Airline pays }\\\text{passenger if delay}}} \times \underbrace{\mu_t}_{\substack{\text{predicted probability}\\\text{of delay}}} - \underbrace{b\mu_t}_{\substack{\text{Passenger pays }\\\text{airline if on-time}}} \times \underbrace{(1-\mu_t)}_{\substack{\text{Predicted probability}\\\text{of on-time}}} = 0$ The passenger can also opt-out by choosing $b=0$. Why this mechanism benefits passengers In our paper (Proposition 2) we show that any passenger can have the same guarantee that Alice’s gets in our numerical example: he or she can always choose some $b \in \mathbb{R}$ such that 1. The choice of $b$ only depends on information available to him or her (which is his/her own utilities, and the airline’s predicted probability $\mu_t$) 2. His or her total utility (utility of flight + utility from betting) always equals the expected utility as if the predicted probability $\mu_t$ is correct. Why this mechanism doesn’t hurt airline (forecaster) In our paper (Theorem 1 and Corollary 1) we show that the airline (or forecaster) has nothing to lose: we design an online algorithm to make the predictions $\mu_t, t=1, 2, \cdots, T$ such that when $T$ is large the betting loss (for the airline) is guaranteed to vanish. The only requirement is that the maximum betting loss at each time step $t$ is bounded, which the airline can easily enforce by limiting the maximum stake $b$. Our guarantee in Theorem 1 holds even when the passengers are adversarial. For example, if some malicious passenger Bob knows that the true probability of delay of a flight is actually 40% while the airline predicts it is 30%, Bob can choose some stake $b$ to maximally profit from the airline’s mistake. Even though initially Bob could be successful, in the long run our online forecasting algorithm can adapt and prevent such exploitation. From a very high level, our algorithm is based on the observation that the airline’s (forecaster’s) cumulative betting loss is upper bounded by a type of regret called swap regret [1]; our prediction algorithm can modify any existing prediction algorithm to have vanishing swap regret (in the long run), so the betting loss also vanishes. Why this mechanism benefits airline (forecaster) If we assume that some passengers are risk averse (the fundamental assumption of the insurance industry), then this mechanism can also benefit the airline. For example, Alice might be hesitant to buy the flight ticket because she doesn’t know whether the decision has positive utility or negative utility. If the airline can convey confidence and assure Alice that she can receive \$40 utility if she takes the flight, Alice should be more willing to purchase the flight ticket. Consequently, the airline can also charge a higher ticket price without reducing sales. In our example above, with the betting mechanism Alice should be willing to pay up to \$140 ticket price (at \$140 her utility is guaranteed to be exactly zero), but without it she might not be willing to pay \$100. The airline can benefit from this mechanism even when only a small percentage of customers are risk averse, and the rest are risk neutral or even risk seeking. The bet is an option — each customer can choose to opt-out by selecting the stake $b=0$. As one of our experiments in the paper, we run a simulation on real flight delay data and vary the proportion of risk averse passengers. By offering the opportunity to bet, we show that the airline can increase both its revenue and the overall utility (i.e. airline’s revenue + total passenger utility). Even when only 20% of passengers are risk averse, the airline’s revenue still increases by around 10%. ### References [1] Blum, Avrim, and Yishay Mansour. “From external to internal regret.” Journal of Machine Learning Research 8, no. 6 (2007).	2021-12-07 21:36:44	{"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": 3, "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": 0.4919509291648865, "perplexity": 1100.858239201669}, "config": {"markdown_headings": true, "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-2021-49/segments/1637964363418.83/warc/CC-MAIN-20211207201422-20211207231422-00030.warc.gz"}
