Oberwolfach Preprints (OWP)

Permanent URI for this collectionhttps://oa.tib.eu/renate/handle/123456789/14626

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Local Existence and Conditional Regularity for the Navier-Stokes-Fourier System Driven by Inhomogeneous Boundary Conditions
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Abbatiello, Anna; Basarić, Danica; Chaudhuri, Nilasis; Feireisl, Eduard
We consider the Navier–Stokes–Fourier system with general inhomogeneous Dirichlet–Neumann boundary conditions. We propose a new approach to the local well-posedness problem based on conditional regularity estimates. By conditional regularity we mean that any strong solution belonging to a suitable class remains regular as long as its amplitude remains bounded. The result holds for general Dirichlet-Neumann boundary conditions provided the material derivative of the velocity field vanishes on the boundary of the physical domain. As a corollary of this result we obtain: Blow up criteria for strong solutions; Local existence of strong solutions in the optimal Lp - Lq framework; Alternative proof of the existing results on local well posedness.
Diameter and Connectivity of Finite Simple Graphs II
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Hibi, Takayuki; Saeedi Madani, Sara
Let $G$ be a finite simple non-complete connected graph on $[n] = \{1, \ldots, n\}$ and $\kappa(G) \geq 1$ its vertex connectivity. Let $f(G)$ denote the number of free vertices of $G$ and $\mathrm{diam}(G)$ the diameter of $G$. The final goal of this paper is to determine all sequences of integers $(n,f,d,k)$ with $n\geq 8$, $f\geq 0$, $d\geq 2$ and $k\geq 1$ for which there exists a finite simple non-complete connected graph on $[n]$ with $f=f(G)$, $d=\mathrm{diam}(G)$ and $k=\kappa(G)$.
The Subgroup Structure of Pseudo-Reductive Groups
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Bate, Michael; Martin, Benjamin; Röhrle, Gerhard; Sercombe, Damian
Let $k$ be a field. We investigate the relationship between subgroups of a pseudo-reductive $k$-group $G$ and its maximal reductive quotient $G'$, with applications to the subgroup structure of $G$. Let $k'/k$ be the minimal field of definition for the geometric unipotent radical of $G$, and let $\pi':G_{k'} \to G'$ be the quotient map. We first characterise those smooth subgroups $H$ of $G$ for which $\pi'(H_{k'})=G'$. We next consider the following questions: given a subgroup $H'$ of $G'$, does there exist a subgroup $H$ of $G$ such that $\pi'(H_{k'})=H'$, and if $H'$ is smooth can we find such a $H$ that is smooth? We find sufficient conditions for a positive answer to these questions. In general there are various obstructions to the existence of such a subgroup $H$, which we illustrate with several examples. Finally, we apply these results to relate the maximal smooth subgroups of $G$ with those of $G'$.
The Alternating Halpern-Mann Iteration for Families of Maps
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Firmino, Paulo; Pinto, Pedro
We generalize the alternating Halpern-Mann iteration to countably infinite families of nonexpansive maps and prove its strong convergence towards a common fixed point in the general nonlinear setting of Hadamard spaces. Our approach is based on a quantitative perspective which allowed to circumvent prevalent troublesome arguments and in the end provide a simple convergence proof. In that sense, discussing both the asymptotic regularity and the strong convergence of the iteration in quantitative terms, we furthermore provide low complexity uniform rates of convergence and of metastability (in the sense of T. Tao). In CAT(0) spaces, we obtain linear and quadratic uniform rates of convergence. Our results are made possible by proof-theoretical insights of the research program proof mining and extend several previous theorems in the literature.
Proof Mining and the Convex Feasibility Problem : the Curious Case of Dykstra's Algorithm
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Pinto, Pedro
In a recent proof mining application, the proof-theoretical analysis of Dykstra's cyclic projections algorithm resulted in quantitative information expressed via primitive recursive functionals in the sense of Gödel. This was surprising as the proof relies on several compactness principles and its quantitative analysis would require the functional interpretation of arithmetical comprehension. Therefore, a priori one would expect the need of Spector's bar-recursive functionals. In this paper, we explain how the use of bounded collection principles allows for a modified intermediate proof justifying the finitary results obtained, and discuss the approach in the context of previous eliminations of weak compactness arguments in proof mining.
