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Let A and B be two nonempty subsets of a normed space X, and let T : A ∪ B → A ∪ B be a cyclic (resp., noncyclic) mapping. The objective of this paper is to establish weak conditions on T that ensure its relative nonexpansiveness. The idea is to recover the results mentioned in two papers by Matkowski (Banach J. Math. Anal. 2:237–244, 2007; J. Fixed Point Theory Appl. 24:70, 2022), by replacing the nonexpansive mapping f : C → C with a cyclic (resp., noncyclic) relatively nonexpansive mapping to obtain the best proximity pair. Additionally, we provide an application to a functional equation. [ABSTRACT FROM AUTHOR]

In this paper, motivated by Reich contraction and tool of measure of noncompactness, some generalizations of Reich, Kannan, Darbo, Sadovskii, and Krasnoselskii type fixed point results are presented by considering a pair of maps A, B on a nonempty closed subset M of a Banach space X into X. The existence of a solution to the equation A x + B x = x , where A is k-set contractive and B is a generalized Reich contraction, is established. As applications, it is established that the main result of this paper can be applied to learn conditions under which a solution of a nonlinear integral equation exists. Further we explain this phenomenon with the help of a practical example to approximate such solutions by using fixed point techniques. The graphs of exact and approximate solutions are also given to attract readers for further research activities. [ABSTRACT FROM AUTHOR]

Inspired by locale theory, pointfree convex geometry was first proposed and studied by Yoshihiro Maruyama. In this paper, we shall continue to his work and investigate the related topics on pointfree convex spaces. Concretely, the following results are obtained: (1) A Hofmann–Lawson-like duality for pointfree convex spaces is established. (2) The M -injective objects in the category of S 0 -convex spaces are proved precisely to be sober convex spaces, where M is the class of strict maps of convex spaces; (3) A convex space X is sober iff there never exists a nontrivial identical embedding i : X ↪ Y such that its dualization is an isomorphism, and a convex space X is S D iff there never exists a nontrivial identical embedding k : Y ↪ X such that its dualization is an isomorphism. (4) A dual adjunction between the category CLat D of continuous lattices with continuous D-homomorphisms and the category CS D of S D -convex spaces with CP-maps is constructed, which can further induce a dual equivalence between CS D and a subcategory of CLat D ; (5) The relationship between the quotients of a continuous lattice L and the convex subspaces of cpt (L) is investigated and the collection Alg (Q (L)) of all algebraic quotients of L is proved to be an algebraic join-sub-complete lattice of Q (L) of all quotients of L, where cpt (L) denote the set of non-bottom compact elements of L. Furthermore, it is shown that Alg (Q (L)) is isomorphic to the collection Sob (P (cpt (L))) of all sober convex subspaces of cpt (L) ; (6) Several necessary and sufficient conditions for all convex subspaces of cpt (L) to be sober are presented. [ABSTRACT FROM AUTHOR]

The paper deals with minimum energy problems in the presence of external fields on a locally compact space X with respect to a function kernel κ satisfying the energy and consistency principles. For quite a general (not necessarily lower semicontinuous) external field f, we establish sufficient and/or necessary conditions for the existence of λ A , f minimizing the Gauss functional ∫ κ (x , y) d (μ ⊗ μ) (x , y) + 2 ∫ f d μ over all positive Radon measures μ with μ (X) = 1 , concentrated on quite a general (not necessarily closed or bounded) set A ⊂ X , thereby giving an answer to a question raised by M. Ohtsuka (J. Sci. Hiroshima Univ., 1961). Such results are specified for the Riesz kernels | x - y | α - n , 0 < α < n , on R n , n ⩾ 2 , and are illustrated by some examples. Furthermore, we provide various alternative characterizations of the minimizer λ A , f , and as a by-product we prove the continuity of both λ A , f and the modified Robin constant under the exhaustion of A by compact K ⊂ A . Along with the advantages of using perfect kernels, the analysis performed is substantially based on the author's recent theory of inner balayage on locally compact spaces (Potential Anal., 2022). The results obtained do hold and are new for many interesting kernels in classical and modern potential theory. [ABSTRACT FROM AUTHOR]

