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In 2020, Olia et al. [Olia, Z. E. D. D., Gordji, M. E. and Bagha, D. E. (2020). Banach fixed point theorem on orthogonal cone metric spaces. FACTA Universitatis (NIS) Ser. Math. Inform, 35, 1239-1250] examined orthogonal cone metric spaces. They assumed that P is a normal cone with normal constant K and that self mapping T is orthogonal continuous on the orthogonal cone metric space X in their study. This study now presentes certain required definitions on orthogonal cone metric spaces that were not previously given in [9]. The examples that show the link between existing and new definitions have also been included. The results are also generalized by eliminating the normalcy condition and utilizing point orthogonal continuity instead of general orthogonal continuity in the major results of [9]. The fundamental finding of the study is then generalized by removing the requirement of orthogonal continuity and introducing normality. In addition, certain outcomes of stated theorems are proven, and some examples are provided to demonstrate these theorems. [ABSTRACT FROM AUTHOR]

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

INTRODUCTION: This paper describes the method for assigning medical diagnostic and treatment capabilities and resources to the database which assists with an updated probabilistic risk analysis (PRA) tool for exploration class medical system planning. The National Aeronautics and Space Administration has used PRA since 2011 to inform mission medical system design, but existing tools are designed only for low Earth orbit. An updated PRA tool was needed to assist with exploration class missions. METHODS: A team of medical experts with a wide range of expertise and experience, including Space Medicine, was assembled to build capability and resource tables for the new PRA tool. This team met over 8 mo and used practice guidelines, literature, and experience to build capability and resource tables (CRTs) for each condition in the new PRA tool database. RESULTS: This process led to CRTs for each condition and a total of 617 distinct capabilities and 839 discrete resources. CONCLUSION: The CRT method is an effective way to translate medical practice guidelines into capabilities and resources usable by PRA tools for exploration class medical system planning. This same method may be used in commercial space ventures and in other applications in which medical predictive analytics are informative. [ABSTRACT FROM AUTHOR]

In two previous papers [Katsanikas & Wiggins, 2021a, 2021b], we developed two methods for the construction of periodic orbit dividing surfaces for Hamiltonian systems with three or more degrees of freedom. We applied the first method (see [Katsanikas & Wiggins, 2021a]) in the case of a quadratic Hamiltonian system in normal form with three degrees of freedom, constructing a geometrical object that is the section of a 4D toroidal structure in the 5D energy surface with the space x = 0. We provide a more detailed geometrical description of this object within the family of 4D toratopes. We proved that this object is a dividing surface and it has the no-recrossing property. In this paper, we extend the results for the case of the full 4D toroidal object in the 5D energy surface. Then we compute this toroidal object in the 5D energy surface of a coupled quadratic normal form Hamiltonian system with three degrees of freedom. [ABSTRACT FROM AUTHOR]

Consider an infinite-dimentional Banach space denoted as X, and designate ∂(X) as the algebra of all bounded linear operators on X. Moreover, let s (A) denote the spectrum of ..., and ...(A)) indicate the boundary of s (A). A map ... is termed a ρ-spectrum if ... (A) for all A ... In this paper, we characterize all surjective maps f1 and f2 on ρ(X) satisfying ... = (AB) for all A,B(X). [ABSTRACT FROM AUTHOR]

Growth models with resources and environmental externalities typically assume that planet Earth is a closed economy. However, private firms like Blue Origin and SpaceX have reduced the cost of rocket launches by a factor of 20 over the last decade. What if these costs continue to decline, making mining from asteroids or the moon feasible? What would be the implications for economic growth and the environment? This paper provides stylized facts about cost trends, geology, and the environmental impact of mining on Earth and potentially in Space. We extend a neoclassical growth model to investigate the transition from mining on Earth to Space. We find that such a transition could potentially allow for continued growth of metal use, while limiting environmental and social costs on Earth. Acknowledging the high uncertainty around the topic, our paper provides a starting point for research on how Space mining could contribute to sustainable growth on Earth. [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]

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]

The object of this paper is to show that non-homotopy finite Poincaré duality spaces are plentiful. Let π be a finitely presented group. Assuming that the reduced Grothendieck group eK 0(Z[π]) has a non-trivial 2-divisible element, we construct a finitely dominated Poincaré space X with fundamental group π such that X is not homotopy finite. The dimension of X can be made arbitrarily large. Our proof relies on a result which says that every finitely dominated space possesses a stable Poincaré duality thickening. [ABSTRACT FROM AUTHOR]

