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This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the Čech cohomology functors ˇH𝑛 on the category of locally compact separable metric spaces each factor into (i) what we term their definable version, a functor ˇH𝑛 def taking values in the category GPC of groups with a Polish cover (a category first introduced in this work's predecessor), followed by (ii) a forgetful functor from GPC to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of d-spheres or d-tori for any 𝑑 ≥ 1, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functors ˇH𝑛 def to show that a seminal problem in the development of algebraic topology - namely, Borsuk and Eilenberg's 1936 problem of classifying, up to homotopy, the maps from a solenoid complement 𝑆3\Σ to the 2-sphere - is essentially hyperfinite but not smooth. Fundamental to our analysis is the fact that the Čech cohomology functors 𝑋 ↦→ ˇH𝑛(𝑋;𝐺) admit two main formulations: a more combinatorial one and a more homotopical formulation as the group [𝑋, 𝑃] of homotopy classes of maps from X to a polyhedral 𝐾(𝐺, 𝑛) space P. We describe the Borel-definable content of each of these formulations and prove a definable version of Huber's theorem reconciling the two. In the course of this work, we record definable versions of Urysohn's Lemma and the simplicial approximation and homotopy extension theorems, along with a definable Milnor-type short exact sequence decomposition of the space Map(𝑋, 𝑃) in terms of its subset of phantom maps; relatedly, we provide a topological characterization of this set for any locally compact Polish space X and polyhedron P. In aggregate, this work may be more broadly construed as laying foundations for the descriptive set theoretic study of the homotopy relation on such spaces Map(𝑋, 𝑃), a relation which, together with the more combinatorial incarnation of ˇH𝑛, embodies a substantial variety of classification problems arising throughout mathematics. We show, in particular, that if P is a polyhedral H-group, then this relation is both Borel and idealistic. In consequence, [𝑋, 𝑃] falls in the category of definable groups, an extension of the category GPC introduced herein for its regularity properties, which facilitate several of the aforementioned computations. [ABSTRACT FROM AUTHOR]

The finite Hilbert transform T$T$, when acting in the classical Zygmund space LlogL$L\textnormal {log} L$ (over (−1,1)$(-1,1)$), was intensively studied in [8]. In this note, an integral representation of T$T$ is established via the L1(−1,1)$L^1(-1,1)$‐valued measure mL1:A↦T(χA)$m_{L^1}: A\mapsto T(\chi _A)$ for each Borel set A⊆(−1,1)$A\subseteq (-1,1)$. This integral representation, together with various non‐trivial properties of mL1$m_{L^1}$, allows the use of measure theoretic methods (not available in [8]) to establish new properties of T$T$. For instance, as an operator between Banach function spaces T$T$ is not order bounded, it is not completely continuous and neither is it weakly compact. An appropriate Parseval formula for T$T$ plays a crucial role. [ABSTRACT FROM AUTHOR]

We provide a definition of a ≺-asymptotic (we suggest the pronunciation "prec-asymptotic") pair in a topological action (X , G) of a countable amenable group G , where ≺ is an order on G of type Z. In the case where for some G -invariant Borel probability measure μ on X , the measure-preserving system (X , μ , G) factors, via a map φ , onto a multiorder (O ˜ , ν , G) , we also introduce the notion of a φ -asymptotic pair. Then we prove that if μ has positive measure-theoretic conditional entropy with respect to the multiorder factor, then the set of points which belong to φ -asymptotic pairs has positive measure μ. This result is a generalization of the Blanchard-Host-Ruette Theorem for classical topological dynamical systems (actions of Z). As a strengthening of our theorem, we show that for any system (X , G) of positive topological entropy, any multiorder (O ˜ , ν , G) and ν -almost every ≺ ∈ O ˜ , there exist ≺-asymptotic pairs in X. Finally, we characterize systems (X , G) of topological entropy zero as factors of topologically multiordered systems (in which case φ is defined μ -almost everywhere for every G -invariant measure μ) with no φ -asymptotic pairs. [ABSTRACT FROM AUTHOR]

We show that every \Delta ^0_2 Polish space admits a computable topological presentation given by an effective indexing of some non-empty open sets in the space. [ABSTRACT FROM AUTHOR]

Let X be a completely regular Hausdorff space, let E and F denote Banach spaces. Let C_b(X,E) denote the space of E-valued bounded continuous functions on X and let \beta be the strict topology on this space. We establish the relationship between nuclear operators T:C_b(X,E)\rightarrow F between the locally convex space (C_b(X,E),\beta) and the Banach space F and their representing operator-valued Borel measures. [ABSTRACT FROM AUTHOR]