https://www.physicsforums.com/threads/throwing-a-javelin-projectile-motion.194085/	# Throwing a Javelin (Projectile Motion) 1. Oct 26, 2007 ### TopCat 1. The problem statement, all variables and given/known data An athlete can throw a javelin 60m from a standing position. If he can run 100m at constant velocity in 10s, how far could he hope to throw the javelin while running? Neglect air resistance and the height of the thrower in the interest of simplicity. 2. Relevant equations I've found the range to be R(θ) = 2 v0/g sinθ (v0 cosθ + vr) where v0 is the initial velocity and vr is the velocity of the thrower's initial run. 3. The attempt at a solution Given the above equation and the premise that the thrower can throw 60m from standing, I solved for v0 using vr=0, R=60, g=9.8, and θ=45° as that is the ideal throwing angle from standing. I found v0 = √(60g). Next I tried to find the ideal angle to throw at when vr = 10 m/s. I did this by taking solving R'(θ)=0 where R'(θ) is the derivative of the range formula. I found: R'(θ) = c v0²/2 cos2θ + c vr cosθ = 0 where c = 2 v0/g. This yielded the equation v0 cos2θ = -vr cosθ. Using an identity for cos2θ, I obtained the following quadratic equation: 2 cos²θ + d cosθ -1 = 0 where d = vr/v0 = 10/√(60g). Solving for θ I got θ = 0.65 rads = 37°. The book, however, lists 52.3° as the angle, and therefore gets a different value of R than I do. I'm not sure where I went wrong with my logic. Last edited: Oct 26, 2007 2. Oct 26, 2007 ### Staff: Mentor I haven't look through the detail yet, but assuming that the runner/thrower throws at the same angle as when standing, the runner's velocity adds to the vx of the javelin, but does not contribute to the vertical velocity. Make sure the correct angle is calculated for the range (60 m) of the throw from stationary position. 3. Oct 26, 2007 ### TopCat That's kind of what's tying me up. The problem stated nothing about the initial angle thrown, so I made the assumption that the thrower threw at the angle to maximize range. From standstill that's pi/4, and with some added vr, I presumed the ideal angle would change and sought to calculate it and use it to calculate the final range. 4. Oct 26, 2007 5. Nov 3, 2007 ### TopCat Okay, so after thinking about this, I've realized the angle clearly should be greater than $$45^\circ$$. This is because the $$v_x$$ is greater than the $$v_y$$ when the thrower is running. That being the case, he must throw at a greater angle so that the effective angle thrown at is $$45^\circ$$. Provided that reasoning correct, I have, for maximum range: $$v_{0x} = v_{oy}$$ $$\Downarrow$$ $$v_0 \sin\theta = v_0 \cos\theta + v_r$$ $$\Downarrow$$ $$\sin\theta - \cos\theta = \frac{v_r}{v_0}$$ Squaring both sides gives $$\sin^2\theta + \cos^2\theta - 2\sin\theta\cos\theta = \frac{v_r^2}{v_0^2}$$ $$\Downarrow$$ $$\sin2\theta = -\frac{v^2_r}{v^2_0}$$ However, when I solve that I don't get the angle of $$52.3^\circ$$ (I get something ludicrous like $$-10^\circ$$). Was my reasoning incorrect? Also, when I take $$R = \frac{2v_0}{g}(v_0 \cos\theta + v_r) \sin\theta$$ and solve R' = 0 I get $$35^\circ$$ as the solution. Where am I going wrong here? Last edited: Nov 3, 2007	2017-06-23 07:04:36	{"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": 0.7187714576721191, "perplexity": 1590.788030387841}, "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-2017-26/segments/1498128320023.23/warc/CC-MAIN-20170623063716-20170623083716-00363.warc.gz"}
http://kamerynjw.net/teaching/2020/math455/syllabus/	# Class Information Course Title Math 455 (Section 001): Logic Instructor Kameryn Williams Email kamerynw [ at ] hawaii ( period ) edu Class Hours Monday/Wednesday/Friday 13:30–14:20 Room Keller Hall 301 Office Hours Monday/Wednesday 14:30–15:30; Friday 9:00–10:00 Office Physical Science Building 305 Textbook Mathematical Logic, 2nd ed., Ebbinghaus, Flum, and Thomas (ISBN: 978-1-4757-2357-1) Course Description A system of first order logic. Formal notions of well-formed formula, proof, and derivability. Semantic notions of model, truth, and validity. Completeness theorem. Prerequisite 321 or graduate standing in a related field or consent. Recommended: 454. There will be three portions of your final grade, each worth one third of the total: one midterm exam, one final exam, and homework. Grades will be assigned based upon the standard F to A scale. I do not anticipate this happening, but I reserve the right