As the International Space Station comes to the end of a transformative era of in-space research, NASA's Commercial Low Earth Orbit (LEO) Destinations (CLD) Program aims to catalyze a new generation of platforms with co-investment from the private sector, preventing a potential gap in research performed in LEO, while building a robust LEO economy. In this paper, we provide insight into the CLD Program focusing on Orbital Reef, describing its operational and technical characteristics as well as new opportunities it may enable. Achieving about a third of the pressurized volume of the ISS with the launch of a single pressurized module and growing to support hundreds of Middeck Locker Equivalents (MLE) in passive and active payloads internally and externally, Orbital Reef will enable government, academic, and commercial institutions to continue and expand upon research and development (R&D) efforts currently performed on ISS. Additionally, it will enable nascent markets to establish their operations in space, by initiating new lines of research and technology development and the implementation of new ventures and visions. Using Blue Origin's New Glenn heavy launch system, Sierra Space's cargo and crew Dream Chaser® vehicles, and Boeing's Starliner crew vehicle, and expertise from Amazon/Amazon Supply Chain, Arizona State University, Genesis Engineering, and Redwire, Orbital Reef is being designed to address ISS-era transportation logistics challenges. Finally, this manuscript describes some of the expected challenges from the ISS-to-CLD transition, and provides guidance on how researchers in academia and industry can shape the future of commercial destinations and work performed in LEO. [ABSTRACT FROM AUTHOR]

In this paper, Jordan–Kronecker invariants are calculated for all nilpotent 6- and 7-dimensional Lie algebras. We consider the Poisson bracket family, depending on the lambda parameter on a Lie coalgebra, i.e., on the linear space dual to a Lie algebra. For some space 픤 proposed in the paper, two skew-symmetric matrices are defined for all points x on this linear space. To understand the behaviour of the matrix pencil (A − λB)(x), we consider Jordan–Kronecker invariants for this pencil and how they change with x (the latter is done for 6-dimensional Lie algebras). [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]

In this paper, we propose numerical methods for the Riesz space distributed-order advection–diffusion equation with delay in 2D. We utilize the fractional backward differential formula method of second order (FBDF2), and weighted and shifted Grünwald difference (WSGD) operators to approximate the Riesz fractional derivative and develop the finite difference method for the RFADED. It has been shown that the obtained schemes are unconditionally stable and convergent with the accuracy of O (h 2 + k 2 + κ 2 + σ 2 + ρ 2) , where h, k and κ are space step for x, y and time step, respectively. Also, numerical examples are constructed to demonstrate the effectiveness of the numerical methods, and the results are found to be in excellent agreement with analytic exact solution. [ABSTRACT FROM AUTHOR]

Let X be a space of homogeneous type. Assume that L is a non negative, selfadjoint operator on L 2 (X) satisfying the sub-Gaussian upper bounds. In this paper, we prove that ‖ (I + L) - n / 2 e i τ L f ‖ 1 , ∞ ≤ C (1 + | τ |) n / 2 ‖ f ‖ 1 , ∀ τ ∈ R. By interpolation, we obtain ‖ (I + L) - n | 1 / p - 1 / 2 | e i τ L f ‖ p ≤ C (1 + | τ |) n | 1 / p - 1 / 2 | ‖ f ‖ p , ∀ τ ∈ R , 1 < p < ∞. [ABSTRACT FROM AUTHOR]

In this paper, we are devoted to the study of a problem proposed in Aujol and Dossal (SIAM J Optim 29:3131–3153, 2019), which concerns the optimal convergence rates for damped inertial gradient dynamics with flat geometries. Fortunately, we find a solution to the problem. More specifically, we obtain an optimal decay rate for the system energy of the second-order differential equation with time-dependent damping in a Hilbert space H x ¨ (t) + γ (t) x ˙ (t) + ∇ F (x (t)) = 0 , where F : H → R is a differentiable convex function possessing at least one minimizer and γ (t) = α / t with α > 0. It is known that this system, with the time-dependent damping coefficient γ (t) , models a nonlinear oscillator with viscous damping, and is associated with the Nesterov acceleration scheme, FISTA, or the accelerated gradient method in the area of numerical optimization. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a notion of shadowing property for a free semigroup action on a compact metric space, which is different of the notion of the shadowing property introduced by Bahabadi, called chain shadowing property. We study the relation between the shadowing property of a free semigroup action on a compact metric space X and the shadowing property of the induced free semigroup action on the hyperspace 2X. Specially, we not only theoretically prove that (F m + , F) ↷ X has the (chain) shadowing property if and only if (F m + , F) ↷ 2 X has the (chain) shadowing property, but also give examples to illustrate it. Finally, we compare the two notions of shadowing for free semigroup actions and obtain an interesting result that if (F m + , F) ↷ X has the shadowing property, then it has the chain shadowing property, but not vice versa. [ABSTRACT FROM AUTHOR]