Let M n = diag [ 3 p n , 3 q n ] with p n , q n ⩾ 1 for all n ⩾ 1 and let D = { (0 , 0) t , (1 , 0) t , (0 , 1) t } . One can generate a Borel probability measure μ { M n } , D = δ M 1 − 1 D ∗ δ (M 2 M 1) − 1 D ∗ δ (M 3 M 2 M 1) − 1 D ∗ ⋯. Such measure μ { M n } , D is called a Moran Sierpinski-type measure. It is known Deng et al (Acta Math. Sin. submitted) that the associated Hilbert space L 2 (μ { M n } , D) has an exponential orthonormal basis. In this paper, we first characterize all the maximal exponential orthogonal sets for L 2 (μ { M n } , D) . For such a maximal orthogonal set, we then give some sufficient conditions to determine whether it is an orthonormal basis of L 2 (μ { M n } , D) or not. [ABSTRACT FROM AUTHOR]

Let 0 < \beta \leq \alpha < 1 and \kappa > 0. I prove that there exists \eta > 0 such that the following holds for every pair of Borel sets A,B \subset \mathbb {R} with \dim _{\mathrm {H}} A = \alpha and \dim _{\mathrm {H}} B = \beta: \begin{equation*} \dim _{\mathrm {H}} \{c \in \mathbb {R}: \dim _{\mathrm {H}} (A + cB) \leq \alpha + \eta \} \leq \tfrac {\alpha - \beta }{1 - \beta } + \kappa. \end{equation*} This extends a result of Bourgain from 2010, which contained the case \alpha = \beta. The paper also contains a \delta-discretised, and somewhat stronger, version of the estimate above, and new information on the size of long sums of the form a_{1}B + \ldots + a_{n}B. [ABSTRACT FROM AUTHOR]

Let X be an infinite Tychonoff space, and Y be a topological subspace of X. In this paper, we study some covering properties of the subspace Cp(Y|X) of Cp(Y) consisting of those functions f ∈ C (Y) which admit a continuous extension to X equipped with the relative topology of Cp(Y). Among other results, we show that (i) Cp(Y|X) has a fundamental bounded resolution if and only if Y is countable; when X is realcompact and Y is closed in X, we have (ii) if Cp(Y|X) admits a resolution of convex compact sets that swallows the local null sequences in Cp(Y|X), then Y is countable and discrete; (iii) if Cp(Y|X) admits a compact resolution that swallows the compact sets, then Y is also countable and discrete, and, as a corollary, we deduce that Cp(Y|X) admits a compact resolution that swallows the compact sets if and only if Cp(Y|X) is a Polish space. We also prove that (iv) for a metrizable space X, Cp(X) is a quasi-(LB)-space if and only if X is σ-compact, and hence for a subspace Y of X, the space Cp(Y|X) is a quasi-(LB)-space. We include some examples and observations that answer natural questions raised in this paper. [ABSTRACT FROM AUTHOR]

In this paper we characterize hyperbolic cosine transforms of (positive) Borel measures ν in terms of exponential convexity (Bernstein's terminology). The case of compactly supported measures ν is also considered. All of this is then applied to (bounded) composition operators C T , ρ : f ↦ f ∘ T on L 2 (R κ , μ ρ) with affine symbols T = A + a , where d μ ρ (x) = ρ (x) d x , ρ (x) = ψ (‖ x ‖) − 1 , ψ is a continuous positive real valued function and ‖ ⋅ ‖ is the Euclidean norm on R κ. The main result states that the map R κ ∋ a ↦ C I + a , ρ is continuous in the strong operator topology and has cosubnormal values if and only if ψ is the hyperbolic cosine transform of a compactly supported Borel measure (I is the identity transformation). The case of affine symbols T that are not translations is also discussed. [ABSTRACT FROM AUTHOR]

In the context of predicting the behaviour of chaotic systems, Schroer, Sauer, Ott and Yorke conjectured in 1998 that if a dynamical system defined by a smooth diffeomorphism T of a Riemannian manifold X admits an attractor with a natural measure μ of information dimension smaller than k , then k time-delayed measurements of a one-dimensional observable h are generically sufficient for μ -almost sure prediction of future measurements of h. In a previous paper we established this conjecture in the setup of injective Lipschitz transformations T of a compact set X in Euclidean space with an ergodic T -invariant Borel probability measure μ. In this paper we prove the conjecture for all (also non-invertible) Lipschitz systems on compact sets with an arbitrary Borel probability measure, and establish an upper bound for the decay rate of the measure of the set of points where the prediction is subpar. This partially confirms a second conjecture by Schroer, Sauer, Ott and Yorke related to empirical prediction algorithms as well as algorithms estimating the dimension and number of required delayed measurements (the so-called embedding dimension) of an observed system. We also prove general time-delay prediction theorems for locally Lipschitz or Hölder systems on Borel sets in Euclidean space. [ABSTRACT FROM AUTHOR]

