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To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of R d . For restrictions to the Euclidean ball in odd dimensions, to the rotation group SO (3) , and to the Grassmannian manifold G 2 , 4 , we compute the kernels' Fourier coefficients and determine their asymptotics. The L 2 -discrepancy is then expressed in the Fourier domain that enables efficient numerical minimization based on the nonequispaced fast Fourier transform. For SO (3) , the nonequispaced fast Fourier transform is publicly available, and, for G 2 , 4 , the transform is derived here. We also provide numerical experiments for SO (3) and G 2 , 4 . [ABSTRACT FROM AUTHOR]

Let X= \{X(t), t \in \mathbb {R}^N\} be a centered Gaussian random field with values in \mathbb {R}^d satisfying certain conditions and let F \subset \mathbb {R}^d be a Borel set. In our main theorem, we provide a sufficient condition for F to be polar for X, i.e. \mathbb P\big (X(t) \in F \text { for some } t \in \mathbb {R}^N\big) = 0, which improves significantly the main result in Dalang et al. [Ann. Probab. 45 (2017), pp. 4700–4751], where the case of F being a singleton was considered. We provide a variety of examples of Gaussian random field for which our result is applicable. Moreover, by using our main theorem, we solve a problem on the existence of collisions of the eigenvalues of random matrices with Gaussian random field entries that was left open in Jaramillo and Nualart [Random Matrices Theory Appl. 9 (2020), p. 26] and Song et al. [J. Math. Anal. Appl. 502 (2021), p. 22]. [ABSTRACT FROM AUTHOR]

Given a Chevalley group \mathcal {G} of classical type and a Borel subgroup \mathcal {B} \subseteq \mathcal {G}, we compute the \Sigma-invariants of the S-arithmetic groups \mathcal {B}(\mathbb {Z}[1/N]), where N is a product of large enough primes. To this end, we let \mathcal {B}(\mathbb {Z}[1/N]) act on a Euclidean building X that is given by the product of Bruhat–Tits buildings X_p associated to \mathcal {G}, where p is a prime dividing N. In the course of the proof we introduce necessary and sufficient conditions for convex functions on CAT(0)-spaces to be continuous. We apply these conditions to associate to each simplex at infinity \tau \subset \partial _\infty X its so-called parabolic building X^{\tau } and to study it from a geometric point of view. Moreover, we introduce new techniques in combinatorial Morse theory, which enable us to take advantage of the concept of essential n-connectivity rather than actual n-connectivity. Most of our building theoretic results are proven in the general framework of spherical and Euclidean buildings. For example, we prove that the complex opposite each chamber in a spherical building \Delta contains an apartment, provided \Delta is thick enough and Aut(\Delta) acts chamber transitively on \Delta. [ABSTRACT FROM AUTHOR]

We construct optimal flat functions in Carleman–Roumieu ultraholomorphic classes associated to general strongly nonquasianalytic weight sequences, and defined on sectors of suitably restricted opening. A general procedure is presented in order to obtain linear continuous extension operators, right inverses of the Borel map, for the case of regular weight sequences in the sense of Dyn'kin. Finally, we discuss some examples (including the well-known q-Gevrey case) where such optimal flat functions can be obtained in a more explicit way. [ABSTRACT FROM AUTHOR]

This work studies the convergence and finite sample approximations of entropic regularized Wasserstein distances in the Hilbert space setting. Our first main result is that for Gaussian measures on an infinite-dimensional Hilbert space, convergence in the 2-Sinkhorn divergence is strictly weaker than convergence in the exact 2-Wasserstein distance. Specifically, a sequence of centered Gaussian measures converges in the 2-Sinkhorn divergence if the corresponding covariance operators converge in the Hilbert–Schmidt norm. This is in contrast to the previous known result that a sequence of centered Gaussian measures converges in the exact 2-Wasserstein distance if and only if the covariance operators converge in the trace class norm. In the reproducing kernel Hilbert space (RKHS) setting, the kernel Gaussian–Sinkhorn divergence, which is the Sinkhorn divergence between Gaussian measures defined on an RKHS, defines a semi-metric on the set of Borel probability measures on a Polish space, given a characteristic kernel on that space. With the Hilbert–Schmidt norm convergence, we obtain dimension-independent convergence rates for finite sample approximations of the kernel Gaussian–Sinkhorn divergence, of the same order as the Maximum Mean Discrepancy. These convergence rates apply in particular to Sinkhorn divergence between Gaussian measures on Euclidean and infinite-dimensional Hilbert spaces. The sample complexity for the 2-Wasserstein distance between Gaussian measures on Euclidean space, while dimension-dependent, is exponentially faster than the worst case scenario in the literature. [ABSTRACT FROM AUTHOR]