In this paper we investigate the following fractional order in time integrodifferential problem D t α u (t) + A u (t) = f (t , u (t)) + ∫ − ∞ t k (t − s) g (s , u (s)) d s , t ∈ R. Here, D t α is the Caputo derivative. We obtain results on the existence and uniqueness of (ω , c) -periodic mild solutions assuming that −A generates an analytic semigroup on a Banach space X and f, g, and k satisfy suitable conditions. Finally, an interesting example that fits our framework is given. [ABSTRACT FROM AUTHOR]

In this paper, we show that the density in energy of Lipschitz functions in a Sobolev space N 1 , p (X) holds for all p ∈ [ 1 , ∞) whenever the space X is complete and separable and the measure is Radon and positive and finite on balls. Emphatically, p = 1 is allowed. We also give a few corollaries and pose questions for future work. The proof is direct and does not involve the usual flow techniques from prior work. It also yields a new approximation technique, which has not appeared in prior work. Notable with all of this work is that we do not use any form of Poincaré inequality or doubling assumption. The techniques are flexible and suggest a unification of a variety of approaches that have appeared in the literature on the topic. [ABSTRACT FROM AUTHOR]

For bounded linear operators A, B, C and D on a Banach space X, we show that if BAC = BDB and CDB = CAC then I — AC is generalized Drazin—Riesz invertible if and only if I — BD is generalized Drazin—Riesz invertible, which gives a positive answer to Question 4.9 in Yan, Zeng and Zhu [Complex Anal. Oper. Theory 14, Paper No. 12 (2020)]. In particular, we show that Jacobson's lemma holds for generalized Drazin—Riesz inverses. [ABSTRACT FROM AUTHOR]

A compact space X is called π -monolithic if for any surjective continuous mapping f : X → K where K is a metrizable compact space there exists a metrizable compact space T ⊆ X such that f (T) = K . A topological space X is Baire if the intersection of any sequence of open dense subsets of X is dense in X. Let C p (X , Y) denote the space of all continuous Y-valued functions C(X,Y) on a Tychonoff space X with the topology of pointwise convergence. In this paper we have proved that for a totally disconnected space X the space C p (X , { 0 , 1 }) is Baire if, and only if, C p (X , K) is Baire for every π -monolithic compact space K. For a Tychonoff space X the space C p (X , R) is Baire if, and only if, C p (X , L) is Baire for each Fréchet space L. We construct a totally disconnected Tychonoff space T such that C p (T , M) is Baire for a separable metric space M if, and only if, M is a Peano continuum. Moreover, C p (T , [ 0 , 1 ]) is Baire but C p (T , { 0 , 1 }) is not. [ABSTRACT FROM AUTHOR]

In this paper, we study a new class of Sobolev type (ω , T) -periodic linear and semilinear impulsive evolution equations, where T denotes a linear isomorphism from Banach space X to itself. We give a sufficient and necessary condition depending on the initial value, periodic boundary value and linear isomorphism to guarantee that the homogeneous linear impulsive problem has a (ω , T) -periodic solution. Next, we give the explicit expression of (ω , T) -periodic solutions for nonhomogeneous linear impulsive problem and derive two important estimations for the certain sum and integration including the Green function. Further, we show the existence and uniqueness of solutions to semilinear impulsive problem, where we remove the compactness of A B - 1 and use the compactness of a mapping depending on the nonlinear term. Finally, examples are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of linear difference equations of fractional order 2 < α ≤ 3 in the sense of Riemann–Liouville. The explicit solution for this model is provided in terms of a fractional resolvent sequence which allows to write the solution to this equation as a variation of constant formula. We also characterize the existence and uniqueness of solutions in ℓ p (N 0 , X) spaces with X being a UMD-space in terms of the R-boundedness of the operator symbol of the model. Moreover, we are able to relax this condition in the case of Hilbert spaces. Finally, we illustrate our results with an example that involves the generator of a contraction semigroup. [ABSTRACT FROM AUTHOR]