It is consistent that the continuum be arbitrary large and no absolute κ -Borel set X of density κ , ℵ 1 < κ < c ,condenses onto a compactum. It is consistent that the continuum be arbitrary large and any absolute κ -Borel set X of density κ , κ ≤ c , containing a closed subspace of the Baire space of weight κ , condenses onto a compactum. In particular, applying Brian's results in model theory, we get the following unexpected result. Given any A ⊆ N with 1 ∈ A , there is a forcing extension in which every absolute ℵ n -Borel set, containing a closed subspace of the Baire space of weight ℵ n , condenses onto a compactum if and only if n ∈ A . [ABSTRACT FROM AUTHOR]

In this paper, we prove that in a hyperconvex domain Ω in ℍn, if a non-negative Borel measure is dominated by a quaternionic Monge–Ampère measure, then it is a quaternionic Monge–Ampère measure of a function in the class . [ABSTRACT FROM AUTHOR]

For which infinite cardinals $\kappa $ is there a partition of the real line ${\mathbb R}$ into precisely $\kappa $ Borel sets? Work of Lusin, Souslin, and Hausdorff shows that ${\mathbb R}$ can be partitioned into $\aleph _1$ Borel sets. But other than this, we show that the spectrum of possible sizes of partitions of ${\mathbb R}$ into Borel sets can be fairly arbitrary. For example, given any $A \subseteq \omega $ with $0,1 \in A$ , there is a forcing extension in which ${A = \{ n :\, \text {there is a partition of } {{\mathbb R}} \text { into }\aleph _n\text { Borel sets}\}}$. We also look at the corresponding question for partitions of ${\mathbb R}$ into closed sets. We show that, like with partitions into Borel sets, the set of all uncountable $\kappa $ such that there is a partition of ${\mathbb R}$ into precisely $\kappa $ closed sets can be fairly arbitrary. [ABSTRACT FROM AUTHOR]

The relations between asymptotic stability, the eventual e-property and the e-property of Markov semigroups, acting on measures defined on general (Polish metric) spaces, are studied. While much attention is usually paid to asymptotic stability (for years the e-property has only served as a tool to establish it), it should be noted that the e-property itself is also important as it, e.g., ensures that numerical errors in simulations are negligible. Here, it is shown that any asymptotically stable Markov–Feller semigroup with an invariant measure such that the interior of its support is non-empty satisfies the eventual e-property. Moreover, we prove that any Markov–Feller semigroup, which is strongly stochastically continuous, and which possesses the eventual (Polish metric), also has the e-property. We also present an example highlighting that the assumption of strong stochastic continuity (given in terms of the supremum norm) cannot be relaxed to its weak form, involving pointwise convergence, unless a state space of a process corresponding to a Markov semigroup is a compact metric space. [ABSTRACT FROM AUTHOR]

This paper extends some results of Mattila (J. Fractal Geom. 66 (2021) 389–401 and Ann. Acad. Sci. Fenn. A Math. 42 (2017) 611–620), in particular, removing assumptions of positive lower density. We give conditions on a general family Pλ:Rn→Rm,λ∈Λ$P_{\lambda }:\mathbb {R}^n\rightarrow \mathbb {R}^m, \lambda \in \Lambda$, of orthogonal projections which guarantee that the Hausdorff dimension formula dimA∩Pλ−1{u}=s−m$\dim A\cap P_{\lambda }^{-1}\lbrace u\rbrace =s-m$ holds generically for measurable sets A⊂Rn$A\subset \mathbb {R}^{n}$ with positive and finite s$s$‐dimensional Hausdorff measure, s>m$s>m$. As an application we prove for Borel sets A,B⊂Rn$A,B\subset \mathbb {R}^{n}$ with positive s$s$‐ and t-dimensional$t{\text{-dimensional}}$ measures that if s+(n−1)t/n>n$s + (n-1)t/n > n$, then dimA∩(g(B)+z)⩾s+t−n$\dim A\cap (g(B)+z) \geqslant s+t - n$ for almost all rotations g$g$ and for positively many z∈Rn$z\in \mathbb {R}^{n}$. We shall also give an application to the estimates of the dimension of the set of exceptional rotations. [ABSTRACT FROM AUTHOR]

