651 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Divisible convex sets with properly embedded cones

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    In this article we construct many examples of properly convex irreducible domains divided by Zariski dense relatively hyperbolic groups in every dimension at least 3. This answers a question of Benoist. Relative hyperbolicity and non-strict convexity are captured by a family of properly embedded cones (convex hulls of points and ellipsoids) in the domain. Our construction is most flexible in dimension 3 where we give a purely topological criterion for the existence of a large deformation space of geometrically controlled convex projective structures with totally geodesic boundary on a compact 3-manifold.Comment: 90 pages. Comments are welcome

    Noncommutative Ergodic Optimization

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    We extend the theory of ergodic optimization and maximizing measures to the non-commutative field of C*-dynamical systems. We then provide a result linking the ergodic optimizations of elements of a C*-dynamical system to the convergence of certain ergodic averages in a suitable seminorm. We also provide alternate proofs of several results in this article using the tools of nonstandard analysis.Comment: This article is derived from a chapter of the author's doctoral dissertation. It combines and extends two articles previously uploaded to the arXiv:2109.10425 and arXiv:2109.13965. This version corrects a few minor typos found in the first versio

    Class numbers of multinorm-one tori

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    We present a formula for the class number of a multinorm one torus TL/kT_{L/k} associated to any \'etale algebra LL over a global field kk. This is deduced from a formula for analogues of invariants introduced by T.~Ono, which are interpreted as a generalization of Gauss genus theory. This paper includes the variants of Ono's invariant for arbitrary SS-ideal class numbers and the narrow version, generalizing results of Katayama, Morishita, Sasaki and Ono.Comment: 21 pages; comments welcom

    L2L^{2}-approach to the Saito vanishing theorem

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    We give an analytic approach to the Saito vanishing theorem by going back to the original idea for the proof of the Kodaira-Nakano vanishing theorem. The key ingredient is the curvature formula for Hodge bundles and the Higgs field estimates for degenerations of Hodge structures

    Temporo-spatial differentiations

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    Temporo-spatial differentiation problems were first introduced in \cite{Assani-Young} under the name of spatial-temporal differentiation problems. Given a probability space (X,μ)(X, \mu) and measurable map T:X→XT : X \to X, a temporo-spatial differentiation problem is concerned with the limiting behavior of the sequence1μ(Ck)∫Ck1k∑j=0k−1f∘Tjdμ,\frac{1}{\mu(C_k)} \int_{C_k} \frac{1}{k} \sum_{j = 0}^{k - 1} f \circ T^j \mathrm{d} \mu ,where f∈L1(X,μ)f \in L^1(X, \mu) and (Ck)k=1∞(C_k)_{k = 1}^\infty is a sequence of measurable subsets of XX with positive measure. These problems were then generalized to the setting of non-autonomous dynamical systems in \cite{Non-AutonomousTSD}. We will present several of the basic aspects of temporo-spatial differentiation problems, including their connections with the field of ergodic optimization. We also present several positive convergence results for ``random temporo-spatial differentiation problems." We then discuss generalizations of temporo-spatial differentiation problems to the setting of actions of groups and semigroups other than Z\mathbb{Z}, as well as the construction of pathological temporo-spatial differentiation problems.Doctor of Philosoph

    Universal α-central extensions of Hom-Leibniz n-algebras

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    We construct homology with trivial coefficients of Hom-Leibniz n-algebras. We introduce and characterize universal (α)-central extensions of Hom-Leibniz n-algebras. In particular, we show their interplay with the zero-th and first homology with trivial coefficients. When n = 2 we recover the corresponding results on universal central extensions of Hom-Leibniz algebras. The notion of non-abelian tensor product of Hom-Leibniz n-algebras is introduced and we establish its relationship with universal central extensions. A generalization of the concept and properties of unicentral Leibniz algebras to the setting of Hom-Leibniz n-algebras is developed.Agencia Estatal de Investigación | Ref. MTM2016-79661-

    The Complexity of Some Geometric Proof Systems

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    In this Thesis we investigate proof systems based on Integer Linear Programming. These methods inspect the solution space of an unsatisfiable propositional formula and prove that this space contains no integral points. We begin by proving some size and depth lower bounds for a recent proof system, Stabbing Planes, and along the way introduce some novel methods for doing so. We then turn to the complexity of propositional contradictions generated uniformly from first order sentences, in Stabbing Planes and Sum-Of-Squares. We finish by investigating the complexity-theoretic impact of the choice of method of generating these propositional contradictions in Sherali-Adams

    Finitist Axiomatic Truth

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    On homogeneous spaces for diagonal ind-groups

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    We study the homogeneous ind-spaces GL(s)/P\mathrm{GL}(\mathbf{s})/\mathbf{P} where GL(s)\mathrm{GL}(\mathbf{s}) is a strict diagonal ind-group defined by a supernatural number s\mathbf{s} and P\mathbf{P} is a parabolic ind-subgroup of GL(s)\mathrm{GL}(\mathbf{s}). We construct an explicit exhaustion of GL(s)/P\mathrm{GL}(\mathbf{s})/\mathbf{P} by finite-dimensional partial flag varieties. As an application, we characterize all locally projective GL(∞)\mathrm{GL}(\infty)-homogeneous spaces, and some direct products of such spaces, which are GL(s)\mathrm{GL}(\mathbf{s})-homogeneous for a fixed s\mathbf{s}. The very possibility for a GL(∞)\mathrm{GL}(\infty)-homogeneous space to be GL(s)\mathrm{GL}(\mathbf{s})-homogeneous for a strict diagonal ind-group GL(s)\mathrm{GL}(\mathbf{s}) arises from the fact that the automorphism group of a GL(∞)\mathrm{GL}(\infty)-homogeneous space is much larger than GL(∞)\mathrm{GL}(\infty)
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