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The famous Ascoli-Arzelà theorem served as a springboard for research into compactness in function spaces, particularly spaces of continuous functions. This paper investigates compact subsets of spaces of continuous functions endowed with topologies between the topology of pointwise convergence and the topology of uniform convergence. More precisely, this paper studies necessary and sufficient conditions for a subset to be compact in Cλ,u(X) for a locally-λ space X when λ ⊇ 퓕(X), for a hemi- λ λf-space X when λ ⊆ 퓟 퓢(X), and for a k-space X when λ ⊇ 퓚(X). This paper also studies that every bounded subset of Cλ,u(X) has compact closure for some classes of topological spaces X. [ABSTRACT FROM AUTHOR]

This paper is devoted to introducing and investigating the bounded variation capacity and the perimeter in the abstract Wiener space X, thereby discovering some related inequalities. Functions of bounded variation in an abstract Wiener space X have been studied by many scholars. As the continuation of this research, we define the corresponding BV capacity cap H ⁡ (⋅) (now called abstract Wiener BV capacity) and investigate its properties. We also investigate some properties of sets of finite γ-perimeter, with γ being a Gaussian measure. Subsequently, the isocapacitary inequality associated with cap H ⁡ (⋅) is presented and we are able to show that it is equivalent to the Gaussian isoperimetric inequality. Finally, we prove that every set of finite γ-perimeter in X has mean curvature in L 1 ⁢ (X , γ) . [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of some classes of topological spaces under the influence of the functor of G-permutation degree 햲햯 G n . We prove: (a) if a space X is an r-space, then so is 햲햯 G n ⁡ X , (b) if X is a cosmic space, then so is 햲햯 G n ⁡ X , (c) if a space X is a C ⁢ (κ) -cosmic, then so is 햲햯 G n ⁡ X , (d) if a space X is an α-space, then so is 햲햯 G n ⁡ X . [ABSTRACT FROM AUTHOR]

In this paper, we define some combinatorial principles to characterize spaces X whose hyperspace satisfies some variation of some classical star selection principle. Specifically, the variations characterized are the selective and absolute versions of the star selection principles for the Menger and Rothberger cases; also, the hyperspaces considered in these characterizations are CL(X), 핂(X), 픽(X) and ℂ핊(X) in both cases, endowed with either the Fell topology or the Vietoris topology. [ABSTRACT FROM AUTHOR]

In 2010, Qinian Jin considered a regularized Levenberg–Marquardt method in Hilbert spaces for getting stable approximate solution for nonlinear ill-posed operator equation F ⁢ (x) = y , where F : D ⁢ (F) ⊂ X → Y is a nonlinear operator between Hilbert spaces X and Y and obtained rate of convergence results under an appropriate source condition. In this paper, we propose a simplified Levenberg–Marquardt method in Hilbert spaces for solving nonlinear ill-posed equations in which sequence of iteration { x n δ } is defined as x n + 1 δ = x n δ - ( α n ⁢ I + F ′ ⁢ (x 0) * ⁢ F ′ ⁢ (x 0) ) - 1 ⁢ F ′ ⁢ (x 0) * ⁢ (F ⁢ (x n δ) - y δ) . Here { α n } is a decreasing sequence of nonnegative numbers which converges to zero, F ′ ⁢ (x 0) denotes the Fréchet derivative of F at an initial guess x 0 ∈ D ⁢ (F) for the exact solution x † and (F ′ ⁢ (x 0)) * denote the adjoint of F ′ ⁢ (x 0) . In our proposed method, we need to calculate Fréchet derivative of F only at an initial guess x 0 . Hence, it is more economic to use in numerical computations than the Levenberg–Marquardt method used in the literature. We have proved convergence of the method under Morozov-type stopping rule using a general tangential cone condition. In the last section of the paper, numerical examples are presented to demonstrate advantages of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we first define ray increasing and decreasing monotonicity of maps. If 푇 is an optimal transport map for the Monge problem with cost function ∥ y - x ∥ sc in R n or 푇 is an optimal transport map for the Monge problem with cost function d ⁢ (x , y) , the geodesic distance, in more general, non-branching geodesic spaces 푋, we show respectively equivalence of some previously introduced monotonicity properties and the property of ray increasing as well as ray decreasing monotonicity which we define in this paper. Then, by solving secondary variational problems associated with strictly convex and concave functions respectively, we show that there exist ray increasing and decreasing optimal transport maps for the Monge problem with cost function ∥ y - x ∥ sc . Finally, we give the classification of optimal transport maps for the Monge problem such that the cost function ∥ y - x ∥ sc further satisfies the uniform smoothness and convexity estimates. That is, all of the optimal transport maps for such Monge problem can be divided into three different classes: the ray increasing map, the ray decreasing map and others. [ABSTRACT FROM AUTHOR]