to make later adjustments to the scale. • [93,100] A; • [90,93) A-; • [87,90) B+; • [83,87) B; • [80,83) B-; • [77,80) C+; • [73,77) C; • [70,73) C-; • [67,70) D+; • [63,67) D; • [60,63) D-; • [0,60) F. # Attendance Policy You are expected to attend every class, for that is an important part of learning the material. I advise you to exchange contact info with classmates so that you can share notes in case you have to miss class. # Exams There will be one midterm and one final exam. The format of the exams will be announced later. • Midterm: (tentatively) Friday, Mar 13 • Final: Monday, May 11, 14:15–16:15 # Homework Assigned homework can be found on the course website. Homework will be collected once a week, on Fridays. The work you turn in is expected to be your own. Each homework assignment will be graded on the following rubric. • 8 points: Mathematical correctness. To get full points you should demonstrate mastery of the concepts. Your proofs should be largely correct, with any errors being minor. • 2 points: Mathematical communication. Doing mathematics is not just about having a valid argument, but also presenting it in a way that a reader can follow. To get full points your arguments should be presented clearly, showing competency at mathematical writing. Occasionally, I may assign an optional, more difficult “reach” exercise as part of a homework assignment. A correct write-up of a solution to the reach exercise will be worth a bonus 1 point for your grade for the homework. Collaboration is both allowed and encouraged. However, you should write your own solutions. It is okay to discuss the ideas with your fellow classmates, but simply copying their solution constitutes cheating. If you do collaborate on a problem, state so at the beginning of your solution. If you consult additional references or books, you should list them as well. Problem sets will not be accepted past their deadline. If you will not be in class, then arrange to turn in your homework by email or through a classmate, or get it to me earlier. If you do not fully finish, then turn in your partial solutions. It may be that you do not see how to completely solve a problem, but you see how it could be solved if you could prove an intermediate result, or you can prove a special case of the problem. In this case, clearly indicate the nature of your partial solution. This may result in partial credit or, for the really tricky problems, full credit. On the other hand, just writing something does not guarantee credit. You must demonstrate understanding of the material. That said, I understand that life gets in the way and sometimes homework doesn’t get done on time. To accommodate this, I will drop your lowest one homework score from the final calculation for your grade. If there is something that will cause you to miss more than one week of homework, please get in contact with me as early as possible. Here is a short example of how I would like to see homework written up. In case you want to use it as a template, here is the .tex source file. Cheating, plagiarism, and other forms of academic dishonesty will not be tolerated. # KOKUA Program and Accessibility Students with disabilities are legally entitled to reasonable accommodations to ensure equal access to education. Any student who feels they may need accommodation based on the impact of a disability should contact the KOKUA Program, the UH Mānoa office for students with disabilities. I am committed to providing students with equal access to this class, and am happy to work with you and KOKUA to ensure reasonable accommodations in my course. Because the accommodations offered are usually forward-looking modifications rather than mitigating poor grades you may have already received due to your disability, it is important to get in touch with the KOKUA Program as soon as you can. Further information and contact details can be found on their website. The ADA defines a disability as a medical condition that substantially limits one or more major life activities—including things like walking sleeping, taking care of yourself, learning, and regulating your emotions—or major bodily functions. If you have a medical condition—including mental health conditions—that significantly interferes with your schoolwork, you probably