This paper embarks on a journey through the history and evolution of space exploration, spanning from its 20th century origins in military and telecommunication endeavors to its current status as a global pursuit in the 21st century. It emphasizes the pressing need for contemporary and adaptable international laws and regulations to govern this increasingly vital field effectively. The competitive landscape of space exploration is highlighted, focusing on major players such as the US, Russia, and China, exploring their motivations and significant achievements in the cosmic race. The paper also examines the growing role of private space companies like SpaceX, Blue Origin, and iSpace, detailing their substantial contributions to commercial space endeavors, colonization efforts, and resource extraction. It sheds light on the escalating complexities arising from challenges like space debris and resource utilization, underscoring the critical importance of robust international regulations to govern the burgeoning commercial space industry. The paper discusses the urgent need for an updated global regulatory framework for space activities, emphasizing the importance of precise celestial object classifications and responsible resource utilization. It calls for strategic public-private partnerships, like NASA's collaboration with TransAstra's Apis program, and explores the potential of alternative fuels. [ABSTRACT FROM AUTHOR]

In the paper [ The Teichmüller theory of harmonic maps , J. Differential Geom. 29 (1989), no. 2, 449–479], Wolf provided a global coordinate system of the Teichmüller space of a closed oriented surface S with the vector space of holomorphic quadratic differentials on a Riemann surface X homeomorphic to S. This coordinate system is via harmonic maps from the Riemann surface X to hyperbolic surfaces. Moreover, he gave a compactification of the Teichmüller space by adding a point at infinity to each endpoint of harmonic map rays starting from X in the space. Wolf also showed this compactification coincides with the Thurston compactification. In this paper, we extend the harmonic map ray compactification to the case of punctured Riemann surfaces and show that it still coincides with the Thurston compactification. [ABSTRACT FROM AUTHOR]

As a step towards establishing homotopy-theoretic foundations for topological data analysis (TDA), we introduce and study homotopy interleavings between filtered topological spaces. These are homotopy-invariant analogues of interleavings, objects commonly used in TDA to articulate stability and inference theorems. Intuitively, whereas a strict interleaving between filtered spaces X and Y certifies that X and Y are approximately isomorphic, a homotopy interleaving between X and Y certifies that X and Y are approximately weakly equivalent. The main results of this paper are that homotopy interleavings induce an extended pseudometric d_{HI} on filtered spaces, and that this is the universal pseudometric satisfying natural stability and homotopy invariance axioms. To motivate these axioms, we also observe that d_{HI} (or more generally, any pseudometric satisfying these two axioms and an additional "homology bounding" axiom) can be used to formulate lifts of several fundamental TDA theorems from the algebraic (homological) level to the level of filtered spaces. Finally, we consider the problem of establishing a persistent Whitehead theorem in terms of homotopy interleavings. We provide a counterexample to a naive formulation of the result. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove the small data global existence of solutions to the semilinear damped wave equation$ \partial_t^2 u + Au + \partial_t u = |u|^{p-1}u $on a measure space $ X $ with a self-adjoint operator $ A $ on $ L^2(X) $. Under a certain decay estimate on the corresponding heat semigroup, we establish the linear estimates which generalize the so-called Matsumura estimates. Our approach is based on a direct spectral analysis analogous to the Fourier analysis. The self-adjoint operators treated in this paper include some important examples such as the Laplace operators on Euclidean spaces, the Dirichlet Laplacian on an arbitrary open set, the Robin Laplacian on an exterior domain, the Schrödinger operator, the elliptic operator, the Laplacian on Sierpinski gasket, and the fractional Laplacian. [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]

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]