One formulation of Marstrand's slicing theorem is the following. Assume that t∈(1,2]$t \in (1,2]$, and B⊂R2$B \subset \mathbb {R}^{2}$ is a Borel set with Ht(B)<∞$\mathcal {H}^{t}(B) < \infty$. Then, for almost all directions e∈S1$e \in S^{1}$, Ht$\mathcal {H}^{t}$ almost all of B$B$ is covered by lines ℓ$\ell$ parallel to e$e$ with dimH(B∩ℓ)=t−1$\dim _{\mathrm{H}}(B \cap \ell) = t - 1$. We investigate the prospects of sharpening Marstrand's result in the following sense: in a generic direction e∈S1$e \in S^{1}$, is it true that a strictly less than t$t$‐dimensional part of B$B$ is covered by the heavy lines ℓ⊂R2$\ell \subset \mathbb {R}^{2}$, namely those with dimH(B∩ℓ)>t−1$\dim _{\mathrm{H}} (B \cap \ell) > t - 1$? A positive answer for t$t$‐regular sets B⊂R2$B \subset \mathbb {R}^{2}$ was previously obtained by the first author. The answer for general Borel sets turns out to be negative for t∈(1,32]$t \in (1,\tfrac{3}{2}]$ and positive for t∈(32,2]$t \in (\tfrac{3}{2},2]$. More precisely, the heavy lines can cover up to a min{t,3−t}$\min \lbrace t,3 - t\rbrace$ dimensional part of B$B$ in a generic direction. We also consider the part of B$B$ covered by the s$s$‐heavy lines, namely those with dimH(B∩ℓ)⩾s$\dim _{\mathrm{H}}(B \cap \ell) \geqslant s$ for s>t−1$s > t - 1$. We establish a sharp answer to the question: how much can the s$s$‐heavy lines cover in a generic direction? Finally, we identify a new class of sets called sub‐uniformly distributed sets, which generalise Ahlfors‐regular sets. Roughly speaking, these sets share the spatial uniformity of Ahlfors‐regular sets, but pose no restrictions on uniformity across different scales. We then extend and sharpen the first author's previous result on Ahlfors‐regular sets to the class of sub‐uniformly distributed sets. [ABSTRACT FROM AUTHOR]

We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa–Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich–Itô correction term. [ABSTRACT FROM AUTHOR]

In many physical problems, it is important to capture exponentially small effects that lie beyond‐all‐orders of an algebraic asymptotic expansion; when collected, the full asymptotic expansion is known as a trans‐series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans‐series expansion, typically for singularly perturbed nonlinear differential or integral equations. Separately to applied exponential asymptotics, there exists a related line of research known as Écalle's theory of resurgence, which, via Borel resummation, describes the connection between trans‐series and a certain class of holomorphic functions known as resurgent functions. Most applications and examples of Écalle's resurgence theory focus mainly on nonparametric asymptotic expansions (i.e., differential equations without a parameter). The relationships between these latter areas with applied exponential asymptotics have not been thoroughly examined—largely due to differences in language and emphasis. In this work, we establish these connections as an alternative framework to the factorial‐over‐power ansatz procedure in applied exponential asymptotics and clarify a number of aspects of applied exponential asymptotic methodology, including Van Dyke's rule and the universality of factorial‐over‐power ansatzes. We provide a number of useful tools for probing more pathological problems in exponential asymptotics and establish a framework for future applications to nonlinear and multidimensional problems in the physical sciences. [ABSTRACT FROM AUTHOR]

It is well known that the lower quantization dimension of a Borel probability measure given on a metric compact set does not exceed the lower box dimension of . We prove the following intermediate value theorem for the lower quantization dimension of probability measures: for any nonnegative number smaller that the dimension of the compact set , there exists a probability measure on with support such that . The number characterizes the asymptotic behavior of the lower box dimension of closed -neighborhoods of zero-dimensional, in the sense of , closed subsets of as . For a wide class of metric compact sets, the equality holds. [ABSTRACT FROM AUTHOR]

This paper addresses a class of discrete-time Markov decision processes in Borel spaces with a finite number of cost constraints. The constrained control model considers costs of discounted type with state-dependent discount factors which are subject to external disturbances. Our objective is to prove the existence of optimal control policies and characterize them according to certain optimality criteria. Specifically, by rewriting appropriately our original constrained problem as a new one on a space of occupation measures, we apply the direct method to show solvability. Next, the problem is defined as a convex program, and we prove that the existence of a saddle point of the associated Lagrangian operator is equivalent to the existence of an optimal control policy for the constrained problem. Finally, we turn our attention to multi-objective optimization problems, where the existence of Pareto optimal policies can be obtained from the existence of saddle-points of the aforementioned Lagrangian or equivalently from the existence of optimal control policies of constrained problems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the large-time behavior of bounded solutions of the Cauchy problem for a reaction-diffusion equation in $ \mathbb{R}^N $ with bistable reaction term. We consider initial conditions that are chiefly indicator functions of bounded Borel sets. We examine how geometric transformations of the supports of these initial conditions affect the propagation or extinction of the solutions at large time. We also consider two fragmentation indices defined in the set of bounded Borel sets and we establish some propagation or extinction results when the initial supports are weakly or highly fragmented. Lastly, we show that the large-time dynamics of the solutions is not monotone with respect to the considered fragmentation indices, even for equimeasurable sets. [ABSTRACT FROM AUTHOR]

