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We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers k ≥ 0 and n ≥ 1 we consider the dimension k Vietoris–Rips persistence diagrams of all subsets of a given metric space with cardinality at most n. We call these invariants persistence sets and denote them as D n , k VR . We first point out that this family encompasses the usual Vietoris–Rips diagrams. We then establish that (1) for certain range of values of the parameters n and k, computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy's inequality. We also identify a rich family of metric graphs for which D 4 , 1 VR fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris–Rips persistence diagrams using Mayer–Vietoris sequences. These yield a geometric algorithm for computing the Vietoris–Rips persistence diagram of a space X with cardinality 2 k + 2 with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction. [ABSTRACT FROM AUTHOR]

Advancing the precision of low-gravity liquid propellant mass gauging is a roadmap challenge for NASA and a significant obstacle to expanding human presence in space. We report flight test data for an implementation of a low-gravity liquid propellant mass gauging technology that is both non-invasive and propellant agnostic. The TRIO payload experiment flew twice on the Blue Origin New Shepard suborbital rocket and validated computational models of liquid distribution in microgravity at three different fill levels. The Propellant Refueling and On-orbit Transfer Operations (PROTO) payload experiment also flew twice on New Shepard and acquired mass gauging data before, during, and after a tank drain and fill that simulated in-space propellant transfers. Propellant volume estimates were derived at 1 Hz using the Modal Propellant Gauging (MPG) system. Analytical, semi-empirical, and finite element (FE) model results for predicted mode frequencies are compared to flight data for mode frequencies across a range of fill levels during the simulated propellant transfer. Flight (0-g), ground (1-g) and FEM mode frequencies for both 0- and 1-g are well-predicted by the semi-empirical models for frequency dependence on tank fill level. • Modal Propellant Gauging (MPG) is a non-invasive and propellant agnostic liquid mass gauging technology. • Demonstrated high-accuracy low-gravity mass gauge during liquid transfer between two tanks. • Added mass effect shifts modal frequencies in predictable ways. • Modal Peak Tracking is capable of identifying liquid mass effects in fluid-loaded tanks. [ABSTRACT FROM AUTHOR]

Let $ (X, T) $ (X , T) be a linear dynamical system defined on a Banach space X and $ {L_T}:{K_{{X^ * }}} \to {K_{{X^ * }}} $ L T : K X ∗ → K X ∗ be defined by $ {L_T}S = TS $ L T S = TS , where $ {K_{{X^ * }}} = \overline {span\{ \langle \cdot,{y^ * } \rangle x:{y^ * } \in {X^ * },x \in X\} } $ K X ∗ = span { 〈 ⋅ , y ∗ 〉 x : y ∗ ∈ X ∗ , x ∈ X } ¯ . This paper first obtains the equivalence of the (syndetically) weak multi-sensitive compactness between $ (X, T) $ (X , T) and $ ({K_{{X^ * }}},{L_T}) $ (K X ∗ , L T). Then, it is shown that (1) there exists a non-mean-sensitive linear dynamical system $ (X ,T) $ (X , T) satisfying that $ ({K_{{X^ * }}},{L_T}) $ (K X ∗ , L T) is mean sensitive; (2) There exists a non-topologically-transitive and weakly multi-sensitive compact system, which is not syndetically weakly multi-sensitive compact. Besides, it is proved that every weakly mixing linear dynamical system defined on a separable Banach space is weakly multi-sensitive compact. [ABSTRACT FROM AUTHOR]

