The Experts below are selected from a list of 963 Experts worldwide ranked by ideXlab platform
Guohua Zhang - One of the best experts on this subject based on the ideXlab platform.
-
Local variational principle concerning entropy of a sofic group action
Journal of Functional Analysis, 2012Co-Authors: Guohua ZhangAbstract:Abstract Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of countable sofic groups admitting a generating Measurable Partition with finite entropy; and then David Kerr and Hanfeng Li developed an operator-algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space, and established the global variational principle concerning measure-theoretic and topological entropy in this sofic context. By localizing these two kinds of entropy, in this paper we prove a local version of the global variational principle for any finite open cover of the space, and show that these local measure-theoretic and topological entropies coincide with their classical counterparts when the acting group is an infinite amenable group.
-
Local variational principle concerning entropy of a sofic group action
arXiv: Dynamical Systems, 2011Co-Authors: Guohua ZhangAbstract:Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of countable sofic groups admitting a generating Measurable Partition with finite entropy; and then David Kerr and Hanfeng Li developed an operator-algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space, and established the global variational principle concerning measure-theoretic and topological entropy in this sofic context. By localizing these two kinds of entropy, in this paper we prove a local version of the global variational principle for any finite open cover of the space, and show that these local measure-theoretic and topological entropy coincide with their classical counterparts when the acting group is an infinite amenable group.
Assaf Naor - One of the best experts on this subject based on the ideXlab platform.
-
Solution of the Propeller Conjecture in $$\mathbb R ^3$$ R 3
Discrete & Computational Geometry, 2013Co-Authors: Steven Heilman, Aukosh Jagannath, Assaf NaorAbstract:It is shown that every Measurable Partition $$\{A_1,\ldots , A_k\}$$ { A 1 , … , A k } of $$\mathbb R ^3$$ R 3 satisfies 1 $$\begin{aligned} \sum _{i=1}^k\big \Vert \int _{A_i} x\mathrm{{e}}^{-\frac{1}{2}\Vert x\Vert _2^2}\mathrm{{d}}x\big \Vert _2^2\leqslant 9\pi ^2. \end{aligned}$$ ∑ i = 1 k ‖ ∫ A i x e - 1 2 ‖ x ‖ 2 2 d x ‖ 2 2 ⩽ 9 π 2 . Let $$\{P_1,P_2,P_3\}$$ { P 1 , P 2 , P 3 } be the Partition of $$\mathbb R ^2$$ R 2 into $$120^{\circ }$$ 120 ∘ sectors centered at the origin. The bound ( 1 ) is sharp, with equality holding if $$A_i=P_i\times \mathbb R $$ A i = P i × R for $$i\in \{1,2,3\}$$ i ∈ { 1 , 2 , 3 } and $$A_i=\emptyset $$ A i = ∅ for $$i\in \{4,\ldots ,k\}$$ i ∈ { 4 , … , k } . This settles positively the $$3$$ 3 -dimensional Propeller Conjecture of Khot and Naor [(Mathematika 55(1-2):129–165, 2009 (FOCS 2008)]. The proof of ( 1 ) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of ( 1 ) is complexity-theoretic: the unique games hardness threshold of the kernel clustering problem with $$4\times 4$$ 4 × 4 centered and spherical hypothesis matrix equals $$\frac{2\pi }{3}$$ 2 π 3 .
-
Solution of the propeller conjecture in $\mathbb{R}^3$
Discrete & Computational Geometry, 2013Co-Authors: Steven Heilman, Aukosh Jagannath, Assaf NaorAbstract:It is shown that every Measurable Partition \(\{A_1,\ldots , A_k\}\) of \(\mathbb R ^3\) satisfies $$\begin{aligned} \sum _{i=1}^k\big \Vert \int _{A_i} x\mathrm{{e}}^{-\frac{1}{2}\Vert x\Vert _2^2}\mathrm{{d}}x\big \Vert _2^2\leqslant 9\pi ^2. \end{aligned}$$ (1) Let \(\{P_1,P_2,P_3\}\) be the Partition of \(\mathbb R ^2\) into \(120^{\circ }\) sectors centered at the origin. The bound (1) is sharp, with equality holding if \(A_i=P_i\times \mathbb R \) for \(i\in \{1,2,3\}\) and \(A_i=\emptyset \) for \(i\in \{4,\ldots ,k\}\). This settles positively the \(3\)-dimensional Propeller Conjecture of Khot and Naor [(Mathematika 55(1-2):129–165, 2009 (FOCS 2008)]. The proof of (1) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of (1) is complexity-theoretic: the unique games hardness threshold of the kernel clustering problem with \(4\times 4\) centered and spherical hypothesis matrix equals \(\frac{2\pi }{3}\).