qualify. You do not need to disclose your condition to your instructors to receive accommodations. # Counseling and Student Development Center The Counseling and Student Development Center offers confidential counseling services to support students with personal, academic, or career concerns. # Food Vault Hawaiʻi Groups on campus have organized a food pantry, free to use for students at UH Mānoa. All registered students with a valid student ID may access the food pantry. Further information, including location and schedule, can be found here or on this facebook page.	2020-02-29 00:58:27	{"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": 0.36420708894729614, "perplexity": 2123.1657902050874}, "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/1581875148163.71/warc/CC-MAIN-20200228231614-20200229021614-00379.warc.gz"}
https://www.physicsforums.com/threads/inflation-and-expansion-of-spacetime.555748/	# Inflation and expansion of spacetime 1. Dec 1, 2011 ### stglyde How come a "positive-energy false vacuum would, according to general relativity, generate an exponential expansion of space"? 2. Dec 2, 2011 ### Chalnoth If we have a smoothly-distributed energy density, then the expansion of space (neglecting spatial curvature) can be written as: $$H(t)^2 = \rho(t)$$ (neglecting constants for clarity) This can be derived directly from the Einstein field equations in General Relativity. Here $H(t)$ is the expansion rate, defined as: $$H(t) = {1 \over a(t)}{d \over dt}a(t)$$ ...and $\rho(t)$ is the energy density of the universe. Now, if the energy comes just from a false vacuum, then that energy is a constant. So if we define $H_0 = H(t=0)$, then we can simply write: $${1 \over a}{da \over dt} = H_0$$ So now we have a simple differential equation. I can then multiply both sides by the scale factor $a$ to put the differential equation in a more familiar form: $${da \over dt} = H_0 a$$ If you know your most basic differential equations, this should look very familiar to you: the rate of change in the scale factor is proportional to the scale factor. This is the equation for exponential growth! $$a(t) = a(t=0) e^{H_0 t}$$ (If you're having difficulty, think compound interest: the amount added to your bank account each month is proportional to your balance, which means that your bank account balance grows exponentially). 3. Dec 2, 2011 ### Ken G And in case that excellent mathematical answer still leaves you some questions about the basic physics at a more descriptive level, I would point out that in general relativity, gravity does not just come from rest mass (and hence rest energy), it also comes from pressure. Usually the pressure contribution is negligible-- like the way the pressure of the Sun contributes to its gravity is totally swamped by the way its rest mass contributes to its gravity. But that's because the Sun is mostly nonrelativistic gas-- vacuum energy would be working in a highly relativistic way, whatever is causing it. Now, in unusual situations (like with vacuum energy), pressure can not only be important to gravity, it can be related to energy in weird ways-- in particular, it can be negative when the energy is positive! The reason for this is that pressure is basically how much energy you can remove from a system when you expand it a given tiny amount, but to expand vacuum, it requires more vacuum-- which if vacuum holds energy, requires that you add energy! So you don't extract energy when vacuum expands, you need to add it instead. That means the pressure of vacuum is negative if there is vacuum energy, and that means its gravity is negative too (or "antigravity"). So two masses placed far enough apart actually experience a kind of repulsion-- due to the vacuum between them (using the cosmological constant model-- so simple vacuum energy). Now, normally this requires a whole lot of vacuum to be in there, so perhaps this is why we are just now starting to see this effect in our expanding universe (the accelerated expansion phase that is just getting going). But if there was a very early phase of the universe where there was a huge energy associated with a "false vacuum", then even with dense positive pressure there could be a huge negative pressure component, just for a short while, that could have created the inflation. After the false vacuum "decayed", it no longer had that huge antigravity, and the normal gravity of all the positive pressure and rest mass would have taken over, until the recent era of acceleration.	2018-05-28 05:38:58	{"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": 0.7586743235588074, "perplexity": 464.4611668203272}, "config": {"markdown_headings": true, "markdown_code": true, "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-2018-22/segments/1526794871918.99/warc/CC-MAIN-20180528044215-20180528064215-00623.warc.gz"}