Let (u, v) be a pair of quasidefinite and symmetric linear functionals with {Pn}n≥0 and {Qn}n≥0 as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials {Rn}n≥0 as follows: P n + 2 ′ x n + 2 + b n P n ′ x n - Q n + 1 x = d n - 1 x , n ≥ 1. We present necessary and sufficient conditions for {Rn}n≥0 to be orthogonal with respect to a quasidefinite linear functional w. In addition, we consider the case where {Pn}n≥0 and {Qn}n≥0 are monic Chebyshev polynomials of the first and second kinds, respectively, and study the relative outer asymptotics of Sobolev polynomials orthogonal with respect to the Sobolev inner product ⟨ p , q ⟩ s = ∫ 1 - 1 p q 1 - x 2 - 1 / 2 d x + λ 1 ∫ 1 - 1 p ′ q ′ 1 - x 2 1 / 2 d x + λ 2 ∫ 1 - 1 p ′ ′ q ′ ′ d μ x , where μ is a positive Borel measure associated with w and λ1, λ2> 0; λ2 is a linear polynomial of λ1. [ABSTRACT FROM AUTHOR]

We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let be non-empty Borel sets. If X is not contained in any line, we prove that If dimHY>1, we have the following improved lower bound: Our results solve conjectures of Lund-Thang-Huong, Liu, and the first author. Another corollary is the following continuum version of Beck's theorem in combinatorial geometry: if is a Borel set with the property that dimH(X ∖ ℓ)=dimHX for all lines , then the line set spanned by X has Hausdorff dimension at least min{2dimHX,2}. While the results above concern , we also derive some counterparts in by means of integralgeometric considerations. The proofs are based on an ϵ-improvement in the Furstenberg set problem, due to the two first authors, a bootstrapping scheme introduced by the second and third author, and a new planar incidence estimate due to Fu and Ren. [ABSTRACT FROM AUTHOR]

Assuming that 0^\dagger does not exist, we prove that if there is a partition of \mathbb R into \aleph _\omega Borel sets, then there is also a partition of \mathbb R into \aleph _{\omega +1} Borel sets. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Markov decision process (MDP) with a Borel state space X⊔ {Δ}, where Δ is an absorbing state (cemetery), and a Borel action space A. We consider the space of finite occupation measures restricted on X\times A and the extreme points in it. It is possible that some strategies have infinite occupation measures. Nevertheless, we prove that every finite extreme occupation measure is generated by a deterministic stationary strategy. Then, for this MDP, we consider a constrained problem with total undiscounted criteria and J constraints, where the cost functions are nonnegative. By assumption, the strategies inducing infinite occupation measures are not optimal. Then our second main result is that, under mild conditions, the solution to this constrained MDP is given by a mixture of no more than J +1 occupation measures generated by deterministic stationary strategies. [ABSTRACT FROM AUTHOR]

This paper presents a mathematical framework for modeling the dynamics of heterogeneous populations. Models describing local and non-local growth and transport processes appear in a variety of applications, such as crowd dynamics, tissue regeneration, cancer development and coagulation-fragmentation processes. The diverse applications pose a common challenge to mathematicians due to the multiscale nature of the structures that underlie the system's self-organization and control. Similar abstract mathematical problems arise when formulating problems in the language of measure evolution on a multi-faceted state space. Motivated by these observations, we propose a general mathematical framework for nonlinear structured population models on abstract metric spaces, which are only assumed to be separable and complete. We exploit the structure of the space of non-negative Radon measures with the dual bounded Lipschitz distance (flat metric), which is a generalization of the Wasserstein distance, capable of addressing non-conservative problems. The formulation of models on general metric spaces allows considering infinite-dimensional state spaces or graphs and coupling discrete and continuous state transitions. This opens up exciting possibilities for modeling single-cell data, crowd dynamics or coagulation-fragmentation processes. [ABSTRACT FROM AUTHOR]

Let Tq = -d2/dx2 +q be a Schr¨odinger operator in the space L2(R). A potential q is called reflectionless if the operator Tq is reflectionless. Let Q be the set of all reflectionless potentials of the Schr¨odinger operator, and let M be the set of nonnegative Borel measures on R with compact support. As shown by Marchenko, each potential q ∈ Q can be associated with a unique measure μ ∈ M. As a result, we get the bijection Θ: Q → M. In this paper, we show that one can define topologies on Q andM, under which the mapping Θ is a homeomorphism. [ABSTRACT FROM AUTHOR]

