FRINK Linked Data Fragments server

Matches in SemOpenAlex for { <https://semopenalex.org/work/W2963126932> ?p ?o ?g. }

• W2963126932 endingPage "1733" @default.
• W2963126932 startingPage "1709" @default.
• W2963126932 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X 1> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>X_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X Subscript n> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>X_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a standard normal sample in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript d> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We compute exactly the expected volume of the Gaussian polytope <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c o n v left-bracket upper X 1 comma ellipsis comma upper X Subscript n Baseline right-bracket> <mml:semantics> <mml:mrow> <mml:mi>conv</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {conv} [X_1,ldots ,X_n]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the symmetric Gaussian polytope <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c o n v left-bracket plus-or-minus upper X 1 comma ellipsis comma plus-or-minus upper X Subscript n Baseline right-bracket> <mml:semantics> <mml:mrow> <mml:mi>conv</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mo>±<!-- ± --></mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:mo>±<!-- ± --></mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {conv} [pm X_1,ldots ,pm X_n]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the Gaussian zonotope <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket 0 comma upper X 1 right-bracket plus midline-horizontal-ellipsis plus left-bracket 0 comma upper X Subscript n Baseline right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>]</mml:mo> <mml:mo>+</mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mo>+</mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>[0,X_1]+cdots +[0,X_n]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by exploiting their connection to the regular simplex, the regular cross-polytope, and the cube with the aid of Tsirelson’s formula. The expected volumes of these random polytopes are given by essentially the same expressions as the intrinsic volumes and external angles of the regular polytopes. For all of these quantities, we obtain asymptotic formulae which are more precise than the results which were known before. More generally, we determine the expected volumes of some heteroscedastic random polytopes including <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c o n v left-bracket l 1 upper X 1 comma ellipsis comma l Subscript n Baseline upper X Subscript n Baseline right-bracket> <mml:semantics> <mml:mrow> <mml:mi>conv</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>l</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {conv}[l_1X_1,ldots ,l_nX_n]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c o n v left-bracket plus-or-minus l 1 upper X 1 comma ellipsis comma plus-or-minus l Subscript n Baseline upper X Subscript n Baseline right-bracket> <mml:semantics> <mml:mrow> <mml:mi>conv</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mo>±<!-- ± --></mml:mo> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:mo>±<!-- ± --></mml:mo> <mml:msub> <mml:mi>l</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {conv} [pm l_1 X_1,ldots , pm l_n X_n]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=l 1> <mml:semantics> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>l_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=l Subscript n Baseline greater-than-or-equal-to 0> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>l_ngeq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are parameters, and the intrinsic volumes of the corresponding deterministic polytopes. Finally, we relate the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>th intrinsic volume of the regular simplex <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Delta Superscript n minus 1> <mml:semantics> <mml:msup> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>Delta ^{n-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the expected maximum of independent standard Gaussian random variables <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi 1> <mml:semantics> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>xi _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi Subscript n> <mml:semantics> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>xi _n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> given that the maximum has multiplicity <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Namely, we show that <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V Subscript k Baseline left-parenthesis normal upper Delta Superscript n minus 1 Baseline right-parenthesis equals StartFraction left-parenthesis 2 pi right-parenthesis Superscript StartFraction k Over 2 EndFraction Baseline Over k factorial EndFraction dot limit Underscript epsilon down-arrow 0 Endscripts epsilon Superscript 1 minus k Baseline double-struck upper E left-bracket max left-brace right-brace comma comma xi 1 comma ellipsis comma xi n double-struck 1 Subscript left-brace xi Sub Subscript left-parenthesis n right-parenthesis Subscript minus xi Sub Subscript left-parenthesis n minus k plus 1 right-parenthesis Subscript less-than-or-equal-to epsilon right-brace Baseline right-bracket comma> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>π<!-- π --></mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mfrac> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> </mml:mrow> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>!</mml:mo> </mml:mrow> </mml:mfrac> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:munder> <mml:mo movablelimits=true form=prefix>lim</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo stretchy=false>↓<!-- ↓ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:munder> <mml:msup> <mml:mi>ε<!-- ε --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>E</mml:mi> </mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mo movablelimits=true form=prefix>max</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn mathvariant=double-struck>1</mml:mn> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msub> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msub> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy=false>]</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>V_k(Delta ^{n-1}) = frac {(2pi )^{frac k2}} {k!} cdot lim _{varepsilon downarrow 0} varepsilon ^{1-k} mathbb {E} [max {xi _1,ldots ,xi _n} mathbb {1}_{{xi _{(n)} - xi _{(n-k+1)}leq varepsilon }}],</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi Subscript left-parenthesis 1 right-parenthesis Baseline less-than-or-equal-to midline-horizontal-ellipsis less-than-or-equal-to xi Subscript left-parenthesis n right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msub> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>xi _{(1)} leq cdots leq xi _{(n)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the order statistics. A similar result holds for the cross-polytope if we replace <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi 1> <mml:semantics> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>xi _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi Subscript n> <mml:semantics> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>xi _n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with their absolute values." @default.