https://www.jobilize.com/course/section/notation-orthonormal-bases-in-real-and-complex-spaces-by-openstax?qcr=www.quizover.com	# 15.11 Orthonormal bases in real and complex spaces Page 1 / 1 This module defines the terms transpose, inner product, and Hermitian transpose and their use in finding an orthonormal basis. ## Notation Transpose operator $A^T$ flips the matrix across it's diagonal. $A=\begin{pmatrix}a_{1, 1} & a_{1, 2}\\ a_{2, 1} & a_{2, 2}\\ \end{pmatrix}$ $A^T=\begin{pmatrix}a_{1, 1} & a_{2, 1}\\ a_{1, 2} & a_{2, 2}\\ \end{pmatrix}$ Column $i$ of $A$ is row $i$ of $A^T$ Recall, inner product $x=\begin{pmatrix}{x}_{0}\\ {x}_{1}\\ ⋮\\ {x}_{n-1}\\ \end{pmatrix}$ $y=\begin{pmatrix}{y}_{0}\\ {y}_{1}\\ ⋮\\ {y}_{n-1}\\ \end{pmatrix}$ $x^Ty=\begin{pmatrix}{x}_{0} & {x}_{1} & \dots & {x}_{n-1}\\ \end{pmatrix}\begin{pmatrix}{y}_{0}\\ {y}_{1}\\ ⋮\\ {y}_{n-1}\\ \end{pmatrix}=\sum x_{i}y_{i}=y\dot x$ on $\mathbb{R}^{n}$ Hermitian transpose $(A)$ , transpose and conjugate $(A)=\overline{A^T}$ $y\dot x=(x)y=\sum x_{i}\overline{y_{i}}$ on $\mathbb{C}^{n}$ Now, let $\{{b}_{0}, {b}_{1}, \dots , {b}_{n-1}\}$ be an orthonormal basis for $\mathbb{C}^{n}$ $\forall i, \colon {b}_{i}\dot {b}_{i}=1$ $(i\neq j, {b}_{i}\dot {b}_{j}=({b}_{j}){b}_{i}=0)$ Basis matrix: $B=\begin{pmatrix}⋮ & ⋮ & & ⋮\\ {b}_{0} & {b}_{1} & \dots & {b}_{n-1}\\ ⋮ & ⋮ & & ⋮\\ \end{pmatrix}$ Now, $(B)B=\begin{pmatrix}\dots & ({b}_{0}) & \dots \\ \dots & ({b}_{1}) & \dots \\ & ⋮ & \\ \dots & ({b}_{n-1}) & \dots \\ \end{pmatrix}\begin{pmatrix}⋮ & ⋮ & & ⋮\\ {b}_{0} & {b}_{1} & \dots & {b}_{n-1}\\ ⋮ & ⋮ & & ⋮\\ \end{pmatrix}=\begin{pmatrix}({b}_{0}){b}_{0} & ({b}_{0}){b}_{1} & \dots & ({b}_{0}){b}_{n-1}\\ ({b}_{1}){b}_{0} & ({b}_{1}){b}_{1} & \dots & ({b}_{1}){b}_{n-1}\\ ⋮\\ ({b}_{n-1}){b}_{0} & ({b}_{n-1}){b}_{1} & \dots & ({b}_{n-1}){b}_{n-1}\\ \end{pmatrix}$ For orthonormal basis with basis matrix $B$ $(B)=B^{(-1)}$ ( $B^T=B^{(-1)}$ in $\mathbb{R}^{n}$ ) $(B)$ is easy to calculate while $B^{(-1)}$ is hard to calculate. So, to find $\{{\alpha }_{0}, {\alpha }_{1}, \dots , {\alpha }_{n-1}\}$ such that $x=\sum {\alpha }_{i}{b}_{i}$ Calculate $(\alpha =B^{(-1)}x)\implies (\alpha =(B)x)$ Using an orthonormal basis we rid ourselves of the inverse operation. #### Questions & Answers Is there any normative that regulates the use of silver nanoparticles? Damian Reply what king of growth are you checking .? Renato What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ? Stoney Reply why we need to study biomolecules, molecular biology in nanotechnology? Adin Reply ? Kyle yes I'm doing my masters in nanotechnology, we are being studying all these domains as well.. Adin why? Adin what school? Kyle biomolecules are e building blocks of every organics and inorganic materials. Joe anyone know any internet site where one can find nanotechnology papers? Damian Reply research.net kanaga sciencedirect big data base Ernesto Introduction about quantum dots in nanotechnology Praveena Reply what does nano mean? Anassong Reply nano basically means 10^(-9). nanometer is a unit to measure length. Bharti do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment? Damian Reply absolutely yes Daniel how to know photocatalytic properties of tio2 nanoparticles...what to do now Akash Reply it is a goid question and i want to know the answer as well Maciej characteristics of micro business Abigail for teaching engĺish at school how nano technology help us Anassong Do somebody tell me a best nano engineering book for beginners? s. Reply there is no specific books for beginners but there is book called principle of nanotechnology NANO what is fullerene does it is used to make bukky balls Devang Reply are you nano engineer ? s. fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball. Tarell