Konstantin Tsiolkovsky famously stated that while the Earth serves as the birthplace of humanity, it is not a place where mankind can indefinitely remain. Perhaps during that period, the prospect of exploring the mysteries of outer space appeared to be an unattainable aspiration. However, in the present day, there are no longer any limitations, not even the sky, since human ingenuity has facilitated access to outer space for humanity. This access is not just for the purposes of research and exploration but also for economic endeavours. Until now, the commercial utilisation of outer space has advanced at a very sluggish rate. However, firms including SpaceX, Orion Span, Virgin Galactic, and Blue Origin have achieved significant advancements in the growth of the space industry. The revenue generated by various space-related endeavours has experienced a significant 73% increase over the last ten years. The global space economy, estimated to be valued at USD one trillion in the coming years, is primarily driven by commercial activities. This presents a formidable challenge to the existing national and international taxation systems. Similar to the open seas, space is also considered res communis omnium, meaning it belongs to the entire community, and presents comparable taxing challenges with potentially uncertain solutions. The three fundamental elements of every taxation regulation, such as the Organisation for Economic Co-operation and Development or the United Nations Model Double Taxation Convention, are the taxpayer's place of residence, the origin of their income, and the methods by which they generate their money. The current tax system does not have the necessary concepts and provisions to adapt to the rapid advancements in commercial space technology. This paper examines the legal issues surrounding commercial activities conducted in space, including the nature and handling of the income generated in various legal systems. It also addresses concerns such as tax avoidance and excessive taxation, emphasising the necessity for a globally coordinated approach to effectively tax commercial activities in space. • Human ingenuity has facilitated access to outer space for humanity, for the purposes of research and exploration and also for economic endeavours. • Space-related endeavours has experienced a significant 73% increase over the last ten years. • Commercial activities conducted in space require concerted action at international level. [ABSTRACT FROM AUTHOR]

Until recently space travel was considered a privilege of governmental astronauts, with the exception perhaps of a few billionaire space tourists, who had been accommodated on the International Space Station on some rare occasions. Nonetheless, with major space pioneers announcing plans for space tourism activities and after the successful launch in orbit of space tourism vehicles from the Virgin Galactic and Blue Origin companies, academic interest about human spaceflight for purposes of leisure and recreation has ignited. Taking into account that space tourism is in fact an unregulated activity, this paper will firstly examine whether space tourism activities fall under the scope of space law or under the scope of international air law, depending on the planned trajectory of the space tourism vehicle. Furthermore, the authors will attempt to draw conclusions concerning the legal implications of space tourism activities through a comparative analysis with the 1974 Athens Convention on the Carriage of Passengers and the Carriage of their Luggage by Sea. This international convention provides a great example on the regulation of touristic ventures, as it refers to commercial touristic activities realized within a res communis omnium , i. e., the high seas, similarly with touristic activities in space. Moreover, said Convention provides for a specialized regime for touristic activities and is of particular interest, since it has achieved stability and a desirable balance of interests between private tourism operators and passengers. In this connection, the authors will evaluate the provisions of the 1974 Athens Convention in comparison with the space treaties applicable to space tourism activities. Furthermore, suggestions will be made as to whether it would be appropriate and feasible to introduce in the space law realm certain rules of law similar to those of the 1974 Athens Convention, in order to facilitate growth in the space tourism sector, without compromising the safety of human spaceflight for space passengers. • The 1974 Athens Convention in comparison with the space treaties applicable to space tourism activities. • Legal analogy between space law and maritime law. • Legal regime governing space tourism activities. [ABSTRACT FROM AUTHOR]

We consider an elliptic variational inequality with unilateral constraints in a Hilbert space X which, under appropriate assumptions on the data, has a unique solution u. We formulate a convergence criterion to the solution u, i.e. we provide necessary and sufficient conditions on a sequence $ \{u_n\}\subset X $ { u n } ⊂ X which guarantee the convergence $ u_n\to u $ u n → u in the space X. Then we illustrate the use of this criterion to recover well-known convergence results and well-posedness results in the sense of Tykhonov and Levitin–Polyak. We also provide two applications of our results, in the study of a heat transfer problem and an elastic frictionless contact problem, respectively. [ABSTRACT FROM AUTHOR]

A useful result of H. Rosenthal and J. Bourgain states that, given a Banach space X, an operator T:L_1[0,1]\to X is completely continuous if and only if its composition with the natural inclusion i_\infty :L_\infty [0,1] \to L_1[0,1] is compact. We extend this result to multilinear mappings on products of L_1[0,1] spaces, and consider also the composition with the natural inclusion i:C[0,1]\to L_1[0,1]. We show that a multilinear mapping on a product of L_1[0,1] spaces is completely continuous if and only if its associated polymeasure has a relatively norm compact range. [ABSTRACT FROM AUTHOR]