We provide a method for constructing (possibly non-trivial) measures on non-locally compact Polish subspaces of infinite-dimensional separable Banach spaces which, under suitable assumptions, are minimizers of causal variational principles in the non-locally compact setting. Moreover, for non-trivial minimizers the corresponding Euler–Lagrange equations are derived. The method is to exhaust the underlying Banach space by finite-dimensional subspaces and to prove existence of minimizers of the causal variational principle restricted to these finite-dimensional subsets of the Polish space under suitable assumptions on the Lagrangian. This gives rise to a corresponding sequence of minimizers. Restricting the resulting sequence to countably many compact subsets of the Polish space, by considering the resulting diagonal sequence, we are able to construct a regular measure on the Borel algebra over the whole topological space. For continuous Lagrangians of bounded range, it can be shown that, under suitable assumptions, the obtained measure is a (possibly non-trivial) minimizer under variations of compact support. Under additional assumptions, we prove that the constructed measure is a minimizer under variations of finite volume and solves the corresponding Euler–Lagrange equations. Afterwards, we extend our results to continuous Lagrangians vanishing in entropy. Finally, assuming that the obtained measure is locally finite, topological properties of spacetime are worked out and a connection to dimension theory is established. [ABSTRACT FROM AUTHOR]

We generalize the notion of analytic/Borel equivalence relations, orbit equivalence relations, and Borel reductions between them to their continuous and quantitative counterparts: analytic/Borel pseudometrics, orbit pseudometrics, and Borel reductions between them. We motivate these concepts on examples and we set some basic general theory. We illustrate the new notion of reduction by showing that the Gromov–Hausdorff distance maintains the same complexity if it is defined on the class of all Polish metric spaces, spaces bounded from below, from above, and from both below and above. Then we show that E 1 is not reducible to equivalences induced by orbit pseudometrics, generalizing the seminal result of Kechris and Louveau. We answer in negative a question of Ben Yaacov, Doucha, Nies, and Tsankov on whether balls in the Gromov–Hausdorff and Kadets distances are Borel. In appendix, we provide new methods using games showing that the distance-zero classes in certain pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and Tsankov. There is a complementary paper of the authors where reductions between the most common pseudometrics from functional analysis and metric geometry are provided. [ABSTRACT FROM AUTHOR]

We introduce and investigate a notion of multivalued λ -dissipative probability vector field (MPVF) in the Wasserstein space P 2 (X) of Borel probability measures on a Hilbert space X . Taking inspiration from the theories of dissipative operators in Hilbert spaces and of Wasserstein gradient flows for geodesically convex functionals, we study local and global well posedness of evolution equations driven by dissipative MPVFs. Our approach is based on a measure-theoretic version of the Explicit Euler scheme, for which we prove novel convergence results with optimal error estimates under an abstract stability condition, which do not rely on compactness arguments and also hold when X has infinite dimension. We characterize the limit solutions by a suitable Evolution Variational Inequality (EVI), inspired by the Bénilan notion of integral solutions to dissipative evolutions in Banach spaces. Existence, uniqueness and stability of EVI solutions are then obtained under quite general assumptions, leading to the generation of a semigroup of nonlinear contractions. [ABSTRACT FROM AUTHOR]

Risk forms are real functionals of two arguments: a bounded measurable function on a Polish space and a probability measure on that space. They are convenient mathematical structures adapting the coherent risk measures to the situation of a variable reference probability measure. We introduce a new class of risk forms called mini-batch forms. We construct them by using a random empirical probability measure as the second argument and by post-composition with the expected value operator. We prove that coherent and law invariant risk forms generate mini-batch risk forms which are well defined on the space of integrable random variables, and we derive their dual representation. We demonstrate how unbiased stochastic subgradients of such risk forms can be constructed. Then, we consider pre-compositions of mini-batch risk forms with nonsmooth and nonconvex functions, which are differentiable in a generalized way, and we derive generalized subgradients and unbiased stochastic subgradients of such compositions. Finally, we study the dependence of risk forms and mini-batch risk forms on perturbation of the probability measure and establish quantitative stability in terms of optimal transport metrics. We obtain finite-sample expected error estimates for mini-batch risk forms involving functions on a finite-dimensional space. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to introduce and study some new subclasses of p-valent functions by making use of a linear q-differential Borel operator.We also deduce some properties, such as inclusion relationships of the newly introduced classes and the integral operator J μ , p . [ABSTRACT FROM AUTHOR]