what is the actual application of fullerenes nowadays? Damian That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes. Tarell what is the Synthesis, properties,and applications of carbon nano chemistry Abhijith Reply Mostly, they use nano carbon for electronics and for materials to be strengthened. Virgil is Bucky paper clear? CYNTHIA carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc NANO so some one know about replacing silicon atom with phosphorous in semiconductors device? s. Reply Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure. Harper Do you know which machine is used to that process? s. how to fabricate graphene ink ? SUYASH Reply for screen printed electrodes ? SUYASH What is lattice structure? s. Reply of graphene you mean? Ebrahim or in general Ebrahim in general s. Graphene has a hexagonal structure tahir On having this app for quite a bit time, Haven't realised there's a chat room in it. Cied what is biological synthesis of nanoparticles Sanket Reply how did you get the value of 2000N.What calculations are needed to arrive at it Smarajit Reply Privacy Information Security Software Version 1.1a Good Got questions? Join the online conversation and get instant answers! Jobilize.com Reply ### Read also: #### Get the best Algebra and trigonometry course in your pocket! Source: OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15 Google Play and the Google Play logo are trademarks of Google Inc. Notification Switch Would you like to follow the 'Signals and systems' conversation and receive update notifications? By By Nick Swain By By By Mldelatte	2019-07-22 01:06:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 30, "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": 0.30946585536003113, "perplexity": 4703.055519781886}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "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-2019-30/segments/1563195527458.86/warc/CC-MAIN-20190722010436-20190722032436-00038.warc.gz"}
http://quant.stackexchange.com/tags/local-volatility/hot?filter=year	# Tag Info There is no difference in information, though the fitting algorithm may increase in complexity. First note that in practice you never have an entire curve or surface of prices $C(K,T)$ of any kind of option. You only have a finite number of observations and even those typically have a bid and an offer. I would therefore argue that the correct picture of ...	2014-04-20 18:49:39	{"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": 0.5252737998962402, "perplexity": 344.201600161303}, "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-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00409-ip-10-147-4-33.ec2.internal.warc.gz"}
https://www.techwhiff.com/issue/the-organic-molecules-known-as-phospholipids-are-key--496022	# The organic molecules known as phospholipids are key components of cell membranes and composed of which molecules? ###### Question: The organic molecules known as phospholipids are key components of cell membranes and composed of which molecules? ### A rectangle has an area of 30 square meters and a perimeter of 34 meters. What are the dimensions of the rectangle? A rectangle has an area of 30 square meters and a perimeter of 34 meters. What are the dimensions of the rectangle?... ### 11. Which of these are characteristics of organisms in the animal kingdom? (Select all that apply.) cells have a cell wall cells contain chlorophyll multi-cellular consumers 11. Which of these are characteristics of organisms in the animal kingdom? (Select all that apply.) cells have a cell wall cells contain chlorophyll multi-cellular consumers... ### Dakota makes a salad dressing by combining 6 1/3 fluid ounces of oil and 2 3/8 fluid ounces of vinegar in a jar. She then pours 2 1/4 fluid ounces of the dressing onto her salad. How much dressing remains in the jar Dakota makes a salad dressing by combining 6 1/3 fluid ounces of oil and 2 3/8 fluid ounces of vinegar in a jar. She then pours 2 1/4 fluid ounces of the dressing onto her salad. How much dressing remains in the jar... ### Ethan is proving that the slope between any two points on a straight line is the same. He has already proved that triangles 1 and 2 are similar. Drag statements and reasons to complete the proof. Ethan is proving that the slope between