In this paper we consider a dual gradient method for solving linear ill-posed problems A x = y , where A : X → Y is a bounded linear operator from a Banach space X to a Hilbert space Y. A strongly convex penalty function is used in the method to select a solution with desired feature. Under variational source conditions on the sought solution, convergence rates are derived when the method is terminated by either an a priori stopping rule or the discrepancy principle. We also consider an acceleration of the method as well as its various applications. [ABSTRACT FROM AUTHOR]

In this paper, we consider covariant functors F : Comp → Comp acting in category of shape-preserving compact sets [2], infinite compact sets, and shape equivalence [6]. Also we study action of compact functors and shape properties of the compact space X consisting of connected components X of the compact X as well as shape identity ShX = ShY of infinite compacts X and Y for the space P(X) of probability measures and its subspaces. [ABSTRACT FROM AUTHOR]

While it is well-known that nonlinear methods of approximation can often perform dramatically better than linear methods, there are still questions on how to measure the optimal performance possible for such methods. This paper studies nonlinear methods of approximation that are compatible with numerical implementation in that they are required to be numerically stable. A measure of optimal performance, called stable manifold widths, for approximating a model class K in a Banach space X by stable manifold methods is introduced. Fundamental inequalities between these stable manifold widths and the entropy of K are established. The effects of requiring stability in the settings of deep learning and compressed sensing are discussed. [ABSTRACT FROM AUTHOR]

Our aim in this article is to study the class of so-called ρ - paraconvex multifunctions from a Banach space X into the subsets of another Banach space Y. These multifunctions are defined in relation with a modulus function ρ : X → [ 0 , + ∞) satisfying some suitable conditions. This class of multifunctions generalizes the class of γ - paraconvex multifunctions with γ > 1 introduced and studied by Rolewicz, in the eighties and subsequently studied by A. Jourani and some others authors. We establish some regular properties of graphical tangent and normal cones to paraconvex multifunctions between Banach spaces as well as a sum rule for coderivatives for such class of multifunctions. The use of subdifferential properties of the lower semicontinuous envelope function of the distance function associated to a multifunction established in the present paper plays a key role in this study. [ABSTRACT FROM AUTHOR]

In this paper, we consider solvability of the Cauchy initial problem d x / dt = f (t , x) , x (0) = x 0 in a Banach space X. As a result, we generalize Peano's existence theorem in the following manner: For every Banach space X, the problem always has a solution x ∈ C 1 ([ 0 , a ] , X) for all a > 0 under the assumption that f : R + ⊕ X → X is weak-to-weak continuous on some bounded set with a relatively weakly compact range. We also show that for any infinite dimensional reflexive Banach space X with an unconditional basis, in particular, ℓ p and L p (Ω , ∑ , μ) ( 1 < p < ∞ and (Ω , ∑ , μ) is σ -finite), and for all a > 0 there is a bounded nowhere locally Lipschitz function f : R + ⊕ X → X which is weak-to-weak continuous on some bounded set so that the Cauchy initial problem has a solution x ∈ C 1 ([ 0 , a ] , X) . [ABSTRACT FROM AUTHOR]

In this paper, we focus on p-sober spaces and prove that (1) the T0 space X is p-sober if and only if the Smyth power space of X is p-sober; (2) the space X has a p-sober dcpo model if and only if X is T1 and p-sober; (3) every non-p-sober T0 space does not have a p-sobrification; (4) the T0 space X is sober if and only if X is p-sober and PD. [ABSTRACT FROM AUTHOR]