We consider an operator { S (t) } t ≥ 0 on a Banach space X with generator A, characterized by being an α -times-integrated C-regularized semigroup. The adjoint family S * (t) : X * → X * is introduced for analysis. { S * (t) } t ≥ 0 maintains the characteristics of an α -times-integrated C-regularized semigroup, though with strong continuity and Bochner integrals being substituted by weak* continuity and weak* integrals, respectively. Our investigation focuses on the closed subspace X ⊙ , where { S * (t) } t ≥ 0 exhibits strong continuity. Additionally, a comparison between the adjoint A * of A and the generator of the adjoint family is conducted. [ABSTRACT FROM AUTHOR]

Building on the recent release of a new Gaia-based blue straggler star catalog in Galactic open star clusters (OCs), we explored the properties of these stars in a cluster sample spanning a wide range in fundamental parameters. We employed Gaia EDR3 to assess the membership of any individual blue or yellow straggler to their parent cluster. We then made use of the ASteCA code to estimate the fundamental parameters of the selected clusters, in particular, the binary fraction. With all this at hand, we critically revisited the relation of the blue straggler population and the latter. For the first time, we found a correlation between the number of blue stragglers and the host cluster binary fraction and binaries. This supports the hypothesis that binary evolution is the most viable scenario of straggler formation in Galactic star clusters. The distribution of blue stragglers in the Gaia color-magnitude diagram was then compared with a suite of composite evolutionary sequences derived from binary evolutionary models that were run by exploring a range of binary parameters: age, mass ratio, period, and so forth. The excellent comparison between the bulk distribution of blue stragglers and the composite evolutionary sequences loci further supports the binary origin of most stragglers in OCs and paves the way for a detailed study of individual blue stragglers. [ABSTRACT FROM AUTHOR]

The drag coefficient Cd for a rigid and uniformly distributed rod canopy covering a sloping channel following the instantaneous collapse of a dam was examined using flume experiments. The measurements included space x and time t high resolution images of the water surface h(x, t) for multiple channel bed slopes So and water depths behind the dam Ho along with drag estimates provided by sequential load cells. Using these data, an analysis of the Saint-Venant equation (SVE) for the front speed was conducted using the diffusive wave approximation. An inferred C d = 0.4 from the h(x, t) data near the advancing front region, also confirmed by load cell measurements, is much reduced relative to its independently measured steady-uniform flow case. This finding suggests that drag reduction mechanisms associated with transients and flow disturbances are more likely to play a dominant role when compared to conventional sheltering or blocking effects on Cd examined in uniform flow. The increased air volume entrained into the advancing wave front region as determined from an inflow–outflow volume balance partly explains the Cd reduction from unity. [ABSTRACT FROM AUTHOR]

We characterize measure spaces such that the canonical map L ∞ → L 1 ∗ is surjective. In case of d -dimensional Hausdorff measure on a complete separable metric space X , we give two equivalent conditions. We give examples of X and d so that whether these conditions are met is undecidable in ZFC, including one with d equals the Hausdorff dimension of X. [ABSTRACT FROM AUTHOR]

Gálvez-Carrillo et al. [Decomposition spaces, incidence algebras and Möbius inversion III: The decomposition space of Möbius intervals, Adv. Math. 334 (2018) 544–584] constructed a decomposition space U of all Möbius intervals, as a recipient of Lawvere's interval construction for Möbius categories, and conjectured that U enjoys a certain universal property: for every Möbius decomposition space X , the space of culf functors from X to U is contractible. In this paper, we work at the level of homotopy 1-types to prove the first case of the conjecture, namely for locally discrete decomposition spaces. This provides also the first substantial evidence for the general conjecture. This case is general enough to cover all locally finite posets, Cartier–Foata monoids, Möbius categories and strict (directed) restriction species. The proof is 2-categorical. First, we construct a local strict model of U , which is then used to show by hand that the Lawvere interval construction, considered as a natural transformation, does not admit other self-modifications than the identity. [ABSTRACT FROM AUTHOR]