any two points on a straight line is the same. He has already proved that triangles 1 and 2 are similar. Drag statements and reasons to complete the proof.... ### Calculate the mass defect for the formation of phosphorus-31. The mass of a phosphorus-31 nucleus is 30.973765 amu. The masses of a proton and a neutron are 1.00728 and 1.00866 amu, respectively. Enter your answer in amu's with five decimal places and no units. Calculate the mass defect for the formation of phosphorus-31. The mass of a phosphorus-31 nucleus is 30.973765 amu. The masses of a proton and a neutron are 1.00728 and 1.00866 amu, respectively. Enter your answer in amu's with five decimal places and no units.... ### PLEASSSE HELLPP Using the graphs, what is the value of g(f(3)) a) 1 b) 5 c) 0 d) 2 Graphs pictured ^ PLEASSSE HELLPP Using the graphs, what is the value of g(f(3)) a) 1 b) 5 c) 0 d) 2 Graphs pictured ^... ### 3 signs of depression 3 signs of depression... ### ANSWER CORRECTLY, ILL GIVE 100 POINTS & BRAINLIEST!! ANSWER CORRECTLY, ILL GIVE 100 POINTS & BRAINLIEST!!... ### Br uh.................................. br uh..................................... ### I need help with the number one I need help with the number one... ### Write one paragraph responding to the prompt (min. 5 sentences) Prompt: How do Louisiana’s natural resources impact Louisiana and the United States economy? Write one paragraph responding to the prompt (min. 5 sentences) Prompt: How do Louisiana’s natural resources impact Louisiana and the United States economy?... ### There is too much traffic in the city. what do you think the authorities should do about it? there is too much traffic in the city. what do you think the authorities should do about it?... ### In Genesis 3:15, God made a promise that a descendant of Eve would _____ all men from bondage to sin and restore the right relationship of man to God. redeem deliver save In Genesis 3:15, God made a promise that a descendant of Eve would _____ all men from bondage to sin and restore the right relationship of man to God. redeem deliver save... ### The discipline, which seeks to understand and compare religious patterns from around the world is known as? The discipline, which seeks to understand and compare religious patterns from around the world is known as?... ### Which statement BEST explains why United States entered World War I in 1917? A) The Japanese launched a surprise attack on the U.S. B) The U.S. Citizens were calling for war against Germany. C) The German navy had sunk the Lusitania that year. D) The U.S. Had evidence of a threat to its national security. Which statement BEST explains why United States entered World War I in 1917? A) The Japanese launched a surprise attack on the U.S. B) The U.S. Citizens were calling for war against Germany. C) The German navy had sunk the Lusitania that year. D) The U.S. Had evidence of a threat to its national sec... ### What is the area of this triangle? What is the area of this triangle?...	2022-12-10 09:16:54	{"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": 0.3205741047859192, "perplexity": 2302.9018551360414}, "config": {"markdown_headings": true, "markdown_code": false, "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-2022-49/segments/1669446710421.14/warc/CC-MAIN-20221210074242-20221210104242-00348.warc.gz"}
http://spindynamics.org/wiki/index.php?title=Fatty_acid.m	fatty_acid.m Spin system approximating that of a fatty acid. Syntax [sys,inter]=fatty_acid(nprotons) Description This is useful in imaging simulations - the spin system returned by this function captures the essential things like chemical shift distribution and J-couplings. Arguments nprotons - the number of protons that the spin system should have Outputs sys, inter - input data structures for Spinach	2018-01-23 19:59:47	{"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": 0.37922167778015137, "perplexity": 5326.408193127369}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2018-05/segments/1516084892238.78/warc/CC-MAIN-20180123191341-20180123211341-00274.warc.gz"}