-
STOC - Solution of the propeller conjecture in R 3
Proceedings of the 44th symposium on Theory of Computing - STOC '12, 2012Co-Authors: Steven Heilman, Aukosh Jagannath, Assaf NaorAbstract:It is shown that every Measurable Partition {A1,..., Ak} of R3 satisfies: ∑i=1k|intAi xe-1/2|x|22dx|22≤ 9π2. Let P1,P2,P3 be the Partition of R2 into 120o sectors centered at the origin. The bound (1) is sharp, with equality holding if Ai=Pi x R for i∈ {1,2,3} and Ai=∅ for i∈ {4,...,k}. This settles positively the 3-dimensional Propeller Conjecture of Khot and Naor (FOCS 2008). The proof of (1) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of (1) is complexity-theoretic: the Unique Games hardness threshold of the Kernel Clustering problem with 4 x 4 centered and spherical hypothesis matrix equals 2π/3.
-
Solution of the propeller conjecture in $\mathbb{R}^3$
arXiv: Computational Complexity, 2011Co-Authors: Steven Heilman, Aukosh Jagannath, Assaf NaorAbstract:It is shown that every Measurable Partition ${A_1,..., A_k}$ of $\mathbb{R}^3$ satisfies $$\sum_{i=1}^k||\int_{A_i} xe^{-\frac12||x||_2^2}dx||_2^2\le 9\pi^2.\qquad(*)$$ Let ${P_1,P_2,P_3}$ be the Partition of $\mathbb{R}^2$ into $120^\circ$ sectors centered at the origin. The bound is sharp, with equality holding if $A_i=P_i\times \mathbb{R}$ for $i\in {1,2,3}$ and $A_i=\emptyset$ for $i\in \{4,...,k\}$ (up to measure zero corrections, orthogonal transformations and renumbering of the sets $\{A_1,...,A_k\}$). This settles positively the 3-dimensional Propeller Conjecture of Khot and Naor (FOCS 2008). The proof of reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of $(*)$ is complexity-theoretic: the Unique Games hardness threshold of the Kernel Clustering problem with $4 \times 4$ centered and spherical hypothesis matrix equals $\frac{2\pi}{3}$.
Steven Heilman - One of the best experts on this subject based on the ideXlab platform.
-
Solution of the Propeller Conjecture in $$\mathbb R ^3$$ R 3
Discrete & Computational Geometry, 2013Co-Authors: Steven Heilman, Aukosh Jagannath, Assaf NaorAbstract:It is shown that every Measurable Partition $$\{A_1,\ldots , A_k\}$$ { A 1 , … , A k } of $$\mathbb R ^3$$ R 3 satisfies 1 $$\begin{aligned} \sum _{i=1}^k\big \Vert \int _{A_i} x\mathrm{{e}}^{-\frac{1}{2}\Vert x\Vert _2^2}\mathrm{{d}}x\big \Vert _2^2\leqslant 9\pi ^2. \end{aligned}$$ ∑ i = 1 k ‖ ∫ A i x e - 1 2 ‖ x ‖ 2 2 d x ‖ 2 2 ⩽ 9 π 2 . Let $$\{P_1,P_2,P_3\}$$ { P 1 , P 2 , P 3 } be the Partition of $$\mathbb R ^2$$ R 2 into $$120^{\circ }$$ 120 ∘ sectors centered at the origin. The bound ( 1 ) is sharp, with equality holding if $$A_i=P_i\times \mathbb R $$ A i = P i × R for $$i\in \{1,2,3\}$$ i ∈ { 1 , 2 , 3 } and $$A_i=\emptyset $$ A i = ∅ for $$i\in \{4,\ldots ,k\}$$ i ∈ { 4 , … , k } . This settles positively the $$3$$ 3 -dimensional Propeller Conjecture of Khot and Naor [(Mathematika 55(1-2):129–165, 2009 (FOCS 2008)]. The proof of ( 1 ) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of ( 1 ) is complexity-theoretic: the unique games hardness threshold of the kernel clustering problem with $$4\times 4$$ 4 × 4 centered and spherical hypothesis matrix equals $$\frac{2\pi }{3}$$ 2 π 3 .
-
Solution of the propeller conjecture in $\mathbb{R}^3$
Discrete & Computational Geometry, 2013Co-Authors: Steven Heilman, Aukosh Jagannath, Assaf NaorAbstract:It is shown that every Measurable Partition \(\{A_1,\ldots , A_k\}\) of \(\mathbb R ^3\) satisfies $$\begin{aligned} \sum _{i=1}^k\big \Vert \int _{A_i} x\mathrm{{e}}^{-\frac{1}{2}\Vert x\Vert _2^2}\mathrm{{d}}x\big \Vert _2^2\leqslant 9\pi ^2. \end{aligned}$$ (1) Let \(\{P_1,P_2,P_3\}\) be the Partition of \(\mathbb R ^2\) into \(120^{\circ }\) sectors centered at the origin. The bound (1) is sharp, with equality holding if \(A_i=P_i\times \mathbb R \) for \(i\in \{1,2,3\}\) and \(A_i=\emptyset \) for \(i\in \{4,\ldots ,k\}\). This settles positively the \(3\)-dimensional Propeller Conjecture of Khot and Naor [(Mathematika 55(1-2):129–165, 2009 (FOCS 2008)]. The proof of (1) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of (1) is complexity-theoretic: the unique games hardness threshold of the kernel clustering problem with \(4\times 4\) centered and spherical hypothesis matrix equals \(\frac{2\pi }{3}\).