We call a real-valued function f on a topological space X a locally constant if each point x ∈ X has a neighborhood on which f is constant. The set of all locally constant functions f: X → ℝ is denoted by E(X). We observe that E(X) is the largest von Neumann regular subring of C(X) containing ℝ. We show that for every space X, there is a zero-dimensional space Y such that E(X) ≅ E(Y) and E(X) determines the topology of X if and only if X is a zero-dimensional space. By an example, it turns out that E(X) is not always of the form C(Y) for some space Y, but whenever X is a locally connected space, we observe that E(X) is a C(Y), where Y is a discrete space. Finally some von Neumann local subrings of C(X) containing ℝ are introduced and topological spaces X are characterized for which these subrings coincide with the smallest one. Note that a ring R is said to be a von Neumann local ring if for each a ∈ R either a or 1 − a has von Neumann inverse. [ABSTRACT FROM AUTHOR]

We consider local solutions u of nonlinear elliptic systems of the type div A (x , D u) = div F in Ω ⊂ R n , where u : Ω → R N is in a weighted W loc 1 , p space, with p ≥ 2 , F is in a weighted W loc 1 , 2 space and x → A (x , ξ) has growth coefficients in the space of functions with bounded mean oscillation. We prove higher differentiability of u in the sense that the nonlinear expression of its gradient V μ (D u) : = (μ 2 + | D u | 2 ) p - 2 4 D u , with 0 < μ ≤ 1 , is weakly differentiable with D (V μ (D u)) in a weighted L loc 2 space. Moreover we derive some local Calderón–Zygmund estimates when the source term is not necessarily differentiable. Global estimates for a suitable Dirichlet problem are also available. [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]

M. Daher [Canad. Math. Bull. 38 (1995), pp. 286–294] showed that if (X_0, X_1) is a regular couple of uniformly convex spaces then the unit spheres of the complex interpolation spaces X_{\theta } and X_{\eta } are uniformly homeomorphic for every 0 < \theta, \eta < 1. We show that this is a rather general phenomenon of the interpolation methods described by the discrete framework of interpolation of Lindemulder and Lorist [ A discrete framework for the interpolation of Banach spaces , https://arxiv.org/abs/2105.08373, 2021]. [ABSTRACT FROM AUTHOR]

Given a metric pair (X,A), i.e. a metric space X and a distinguished closed set A\subset X, one may construct in a functorial way a pointed pseudometric space \mathcal {D}_\infty (X,A) of persistence diagrams equipped with the bottleneck distance. We investigate the basic metric properties of the spaces \mathcal {D}_\infty (X,A) and obtain characterizations of their metrizability, completeness, separability, and geodesicity. [ABSTRACT FROM AUTHOR]

We give a pointwise version of sensitivity in terms of open covers for a semiflow (T, X) of a topological semigroup T on a Hausdorff space X and call it a Hausdorff sensitive point. If (X , U) is a uniform space with topology τ , then the definition of Hausdorff sensitivity for (T , (X , τ)) gives a pointwise version of sensitivity in terms of uniformity and we call it a uniformly sensitive point. For a semiflow (T, X) on a compact Hausdorff space X, these notions (i.e. Hausdorff sensitive point and uniformly sensitive point) are equal and they are T-invariant if T is a C-semigroup. They are not preserved by factor maps and subsystems, but behave slightly better with respect to lifting. We give the definition of a topologically equicontinuous pair for a semiflow (T, X) on a topological space X and show that if (T, X) is a topologically equicontinuous pair in (x, y), for all y ∈ X , then Tx ¯ = D T (x) where D T (x) = ⋂ { TU ¯ : for all open neighborhoods U of x }. We prove for a topologically transitive semiflow (T, X) of a C-semigroup T on a regular space X with a topologically equicontinuous point that the set of topologically equicontinuous points coincides with the set of transitive points. This implies that every minimal semiflow of C-semigroup T on a regular space X with a topologically equicontinuous point is topologically equicontinuous. Moreover, we show that if X is a regular space and (T, X) is not a topologically equicontinuous pair in (x, y), then x is a Hausdorff sensitive point for (T, X). Hence, a minimal semiflow of a C-semigroup T on a regular space X is either topologically equicontinuous or topologically sensitive. [ABSTRACT FROM AUTHOR]