We study zero-sum stochastic games for controlled discrete time Markov chains with risk-sensitive average cost criterion with countable/compact state space and Borel action spaces. The payoff function is nonnegative and possibly unbounded for countable state space case and for compact state space case it is a real-valued and bounded function. For countable state space case, under a certain Lyapunov type stability assumption on the dynamics we establish the existence of the value and a saddle point equilibrium. For compact state space case we establish these results without any Lyapunov type stability assumptions. Using the stochastic representation of the principal eigenfunction of the associated optimality equation, we completely characterize all possible saddle point strategies in the class of stationary Markov strategies. Also, we present and analyze an illustrative example. [ABSTRACT FROM AUTHOR]

This paper extends the previous results existing in the literature to a class of generalized Riccati equations acting on ordered Banach spaces of measurable functions defined on standard Borel spaces and taking values in infinite dimensional spaces. We investigate the global solvability of these Riccati equations and we establish necessary and sufficient conditions for the existence of some of their global solutions such as the maximal and the stabilizing solutions. [ABSTRACT FROM AUTHOR]

We consider the isometry group of the infinite-dimensional separable hyperbolic space with its Polish topology. This topology is given by pointwise convergence. For non-locally compact Polish groups, some striking phenomena like automatic continuity or extreme amenability may happen. Our leading idea is to compare this topological group with usual Lie groups on one side and with non-Archimedean infinite-dimensional groups like S∞, the group of all permutations of a countable set on the other side. Our main results are • automatic continuity (any homomorphism to a separable group is continuous); • minimality of the Polish topology; • identification of its universal Furstenberg boundary as the closed unit ball of a separable Hilbert space with its weak topology; • identification of its universal minimal flow as the completion of some suspension of the action of the additive group of the reals R on its universal minimal flow. All along the text, we lead a parallel study with the sibling group Isom(H), where H is a separable Hilbert space. [ABSTRACT FROM AUTHOR]

In this work, we focus on the centered Hausdorff measure, the packing measure, and the Hewitt–Stromberg measure that determines the modified lower box dimension Moran fractal sets. The equivalence of these measures for a class of Moran is shown by having a strong separation condition. We give a sufficient condition for the equality of the Hewitt–Stromberg dimension, Hausdorff dimension, and packing dimensions. As an application, we obtain some relevant conclusions about the Hewitt–Stromberg measures and dimensions of the image measure of a τ -invariant ergodic Borel probability measures. Moreover, we give some statistical interpretation to dimensions and corresponding geometrical measures. [ABSTRACT FROM AUTHOR]

We show that the "stabilization" of any countable ergodic probability measure preserving (p.m.p.) equivalence relation which is not Schmidt, i.e. admits no central sequences in its full group, always gives rise to a stable equivalence relation with a unique stable decomposition, providing the first non-strongly ergodic such examples. In the proof, we moreover establish a new local characterization of the Schmidt property. We also prove some new structural results for product equivalence relations and orbit equivalence relations of diagonal product actions. [ABSTRACT FROM AUTHOR]

We present a novel analogue for finite exchangeable sequences of the de Finetti, Hewitt and Savage theorem and investigate its implications for multi-marginal optimal transport (MMOT) and Bayesian statistics. If (Z 1 , ... , Z N) is a finitely exchangeable sequence of N random variables taking values in some Polish space X, we show that the law μ k of the first k components has a representation of the form μ k = ∫ P 1 N (X) F N , k (λ) d α (λ) for some probability measure α on the set of 1 N -quantized probability measures on X and certain universal polynomials F N , k . The latter consist of a leading term N k - 1 / ∏ j = 1 k - 1 (N - j) λ ⊗ k and a finite, exponentially decaying series of correlated corrections of order N - j ( j = 1 , ... , k ). The F N , k (λ) are precisely the extremal such laws, expressed via an explicit polynomial formula in terms of their one-point marginals λ . Applications include novel approximations of MMOT via polynomial convexification and the identification of the remainder which is estimated in the celebrated error bound of Diaconis and Freedman (Ann Probab 8(4):745–764, 1980) between finite and infinite exchangeable laws. [ABSTRACT FROM AUTHOR]