Cauchy problems of the form d ⁢ u d ⁢ t = q ⁢ (A) ⁢ u + h ⁢ (t) , 0 < t < T , u ⁢ (0) = φ , are studied in a Banach space X where A is a strong strip-type operator and q ⁢ (A) is a complex polynomial in A. In this case, the spectrum of A lies within a horizontal strip of height θ, and so potentially neither A nor - A generates a strongly continuous semigroup on X. Therefore, depending on the definition of q ⁢ (A) , the original problem may be severely ill-posed. We utilize a functional calculus for strip-type operators in order to define an approximate operator f β ⁢ (A) such that f β ⁢ (A) is bounded for each β > 0 and f β ⁢ (A) ⁢ χ → q ⁢ (A) ⁢ χ as β → 0 for χ in a suitable domain. We show that this approximation gives rise to regularization for the original problem with respect to a graph norm related to C-regularized semigroups. We also fit the theory of the paper into a special case where iA generates a bounded, strongly continuous group on X. Under this assumption, which implies that A is a strong strip-type operator of height 0, results follow for a wide variety of ill-posed PDEs in L p spaces. [ABSTRACT FROM AUTHOR]

A space X is said to have the set-star-K-Menger property if for each nonempty subset A of X and for each sequence (퓤n : n ∈ ℕ) of collections of open sets in X such that for each n ∈ ℕ, A ⊆ ⋃ 퓤n, there is a sequence (Kn : n ∈ ℕ) of compact subsets of X such that A ⊆ ⋃ n ∈ N St(Kn, 퓤n). In this paper, we prove that: There exists a T1 set-star-Menger space which is not set-star-K-Menger and there exists a Tychonoff set-star-K-Menger space that is not set-star-Menger. Assuming 픡 = 픠, there exists a Tychonoff set-star-K-Menger space having a regular-closed Gδ-subspace which is not set-star-K-Menger. If the Alexandroff duplicate of a space X is set-star-K-Menger, then X is set-star-K-Menger. The product of set-star-K-Menger space and a compact space is rectangular set-star-K-Menger space. The above-mentioned results answer to Problem 2.5 and Problem 3.6, and give a partial answer to Problem 3.11 in [SINGH, S.: On set-star-K-Menger spaces, Publ. Math. Debrecen 100 (2022), 87–100]. Further, we continue to study the topological properties of set-star-K-Menger spaces. [ABSTRACT FROM AUTHOR]

Recently, the authors interested some new problems on multiplier spaces of Lorentz' almost convergence and fλ-convergence as a generalization of almost convergence. fλ-convergence is firstly introduced by Karakuş and Başar, and used for some new characterizations of completeness and barrelledness of the spaces through weakly unconditionally Cauchy series in a normed space X and its continuous dual X*. In the present paper, we deal with fλ-convergence to have some inclusion relations between the vector valued spaces obtained from this type convergence and corresponding classical sequence spaces, and to give new characterizations of some classical properties like completeness, reflexivity, Schur property and Grothendieck property of normed spaces. By the way, we give a characterization of finite-dimensional normed spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study some properties of homotopical closeness for paths. We define the quasi-small loop group as the subgroup of all classes of loops that are homotopically close to null-homotopic loops, denoted by π 1 q s (X , x) for a pointed space (X, x). Then we prove that, unlike the small loop group, the quasi-small loop group π 1 q s (X , x) does not depend on the base point, and that it is a normal subgroup containing π 1 s g (X , x) , the small generated subgroup of the fundamental group. Also, we show that a space X is homotopically path Hausdorff if and only if π 1 q s (X , x) is trivial. Finally, as consequences, we give some relationships between the quasi-small loop group and the quasi-topological fundamental group. [ABSTRACT FROM AUTHOR]

