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Towards a Calculus for NonLinear Spectral Gaps [Extended Abstract]
"... Given a finite regular graph ..."
Expanders with respect to Hadamard spaces and random graphs
, 2012
"... Abstract. It is shown that there exists a sequence of 3regular graphs {Gn} ∞ n=1 and a Hadamard space X such that {Gn} ∞ n=1 forms an expander sequence with respect to X, yet random regular graphs are not expanders with respect to X. This answers a question of [NS11]. {Gn} ∞ n=1 are also shown t ..."
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Abstract. It is shown that there exists a sequence of 3regular graphs {Gn} ∞ n=1 and a Hadamard space X such that {Gn} ∞ n=1 forms an expander sequence with respect to X, yet random regular graphs are not expanders with respect to X. This answers a question of [NS11]. {Gn} ∞ n=1 are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear time constant factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of
NONLINEAR SPECTRAL CALCULUS AND SUPEREXPANDERS
"... Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesàro averages. Nonlinear spectral gaps of graphs are also shown to behave submultiplicatively under zigzag products. These results yield a ..."
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Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesàro averages. Nonlinear spectral gaps of graphs are also shown to behave submultiplicatively under zigzag products. These results yield a combinatorial construction of superexpanders, i.e., a sequence of 3regular graphs that does not admit a coarse embedding into any uniformly convex normed space.
A NOTE ON DICHOTOMIES FOR METRIC TRANSFORMS
"... Abstract. We show that for every nondecreasing concave function ω: [0, ∞) → [0, ∞) with ω(0) = 0, either every finite metric space embeds with distortion arbitrarily close to 1 into a metric space of the form (X, ω◦d) for some metric d on X, or there exists α = α(ω)> 0 and n0 = n0(ω) ∈ N such tha ..."
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Abstract. We show that for every nondecreasing concave function ω: [0, ∞) → [0, ∞) with ω(0) = 0, either every finite metric space embeds with distortion arbitrarily close to 1 into a metric space of the form (X, ω◦d) for some metric d on X, or there exists α = α(ω)> 0 and n0 = n0(ω) ∈ N such that for all n � n0, any embedding of {0,..., n} ⊆ R into a metric space of the form (X, ω ◦ d) incurs distortion at least n α. 1.
Rearrangements with supporting Trees, Isomorphisms and Combinatorics of coloured dyadic Intervals ∗
, 2009
"... We determine a class of rearrangement operators acting on dyadic intervals that admit a supporting tree. This condition implies that the associated rearrangement operator has a bounded vector valued extension to L p E, where E is a UMD space. We prove the existence of a large subspace Xp ⊂ Lp on whi ..."
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We determine a class of rearrangement operators acting on dyadic intervals that admit a supporting tree. This condition implies that the associated rearrangement operator has a bounded vector valued extension to L p E, where E is a UMD space. We prove the existence of a large subspace Xp ⊂ Lp on which a bounded rearrangement operator acts as an isomorphism. Moreover, we study winning strategies for a combinatorial two person game played
We study pairs of metric spaces (X, dX) and (Y, dY) for which there
"... Abstract. Let A = (aij) ∈ Mn(R) be an n by n symmetric stochastic matrix. For p ∈ [1, ∞) and a metric space (X, dX), let γ(A, d p X) be the infimum over those γ ∈ (0, ∞] for which every x1,..., xn ∈ X satisfy 1 n2 n ∑ n∑ dX(xi, xj) p � γ n ∑ n∑ aijdX(xi, xj) n p. i=1 j=1 i=1 j=1 Thus γ(A, d p X) me ..."
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Abstract. Let A = (aij) ∈ Mn(R) be an n by n symmetric stochastic matrix. For p ∈ [1, ∞) and a metric space (X, dX), let γ(A, d p X) be the infimum over those γ ∈ (0, ∞] for which every x1,..., xn ∈ X satisfy 1 n2 n ∑ n∑ dX(xi, xj) p � γ n ∑ n∑ aijdX(xi, xj) n p. i=1 j=1 i=1 j=1 Thus γ(A, d p X) measures the magnitude of the nonlinear spectral gap of the matrix A with respect to the kernel d p
URL: www.emis.de/journals/AFA/ ON A FORMULA OF LE MERDY FOR THE COMPLEX INTERPOLATION OF TENSOR PRODUCTS
"... 719] proved the following complex interpolation formula for injective tensor products: [ℓ2 ˜⊗εℓ1, ℓ2 ˜⊗εℓ∞] 1 2 = S4. We investigate whether related formulas, and give a partially positive answer for θ < 1 1 2 and a negative answer for θ> 2. Furthermore, we briefly discuss the more general case when ..."
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719] proved the following complex interpolation formula for injective tensor products: [ℓ2 ˜⊗εℓ1, ℓ2 ˜⊗εℓ∞] 1 2 = S4. We investigate whether related formulas, and give a partially positive answer for θ < 1 1 2 and a negative answer for θ> 2. Furthermore, we briefly discuss the more general case when ℓ2 is replaced by ℓq, 1 < q < 2, and and ℓp1, respectively. hold when considering arbitrary 0 < θ < 1 instead of 1 2 ℓ1 and ℓ ∞ by ℓp0