Let \Gamma be a sofic group, \Sigma be a sofic approximation sequence of \Gamma and X be a \Gamma-subshift with non-negative sofic topological entropy with respect to \Sigma. Further assume that X is a shift of finite type, or more generally, that X satisfies the topological Markov property. We show that for any sufficiently regular potential f \colon X \to \mathbb {R}, any translation-invariant Borel probability measure on X which maximizes the measure-theoretic sofic pressure of f with respect to \Sigma is a Gibbs state with respect to f. This extends a classical theorem of Lanford and Ruelle, as well as previous generalizations of Moulin Ollagnier, Pinchon, Tempelman and others, to the case where the group is sofic. As applications of our main result we present a criterion for uniqueness of an equilibrium measure, as well as sufficient conditions for having that the equilibrium states do not depend upon the chosen sofic approximation sequence. We also prove that for any group-shift over a sofic group, the Haar measure is the unique measure of maximal sofic entropy for every sofic approximation sequence, as long as the homoclinic group is dense. On the expository side, we present a short proof of Chung's variational principle for sofic topological pressure. [ABSTRACT FROM AUTHOR]

We prove that the lower density operator associated with the Baire category density points in the real line has Borel values of class Π 3 0 which is analogous to the measure case. We also introduce the notion of the Baire category density point of a subset with the Baire property of the Cantor space, and we prove that it generates a lower density operator with Borel values of class Π 3 0 . [ABSTRACT FROM AUTHOR]

A Picard–Borel algebra is a commutative unital complex algebra A$A$ such that every family of pairwise linearly independent invertible elements of A$A$ is linearly independent, and a Picard–Borel ideal I$I$ in a commutative complex unital algebra A$A$ is an ideal of I$I$ of A$A$ such that the quotient algebra A/I$A/I$ is a Picard–Borel algebra. In a preliminary paper, the author proved that every commutative unital Fréchet algebra which is a Picard–Borel algebra is an integral domain. The main result of the present paper is that all Picard–Borel ideals in commutative unital Fréchet algebras are prime. This result seems to be relevant for Michael's problem, since dense Picard–Borel ideals could play a role in the construction of discontinuous characters on some commutative unital Fréchet algebras. [ABSTRACT FROM AUTHOR]

In this paper we consider the following moment problem: find a positive Borel measure μ on C subject to conditions ∫ zndμ = sn, n ∈ Z+, where sn are prescribed complex numbers (moments). This moment problem may be viewed (informally) as an extension of the Stieltjes and Hamburger moment problems to the complex plane. A criterion for the moment problem for the existence of a compactly supported solution is given. In particular, such moment problems appear naturally in the domain of complex Jacobi matrices. For every bounded complex Jacobi matrix its associated functional S has the following integral representation: S(p) = ∫ C p(z)dμ, with a positive Borel measure μ in the complex plane. An interrelation of the associated to the complex Jacobi matrix operator A0, acting in l² on finitely supported vectors, and the multiplication by z operator in L² μ is discussed. [ABSTRACT FROM AUTHOR]

It is well known that all Borel subgroups of a linear algebraic group are conjugate. Berest, Eshmatov, and Eshmatov have shown that this result also holds for the automorphism group Aut ⁢ (픸 2) of the affine plane. In this paper, we describe all Borel subgroups of the complex Cremona group Bir ⁢ (ℙ 2) up to conjugation, proving in particular that they are not necessarily conjugate. In principle, this fact answers a question of Popov. More precisely, we prove that Bir ⁢ (ℙ 2) admits Borel subgroups of any rank r ∈ { 0 , 1 , 2 } and that all Borel subgroups of rank r ∈ { 1 , 2 } are conjugate. In rank 0, there is a one-to-one correspondence between conjugacy classes of Borel subgroups of rank 0 and hyperelliptic curves of genus ℊ ≥ 1 . Hence, the conjugacy class of a rank 0 Borel subgroup admits two invariants: a discrete one, the genus ℊ , and a continuous one, corresponding to the coarse moduli space of hyperelliptic curves of genus ℊ . This moduli space is of dimension 2 ⁢ ℊ - 1 . [ABSTRACT FROM AUTHOR]

If X is a Hilbert space, one can consider the space cabv (X) of X valued measures defined on the Borel sets of a compact metric space, having a bounded variation. On this vector measures space was already introduced a Monge–Kantorovich type norm. Our first goal was to introduce a Monge–Kantorovich type norm on cabv (X) , where X is a Banach space, but not necessarily a Hilbert space. Thus, we introduced here the Monge–Kantorovich type norm on cabv L q ([ 0 , 1 ]) , (1 < q < ∞) . We obtained some properties of this norm and provided some examples. Then, we used the Monge–Kantorovich norm on cabv K n (K being R or C) to obtain convergence properties for sequences of fractal sets and fractal vector measures associated to a sequence of iterated function systems. [ABSTRACT FROM AUTHOR]