• W2963126932 created "2019-07-30" @default.
• W2963126932 creator A5010851411 @default.
• W2963126932 creator A5040298019 @default.
• W2963126932 date "2018-11-27" @default.
• W2963126932 modified "2023-09-23" @default.
• W2963126932 title "Expected volumes of Gaussian polytopes, external angles, and multiple order statistics" @default.
• W2963126932 cites W1607203414 @default.
• W2963126932 cites W1981546655 @default.
• W2963126932 cites W1983121680 @default.
• W2963126932 cites W1996314162 @default.
• W2963126932 cites W1998315626 @default.
• W2963126932 cites W2004843172 @default.
• W2963126932 cites W2016299362 @default.
• W2963126932 cites W2020922669 @default.
• W2963126932 cites W2038176554 @default.
• W2963126932 cites W2038953884 @default.
• W2963126932 cites W2042837540 @default.
• W2963126932 cites W2045795069 @default.
• W2963126932 cites W2045924955 @default.
• W2963126932 cites W2050032151 @default.
• W2963126932 cites W2057068792 @default.
• W2963126932 cites W2057452040 @default.
• W2963126932 cites W2066133757 @default.
• W2963126932 cites W2077846610 @default.
• W2963126932 cites W2080800043 @default.
• W2963126932 cites W2084845136 @default.
• W2963126932 cites W2084911775 @default.
• W2963126932 cites W2090849835 @default.
• W2963126932 cites W2109246257 @default.
• W2963126932 cites W2117218580 @default.
• W2963126932 cites W2163472056 @default.
• W2963126932 cites W2491018796 @default.
• W2963126932 cites W2962807615 @default.
• W2963126932 cites W2963167699 @default.
• W2963126932 cites W4246008375 @default.
• W2963126932 cites W4253948983 @default.
• W2963126932 cites W565214016 @default.
• W2963126932 doi "https://doi.org/10.1090/tran/7708" @default.
• W2963126932 hasPublicationYear "2018" @default.
• W2963126932 type Work @default.
• W2963126932 sameAs 2963126932 @default.
• W2963126932 citedByCount "5" @default.
• W2963126932 countsByYear W29631269322017 @default.
• W2963126932 countsByYear W29631269322020 @default.
• W2963126932 countsByYear W29631269322021 @default.
• W2963126932 countsByYear W29631269322023 @default.
• W2963126932 crossrefType "journal-article" @default.
• W2963126932 hasAuthorship W2963126932A5010851411 @default.
• W2963126932 hasAuthorship W2963126932A5040298019 @default.
• W2963126932 hasBestOaLocation W29631269321 @default.
• W2963126932 hasConcept C10138342 @default.
• W2963126932 hasConcept C105795698 @default.
• W2963126932 hasConcept C114614502 @default.
• W2963126932 hasConcept C121332964 @default.
• W2963126932 hasConcept C145691206 @default.
• W2963126932 hasConcept C162324750 @default.
• W2963126932 hasConcept C163716315 @default.
• W2963126932 hasConcept C182306322 @default.
• W2963126932 hasConcept C33923547 @default.
• W2963126932 hasConcept C44082924 @default.
• W2963126932 hasConcept C62520636 @default.
• W2963126932 hasConceptScore W2963126932C10138342 @default.
• W2963126932 hasConceptScore W2963126932C105795698 @default.
• W2963126932 hasConceptScore W2963126932C114614502 @default.
• W2963126932 hasConceptScore W2963126932C121332964 @default.
• W2963126932 hasConceptScore W2963126932C145691206 @default.
• W2963126932 hasConceptScore W2963126932C162324750 @default.
• W2963126932 hasConceptScore W2963126932C163716315 @default.
• W2963126932 hasConceptScore W2963126932C182306322 @default.
• W2963126932 hasConceptScore W2963126932C33923547 @default.
• W2963126932 hasConceptScore W2963126932C44082924 @default.
• W2963126932 hasConceptScore W2963126932C62520636 @default.
• W2963126932 hasIssue "3" @default.
• W2963126932 hasLocation W29631269321 @default.
• W2963126932 hasLocation W29631269322 @default.
• W2963126932 hasOpenAccess W2963126932 @default.
• W2963126932 hasPrimaryLocation W29631269321 @default.
• W2963126932 hasRelatedWork W1741741233 @default.
• W2963126932 hasRelatedWork W1867676345 @default.
• W2963126932 hasRelatedWork W1978042415 @default.
• W2963126932 hasRelatedWork W1989428327 @default.
• W2963126932 hasRelatedWork W2044070491 @default.
• W2963126932 hasRelatedWork W2051221975 @default.
• W2963126932 hasRelatedWork W2093466950 @default.
• W2963126932 hasRelatedWork W2141148003 @default.
• W2963126932 hasRelatedWork W2218396533 @default.
• W2963126932 hasRelatedWork W3000043859 @default.
• W2963126932 hasVolume "372" @default.
• W2963126932 isParatext "false" @default.
• W2963126932 isRetracted "false" @default.
• W2963126932 magId "2963126932" @default.
• W2963126932 workType "article" @default.