Suppose that ν is an ergodic invariant random subgroup of a countable group G such that [NG​(H):H]=n<∞ for ν-a.e. H∈Sub. In this paper, we consider the question of whether ν can be realized as the stabilizer distribution of an ergodic action G↷(X,μ) on a standard Borel probability space such that the stabilizer map x↦Gx​ is n-to-one. [ABSTRACT FROM AUTHOR]

The purpose of local spatial development plans in Poland is to shape spatial policy at the local level by establishing local law. On this basis, the intended use of land is determined, as well as the manner of its development. Some of the planning tools used are urban planning parameters and indicators, which are defined in planning documents at the local (commune) level. This article discusses the analyses of two selected urban indicators: the maximum built-up area and the minimum share of biologically active area, which are obligatorily determined in local plans. The issue here is a certain discrepancy between the planning provisions used in theory and their practical application. This paper uses a descriptive, computational, and comparative analytical research method to interpret the planning provisions used in selected Local Spatial Development Plans. This research will allow for answering of the following questions: Are the urban indicators defined by architects adapted to spatial needs? Do they allow for optimal shaping of this space, taking into account the provisions of Polish law? And is their skillful designation likely to be one of the sustainable assumptions of the New European Bauhaus? [ABSTRACT FROM AUTHOR]

Let μ$\mu$ be an even Borel probability measure on R${\mathbb {R}}$. For every N>n$N>n$, consider N$N$ independent random vectors X⃗1,...,X⃗N$\vec{X}_1,\ldots ,\vec{X}_N$ in Rn${\mathbb {R}}^n$, with independent coordinates having distribution μ$\mu$. We establish a sharp threshold for the product measure μn$\mu _n$ of the random polytope KN:=conv{X⃗1,...,X⃗N}$K_N:=\operatorname{conv}\lbrace \vec{X}_1,\ldots ,\vec{X}_N\rbrace$ in Rn${\mathbb {R}}^n$ under the assumption that the Legendre transform Λμ*$\Lambda _{\mu }^{\ast }$ of the logarithmic moment generating function of μ$\mu$ satisfies the condition limx↑x*−lnμ([x,∞))Λμ*(x)=1,$$\begin{equation*} \hspace*{6pc}\lim \limits _{x\uparrow x^{\ast }}\dfrac{-\ln \mu ([x,\infty))}{\Lambda _{\mu }^{\ast }(x)}=1,\hspace*{-6pc} \end{equation*}$$where x*=sup{x∈R:μ([x,∞))>0}$x^{\ast }=\sup \lbrace x\in \mathbb {R}\colon \mu ([x,\infty))>0\rbrace$. An application is a sharp threshold for the case of the product measure νpn=νp⊗n$\nu _p^n=\nu _p^{\otimes n}$, p⩾1$p\geqslant 1$ with density (2γp)−nexp(−∥x∥pp)$(2\gamma _p)^{-n}\exp (-\Vert x\Vert _p^p)$, where ∥·∥p$\Vert \cdot \Vert _p$ is the ℓpn$\ell _p^n$‐norm and γp=Γ(1+1/p)$\gamma _p=\Gamma (1+1/p)$. [ABSTRACT FROM AUTHOR]

We compare several natural notions of effective presentability of a topological space up to homeomorphism. We note that every left-c.e. (lower-semicomputable) Stone space is homeomorphic to a computable one. In contrast, we produce an example of a locally compact, left-c.e. space that is not homeomorphic to any computable Polish space. We apply a similar technique to produce examples of computable topological spaces not homeomorphic to any right-c.e. (upper-semicomputable) Polish space, and indeed to any arithmetical or even analytical Polish space. We then apply our techniques to totally disconnected locally compact (tdlc) groups. We prove that every computably locally compact tdlc group is topologically isomorphic to a computable tdlc group. [ABSTRACT FROM AUTHOR]

We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbf {\Sigma }^0_2$ , then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth–Kechris about discrete structures. As a different application, we also give a new proof of Kechris's theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable. [ABSTRACT FROM AUTHOR]

This article starts with a study of the congruence of soft sets modulo soft ideals. Different types of soft ideals in soft topological spaces are used to introduce new weak classes of soft open sets. Namely, soft open sets modulo soft nowhere dense sets and soft open sets modulo soft sets of the first category. The basic properties and representations of these classes are established. The class of soft open sets modulo the soft nowhere dense sets forms a soft algebra. Elements in this soft algebra are primarily the soft sets whose soft boundaries are soft nowhere dense sets. The class of soft open sets modulo soft sets of the first category, known as soft sets of the Baire property, is a soft σ -algebra. In this work, we mainly focus on the soft σ -algebra of soft sets with the Baire property. We show that soft sets with the Baire property can be represented in terms of various natural classes of soft sets in soft topological spaces. In addition, we see that the soft σ -algebra of soft sets with the Baire property includes the soft Borel σ -algebra. We further show that soft sets with the Baire property in a certain soft topology are equal to soft Borel sets in the cluster soft topology formed by the original one. [ABSTRACT FROM AUTHOR]