We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group Γ we introduce the Γ -jump. We study the elementary properties of the Γ -jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups Γ , the Γ -jump is proper in the sense that for any Borel equivalence relation E the Γ -jump of E is strictly higher than E in the Borel reducibility hierarchy. On the other hand, there are examples of groups Γ for which the Γ -jump is not proper. To establish properness, we produce an analysis of Borel equivalence relations induced by continuous actions of the automorphism group of what we denote the full Γ -tree, and relate these to iterates of the Γ -jump. We also produce several new examples of equivalence relations that arise from applying the Γ -jump to classically studied equivalence relations and derive generic ergodicity results related to these. We apply our results to show that the complexity of the isomorphism problem for countable scattered linear orders properly increases with the rank. [ABSTRACT FROM AUTHOR]

Let \gamma \in (0,2), let h be the planar Gaussian free field, and let D_h be the associated \gamma-Liouville quantum gravity (LQG) metric. We prove that for any random Borel set X \subset \mathbb {C} which is independent from h, the Hausdorff dimensions of X with respect to the Euclidean metric and with respect to the \gamma-LQG metric D_h are a.s. related by the (geometric) KPZ formula. As a corollary, we deduce that the Hausdorff dimension of the continuum \gamma-LQG metric is equal to the exponent d_\gamma > 2 studied by Ding and Gwynne (2018), which describes distances in discrete approximations of \gamma-LQG such as random planar maps. We also derive "worst-case" bounds relating the Euclidean and \gamma-LQG dimensions of X when X and h are not necessarily independent, which answers a question posed by Aru (2015). Using these bounds, we obtain an upper bound for the Euclidean Hausdorff dimension of a \gamma-LQG geodesic which equals 1.312\dots when \gamma = \sqrt {8/3}; and an upper bound of 1.9428\dots for the Euclidean Hausdorff dimension of a connected component of the boundary of a \sqrt {8/3}-LQG metric ball. We use the axiomatic definition of the \gamma-LQG metric, so the paper can be understood by readers with minimal background knowledge beyond a basic level of familiarity with the Gaussian free field. [ABSTRACT FROM AUTHOR]

Motivated by Kloeckner's result on the isometry group of the quadratic Wasserstein space W2(Rn)$\mathcal {W}_2(\mathbb {R}^n)$, we describe the isometry group Isom(Wp(E))$\mathrm{Isom}(\mathcal {W}_p(E))$ for all parameters 01$p>1$, by solving Kloeckner's problem affirmatively on the existence of mass‐splitting isometries. As a byproduct of our methods, we also obtain the isometric rigidity of Wp(X)$\mathcal {W}_p(X)$ for all complete and separable ultrametric spaces X$X$ and parameters 0

In this paper, we introduce a map Φ , which we call zonoid map, from the space of all non-negative, finite Borel measures on R n with finite first moment to the space of zonoids of R n . This map, connecting Borel measure theory with zonoids theory, allows to slightly generalize the Gini volume introduced, in the context of Industrial Economics, by Dosi (J Ind Econ 4:875–907, 2016). This volume, based on the geometric notion of zonoid, is introduced as a measure of heterogeneity among firms in an industry and it turned out to be a quite interesting index as it is a multidimensional generalization of the well-known and broadly used Gini index. By exploiting the mathematical context offered by our definition, we prove the continuity of the map Φ which, in turn, allows to prove the validity of a SLLN-type theorem for our generalized Gini index and, hence, for the Gini volume. Both results, the continuity of Φ and the SLLN theorem, are particularly useful when dealing with a huge amount of multidimensional data. [ABSTRACT FROM AUTHOR]

Let n ∈ N and for every w > 0 let Kw: (0;∞)n - R be Borel measurable kernels: Under suitable assumptions, the multivariate Mellin con-volution operator is defined by... In the paper we find the necessary and sufficient conditions for the convergence of the multivariate Mellin convolution operators. In the case of differentiable or twice differentiable functions we give the asymptotic evaluations for the multi-variate Mellin convolution operators. We apply these general results for specific multivariate Mellin-Gauss-Weierstrass, Mellin-Poisson-Cauchy or Hadamard type convolution operators. [ABSTRACT FROM AUTHOR]