We prove that: I. If L is a T 1 space, | L | > 1 and d (L) ≤ κ ≥ ω , then there is a submaximal dense subspace X of L 2 κ such that | X | = Δ (X) = κ . II. If c ≤ κ = κ ω < λ and 2 κ = 2 λ , then there is a Tychonoff pseudocompact globally and locally connected space X such that | X | = Δ (X) = λ and X is not κ + -resolvable. III. If ω 1 ≤ κ < λ and 2 κ = 2 λ , then there is a regular space X such that | X | = Δ (X) = λ , all continuous real-valued functions on X are constant (so X is connected) and X is not κ + -resolvable. [ABSTRACT FROM AUTHOR]

For a quasinilpotent operator T on a Banach space X , Douglas and Yang defined k x = lim sup z → 0 ln ⁡ ‖ (z − T) − 1 x ‖ ln ⁡ ‖ (z − T) − 1 ‖ for each nonzero vector x ∈ X , and call Λ (T) = { k x : x ≠ 0 } the power set of T. They proved that the power set have a close link with T 's lattice of hyperinvariant subspaces. In this paper, we compute the power set of some weighted shifts on ℓ p for 1 ≤ p < ∞. The following results are obtained: (1) If T is an injective quasinilpotent forward unilateral weighted shift on ℓ p (N) , then Λ (T) = { 1 } when k e 0 = 1 , where { e n } n = 0 ∞ be the canonical basis for ℓ p (N) ; (2) There is a class of backward unilateral weighted shifts on ℓ p (N) whose power set is [ 0 , 1 ] ; (3) There exists a bilateral weighted shift on ℓ p (Z) with power set [ 1 2 , 1 ]. [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]

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]

We define the concept of small subsets of asymptotic resemblance spaces and obtain some of their properties. In addition, we show that if the ideal of all small subsets of a metric space X coincides with the ideal of all bounded subsets of X, then the asymptotic dimension of X is equal to zero. We also show that a subset A of a proper metric space X is small, if and only if, the interior of the boundary of A in the Higson corona of X is empty. [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]

In this paper, we investigate the uniform exponential stability of the system dx(t)dt=Ax(t)−ρBx(t),(ρ>0),$$ \frac{dx(t)}{dt}&#x0003D; Ax(t)-\rho Bx(t),\kern3.0235pt \left(\rho >0\right), $$ where the unbounded operator A$$ A $$ is the infinitesimal generator of a linear C0−$$ {C}_0- $$semigroup of contractions S(t)$$ S(t) $$ in a Hilbert space X$$ X $$ and B$$ B $$ is a Desch–Schappacher operator. Then we give sufficient conditions for exponential stability of the above system. The obtained stability result is then applied to prove the uniform exponential stabilization of some bilinear partial differential equations. [ABSTRACT FROM AUTHOR]

We prove that any region \Gamma in a homogeneous n-dimensional and locally compact separable metric space X, where n\geq 2, cannot be irreducibly separated by a closed (n-1)-dimensional subset C with the following property: C is acyclic in dimension n-1 and there is a point b\in C\cap \Gamma having a special local base \mathcal B_C^b in C such that the boundary of each U\in \mathcal B_C^b is acyclic in dimension n-2. In case X is strongly locally homogeneous, it suffices to have a point b\in C\cap \Gamma with an ordinary base \mathcal B_C^b satisfying the above condition. The acyclicity means triviality of the corresponding Čech cohomology groups. This implies all known results concerning the separation of regions in homogeneous connected locally compact spaces. [ABSTRACT FROM AUTHOR]