On Overgroups of Distinguished Unipotent Elements in Reductive Groups and Finite Groups of Lie Type
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Bate, Michael; Böhm, Sören; Martin, Benjamin; Röhrle, Gerhard
Suppose G is a simple algebraic group defined over an algebraically closed field of good characteristic p. In 2018 Korhonen showed that if H is a connected reductive subgroup of G which contains a distinguished unipotent element u of G of order p, then H is G-irreducible in the sense of Serre. We present a short and uniform proof of this result using so-called good A1 subgroups of G, introduced by Seitz. We also formulate a counterpart of Korhonen's theorem for overgroups of u which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of u under a mild condition on p depending on the rank of G, and we present an analogue of Korhonen's theorem for Lie algebras.
Ky Fan Theorem for Sphere Bundles
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Panina, Gaiane; Živaljević, Rade
The classic Ky Fan theorem is a combinatorial equivalent of Borsuk-Ulam theorem. It is a generalization and extension of Tucker's lemma and, just like its predecessor, it pinpoints important properties of antipodal colorings of vertices of a triangulated sphere Sn. Here we describe generalizations of Ky Fan theorem for the case when the sphere is replaced by the total space of a triangulated sphere bundle.
A Gentle Introduction to Interpolation on the Grassmann Manifold
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Ciaramella, Gabriele; Gander, Martin J.; Vanzan, Tommaso
[no abstract available]
On Dykstra's Algorithm with Bregman Projections
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Pinto, Pedro; Pischke, Nicholas
We provide quantitative results on the asymptotic behavior of Dykstra's algorithm with Bregman projections, a combination of the well-known Dykstra's algorithm and the method of cyclic Bregman projections, designed to find best approximations and solve the convex feasibility problem in a non-Hilbertian setting. The result we provide arise through the lens of proof mining, a program in mathematical logic which extracts computational information from non-effective proofs. Concretely, we provide a highly uniform and computable rate of metastability of low complexity and, moreover, we also specify general circumstances in which one can obtain full and effective rates of convergence. As a byproduct of our quantitative analysis, we also for the first time establish the strong convergence of Dykstra's method with Bregman projections in infinite dimensional (reflexive) Banach spaces.
Arm Exponent for the Gaussian Free Field on Metric Graphs in Intermediate Dimensions
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Drewitz, Alexander; Prévost, Alexis; Rodriguez, Pierre-François
We investigate the bond percolation model on transient weighted graphs G induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in G have polynomial volume growth with growth exponent α and that the Green's function for the random walk on G exhibits a power law decay with exponent ν, in the regime 1≤ν≤α/2. In particular, this includes the cases of G=Z3, for which ν=1, and G=Z4, for which ν=α2=2. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance R like R−ν2+o(1). Our results are in fact more precise and yield logarithmic corrections when ν>1 as well as corrections of order loglogR when ν=1. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when ν>1 and essentially optimal when ν=1. This extends previous results from [16].
Hypergroups and Twin Buildings, I
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) French, Christopher; Zieschang, Paul-Hermann
We discuss a conjecture on thick twin buildings the verification of which is needed in order to show that thick twin buildings are mathematically equivalent to regular actions of certain twin Coxeter hypergroups. (A corresponding result for buildings is shown in [5; Sections 10.2, 10.3].) We prove that the conjecture holds in the case where the support of its sagittal has cardinality 2 and in the case where its sagittal has length at most 3. (Sagittals are defined in Section 1.) Our exposition is based on an earlier treatment of the subject; cf. [3].
Lax Comma Categories of Ordered Sets
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Clementino, Maria Manuel; Lucatelli Nunes, Fernando
Let Ord be the category of (pre)ordered sets. Unlike Ord/X, whose behaviour is well-known, not much can be found in the literature about the lax comma 2-category Ord//X. In this paper we show that the forgetful functor Ord//X→Ord is topological if and only if X is complete. Moreover, under suitable hypothesis, Ord//X is complete and cartesian closed if and only if X is. We end by analysing descent in this category. Namely, when X is complete and cartesian closed, we show that, for a morphism in Ord//X, being pointwise effective for descent in Ord is sufficient, while being effective for descent in Ord is necessary, to be effective for descent in Ord//X.
