1,002 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    On certain geometric maximal functions in harmonic analysis

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    The broad theme of the thesis is of geometric maximal functions associated to curved surfaces. We produce novel results about two maximal functions of different types, presented in two parts of the thesis. In the first part (Chapter 2), we study the Lᵖ → Lᵖ boundedness of a lacunary maximal function on a graded homogeneous group. The main theorem of this part generalises the existing maximal results in specific homogeneous groups, such as the Euclidean space and the Heisenberg group. Using an iteration scheme, we estimate the maximal function, assuming that the measure associated to the maximal function satisfies a curvature condition. This second part of this thesis (Chapters 3 and 4) deals with the problem of Lᵖ → Lᵖ boundedness of a Nikodym maximal function in the Euclidean space. The maximal function is defined using a one-parameter family of tubes in Rᵈ⁺¹, whose directions are determined by a non-degenerate curve in Rᵈ. These operators naturally arise in the analysis of maximal averages over space curves. The main theorem generalises the known results for d = 2 and d = 3 to general dimensions

    Σ1\Sigma_1 gaps as derived models and correctness of mice

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    Assume ZF + AD + V=L(R). Let [α,β][\alpha,\beta] be a Σ1\Sigma_1 gap with Jα(R)J_\alpha(R) admissible. We analyze Jβ(R)J_\beta(R) as a natural form of ``derived model'' of a premouse PP, where PP is found in a generic extension of VV. In particular, we will have P(R)Jβ(R)=P(R)D\mathcal{P}(R)\cap J_\beta(R)=\mathcal{P}(R)\cap D, and if Jβ(R)J_\beta(R)\models``Θ\Theta exists'', then Jβ(R)J_\beta(R) and DD in fact have the same universe. This analysis will be employed in further work, yet to appear, toward a resolution of a conjecture of Rudominer and Steel on the nature of (L(R))M(L(R))^M, for ω\omega-small mice MM. We also establish some preliminary work toward this conjecture in the present paper.Comment: 128 page

    An Infinite Needle in a Finite Haystack: Finding Infinite Counter-Models in Deductive Verification

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    First-order logic, and quantifiers in particular, are widely used in deductive verification. Quantifiers are essential for describing systems with unbounded domains, but prove difficult for automated solvers. Significant effort has been dedicated to finding quantifier instantiations that establish unsatisfiability, thus ensuring validity of a system's verification conditions. However, in many cases the formulas are satisfiable: this is often the case in intermediate steps of the verification process. For such cases, existing tools are limited to finding finite models as counterexamples. Yet, some quantified formulas are satisfiable but only have infinite models. Such infinite counter-models are especially typical when first-order logic is used to approximate inductive definitions such as linked lists or the natural numbers. The inability of solvers to find infinite models makes them diverge in these cases. In this paper, we tackle the problem of finding such infinite models. These models allow the user to identify and fix bugs in the modeling of the system and its properties. Our approach consists of three parts. First, we introduce symbolic structures as a way to represent certain infinite models. Second, we describe an effective model finding procedure that symbolically explores a given family of symbolic structures. Finally, we identify a new decidable fragment of first-order logic that extends and subsumes the many-sorted variant of EPR, where satisfiable formulas always have a model representable by a symbolic structure within a known family. We evaluate our approach on examples from the domains of distributed consensus protocols and of heap-manipulating programs. Our implementation quickly finds infinite counter-models that demonstrate the source of verification failures in a simple way, while SMT solvers and theorem provers such as Z3, cvc5, and Vampire diverge

    Martin's conjecture for regressive functions on the hyperarithmetic degrees

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    We answer a question of Slaman and Steel by showing that a version of Martin's conjecture holds for all regressive functions on the hyperarithmetic degrees. A key step in our proof, which may have applications to other cases of Martin's conjecture, consists of showing that we can always reduce to the case of a continuous function.Comment: 12 page

    Learning algebraic structures with the help of Borel equivalence relations

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    We study algorithmic learning of algebraic structures. In our framework, a learner receives larger and larger pieces of an arbitrary copy of a computable structure and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. The learning is successful if the conjectures eventually stabilize to a correct guess. We prove that a family of structures is learnable if and only if its learning domain is continuously reducible to the relation E0 of eventual agreement on reals. This motivates a novel research program, that is, using descriptive set theoretic tools to calibrate the (learning) complexity of nonlearnable families. Here, we focus on the learning power of well-known benchmark Borel equivalence relations (i.e., E1, E2, E3, Z0, and Eset)

    Structural theory of trees II. Completeness and completions of trees

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    Trees are partial orderings where every element has a linearly ordered set of smaller elements. We define and study several natural notions of completeness of trees, extending Dedekind completeness of linear orders and Dedekind-MacNeille completions of partial orders. We then define constructions of tree completions that extend any tree to a minimal one satisfying the respective completeness property

    The tangent cone, the dimension and the frontier of the medial axis

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    This paper establishes a relation between the tangent cone of the medial axis of X at a given point a ∈ Rn and the medial axis of the set of points m(a) in X realising the Euclidean distance d(a, X). As a consequence, a lower bound for the dimension of the medial axis of X in terms of the dimension of the medial axis of m(a) is obtained. This formula appears to be the missing link to the full description of the medial axis’ dimension. An extended study of potentially troublesome points on the frontier of the medial axis is also provided, resulting in their characterisation by the recently introduced by Birbrair and Denkowski reaching radius whose definition we simplify

    Local Definability of HOD\mathsf{HOD} in L(R)L(\mathbb{R})

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    We show that in L(R)L(\mathbb{R}), assuming large cardinals, HODη+HOD\mathsf{HOD} {\parallel}\eta^{+\mathsf{HOD}} is locally definable from HODη\mathsf{HOD} {\parallel}\eta for all HOD\mathsf{HOD}-cardinals η[δ12,Θ)\eta\in [\boldsymbol{\delta}^2_1,\Theta). This is a further elaboration of the statement "HODL(R)\mathsf{HOD}^{L(\mathbb{R})} is a core model below Θ\Theta" made by John Steel

    Perturbation theory of polynomials and linear operators

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    This survey revolves around the question how the roots of a monic polynomial (resp. the spectral decomposition of a linear operator), whose coefficients depend in a smooth way on parameters, depend on those parameters. The parameter dependence of the polynomials (resp. operators) ranges from real analytic over CC^\infty to differentiable of finite order with often drastically different regularity results for the roots (resp. eigenvalues and eigenvectors). Another interesting point is the difference between the perturbation theory of hyperbolic polynomials (where, by definition, all roots are real) and that of general complex polynomials. The subject, which started with Rellich's work in the 1930s, enjoyed sustained interest through time that intensified in the last two decades, bringing some definitive optimal results. Throughout we try to explain the main proof ideas; Rellich's theorem and Bronshtein's theorem on hyperbolic polynomials are presented with full proofs. The survey is written for readers interested in singularity theory but also for those who intend to apply the results in other fields.Comment: 65 page
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