http://hal.in2p3.fr/view_by_stamp.php?label=GANIL&langue=fr&action_todo=view&id=in2p3-00276198&version=1	610 articles – 3052 Notices [english version] HAL : in2p3-00276198, version 1 Physical Review C 77 (2008) 044303 In-beam gamma-ray spectroscopy of the neutron-rich nitrogen isotopes $^{19-22}$N (2008) The structure of $^{19-22}$N nuclei was investigated by means of in-beam gamma-ray spectroscopic technique using fragmentation reactions of both stable and radioactive beams. Based on particle-gamma and particle-gammagamma coincidence data, level schemes are constructed for the neutron-rich nitrogen nuclei. The experimental results are compared with shell model calculations. The strength of the N=14 and Z=8 shell closures and the weakening of the shell model interaction WBT are discussed. équipe(s) de recherche : Structure nucléaire Thème(s) : Physique/Physique Nucléaire Expérimentale in2p3-00276198, version 1 http://hal.in2p3.fr/in2p3-00276198 oai:hal.in2p3.fr:in2p3-00276198 Contributeur : Sandrine Guesnon <> Soumis le : Lundi 28 Avril 2008, 17:32:26 Dernière modification le : Mardi 15 Septembre 2009, 15:39:13	2014-07-29 21:06:22	{"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": 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": 0.40306827425956726, "perplexity": 9758.929972677879}, "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-2014-23/segments/1406510267865.20/warc/CC-MAIN-20140728011747-00427-ip-10-146-231-18.ec2.internal.warc.gz"}
https://www.techwhiff.com/issue/what-was-the-unusual-role-of-mary-musgrove-matthews--333745	# What was the unusual role of Mary Musgrove Matthews? ###### Question: What was the unusual role of Mary Musgrove Matthews? ### Why does Odysseus wish to explore the cave of the Cyclops? Why does Odysseus wish to explore the cave of the Cyclops? ... ### Describe how neurons communicate electrically Describe how neurons communicate electrically... ### What kinds of incentives were given to warlords in Africa for their participation in the slave trade? What kinds of incentives were given to warlords in Africa for their participation in the slave trade?... ### Which of the following is the plant virus? [1 marks] TMV Rabies Rhino virus Bacteriophase Virus Which of the following is the plant virus? [1 marks] TMV Rabies Rhino virus Bacteriophase Virus... ### Change the fraction to a percent:17/50 Change the fraction to a percent:17/50... ### Simplify (check picture) simplify (check picture)... ### Write the number which is exactly a third of the way from 2.6 to 3.5. Please explain how you get the answer.Thanks Write the number which is exactly a third of the way from 2.6 to 3.5. Please explain how you get the answer.Thanks... ### What is the circumference of the circle with 22 inches What is the circumference of the circle with 22 inches... ### How long did it take to make the power loom in the 1800s? How long did it take to make the power loom in the 1800s?... ### 1/3x+3=13 what is the answer for x 1/3x+3=13 what is the answer for x... ### Alex and Scott are out at sea on a small fishing boat when Alex suddenly is suffering a heart attack. Scott is trying his best to reanimate the now unconscious friend. Will he be successful before time is running out for Alex? * Character versus Self; Protagonist Alex; Antagonist: himself Character versus Nature; Protagonist Alex; Antagonist: Scott Character versus Nature; Protagonist Scott & Alex; Antagonist: nature Character versus Character: Protagonist Alex; Antagonist Scott Character ve Alex and Scott are out at sea on a small fishing boat when Alex suddenly is suffering a heart attack. Scott is trying his best to reanimate the now unconscious friend. Will he be successful before time is running out for Alex? * Character versus Self; Protagonist Alex; Antagonist: himself Character ... ### Pls, help with this! I'm having trouble. will give points!! Pls, help with this! I'm having trouble. will give points!!... ### What is 12,430,000 in expanded form what is 12,430,000 in expanded form... ### 8.0 ft15.0 ft8.0 ftWhat is the maximum number of 1 ft boxes that can be hauled by the cargo truck? 8.0 ft15.0 ft8.0 ftWhat is the maximum number of 1 ft boxes that can be hauled by the cargo truck?...	2022-10-01 02:42:14	{"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": 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": 0.34639355540275574, "perplexity": 6401.802545462823}, "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/1664030335514.65/warc/CC-MAIN-20221001003954-20221001033954-00250.warc.gz"}