Let X , Y be two Banach spaces and f : X → Y be a standard coarse isometry. In this paper, we first show a sufficient and necessary condition for the coarse left-inverse operator of general Banach spaces to admit a linearly isometric right inverse. Furthermore, by using the well-known simultaneous extension operator, we obtain an asymptotical stability result when Y is a space of continuous functions. In addition, we also prove that every coarse left-inverse operator does admit a linear isometric right inverse without other assumptions when Y is a L p (1 < p < ∞) space, or both X and Y are finite dimensional spaces of the same dimension. Making use of the results mentioned above, we generalize several results of isometric embeddings and give a stability result of coarse isometries between Banach spaces. [ABSTRACT FROM AUTHOR]

Recently, Ludewig and Thiang introduced a notion of a uniformly localized Wannier basis with localization centers in an arbitrary uniformly discrete subset D in a complete Riemannian manifold X. They show that, under certain geometric conditions on X, the class of the orthogonal projection onto the span of such a Wannier basis in the K-theory of the Roe algebra C*(X) is trivial. In this paper, we clarify the geometric conditions on X, which guarantee triviality of the K-theory class of any Wannier projection. We show that this property is equivalent to triviality of the unit of the uniform Roe algebra of D in the K-theory of its Roe algebra, and provide a geometric criterion for that. As a consequence, we prove triviality of the K-theory class of any Wannier projection on a connected proper measure space X of bounded geometry with a uniformly discrete set of localization centers. [ABSTRACT FROM AUTHOR]

We determine the sharp Bohr inequality for holomorphic functions defined on a balanced domain G of a complex Banach space X and takes values from the unit disk. Also, we investigate the Bohr-type inequalities for harmonic functions $ f=h+\bar {g} $ f = h + g ¯ defined on a balanced domain $ G\subset X $ G ⊂ X. [ABSTRACT FROM AUTHOR]

Let L be a truncated Archimedean vector lattice whose truncation is denoted by *. In a recent paper, we proved that there exists a locally compact Hausdorff space X such that L is a lattice isomorphic with a truncated vector lattice of functions in C∞(X) whose truncation is provided by meet with some characteristic function on X. This representation, no matter how interesting it is, has a major drawback, namely, C∞(X) need not be a vector lattice, unless X is extremally disconnected. The main purpose of this paper is to remedy this shortcoming by proving that, indeed, an extremally disconnected locally compact space X can be found such that L can be seen as an order dense vector sublattice of C∞(X) whose truncation is provided, again, by meet with some characteristic function on X. [ABSTRACT FROM AUTHOR]

It is well known that the bidual of C(X) for a compact space X, supplied with the Arens product, is isometrically isomorphic as a Banach algebra to for some compact space. The space is unique up to homeomorphism. We establish a similar result for realcompact spaces: The order bidual of C(X) for a realcompact space X, when supplied with the Arens product, is isomorphic as an f-algebra to for some realcompact space. The space is unique up to homeomorphism. [ABSTRACT FROM AUTHOR]

We continue to study (strong) property-(R1) in Banach spaces. As discussed by Pai & Nowroji in [On restricted centers of sets, J. Approx. Theory, 66(2), 170–189 (1991)], this study corresponds to a triplet , where X is a Banach space, V is a closed convex set, and is a subfamily of closed, bounded subsets of X. It is observed that if X is a Lindenstrauss space then has strong property-(R1), where represents the compact subsets of X. It is established that for any. This extends the well-known fact that a compact subset of a Lindenstrauss space X admits a nonempty Chebyshev center in X. We extend our observation that is Lipschitz continuous in if X is a Lindenstrauss space. If Y is a subspace of a Banach space X and represents the set of all finite subsets of BX then we observe that BY exhibits the condition for simultaneously strongly proximinal (viz. property-(P1)) in X for if satisfies strong property-(R1), where represents the set of all finite subsets of X. It is demonstrated that if P is a bi-contractive projection in ℓ∞, then exhibits the strong property-(R1), where represents the set of all compact subsets of ℓ∞. Furthermore, stability results for these properties are derived in continuous function spaces, which are then studied for various sums in Banach spaces. [ABSTRACT FROM AUTHOR]