In a recent publication [D. Govc, W. A. Marzantowicz and P. Pavešić, Estimates of covering type and the number of vertices of minimal triangulations, Discrete Comput. Geom. 63 2020, 1, 31–48], we have introduced a new method, based on the Lusternik–Schnirelmann category and the cohomology ring of a space X, that yields lower bounds for the size of a triangulation of X. In this current paper, we present an important extension that takes into account the fundamental group of X. In fact, if π 1 ⁢ (X) contains elements of finite order, then one can often find cohomology classes of high 'category weight', which in turn allow for much stronger estimates of the size of triangulations of X. We develop several weighted estimates and then apply our method to compute explicit lower bounds for the size of triangulations of orbit spaces of cyclic group actions on a variety of spaces including products of spheres, Stiefel manifolds, Lie groups and highly-connected manifolds. [ABSTRACT FROM AUTHOR]

This paper continues our previous program to study the restriction estimates in a class of conical singular spaces X = C ⁢ (Y) = (0 , ∞) r × Y equipped with the metric g = d ⁢ r 2 + r 2 ⁢ h , where the cross section Y is a compact (n - 1) -dimensional closed Riemannian manifold (Y , h) . Assuming the initial data possesses additional regularity in the angular variable θ ∈ Y , we prove some linear restriction estimates for the solutions of Schrödinger equations on the cone X. The smallest positive eigenvalue of the operator Δ h + V 0 + (n - 2) 2 / 4 plays an important role in the result. As applications, we prove local energy estimates and Keel–Smith–Sogge estimates for the Schrödinger equation in this setting. [ABSTRACT FROM AUTHOR]

We show that if 퐺 is an admissible group acting geometrically on a CAT ⁡ (0) space 푋, then 퐺 is a hierarchically hyperbolic space and its 휅-Morse boundary (∂ κ G , ν) is a model for the Poisson boundary of (G , μ) , where 휈 is the hitting measure associated to the random walk driven by 휇. [ABSTRACT FROM AUTHOR]

Let X and Y be pathwise connected and paracompact Hausdorff spaces equipped with free involutions T : X → X and S : Y → Y, respectively. Suppose that there exists a sequence (Xi, Ti) ⁢ ⟶ hi (Xi+1, Ti+1) for ⁢ 1 ≤ i ≤ k, where, for each i, Xi is a pathwise connected and paracompact Hausdorff space equipped with a free involution Ti, such that Xk+1 = X, and hi : Xi → Xi+1 is an equivariant map, for all 1 ≤ i ≤ k. To achieve Borsuk–Ulam-type theorems, in several results that appear in the literature, the involved spaces X in the statements are assumed to be cohomological n-acyclic spaces. In this paper, by considering a more wide class of topological spaces X (which are not necessarily cohomological n-acyclic spaces), we prove that there is no equivariant map ƒ : (X , T) → (Y , S) and we present some interesting examples to illustrate our results. [ABSTRACT FROM AUTHOR]