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and ρ , τ : I ⟶ A , S , T : I ⟶ B be maps on a non-empty set I whose ranges are closed under multiplication and contain exponential functions. In this paper, we first show that if ‖ S (p) ‖ Y = ‖ ρ (p) ‖ X , ‖ T (p) ‖ Y = ‖ τ (p) ‖ X and ‖ S (p) T (q) ‖ Y = ‖ ρ (p) τ (q) ‖ X , for all p , q ∈ I , then there exists a homeomorphism φ from Šilov boundary ∂ A of A onto ∂ B such that for each x ∈ ∂ A and p ∈ I , | S (p) (φ (x)) | = | ρ (p) (x) | and | T (p) (φ (x)) | = | τ (p) (x) | . Then we prove that, if for some ε ≥ 0 , Ran π (T (p) S (q)) is contained in an ε ‖ τ (p) ρ (q) ‖ -neighborhood of Ran π (τ (p) ρ (q)) , for all p , q ∈ I , then, under a certain condition, there exist continuous functions α , β ∈ B such that | α (φ (x)) T (p) (φ (x)) - τ (p) (x) | ⩽ 4 ε | τ (p) (x) | and | β (φ (x)) S (p) (φ (x)) - ρ (p) (x) | ⩽ 4 ε | ρ (p) (x) | , for all p ∈ I and x ∈ ∂ (A) . Our results can be applied for Banach algebras of Lipschitz functions. [ABSTRACT FROM AUTHOR]

In this paper we introduce a discrete fractional resolvent family { S α , β n } n ∈ N 0 $\begin{array}{} \displaystyle \{S_{\alpha,\beta}^n\}_{n\in\mathbb{N}_0} \end{array}$ generated by a closed linear operator in a Banach space X for a given α, β > 0. Moreover, we study its main properties and, as a consequence, we obtain a method to study the existence and uniqueness of the solutions to discrete fractional difference equations in a Banach space. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we present some investigations on the class of almost (L) limited operators. We show that an operator T : X → E , from a Banach space X to a Banach lattice E, is almost (L) limited iff its adjoint carries disjoint almost L-sequences to norm null ones. In addition, we improve several results obtained by Oughajji et al. In its second part, we study the relationship between the class of weakly precompact operators and that of order weakly compact (resp. b-weakly compact) operators. Among other things, we show that for a Banach lattice E and a Banach space X the following statements are equivalent: Every order weakly compact (resp. b-weakly compact) operator T : E → X is weakly precompact; The norm of E ′ is order continuous or X does not contain any isomorphic copy of ℓ 1 . [ABSTRACT FROM AUTHOR]

In this paper, we prove that if K is a nonempty weakly compact set in a Banach space X, T : K → K is a nonexpansive map satisfying x + T x 2 ∈ K for all x ∈ K , then T has a fixed point whenever X is uniformly rotund with respect to every k-dimensional subspace or X satisfies the property (P). These results improve the results of Veeramani and Radhakrishnan et al. [ABSTRACT FROM AUTHOR]

In this paper, we give a number of characterizations for a Banach space X which is isometric to a subspace of c0, or, c0(Γ), successively, in terms of extreme points of its dual unit ball BX*, Fréchet and Gâteaux derivatives of its norm, or, in terms of w*-strongly exposed points and w*-exposed points of BX*. [ABSTRACT FROM AUTHOR]

Given a functionally countable space X, if K is a scattered Corson compact space, then the product X × K is functionally countable. If the space X is, additionally, weakly ω 1 -Lindelöf, then X × L is functionally countable for any scattered Lindelöf space L. In particular, if X is a weakly Lindelöf functionally countable space, then the product X × L is functionally countable whenever L is a scattered Lindelöf space. We also establish that any finite product of countably compact functionally countable spaces is functionally countable and exponential separability of countably compact spaces X and Y implies that X × Y is exponentially separable if one of them is sequential. If the product X = X 1 × ⋯ × X n of functionally countable spaces X 1 ,... , X n is pseudocompact, then X is functionally countable. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective curve over an algebraically closed field k. Let G be an almost simple simply-connected group over k. Let G be a Bruhat–Tits group scheme on X which is generically the trivial group scheme with fibers G. We show that the étale fundamental group of the moduli stack M X (G) of torsors under G is isomorphic to that of the moduli stack M X (G) of principal G-bundles. Our main goal is to prove that for any smooth, noetherian and irreducible stack X , the inclusion of any non-empty open substack X ∘ , whose complement has codimension at least two induces an isomorphism of étale fundamental group. Over C , we show that the open substack of regularly stable torsors in M X (G) has complement of codimension at least two when g X ≥ 3 . As an application, we show that over C the moduli space M X (G) of G -torsors is simply-connected. [ABSTRACT FROM AUTHOR]