We study the well-posedness of the fractional differential equations with finite delay: Dαu(t) + BDβu(t) = Au(t) + Fut + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces and periodic Besov spaces , where A and B are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(B), 0 ≤ β < α, α ≥ 1 and F is a bounded linear operator from Lp ([−2π, 0]; X) (resp. into X. We give necessary and sufficient conditions for the Lp -well-posedness and the -well-posedness of above equations by using known operator-valued Fourier multiplier theorems. [ABSTRACT FROM AUTHOR]

If u : Ω ⊂ R d → X is a harmonic map valued in a metric space X and E : X → R is a convex function, in the sense that it generates an EVI 0 -gradient flow, we prove that the pullback E ∘ u : Ω → R is subharmonic. This property was known in the smooth Riemannian manifold setting or with curvature restrictions on X , while we prove it here in full generality. In addition, we establish generalized maximum principles, in the sense that the L q norm of E ∘ u on ∂ Ω controls the L p norm of E ∘ u in Ω for some well-chosen exponents p ≥ q , including the case p = q = + ∞ . In particular, our results apply when E is a geodesically convex entropy over the Wasserstein space, and thus settle some conjectures of Brenier (Optimal transportation and applications (Martina Franca, 2001), volume 1813 of lecture notes in mathematics, Springer, Berlin, pp 91–121, 2003). [ABSTRACT FROM AUTHOR]

We investigate existence, uniqueness and approximation of solutions to stochastic delay differential equations (SDDEs) under Carathéodory-type drift coefficients. Moreover, we also assume that both drift f = f (t , x , z) and diffusion g = g (t , x , z) coefficient are Lipschitz continuous with respect to the space variable x , but only Hölder continuous with respect to the delay variable z. We provide a construction of randomized Euler scheme for approximation of solutions of Carathéodory SDDEs, and investigate its upper error bound. Finally, we report results of numerical experiments that confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

A subset A of a vector space X is called \alpha-lineable whenever A contains, except for the null vector, a subspace of dimension \alpha. If X has a topology, then A is \alpha-spaceable if such subspace can be chosen to be closed. The vast existing literature on these topics has shown that positive results for lineability and spaceability are quite common. Recently, the stricter notions of (\alpha,\beta)-lineability/spaceability were introduced as an attempt to shed light on more subtle issues. In this paper, among other results, we prove some general criteria for the notion of (\alpha,\beta)-lineability/spaceability and, as applications, we extend recent results of different authors. [ABSTRACT FROM AUTHOR]

We exhibit a Banach space Z failing the approximation property, for which there is an uncountable family \mathscr F of closed subideals contained in the Banach algebra \mathcal K(Z) of the compact operators on Z, such that the subideals in \mathscr F are mutually isomorphic as Banach algebras. This contrasts with the behaviour of closed ideals of the algebras \mathcal L(X) of bounded operators on X, where closed ideals \mathcal I \neq \mathcal J are never isomorphic as Banach algebras. We also construct families of non-trivial closed subideals contained in the strictly singular operators \mathcal S(X) for classical spaces such as X = L^p with p \neq 2, where pairwise isomorphic as well as pairwise non-isomorphic subideals occur. [ABSTRACT FROM AUTHOR]

In this paper we give a quantitative analysis of an explicit iteration method due to C.E. Chidume for the approximation of a zero of an m-accretive operator A : X → 2 X in Banach spaces which does not involve the computation of the resolvent of A. [ABSTRACT FROM AUTHOR]

Let (X , T) be a linear dynamical system, where X is a separable Banach space and T : X → X is a bounded linear operator. We show that if (X , T) is invertible, then (X , T) is weakly sensitive compact if and only if (X , T) is thickly weakly sensitive compact; and that there exists a system (X × Y , T × S) such that: (1) (X × Y , T × S) is cofinitely weakly sensitive compact; (2) (X , T) and (Y , S) are weakly sensitive compact; and (3) (X , T) and (Y , S) are not syndetically weakly sensitive compact. We also show that if (X , T) is weakly sensitive compact, where X is a complex Banach space, then the spectrum of T meets the unit circle. [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]

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]