We show that every two-player stochastic game with finite state and action sets, and bounded, Borel-measurable, and shift-invariant payoffs, admits an ε -equilibrium for all ε > 0. • We study two-player stochastic games with shift-invariant payoffs. • We show that such games admit an ε -equilibrium for all ε > 0. • This result extends earlier work of Vieille (2000) on stochastic games with the long-term average payoff. • This result extends Chatterjee (2005) who studied stochastic games with ω -regular objectives expressed as parity objectives. [ABSTRACT FROM AUTHOR]

This paper is concerned with the proper way to effectivize the notion of a Polish space. A theorem is proved that shows the recursive Polish space structure is not found in the effectively open subsets of a space X , and we explore strong evidence that the effective structure is instead captured by the effectively open subsets of the product space N × X . [ABSTRACT FROM AUTHOR]

Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class Γ , then its Γ -code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau's theorem to Borel functions: If a Borel function on a Polish space happens to be a Σ ~ t -function, then one can find its Σ ~ t -code hyperarithmetically relative to its Borel code. More generally, we prove extension-type, domination-type, and decomposition-type variants of Louveau's theorem for Borel functions. [ABSTRACT FROM AUTHOR]

Let X be a Polish space with Borel probability measure \mu, and let G be a Borel graph on X with no odd cycles and maximum degree \Delta (G). We show that the Baire measurable edge chromatic number of G is at most \Delta (G)+1, and if G is \mu-hyperfinite then the \mu-measurable edge chromatic number obeys the same bound. More generally, we show that G has Borel edge chromatic number at most \Delta (G) plus its asymptotic separation index. [ABSTRACT FROM AUTHOR]

Overt choice was recently introduced and thoroughly studied by de Brecht, Pauly and Schröder. They give estimates on the Weihrauch degree of overt choice on various spaces, and relate it to the topological properties of the space. In this article, we pursue this line of research, answering some of the questions that were left open. We show that overt choice on the rationals is not limit-computable. We identify the Weihrauch degree of overt choice on the space of natural numbers with the co-finite topology. We prove that the quasi-Polish spaces are the countably-based T 0 -spaces on which a variant of overt choice, called Π ~ 2 0 overt choice, is continuous. It extends a previous result that holds in the class of T 1 -spaces. We also prove an effective version of this equivalence. [ABSTRACT FROM AUTHOR]

We prove a version of Bourgain's projection theorem for parametrized families of C2 maps, which refines the original statement even in the linear case by requiring non-concentration only at a single natural scale. As one application, we show that if A is a Borel set of Hausdorff dimension close to 1 in R² or close to 3=2 in R³, then for y ∈ A outside of a very sparse set, the pinned distance set {|x - y|: x ∈ A} has Hausdorff dimension at least 1=2 C c, where c is universal. Furthermore, the same holds if the distances are taken with respect to a C² norm of positive Gaussian curvature. As further applications, we obtain new bounds on the dimensions of spherical projections, and an improvement over the trivial estimate for incidences between ı-balls and δ-neighborhoods of curves in the plane, under fairly general assumptions. The proofs depend on a new multiscale decomposition of measures into "Frostman pieces" that may be of independent interest. [ABSTRACT FROM AUTHOR]

Let $\mu$ be a Radon measure on the n th Heisenberg group ${\mathbb{H}}^n$. In this note we prove that if the $(2n+1)$ -dimensional (Heisenberg) Riesz transform on ${\mathbb{H}}^n$ is $L^2(\mu)$ -bounded, and if $\mu(F)=0$ for all Borel sets with ${\text{dim}}_H(F)\leq 2$ , then $\mu$ must have $(2n+1)$ -polynomial growth. This is the Heisenberg counterpart of a result of Guy David from [ Dav91 ]. [ABSTRACT FROM AUTHOR]

Answering a question of Hrušák, we show that every analytic tall ideal on \omega contains an F_\sigma tall ideal. We also give an example of an F_\sigma tall ideal without a Borel selector. [ABSTRACT FROM AUTHOR]