-
STOC - Solution of the propeller conjecture in R 3
Proceedings of the 44th symposium on Theory of Computing - STOC '12, 2012Co-Authors: Steven Heilman, Aukosh Jagannath, Assaf NaorAbstract:It is shown that every Measurable Partition {A1,..., Ak} of R3 satisfies: ∑i=1k|intAi xe-1/2|x|22dx|22≤ 9π2. Let P1,P2,P3 be the Partition of R2 into 120o sectors centered at the origin. The bound (1) is sharp, with equality holding if Ai=Pi x R for i∈ {1,2,3} and Ai=∅ for i∈ {4,...,k}. This settles positively the 3-dimensional Propeller Conjecture of Khot and Naor (FOCS 2008). The proof of (1) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of (1) is complexity-theoretic: the Unique Games hardness threshold of the Kernel Clustering problem with 4 x 4 centered and spherical hypothesis matrix equals 2π/3.
-
Solution of the propeller conjecture in $\mathbb{R}^3$
arXiv: Computational Complexity, 2011Co-Authors: Steven Heilman, Aukosh Jagannath, Assaf NaorAbstract:It is shown that every Measurable Partition ${A_1,..., A_k}$ of $\mathbb{R}^3$ satisfies $$\sum_{i=1}^k||\int_{A_i} xe^{-\frac12||x||_2^2}dx||_2^2\le 9\pi^2.\qquad(*)$$ Let ${P_1,P_2,P_3}$ be the Partition of $\mathbb{R}^2$ into $120^\circ$ sectors centered at the origin. The bound is sharp, with equality holding if $A_i=P_i\times \mathbb{R}$ for $i\in {1,2,3}$ and $A_i=\emptyset$ for $i\in \{4,...,k\}$ (up to measure zero corrections, orthogonal transformations and renumbering of the sets $\{A_1,...,A_k\}$). This settles positively the 3-dimensional Propeller Conjecture of Khot and Naor (FOCS 2008). The proof of reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of $(*)$ is complexity-theoretic: the Unique Games hardness threshold of the Kernel Clustering problem with $4 \times 4$ centered and spherical hypothesis matrix equals $\frac{2\pi}{3}$.
Yashar Memarian - One of the best experts on this subject based on the ideXlab platform.
-
A Lower Bound for the Mahler Volume of Symmetric Convex Sets.
arXiv: Metric Geometry, 2015Co-Authors: Yashar MemarianAbstract:The goal of this paper is to present a lower bound for the Mahler volume of at least 4-dimensional symmetric convex bodies. We define a computable dimension dependent constant through a 2-dimensional variational (max-min) procedure and demonstrate that the Mahler volume of every (at least 4-dimensional) symmetric convex body is greater than a (simple) function of this constant. Similar to the proof of Gromov's Waist of the Sphere Theorem in [18], our result is proved via localisation-type arguments obtained from a suitable Measurable Partition (or Partitions) of the canonical sphere.
-
A Lower Bound for the Mahler Volume of at Least Four-Dimensional Symmetric Convex Sets
arXiv: Metric Geometry, 2015Co-Authors: Yashar MemarianAbstract:The goal of this paper is to present a lower bound for the Mahler volume of at least 4-dimensional symmetric convex bodies. I define a computable constant (depending on the dimension) through a 2-dimensional variational (max-min) procedure and demonstrate that the Mahler volume of every (at least 4-dimensional) symmetric convex body is greater than the product of this constant times the volume of the (n-1)-dimensional canonical sphere to the power of two. Similar to the proof of Gromov's Waist of the Sphere Theorem in [18], my result is proved via localisation-type arguments obtained from a suitable Measurable Partition (or Partitions) of the canonical sphere.
Ian Williamson - One of the best experts on this subject based on the ideXlab platform.
-
Octanol-water Partitioning of chemical constituents in river water and treated sewage effluent.
Water research, 2005Co-Authors: Andrew Turner, Ian WilliamsonAbstract:Abstract A shake-flask approach has been employed to determine the n -octanol–water Partitioning of chemical constituents in various river waters and in treated sewage effluent. The bulk inorganic composition of the water samples (conductivity, pH and the concentrations of major solutes: Ca, K, Mg, Na) was unaffected by the presence of solvent. Boron, however, exhibited increasing Partition with decreasing sample pH, because its dominant form in freshwaters, B(OH) 3 , is neutral, covalent and acidic. Constituents having significant association with dissolved organic matter (DOM), including components of DOM itself (C, S) and trace metals that form complexes with organic ligands (Al, Cu, Fe, Pb, Zn), exhibited Measurable Partition into the solvent in most cases, with conditional Partition coefficients, D ow , in the region 0.03–2.5. Significant differences in the Partitioning among these constituents and among the environments studied did not appear to be related to bulk sample characteristics or the degree of association of the constituent with DOM. These observations suggest that Partition is sensitive to the nature of the organic matter (C, S) and the availability of specific binding ligands (trace metals). Thus, although D ow is critical for defining the biogeochemical behaviour and potential impacts of chemical constituents in the environment, it appears to be a difficult parameter to model or predict.