Let 프 be a standard operator algebra on a complex Banach space 픛 , dim ⁡ 픛 > 1 , and p n ⁢ (T 1 , T 2 , ... , T n) the (n - 1) th-commutator of elements T 1 , T 2 , ... , T n ∈ 프 . Then every map ξ : 프 → 프 (not necessarily linear) satisfying ξ ⁢ (p n ⁢ (T 1 , T 2 , ... , T n)) = ∑ i = 1 n p n ⁢ (T 1 , T 2 , ... , T i - 1 , ξ ⁢ (T i) , T i + 1 , ... , T n) for all T 1 , T 2 , ... , T n ∈ 프 is of the form ξ = Ω + Γ , where Ω : 프 → 프 is an additive derivation and Γ : 프 → ℂ ⁢ I is a map that vanishes at each (n - 1) th-commutator p n ⁢ (T 1 , T 2 , ... , T n) for all T 1 , T 2 , ... , T n ∈ 프 . In addition, if the map ξ is linear and satisfies the above relation, then there exist an operator S ∈ 프 and a linear map Γ : 프 → ℂ ⁢ I satisfying Γ ⁢ (p n ⁢ (T 1 , T 2 , ... , T n)) = 0 for all T 1 , T 2 , ... , T n ∈ 프 , such that ξ ⁢ (T) = [ T , S ] + Γ ⁢ (T) for all T ∈ 프 . [ABSTRACT FROM AUTHOR]

We obtain results about fundamental groups of RCD ∗ ⁢ (K , N) spaces previously known under additional conditions such as smoothness or lower sectional curvature bounds. For fixed K ∈ ℝ , N ∈ [ 1 , ∞) , D > 0 , we show the following: • There is C > 0 such that for each RCD ∗ ⁢ (K , N) space X of diameter ≤ D , its fundamental group π 1 ⁢ (X) is generated by at most C elements. • There is D ~ > 0 such that for each RCD ∗ ⁢ (K , N) space X of diameter ≤ D with compact universal cover X ~ , one has diam ⁡ (X ~) ≤ D ~ . • If a sequence of RCD ∗ ⁢ (0 , N) spaces X i of diameter ≤ D and rectifiable dimension n is such that their universal covers X ~ i converge in the pointed Gromov–Hausdorff sense to a space X of rectifiable dimension n, then there is C > 0 such that for each i, the fundamental group π 1 ⁢ (X i) contains an abelian subgroup of index ≤ C . • If a sequence of RCD ∗ ⁢ (K , N) spaces X i of diameter ≤ D and rectifiable dimension n is such that their universal covers X ~ i are compact and converge in the pointed Gromov–Hausdorff sense to a space X of rectifiable dimension n, then there is C > 0 such that for each i, the fundamental group π 1 ⁢ (X i) contains an abelian subgroup of index ≤ C . • If a sequence of RCD ∗ ⁢ (K , N) spaces X i with first Betti number ≥ r and rectifiable dimension n converges in the Gromov–Hausdorff sense to a compact space X of rectifiable dimension m, then the first Betti number of X is at least r + m - n . The main tools are the splitting theorem by Gigli, the splitting blow-up property by Mondino and Naber, the semi-locally-simple-connectedness of RCD ∗ ⁢ (K , N) spaces by Wang, the isometry group structure by Guijarro and the first author, and the structure of approximate subgroups by Breuillard, Green and Tao. [ABSTRACT FROM AUTHOR]

We develop the notion of a geometric covering of a rigid space 푋, which yields a larger class of covering spaces than that studied previously by de Jong. Geometric coverings are closed under disjoint unions and are étale local on 푋. If 푋 is connected, its geometric coverings form a tame infinite Galois category and hence are classified by a topological group. The definition is based on the property of lifting of "geometric arcs" and is meant to be an analogue of the notion developed for schemes by Bhatt and Scholze. [ABSTRACT FROM AUTHOR]

We prove that the perfect loop functor LX of a quasi-projective scheme X over a local non-archimedean field k satisfies arc-descent, strengthening a result of Drinfeld. Then we prove that for an unramified reductive group G, the map L ⁢ G → L ⁢ (G / B) is a v-surjection. This gives a mixed characteristic version (for v-topology) of an equal characteristic result (in étale topology) of Bouthier–Česnavičius. In the second part of the article, we use the above results to introduce a well-behaved notion of Deligne–Lusztig spaces X w ⁢ (b) attached to unramified p-adic reductive groups. We show that in various cases these sheaves are ind-representable, thus partially solving a question of Boyarchenko. Finally, we show that the natural covering spaces X ˙ w ˙ ⁢ (b) are pro-étale torsors over clopen subsets of X w ⁢ (b) , and analyze some examples. [ABSTRACT FROM AUTHOR]