We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well-known examples from combinatorial group theory, combined with the Baire category theorem, we obtain a plethora of results demonstrating that several phenomena in group theory are generic. In effect, we provide a new topological framework for the analysis of various well known problems in group theory. We also provide a connection between genericity in these spaces, the word problem for finitely generated groups and model-theoretic forcing. Using these connections, we investigate a natural question raised by Osin: when does a certain space of enumerated groups contain a comeager isomorphism class? We obtain a sufficient condition that allows us to answer Osin's question in the negative for the space of all enumerated groups and the space of left orderable enumerated groups. We document several open questions in connection with these considerations. [ABSTRACT FROM AUTHOR]

We show that the algebra of cylinder functions in the Wasserstein Sobolev space H 1 , q (P p (X , d) , W p , d , m) generated by a finite and positive Borel measure m on the (p , d) -Wasserstein space (P p (X , d) , W p , d) on a complete and separable metric space (X , d) is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space B , then the Wasserstein Sobolev space is reflexive (resp. uniformly convex) if B is reflexive (resp. if the dual of B is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson's type inequalities in the Wasserstein Sobolev space. [ABSTRACT FROM AUTHOR]

We characterize the completely determined Borel subsets of HYP as exactly the Δ 1 (L ω 1 c k ) subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalbán, Solomon and the third author. [ABSTRACT FROM AUTHOR]

We prove the dynamic programming principle (DPP) in a class of problems where an agent controls a d-dimensional diffusive dynamics via both classical and singular controls and, moreover, is able to terminate the optimisation at a time of her choosing, prior to a given maturity. The time-horizon of the problem is random and it is the smallest between a fixed terminal time and the first exit time of the state dynamics from a Borel set. We consider both the cases in which the total available fuel for the singular control is either bounded or unbounded. We build upon existing proofs of DPP and extend results available in the traditional literature on singular control (Haussmann and Suo in SIAM J Control Optim 33(3):916–936, 1995; SIAM J Control Optim 33(3):937–959, 1995) by relaxing some key assumptions and including the discretionary stopping feature. We also connect with more general versions of the DPP (e.g., Bouchard and Touzi in SIAM J Control Optim 49(3):948–962, 2011) by showing in detail how our class of problems meets the abstract requirements therein. [ABSTRACT FROM AUTHOR]

The paper extended the concepts of expansive measures and N-expansive measures to group actions on compact metric spaces and proved that a group action is countably-expansive if and only if it is measure-expansive. It was shown that a group action of a compact metric space is N-expansive if and only if every Borel probability measure is N-expansive. These represented a group action version of the existing results. If the convex combination of Borel measures μ1,…,μn is 1-expansive, then μi is 1-expansive for every i=1,…,n. Finally the 1-expansive measures for continuous action was characterized as the convex sum of finitely many Dirac measures supported on expansive points. [ABSTRACT FROM AUTHOR]

It is shown that ifW OEa; b? ! S2 is C3 with det. 0; ¤ 0, and if A R3 is a Borel set, then dim in?2; dim A; dimA=2C3=4° for a.e. 2 OEa; b?, where denotes projection onto the orthogonal complement of and "dim" refers to Hausdorff dimension. This partially resolves a conjecture of Fässler and Orponen in the range 1 < dim A 3=2, which was previously known only for non-great circles. For 3=2 < dimA < 5=2, this improves the known lower bound for this problem. [ABSTRACT FROM AUTHOR]

Consider the following time-dependent stable-like operator with drift: L t φ (x) = ∫ R d [ φ (x + z) − φ (x) − z (α) ⋅ ∇ φ (x) ] σ (t , x , z) ν α (d z) + b (t , x) ⋅ ∇ φ (x) , where d ⩾ 1 , ν α is an α -stable type Lévy measure with α ∈ (0 , 1 ] and z (α) = 1 α = 1 1 | z | ⩽ 1 z , σ is a real-valued Borel function on R + × R d × R d and b is an R d -valued Borel function on R + × R d. By using the Littlewood-Paley theory, we establish the well-posedness for the martingale problem associated with L t under the sharp balance condition α + β ⩾ 1 , where β is the Hölder index of b with respect to x. Moreover, we also study a class of stochastic differential equations driven by Markov processes with generators of the form L t. We prove the pathwise uniqueness of strong solutions for such equations when the coefficients are in certain Besov spaces. [ABSTRACT FROM AUTHOR]

We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal δ -separated sets in metric and pseudometric spaces from the point of view the Axiom of Choice and its weaker forms. [ABSTRACT FROM AUTHOR]

A σ -ideal I on a Polish group (X , +) has the Smital Property if for every dense set D and a Borel I -positive set B the algebraic sum D + B is a complement of a set from I . We consider several variants of this property and study their connections with the countable chain condition, maximality and how well they are preserved via Fubini products. In particular we show that there are c many maximal invariant σ -ideals with Borel bases on the Cantor space 2 ω . [ABSTRACT FROM AUTHOR]