We consider an irreducible positive-recurrent discrete-time Markov process on the state space X = ℤ + M × J , where ℤ + is the set of non-negative integers and J = { 1 , 2 , ... , n } . The number of states in J may be either finite or infinite. We assume that the process is a homogeneous quasi-birth-and-death process (QBD). It means that the one-step transition probability between non-boundary states (k , i) and (n , j) may depend on i , j , and n − k but not on the specific values of k and n . It is shown that the stationary probability vector of the process is expressed through square matrices of order n , which are the minimal non-negative solutions to nonlinear matrix equations. [ABSTRACT FROM AUTHOR]

We study the question for which Tychonoff spaces X and locally convex spaces E the space Cp(X,E)$C_p(X,E)$ of continuous E‐valued functions on X contains a complemented copy of the space (c0)p={x∈Rω:x(n)→0}$(c_0)_p=\lbrace x\in \mathbb {R}^\omega : x(n)\rightarrow 0\rbrace$, both endowed with the pointwise topology. We provide a positive answer for a vast class of spaces, extending classical theorems of Cembranos, Freniche, and Domański and Drewnowski, proved for the case of Banach and Fréchet spaces Ck(X,E)$C_k(X,E)$. Also, for given infinite Tychonoff spaces X and Y, we show that Cp(X,Cp(Y))$C_p(X,C_p(Y))$ contains a complemented copy of (c0)p$(c_0)_p$ if and only if any of the spaces Cp(X)$C_p(X)$ and Cp(Y)$C_p(Y)$ contains such a subspace. [ABSTRACT FROM AUTHOR]

A study of the class \Delta consisting of topological \Delta-spaces was originated by Jerzy Ka̧kol and Arkady Leiderman [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99; Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 267–280]. The main purpose of this paper is to introduce and investigate new classes \Delta _2 \subset \Delta _1 properly containing \Delta. We observe that for every first-countable X the following equivalences hold: X\in \Delta _1 iff X\in \Delta _2 iff each countable subset of X is G_{\delta }. Thus, new proposed concepts provide a natural extension of the family of all \lambda-sets beyond the separable metrizable spaces. We prove that (1) A pseudocompact space X belongs to the class \Delta _1 iff countable subsets of X are scattered. (2) Every regular scattered space belongs to the class \Delta _2. We investigate whether the classes \Delta _1 and \Delta _2 are invariant under the basic topological operations. Similarly to \Delta, both classes \Delta _1 and \Delta _2 are invariant under the operation of taking countable unions of closed subspaces. In contrast to \Delta, they are not preserved by closed continuous images. Let Y be l-dominated by X, i.e. C_p(X) admits a continuous linear map onto C_p(Y). We show that Y \in \Delta _1 whenever X \in \Delta _1. Moreover, we establish that if Y is l-dominated by a compact scattered space X, then Y is a pseudocompact space such that its Stone–Čech compactification \beta Y is scattered. [ABSTRACT FROM AUTHOR]

We consider the large class G of locally convex spaces that includes, among others, the classes of (D F) -spaces and (L F) -spaces. For a space E in class G we have characterized that a subspace Y of (E , σ (E , E ′)) , endowed with the induced topology, is analytic if and only if Y has a σ (E , E ′) -compact resolution and is contained in a σ (E , E ′) -separable subset of E. This result is applied to reprove a known important result (due to Cascales and Orihuela) about weak metrizability of weakly compact sets in spaces of class G. The mentioned characterization follows from the following analogous result: The space C (X) of continuous real-valued functions on a completely regular Hausdorff space X endowed with a topology ξ stronger or equal than the pointwise topology τ p of C (X) is analytic iff (C (X) , ξ) is separable and is covered by a compact resolution. [ABSTRACT FROM AUTHOR]

Let C be a nonempty closed bounded (not necessary convex) subset of a Banach space X and let T : C → C be an (αβ)-nonexpansive mapping with α > 0, β > 0 and α + β < 1. In this paper, we show that T has a unique fixed point. Moreover, T is a Picard operator if and only if T is asymptotically regular. [ABSTRACT FROM AUTHOR]