For a zero-dimensional Hausdorff space X, denote, as usual, by C(X, ℤ) the ring of continuous integer-valued functions on X. If f ∈ C(X, ℤ), denote by Z(f) the set of all points of X that are mapped to 0 by f. The set C K (X , Z) = { f ∈ C (X , Z) ∣ cl X (X ∖ Z (f)) is compact } is the integer-valued analogue of the ideal of functions with compact support in C(X). By first working in the category of locales and then interpreting the results in spaces, we characterize this ideal in several ways. Writing ζX for the Banaschewski compactification of X, we also explore some properties of ideals of C(X, ℤ) associated with subspaces of ζX analogously to how one associates, for any Tychonoff space Y, subsets of βY with ideals of C(Y). [ABSTRACT FROM AUTHOR]

Let (핏 , d , μ) be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and let X ⁢ (핏) be a ball quasi-Banach function space on 핏 . In this article, the authors introduce the weak Hardy space W ⁢ H ~ X ⁢ (핏) associated with X ⁢ (핏) via the Lusin area function. Then the authors characterize W ⁢ H ~ X ⁢ (핏) by the molecule, the grand maximal function, and the Littlewood–Paley g-function and g λ * -function. Moreover, all these results have a wide generality. Particularly, the results of this article are also new even when they are applied, respectively, to weighted Lebesgue spaces, Orlicz spaces, and variable Lebesgue spaces, which actually are new even on RD-spaces (that is, spaces of homogeneous type with additional reverse doubling condition). The main novelties of this article exist in that the authors take full advantage of the geometrical properties of 핏 expressed by both the dyadic cubes and the exponential decay of the approximations of the identity to overcome the difficulties caused by the deficiencies of both the explicit expression of the quasi-norm of X ⁢ (핏) and the reverse doubling condition of μ, and that the authors use the tent space on 핏 × ℤ to characterize W ⁢ H ~ X ⁢ (핏) by the Littlewood–Paley g λ * -function, where the range of λ might be best possible in some cases. [ABSTRACT FROM AUTHOR]

We prove that a pseudoconvex complex space X of pure dimension n + 1 {n+1} is holomorphically convex provided that its singular set has only compact connected components (e.g., X has isolated singularities) and rank x ⁡ 풪 X = n {\operatorname{rank}_{x}\mathscr{O}_{X}=n} for all smooth points x outside a compact set of X. This extends a known result due to Ohsawa asserting that a weakly 1-complete, smooth complex surface is holomorphically convex if it admits a globally defined non-constant holomorphic function. We also revise an example of Markoe concerning non-invariance of holomorphic convexity under normalization. [ABSTRACT FROM AUTHOR]

Let X be an arithmetic variety over the ring of integers of a number field K, with smooth generic fiber X K {X_{K}}. We give a formula that relates the dimension of the first Arakelov–Chow vector space of X with the Mordell–Weil rank of the Albanese variety of X K {X_{K}} and the rank of the Néron–Severi group of X K {X_{K}}. This is a higher-dimensional and arithmetic version of the classical Shioda–Tate formula for elliptic surfaces. Such an analogy is strengthened by the fact that we show that the numerically trivial arithmetic ℝ {\mathbb{R}} -divisors on X are exactly the linear combinations of principal ones. This result is equivalent to the non-degeneracy of the arithmetic intersection pairing in the argument of divisors, partially confirming a conjecture by H. Gillet and C. Soulé. [ABSTRACT FROM AUTHOR]

Let F denote the factorable matrix and X ∈ {ℓp, c0, c, ℓ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓp(F), ℓ∞), (ℓp(F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix. [ABSTRACT FROM AUTHOR]

Given a Kähler fiber space p : X → Y whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric induces a semipositively curved metric on the relative canonical bundle KX/Y of p. We also propose a conjectural generalization of this result for relative twisted Kähler–Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension). [ABSTRACT FROM AUTHOR]