Automatic Differentiation for ML-Familiy Languages: Correctness via Logical Relations
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Lucatelli Nunes, Fernando; Vákár, Matthijs
[no abstract available]
Logical Relations for Partial Features and Automatic Differentiation Correctness
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Lucatelli Nunes, Fernando; Vákár, Matthijs
We present a simple technique for semantic, open logical relations arguments about languages with recursive types, which, as we show, follows from a principled foundation in categorical semantics. We demonstrate how it can be used to give a very straightforward proof of correctness of practical forward- and reverse-mode dual numbers style automatic differentiation (AD) on ML-family languages. The key idea is to combine it with a suitable open logical relations technique for reasoning about differentiable partial functions (a suitable lifting of the partiality monad to logical relations), which we introduce.
Cauchy Completeness, Lax Epimorphisms and Effective Descent for Split Fibrations
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Lucatelli Nunes, Fernando; Prezado, Rui; Sousa, Lurdes
For any suitable base category V, we find that V-fully faithful lax epimorphisms in V-Cat are precisely those V-functors F:A→B whose induced V-functors CauchyF:CauchyA→CauchyB between the Cauchy completions are equivalences. For the case V=Set, this is equivalent to requiring that the induced functor CAT(F,Cat) between the categories of split (op)fibrations is an equivalence. By reducing the study of effective descent functors with respect to the indexed category of split (op)fibrations F to the study of the codescent factorization, we find that these observations on fully faithful lax epimorphisms provide us with a characterization of (effective) F-descent morphisms in the category of small categories Cat; namely, we find that they are precisely the (effective) descent morphisms with respect to the indexed categories of discrete opfibrations -- previously studied by Sobral. We include some comments on the Beck-Chevalley condition and future work.
Semantic Factorization and Descent
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Lucatelli Nunes, Fernando
Let A be a 2-category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism p exists and is preserved by a suitable morphism, the factorization given by the lax descent object of the higher cokernel of p is up to isomorphism the same as the semantic factorization of p, either one existing if the other does. The result can be seen as a counterpart account to the celebrated Bénabou-Roubaud theorem. This leads in particular to a monadicity theorem, since it characterizes monadicity via descent. It should be noted that all the conditions on the codensity monad of p trivially hold whenever p has a left adjoint and, hence, in this case, we find monadicity to be a 2-dimensional exact condition on p, namely, to be an effective faithful morphism of the 2-category A.
Edifices: Building-like Spaces Associated to Linear Algebraic Groups; In memory of Jacques Tits
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Bate, Michael; Martin, Benjamin; Röhrle, Gerhard
Given a semisimple linear algebraic k-group G, one has a spherical building ΔG, and one can interpret the geometric realisation ΔG(R) of ΔG in terms of cocharacters of G. The aim of this paper is to extend this construction to the case when G is an arbitrary connected linear algebraic group; we call the resulting object ΔG(R) the spherical edifice of G. We also define an object VG(R) which is an analogue of the vector building for a semisimple group; we call VG(R) the vector edifice. The notions of a linear map and an isomorphism between edifices are introduced; we construct some linear maps arising from natural group-theoretic operations. We also devise a family of metrics on VG(R) and show they are all bi-Lipschitz equivalent to each other; with this extra structure, VG(R) becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits Centre Conjecture.
Computer Algebra with GAP
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Piterman, Kevin I.; Vendramin, Leandro
This monograph includes the following topics: a basic introduction to the language, basic arithmetic, permutations, matrices, polynomial rings, finite fields, finite and finitely presented groups, small groups, group representations and character theory, and simple groups. Advanced topics include testing several open conjectures and theorems. In addition, each chapter ends with an extensive list of problems. We hope the reader will find some problems challenging and exciting as they are based on outstanding research papers. Selected solutions can be found at the end of the book.
Ground State of Bose Gases Interacting through Singular Potentials
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Boßmann, Lea; Leopold, Nikolai; Petrat, Sören; Rademacher, Simone
We consider a system of N bosons on the three-dimensional unit torus. The particles interact through repulsive pair interactions of the form N3β−1v(Nβx) for β∈(0,1). We prove the next order correction to Bogoliubov theory for the ground state and the ground state energy.
Bochner-Riesz Means at the Critical Index: Weighted and Sparse Bounds
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Beltran, David; Roos, Joris; Seeger, Andreas
We consider Bochner-Riesz means on weighted Lp spaces, at the critical index λ(p)=d(1/p−1/2)−1/2. For every A₁-weight we obtain an extension of Vargas' weak type (1,1) inequality in some range of p>1. To prove this result we establish new endpoint results for sparse domination. These are almost optimal in dimension d=2; partial results as well as conditional results are proved in higher dimensions. For the means of index λ∗=(d−1)/(2d+2) we prove fully optimal sparse bounds.

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