Given a sequence U = { U n : n ∈ ω } of non-empty open subsets of a space X, a family { V n : n ∈ ω } is a shrinking of U if Vn is a non-empty open set and V n ⊂ U n for every n ∈ ω . A space X has the disjoint (discrete) shrinking property if every sequence of non-empty open subsets of X has a disjoint (discrete) shrinking. We present a topological equivalent of the disjoint shrinking property in general spaces and apply it to characterize the disjoint shrinking property in topological groups. We also show that a locally convex space L has the disjoint shrinking property if and only if w (L) > ω . The space L has the discrete shrinking property if and only if L is not metrizable. We also characterize discrete selectivity in C p (X , [ 0 , 1 ]) for spaces X with a unique non-isolated point. Our results solve several published open questions. [ABSTRACT FROM AUTHOR]

We prove that: I. For every regular Lindelöf space X if | X | = Δ (X) and cf | X | ≠ ω , then X is maximally resolvable; II. For every regular countably compact space X if | X | = Δ (X) and cf | X | = ω , then X is maximally resolvable. Here Δ (X) , the dispersion character of X, is the minimum cardinality of a nonempty open subset of X. Statements I and II are corollaries of the main result: for every regular space X if | X | = Δ (X) and every set A ⊆ X of cardinality cf | X | has a complete accumulation point, then X is maximally resolvable. Moreover, regularity here can be weakened to π -regularity, and the Lindelöf property can be weakened to the linear Lindelöf property. [ABSTRACT FROM AUTHOR]

For a Hausdorff space X, a free involution τ : X → X and a Hausdorff space Y, we discover a connection between the sectional category of the double covers q : X → X / τ and q Y : F (Y , 2) → D (Y , 2) from the ordered configuration space F(Y, 2) to its unordered quotient D (Y , 2) = F (Y , 2) / Σ 2 , and the Borsuk–Ulam property (BUP) for the triple (X , τ) ; Y . Explicitly, we demonstrate that the triple (X , τ) ; Y satisfies the BUP if the sectional category of q is bigger than the sectional category of q Y . This property connects a standard problem in Borsuk–Ulam theory to current research trends in sectional category. As an application of our results, we present a new lower bound for the index in terms of sectional category. We present several examples for whom the lower bound coincides with sectional category minus 1. We conjecture that the index of (M , τ) coincides with the sectional category of the quotient map q : M → M / τ minus 1 for any CW complex M. [ABSTRACT FROM AUTHOR]

We prove a topological completeness theorem for the modal logic GLP containing operators { ⟨ ξ ⟩ : ξ ∈ Ord } intended to capture a wellordered sequence of consistency operators increasing in strength. More specifically, we prove that, given a tall-enough scattered space X, any sentence ϕ consistent with GLP can be satisfied on a polytopological space based on finitely many Icard topologies constructed over X and corresponding to the finitely many modalities that occur in ϕ . [ABSTRACT FROM AUTHOR]

Using weaker versions of the cardinal function ψ c (X) , we derive a series of new bounds for the cardinality of Hausdorff spaces and regular spaces that do not involve ψ c (X) nor its variants at all. For example, we show if X is regular then | X | ≤ 2 c (X) π χ (X) and | X | ≤ 2 c (X) π χ (X) o t (X) , where the cardinal function o t (X) , introduced by Tkachenko, has the property o t (X) ≤ min { t (X) , c (X) } . It follows from the latter that a regular space with cellularity at most c and countable π -character has cardinality at most 2 c . For a Hausdorff space X we show | X | ≤ 2 d (X) π χ (X) , | X | ≤ d (X) π χ (X) o t (X) , and | X | ≤ 2 π w (X) d o t (X) , where d o t (X) ≤ min { o t (X) , π χ (X) } . None of these bounds involve ψ c (X) or ψ (X) . By introducing the cardinal functions w ψ c (X) and d ψ c (X) with the property w ψ c (X) d ψ c (X) ≤ ψ c (X) for a Hausdorff space X, we show | X | ≤ π χ (X) c (X) w ψ c (X) if X is regular and | X | ≤ π χ (X) c (X) d ψ c (X) w ψ c (X) if X is Hausdorff. This improves results of Šapirovskiĭ and Sun. It is also shown that if X is Hausdorff then | X | ≤ 2 d (X) w ψ c (X) , which appears to be new even in the case where w ψ c (X) is replaced with ψ c (X) . Compact examples show that ψ (X) cannot be replaced with d ψ c (X) w ψ c (X) in the bound 2 ψ (X) for the cardinality of a compact Hausdorff space X. Likewise, ψ (X) cannot be replaced with d ψ c (X) w ψ c (X) in the Arhangel׳skiĭ-Šapirovskiĭ bound 2 L (X) t (X) ψ (X) for the cardinality of a Hausdorff space X. Finally, we make several observations concerning homogeneous spaces in this connection. [ABSTRACT FROM AUTHOR]