It is still an open problem to determine the conjugacy classes of Borel subalgebras of non-classical type Lie algebras. In this paper, we prove that there are at least 2 conjugacy classes of Borel subalgebras as well as maximal triangulable subalgebras of restricted Cartan type Lie algebras of type W, S and H. We are particularly interested in maximal triangulable subalgebras of W (n) under some conditions which is called B -subalgebras (Definition 3.1). We classify the conjugacy classes of B -subalgebras for W (n) and determine their representatives. This paper and its sequel [Z. Lin, K. Ou and B. Shu, Geometric Setting of Jacobson–Witt Algebras, preprint] attempt to establish both algebraic and geometric setting for geometric representation theory of W (n). [ABSTRACT FROM AUTHOR]

We construct the s -Gauss probability space by introducing the s -Gaussian density function in R n for s ≥ 0 , a generalization of the classic Gaussian density function. Based on the s -Gaussian density function, we propose the (s , k) -Ehrhard symmetrization which is an extension of the traditional Ehrhard symmetrization for sets in R n. In particular, we establish the s -Gaussian isoperimetric inequality with respect to s -Gaussian measure in R 2. Furthermore, we propose and prove the s -Ehrhard-Borell inequalities for s > 0 when one of the two sets is a Borel set whilst the other being a convex set as well as the case when two sets are convex in R 1 with different methods. [ABSTRACT FROM AUTHOR]

We prove that the arc-wise connection relation in a G δ subset of the plane is Borel. [ABSTRACT FROM AUTHOR]

The topological views of a measure space provide deep insights. In this paper, the sigma-set algebraic structure is extended in a Hausdorff topological space based on the locally compactable neighborhood systems without considering strictly (metrized) Borel variety. The null extension gives rise to a quasi sigma-semiring based on sigma-neighborhoods, which are rectifiable in view of Dieudonné measure in n-space. The concepts of symmetric signed measure, uniformly pushforward measure, and its interval-valued Lebesgue variety within a topological measure space are introduced. The symmetric signed measure preserves the total ordering on the real line; however, the collapse of symmetry admits Dieudonné measure within the topological space. The locally constant measures in compact supports in sigma-neighborhood systems are invariant under topological deformation retraction in a simply connected space where the sequence of deformation retractions induces a strongly convergent sequence of measures. Moreover, the extended sigma-structures in an automorphic and isomorphic topological space preserve the properties of uniformly pushforward measure. The Haar-measurable group algebraic structures equivalent to additive integer groups arise under the locally constant and signed measures as long as the topological space is non-compact and the null-extended sigma-neighborhood system admits compact groups. The comparative analyses of the proposed concepts with respect to existing results are outlined. [ABSTRACT FROM AUTHOR]

Descriptive set theory and computability theory are closely-related fields of logic; both are oriented around a notion of descriptive complexity. However, the two fields typically consider objects of very different sizes; computability theory is principally concerned with subsets of the naturals, while descriptive set theory is interested primarily in subsets of the reals. In this paper, we apply a generalization of computability theory, admissible recursion theory, to consider the relative complexity of notions that are of interest in descriptive set theory. In particular, we examine the perfect set property, determinacy, the Baire property, and Lebesgue measurability. We demonstrate that there is a separation of descriptive complexity between the perfect set property and determinacy for analytic sets of reals; we also show that the Baire property and Lebesgue measurability are both equivalent in complexity to the property of simply being a Borel set, for $\boldsymbol {\Sigma ^{1}_{2}}$ sets of reals. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the complexity of random dynamical systems. Assume that ϕ is a continuous random dynamical system on Polish space X over an ergodic Polish system (Ω , F , P , θ). Let K be a ϕ -invariant random set with compact ω -section K (ω) such that ϕ (ω) (K (ω)) = K (θ ω). We proved that if h top (ϕ , K) > 0 , then there exists chaotic phenomenon (strong second type of distributional chaos) in this random dynamical system. [ABSTRACT FROM AUTHOR]

We show that the bounded Borel class of any dense representation |$\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$| is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group |$G$|⁠. When |$n=2$|⁠ , the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation |$\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$| is uniformly separated in semi-norm from any other representation |$\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$| for which there is a subgroup |$H\le G$| on which |$\rho $| is still dense but |$\rho ^{\prime}$| is discrete or indiscrete but stabilizes a point, line, or plane in |${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$|⁠. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of |${\operatorname{PSL}}_2{\mathbb{R}}$|⁠ , whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties. [ABSTRACT FROM AUTHOR]

We prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler–Lagrange equations are derived. The method is to first prove the existence of minimizers of the causal variational principle restricted to compact subsets for a lower semi-continuous Lagrangian. Exhausting the underlying topological space by compact subsets and rescaling the corresponding minimizers, we obtain a sequence which converges vaguely to a regular Borel measure of possibly infinite total volume. It is shown that, for continuous Lagrangians of compact range, this measure solves the Euler–Lagrange equations. Furthermore, we prove that the constructed measure is a minimizer under variations of compact support. Under additional assumptions, it is proven that this measure is a minimizer under variations of finite volume. We finally extend our results to continuous Lagrangians decaying in entropy. [ABSTRACT FROM AUTHOR]

Let Ω be an open subset of Rn. Let f be a Borel measurable function from R to R. We prove necessary and sufficient conditions on f in order that the composite function Tf OEg? D f ı g belongs to a generalized Morrey space Mw p Ω whenever g belongs to Mw p .Ω. Then we prove necessary conditions and sufficient conditions on f in order that the composition operator Tf OE be continuous, uniformly continuous, Hölder continuous and Lipschitz continuous in Mw p .Ω. We also consider its 'vanishing' generalized Morrey subspace M w;0 p .Ω and prove the related results for the composition operator as operator acting from M w;0 p .Ω to Mw p .Ω and also between the spaces M w;0 p.Ω. For the uniform, Hölder and Lipschitz continuity, we also have conditions that are both necessary and sufficient. We also have both necessary and sufficient conditions for the continuity under certain additional natural assumptions. We also consider the most commonly used Morrey classes that are related to power-type weights in the context of a discussion of some of the conditions that we impose on the weights. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of the soft topological group as a general topological group of soft elements and study their properties. Some interesting results and relation between soft topological group and ordinary topological group of soft elements are studied. Also, soft Borel set and soft Borel measure have been depicted and the relationship between the soft Borel measure and the corresponding Borel measure has been drawn. [ABSTRACT FROM AUTHOR]

We embed the Aubry set and Mather set in the Tonelli Hamiltonian system to the contact Hamiltonian system. We find the embedded Aubry set is the set of non-wandering points of the contact Hamiltonian system and the Mather set is the support of set of the invariant Borel probability measures for the contact Euler—Lagrange flow. From this viewpoint, we can conclude that Aubry set and Mather set are symplectic invariants. [ABSTRACT FROM AUTHOR]

This paper develops a technique to represent linear functionals on spaces of homogeneous polynomials between Banach spaces. This technique applies to cases that are not covered by the classical polynomial Borel transform. We provide applications to represent linear functionals on spaces of approximable, compact, hyper-nuclear and hyper- σ (p) -nuclear polynomials. An unexpected difference between the polynomial and multilinear theories is disclosed. [ABSTRACT FROM AUTHOR]

For a continuous self-map f on a compact metric space X, we provide two simple examples: the first confirms that shadowing of (X, f) is not inherited by (M (X) , f ^) in general, and the other satisfies that both (X, f) and (M (X) , f ^) have no Li–Yorke pair, where M (X) be the space of all Borel probability measures on X. Then we prove that chain transitivity of (X, f) implies chain mixing of (M (X) , f ^) , and provide an example to deny the converse. For a non-autonomous system (X , f 0 , ∞) , we prove that weak mixing of (M (X) , f ^ 0 , ∞) implies that of (X , f 0 , ∞) , and give an example to deny the converse, where f 0 , ∞ = { f n } n = 0 ∞ is a sequence of continuous self-maps on X. We also prove that if f n is surjective for all n ≥ 0 , then chain mixing of (M (X) , f ^ 0 , ∞) always holds true, and shadowing of (M (X) , f ^ 0 , ∞) implies mixing of (X , f 0 , ∞) . If X = I is an interval, we obtain a sharp condition such that transitivity is equivalent between (I, f) and (M (I) , f ^) . Although (M (I) , f ^) has infinite topological entropy for any transitive system (I, f), we give an example such that (I , f 0 , ∞) is transitive but (M (I) , f ^ 0 , ∞) has zero topological entropy. [ABSTRACT FROM AUTHOR]

Anna Melnikov provided a parametrization of Borel orbits in the affine variety of square-zero n × n matrices by the set of involutions in the symmetric group. A related combinatorics leads to a construction a Bott-Samelson type resolution of the orbit closures. This allows to compute cohomological and K-theoretic invariants of the orbits: fundamental classes, Chern-Schwartz-MacPherson classes and motivic Chern classes in torus-equivariant theories. The formulas are given in terms of Demazure-Lusztig operations. The case of square-zero upper-triangular matrices is rich enough to include information about cohomological and K-theoretic classes of the double Borel orbits in Hom (C k , C m) for k + m = n. We recall the relation with double Schubert polynomials and show analogous interpretation of Rimányi-Tarasov-Varchenko trigonometric weight function. [ABSTRACT FROM AUTHOR]

