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Towards computing the grothendieck constant
 In SODA ’09: Proceedings of the 20th Annual ACMSIAM Symposium on Discrete Algorithms
, 2009
"... The Grothendieck constant KG is the smallest constant such that for every d ∈ N and every matrix A = (aij), sup u i,v j ∈B (d) X aij〈ui, vj 〉 � KG · ij sup x i,y j ∈[−1,1] X ij aijxiyj, where B (d) is the unit ball in R d. Despite several efforts [15, 23], the value of the constant KG remains unkno ..."
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The Grothendieck constant KG is the smallest constant such that for every d ∈ N and every matrix A = (aij), sup u i,v j ∈B (d) X aij〈ui, vj 〉 � KG · ij sup x i,y j ∈[−1,1] X ij aijxiyj, where B (d) is the unit ball in R d. Despite several efforts [15, 23], the value of the constant KG remains
Computing the Grothendieck constant of some graph classes
, 2011
"... Given a graph G = ([n], E) and w ∈ RE ∑, consider the integer program maxx∈{±1} n ij∈E wijxixj and its canonical semidefinite programming relaxation max ∑ ij∈E wijvT i vj, where the maximum is taken over all unit vectors vi ∈ Rn. The integrality gap of this relaxation is known as the Grothendieck co ..."
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constant κ(G) of G. We present a closedform formula for the Grothendieck constant of K5minor free graphs and derive that it is at most 3/2. Moreover, we show that κ(G) ≤ κ(Kk) if the cut polytope of G is defined by inequalities supported by at most k points. Lastly, since the Grothendieck constant of Kn
Computing the Grothendieck constant of some graph classes
, 2011
"... Given a graph G = ([n], E) and w ∈ RE ∑, consider the integer program maxx∈{±1} n ij∈E wijxixj and its canonical semidefinite programming relaxation max ∑ ij∈E wijvT i vj, where the maximum is taken over all unit vectors vi ∈ Rn. The integrality gap of this relaxation is known as the Grothendieck co ..."
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constant κ(G) of G. We present a closedform formula for the Grothendieck constant of K5minor free graphs and derive that it is at most 3/2. Moreover, we show that κ(G) ≤ κ(Kk) if the cut polytope of G is defined by inequalities supported by at most k points. Lastly, since the Grothendieck constant of Kn
The Grothendieck constant is strictly smaller than Krivine’s bound
 IN 52ND ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE. PREPRINT AVAILABLE AT HTTP://ARXIV.ORG/ABS/1103.6161
, 2011
"... The (real) Grothendieck constant KG is the infimum over those K ∈ (0, ∞) such that for every m, n ∈ N and every m × n real matrix (aij) we have m ∑ n∑ m ∑ n∑ aij〈xi, yj 〉 � K max aijεiδj. max {xi} m i=1,{yj}n j=1 ⊆Sn+m−1 i=1 j=1 {εi} m i=1,{δj}n j=1⊆{−1,1} i=1 j=1 2 log(1+ √ 2) The classical Groth ..."
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The (real) Grothendieck constant KG is the infimum over those K ∈ (0, ∞) such that for every m, n ∈ N and every m × n real matrix (aij) we have m ∑ n∑ m ∑ n∑ aij〈xi, yj 〉 � K max aijεiδj. max {xi} m i=1,{yj}n j=1 ⊆Sn+m−1 i=1 j=1 {εi} m i=1,{δj}n j=1⊆{−1,1} i=1 j=1 2 log(1+ √ 2) The classical
Lower bounds for Grothendieck problems
"... Given a graph G = (V;E), consider the following problem: The input is a function A: E! R, and the goal is to maximize P (u;v)2E A(u; v)f(u)f(v) over all functions f: V! f¡1; 1g. This problem was formalized by Alon, Makarychev, Makarychev and Naor [AMMN05]; it is a weighted version of the 2cluster \ ..."
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hardness result at versus ( = log(1=)) for any constant > 0. This matches the SDP algorithm of Charikar and Wirth [CW04]. Our lower bounds in the bipartite KN;N hold even in the case of CutNorm for zerosum matrices; we show how to translate the best known lower bound on Grothendieck's constant (due
On A Theorem Of Grothendieck
, 1998
"... It is considered a smooth projective morphism p : T ! S to a smooth variety S. It is proved, in particular, the following result. The total direct image Rp (Z=nZ) of the constant 'etale sheaf Z=nZis locally for Zarisky topology quasiisomorphic to a bounded complex L on S consisting of loc ..."
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It is considered a smooth projective morphism p : T ! S to a smooth variety S. It is proved, in particular, the following result. The total direct image Rp (Z=nZ) of the constant 'etale sheaf Z=nZis locally for Zarisky topology quasiisomorphic to a bounded complex L on S consisting
Grothendieck Inclusion Systems
 APPLIED CATEGORICAL STRUCTURES
"... Inclusion systems have been introduced in algebraic specification theory as a categorical structure supporting the development of a general abstract logicindependent approach to the algebra of specification (or programming) modules. Here we extend the concept of indexed categories and their Grothe ..."
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and their Grothendieck flattenings to inclusion systems. An important practical significance of the resulting Grothendieck inclusion systems is that they allow the development of module algebras for multilogic heterogeneous specification frameworks. At another level, we show that several inclusion systems in use
Factorial Grothendieck Polynomials
, 2005
"... In this paper, we study Grothendieck polynomials from a combinatorial viewpoint. We introduce the factorial Grothendieck polynomials, analogues of the factorial Schur functions and present some of their properties, and use them to produce a generalisation of a LittlewoodRichardson rule for Grothend ..."
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In this paper, we study Grothendieck polynomials from a combinatorial viewpoint. We introduce the factorial Grothendieck polynomials, analogues of the factorial Schur functions and present some of their properties, and use them to produce a generalisation of a LittlewoodRichardson rule
Results 1  10
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8,539