We investigate the impact of polarons on Mo around the oxygen vacancy on the electronic structures and optical properties of SrMoO4 crystals. The screened hybrid density function within the Kohn–Sham density functional theory combined with the finite-size correction are employed in our calculations. Results indicate that the trapped electrons in the form of polarons are all localized on the same Mo atom which is nearest to oxygen vacancy V O × or V O ′ , not on the oxygen vacancy. Optical spectra of polarons around the oxygen vacancy are also obtained. The calculated absorption peaks (4.38 eV) and emission peaks (2.85 eV) are in line with the experimental results, thus providing insight into the origin of the blue luminescence spectra of the SrMoO4. [ABSTRACT FROM AUTHOR]

A ring is called a zip ring (Carl Faith coined this term) if every faithful ideal contains a finitely generated faithful ideal. By first proving that a reduced ring is a zip ring if and only if every dense element of the frame of its radical ideals is above a compact dense element, we study algebraic frames with the property stated in the title. We call them zipped. They generalize the coherent frames of radical ideals of zip rings, but (unlike coherent frames) they need not be compact. The class of zipped algebraic frames is closed under finite products, but not under infinite products. If the coproduct of two algebraic frames is zipped, then each cofactor is zipped. If the ring is not necessarily reduced, then its frame of radical ideals is zipped precisely when the ring satisfies what in the literature is called the weak zip property. For a Tychonoff space X, we show that C(X) is a zip ring if and only if X is a finite discrete space. [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]

We present a new elementary algorithm that takes time O ϵ x 3 5 (log x) 8 5 + ϵ and space O x 3 10 (log x) 13 10 (measured bitwise) for computing M (x) = ∑ n ≤ x μ (n) , where μ (n) is the Möbius function. This is the first improvement in the exponent of x for an elementary algorithm since 1985. We also show that it is possible to reduce space consumption to O (x 1 / 5 (log x) 5 / 3) by the use of (Helfgott in: Math Comput 89:333–350, 2020), at the cost of letting time rise to the order of x 3 / 5 (log x) 2 log log x . [ABSTRACT FROM AUTHOR]

We characterise rectifiable subsets of a complete metric space X in terms of local approximation, with respect to the Gromov–Hausdorff distance, by an n-dimensional Banach space. In fact, if E ⊂ X with H n (E) < ∞ and has positive lower density almost everywhere, we prove that it is sufficient that, at almost every point and each sufficiently small scale, E is approximated by a bi-Lipschitz image of Euclidean space. We also introduce a generalisation of Preiss's tangent measures that is suitable for the setting of arbitrary metric spaces and formulate our characterisation in terms of tangent measures. This definition is equivalent to that of Preiss when the ambient space is Euclidean, and equivalent to the measured Gromov–Hausdorff tangent space when the measure is doubling. [ABSTRACT FROM AUTHOR]

The alpha complex efficiently computes persistent homology of a point cloud X in Euclidean space when the dimension d is low. Given a subset A of X , relative Čech persistent homology can be computed as the persistent homology of the relative Čech complex C ˇ (X , A) . However, this is not computationally feasible for larger point clouds X . The aim of this note is to present a method for efficient computation of relative Čech persistent homology in low dimensional Euclidean space. We introduce the relative Delaunay–Čech complex Del C ˇ (X , A) whose homology is the relative Čech persistent homology. It is constructed from the Delaunay complex of an embedding of X in (d + 1) -dimensional Euclidean space. [ABSTRACT FROM AUTHOR]