In this brief paper, we studied the residual continuity of global attractors A λ in varying parameters λ ∈ Λ with Λ a bounded Borel set in R d . We first reviewed the well-known residual continuity result of global attractors and then showed that this residual continuity is equivalent to the dense continuity. Then, we proved an analogue continuity result in measure sense that, under certain conditions, the set-valued map λ ↦ A λ is almost (in the Lebesgue measure sense) uniformly continuous: for any small ε > 0 there exists a closed subset C ε ⊂ Λ with Lebesgue measure m (C ε) > μ (Λ) − ε such that the set-valued map ε ↦ A ε is uniformly continuous on C ε . This, in return, indicates that the selected attractors { A λ : λ ∈ C ε } can be equi-attracting. [ABSTRACT FROM AUTHOR]

In this research work, a general theorem on a new application of summation of Jacobi series by B-method or Borel method has been established. The main theorem includes as special cases, a number of well known results. Several new results can also be deduced from it. The result is an extension of classical results of [1]. [ABSTRACT FROM AUTHOR]

A set E in a Banach space X is compactivorous if for every compact set K in X there is a nonempty, (relatively) open subset of K which can be translated into E. In a separable Banach space, this is a sufficient condition which guarantees the Haar nonnegligibility of Borel subsets. We give some characterisations of this property in both separable and nonseparable Banach spaces and prove an extension of the main theorem to countable products of locally compact Polish groups. [ABSTRACT FROM AUTHOR]

This paper is a natural continuation of Krylov (2020), where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in L d + 1 (R d + 1). Here we study some properties of these processes such as higher summability of Green's functions, a maximum principle for related transport equations with irregular coefficients, boundedness of resolvent operators in Lebesgue spaces, establish Itô's formula, and so on. [ABSTRACT FROM AUTHOR]

In this paper, for one thing, we introduce positively expansive system and positively expansive measure for nonautonomous dynamical systems. And we study the relation between positively expansive measure and the maximal cardinalities of separated set. We obtain the following main results. Firstly, we prove that every nonautonomous dynamical system with positively expansive measure implies there are e > 0 and a sequence of (n, e)-separated sets whose cardinalities go to infinite as n → ∞ . Secondly, we point out that every positively expansive system also implies the above properties. Thirdly, for each nonautonomous dynamical system that satisfies the above properties, there is a Borel probability measure such that the measure of the set of stable points of nonautonomous dynamical systems is zero. For another, we introduce positively countably expansive and positively measure expansive for nonautonomous dynamical systems. In addition, we prove that a nonautonomous dynamical system is positively measure expansive if and only if it is positively countably expansive. Finally, we introduce stable points and equicontinuous for a nonautonomous dynamical system and study some relevant properties. This paper primarily generalizes the main results obtained by Morales to nonautonomous dynamical systems. [ABSTRACT FROM AUTHOR]

Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel's conjecture, and simply connected manifolds where rigidity is a consequence of Freedman's classification results. [ABSTRACT FROM AUTHOR]

For 0 < p < ∞ , we completely characterize the Schatten p -class embedding and fractional embedding of the Bergman-type space A α 2 on the unit ball of C m. The main results of this paper are twofold: (1) For α , β ∈ R , we show that the necessary and sufficient condition for the embedding operator I : A α 2 → A β 2 is in the Schatten p -class is that p (β − α) − 2 m > 0. (2) For a positive Borel measure μ on B m and a real number τ , we prove that the fractional embedding operator I α + τ τ : A α 2 → L 2 (B m , d μ) is in the Schatten p -class if and only if the Berezin transform of μ , denoted by μ ˜ α α + τ , is in L p 2 (B m , d λ) , where d λ (z) = (1 − | z | 2) − m − 1 d v (z) is the Möbius invariant volume measure and d v is the Lebesgue measure on B m , or equivalently, the averaging function of μ , denoted by μ ˆ r α , is in L p 2 (B m , d λ). • For p > 0 , we completely characterize the Schatten p -class embedding and fractional embedding of the Bergman-type space on the unit ball. • The range 0 < p < 2 is an open problem left by H. T. Kaptanoğlu (see [5]). • We prove that the fractional embedding operator is in Schatten p -class iff the Berezin transform of the positive Borel measure is in L − space. [ABSTRACT FROM AUTHOR]