Purpose: This paper aims to discuss the complexities and foresight of Mars colonization. There are many pioneers competing in a space race to Mars, for example, Elon Musk – SpaceX, Jeff Bezos – Blue Origin and Richard Branson – Virgin Orbit. The analyses are focused on the aerospace industry – the process of space adventures to Mars. Design/methodology/approach: This study offers new methodological approaches – the development of a complexity metric and system innovation mode – to analyze how the complexities relate to the systemic nature of innovation. The complexity metric and system innovation model can be applied in various industries. These analysis tools can help gain insights into the strategies for achieving the diffusion of commercial space. Findings: The analyses of findings have shown that, despite various attempts among the pioneers in a space race to colonize Mars (Elon Musk – SpaceX, Jeff Bezos – Blue Origin and Richard Branson – Virgin Orbit, among others), the aerospace industry has not yet reached a stage of commercialization. The commercial space to Mars is of low systemic nature at present. Many companies compete in a space race to develop technologies on a proprietary basis. However, the highest complexity level suggests a multinational and intergovernmental collaboration to achieve economies of scale and economies of scope as well as accelerate the process of technology diffusion – successful commercial space for the interplanetary settlement. Originality/value: The main contribution that shows originality and value of this paper is the development of a complexity metric and system innovation model which can be used to explore how the complexities relate to the systemic nature of innovation and how they relate to the strategies in managing technological innovations. The new methodological approaches can be used and applied to various industries. [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]

In this paper, we first prove that the space (X , ‖ ⋅ ‖) is separable if and only if for every ε ∈ (0 , 1) , there is a dense subset G of X ∗ and a w ∗ -lower semicontinuous norm ‖ ⋅ ‖ 0 of X ∗ so that (1) the norm ‖ ⋅ ‖ 0 is Frechet differentiable at every point of G and d F ‖ x ∗ ‖ 0 ∈ X is a w ∗ -strongly exposed point of B (X ∗ ∗ , ‖ ⋅ ‖ 0) whenever x ∗ ∈ G ; (2) (1 + ε 2 ) ‖ x ∗ ∗ ∗ ‖ 0 ≤ ‖ x ∗ ∗ ∗ ‖ ≤ (1 + ε) ‖ x ∗ ∗ ∗ ‖ 0 for each x ∗ ∗ ∗ ∈ X ∗ ∗ ∗ ; (3) there exists { x i ∗ } i = 1 ∞ ⊂ G such that ball-covering { B (x i ∗ , r i) } i = 1 ∞ of (X ∗ , ‖ ⋅ ‖ 0) is (1 + ε) − 1 -off the origin and S (X ∗ , ‖ ⋅ ‖) ⊂ ∪ i = 1 ∞ B (x i ∗ , r i) . Moreover, we also prove that if space X is weakly locally uniform convex, then the space X is separable if and only if X ∗ has the ball-covering property. As an application, we get that Orlicz sequence space l M has the ball-covering property. In this paper, we first prove that the space is separable if and only if for every , there is a dense subset of and a -lower semicontinuous norm of so that (1) the norm is Frechet differentiable at every point of and is a -strongly exposed point of whenever ; (2) for each ; (3) there exists such that ball-covering of is -off the origin and . Moreover, we also prove that if space is weakly locally uniform convex, then the space is separable if and only if has the ball-covering property. As an application, we get that Orlicz sequence space has the ball-covering property. [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]

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]

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]

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]

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]

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]

In this paper we investigate the homogeneous linear differential equation v'(t) = A(t)v(t) and the semi-linear differential equation v'(t) = A(t)v(t) + g(t, v(t)) in Banach space X, in which A: R → L(X) is a strongly continuous function, g: R × X → X is continuous and satisfies φ-Lipschitz condition. The first we characterize the exponential dichotomy of the associated evolution family with the homogeneous linear differential equation by space pair (E, E1), this is a Perron type result. Applying the achieved results, we establish the robustness of exponential dichotomy. The next we show the existence of stable and unstable manifolds for the semi-linear differential equation and prove that each a fiber of these manifolds is differentiable submanifold of class C¹. [ABSTRACT FROM AUTHOR]

When we wanted to further investigate the results of Bourgain–Pavlović, Koch–Tataru and Li–Xiao–Yang, we encountered the problem of stability of the parameter flow. For initial data in some space X , norm inflation implies that the solution does not belong to C (X). For BMO − 1 , Auscher–Dubois–Tchamitchian proved that Koch–Tataru's solution is stable in their defined space C 0 . In this paper, we consider Koch–Tataru's iteration space and construct a non-Gauss flow function to show that C 0 -stability may have meaning different to L ∞ (( BMO − 1) n) , which supports Chemin and Gallagher's point: well-posedness and norm inflation may have no conflict. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between a space X satisfying certain generalized metric properties and the Pixley-Roy hyperspace F[X] over X satisfying the same properties. We prove that if X has a s-point-finite cn-network (resp., ck-network), then F[X] also has a s-point-finite cn-network (resp., ck-network). [ABSTRACT FROM AUTHOR]