A coarse space X endowed with a linear order compatible with the coarse structure of X is called linearly ordered. We prove that every linearly ordered coarse space X is locally convex and the asymptotic dimension of X is either 0 or 1. If X is metrizable, then the family of all right bounded subsets of X has a selector. [ABSTRACT FROM AUTHOR]

We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space R n , where n ≥ 2 , assuming that the diffusion matrix depends on the space variable x and has a finite limit along any ray as | x | → ∞ . Under suitable smallness conditions in the nonlinear case, we prove convergence to a self-similar solution whose profile is entirely determined by the asymptotic diffusion matrix. Examples are given which show that the profile can be a rather general Gaussian-like function, and that the approach to the self-similar solution can be arbitrarily slow depending on the continuity and coercivity properties of the asymptotic matrix. The proof of our results relies on appropriate energy estimates for the diffusion equation in self-similar variables. The new ingredient consists in estimating not only the difference w between the solution and the self-similar profile, but also an antiderivative W obtained by solving a linear elliptic problem which involves w as a source term. Hence, a good part of our analysis is devoted to the study of linear elliptic equations whose coefficients are homogeneous of degree zero. [ABSTRACT FROM AUTHOR]

We prove that the existence of a 1-Lipschitz retraction (a contraction) from a space X onto its subspace A implies the persistence diagram of A embeds into the persistence diagram of X. As a tool we introduce tight injections of persistence modules as maps inducing the said embeddings. We show contractions always exist onto shortest loops in metric graphs and conjecture the existence of contractions in planar metric graphs onto all loops of a shortest homology basis. Of primary interest are contractions onto loops in geodesic spaces. These act as ideal circular coordinates. Furthermore, as the Theorem of Adamaszek and Adams describes the pattern of persistence diagram of S 1 , a contraction X → S 1 implies the same pattern appears in persistence diagram of X. [ABSTRACT FROM AUTHOR]

Denote by X a Banach space and by T : X → X a bounded linear operator with non-trivial kernel satisfying suitable conditions. We consider the concepts of entropy—for T-invariant probabilities—and pressure for α -Hölder continuous potentials. We also prove the existence of ground states associated with those potentials (the limit when the temperature goes to zero), when the Banach space X is equipped with a Schauder basis. We also produce an example concerning weighted shift operators defined on the Banach spaces c 0 (R) and l p (R) , 1 ≤ p < + ∞ , where our results do apply. In addition, we prove the existence of calibrated sub-actions when the potential satisfies certain regularity conditions using properties of the so-called Mañé potential and we exhibit examples of selection at zero temperature and explicit sub-actions in the class of α -Hölder continuous potentials. [ABSTRACT FROM AUTHOR]

A wide new class of subsets of a Banach space X named coarse p-limited sets ( 1 ≤ p < ∞ ) is introduced by considering weak* p-summable sequences in X ′ instead of weak* null sequences. We study its basic properties and compare it with the class of compact and weakly compact sets. Results concerning the relationship of coarse p-limited sets with operators are obtained. [ABSTRACT FROM AUTHOR]

We study extreme values of group-indexed stable random fields for discrete groups G acting geometrically on spaces X in the following cases: (1) G acts properly discontinuously by isometries on a CAT(-1) space X, (2) G is a lattice in a higher rank Lie group, acting on a symmetric space X, and (3) G is the mapping class group of a surface acting on its Teichmüller space. The connection between extreme values and the geometric action is mediated by the action of the group G on its limit set equipped with the Patterson–Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth which measures the distortion of measures on the boundary in comparison to the movement of points in the space X and show that its non-vanishing is equivalent to finiteness of the Bowen–Margulis measure for the associated unit tangent bundle U(X/G) provided X/G has non-arithmetic length spectrum. As a consequence, we establish a dichotomy for the growth-rate of a partial maxima sequence of stationary symmetric α -stable ( 0 < α < 2 ) random fields indexed by groups acting on such spaces. We also establish analogous results for normal subgroups of free groups. [ABSTRACT FROM AUTHOR]

Let X be a complex Banach space with dim X ≥ 2 , and A ⊆ B (X) be a standard operator algebra. We show that a linear mapping δ : A → A is anti-derivable at zero (i.e., x y = 0 in A implies y δ (x) + δ (y) x = 0 ) if and only if δ = 0 . [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]