1,002 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
On certain geometric maximal functions in harmonic analysis
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
gaps as derived models and correctness of mice
Assume ZF + AD + V=L(R). Let be a gap with
admissible. We analyze as a natural form of
``derived model'' of a premouse , where is found in a generic extension
of . In particular, we will have , and if ``
exists'', then and 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 , for
-small mice . 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
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
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
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
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
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 in
We show that in , assuming large cardinals, is locally definable from for all -cardinals . This is a further elaboration of the
statement " is a core model below " made
by John Steel
Perturbation theory of polynomials and linear operators
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
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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