Space has always fascinated people. Many years have passed since the first spaceflight, and in addition to the enormous technological progress, the level of understanding of human physiology in space is also increasing. The presented paper aims to summarize the recent research findings on the influence of the space environment (microgravity, pressure differences, cosmic radiation, etc.) on the human body systems during short-term and long-term space missions. The review also presents the biggest challenges and problems that must be solved in order to extend safely the time of human stay in space. In the era of increasing engineering capabilities, plans to colonize other planets, and the growing interest in commercial space flights, the most topical issues of modern medicine seems to be understanding the effects of long-term stay in space, and finding solutions to minimize the harmful effects of the space environment on the human body. [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 equivariant coarse Baum--Connes conjecture interpolates between the Baum--Connes conjecture for a discrete group and the coarse Baum--Connes conjecture for a proper metric space. In this paper, we study this conjecture under certain assumptions. More precisely, assume that a countable discrete group Γ acts properly and isometrically on a discrete metric space X with bounded geometry, not necessarily cocompact. We show that if the quotient space X/Γ admits a coarse embedding into Hilbert space and Γ is amenable, and that the Γ-orbits in X are uniformly equivariantly coarsely equivalent to each other, then the equivariant coarse Baum--Connes conjecture holds for (X; Γ). Along the way, we prove a K-theoretic amenability statement for the Γ-space X under the same assumptions as above; namely, the canonical quotient map from the maximal equivariant Roe algebra of X to the reduced equivariant Roe algebra of X induces an isomorphism on K-theory. [ABSTRACT FROM AUTHOR]

This paper provides two new Velichko-like theorems for the weak counterpart of the locally convex space C k X of all real-valued functions defined on a Tychonoff space X equipped with the compact-open topology τ k . [ABSTRACT FROM AUTHOR]

If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is hyperbolic if there exists a constant δ ≥ 0 such that any side of any geodesic triangle in X is contained in the d-neighborhood of the union of the two other sides. In this paper, we study the hyperbolicity of an important kind of Euclidean graphs called Delaunay triangulations. Furthermore, we characterize the Delaunay triangulations contained in the Euclidean plane that are hyperbolic. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the class of n-quasi-A-(m, q)-isometry operators on a Banach space X, which represents a generalization of the n-quasi-(m, q)-isometry on a Banach space and the nquasi-(A, m)-isometry on a Hilbert space. After giving some basic properties of this class of operators, we study the product and the power of such operators in this class. [ABSTRACT FROM AUTHOR]

The main result of this paper is that, under PFA, for every regular space X with F(X) = \omega we have |X| \le w(X)^\omega; in particular, w(X) \le \mathfrak {c} implies |X| \le \mathfrak {c}. This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces X with F(X) = \omega such that w(X) = \mathfrak {c} and |X| = 2^\mathfrak {c}. We also show that regularity cannot be weakened to the Hausdorff property in this result because we can find in ZFC a Hausdorff space X with F(X) = \omega such that w(X) = \mathfrak {c} and |X| = 2^\mathfrak {c}. In fact, this space X has the strongly anti-Urysohn (SAU) property that any two infinite closed sets in X intersect, which is much stronger than F(X) = \omega. Moreover, any non-empty open set in X also has size 2^\mathfrak {c}, and thus our example answers one of the main problems of both Juhász, Soukup, and Szentmiklóssy [Topology Appl. 213 (2016), pp. 8–23] and Juhász, Shelah, Soukup, and Szentmiklóssy [Topology Appl. 323 (2023), Paper No. 108288, 15 pp.] by providing in ZFC a SAU space with no isolated points. [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]

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]

For a Tychonoff space X, the lattice UX was introduced in L. A. Pérez-Morales, G. Delgadillo-Piñón, and R. Pichardo-Mendoza, The lattice of uniform topologies on C(X), Questions and Answers in General Topology 39 (2021), 65-71. In the present paper we continue the study of UX. To be specific, the present paper deals, in its first half, with structural and categorical properties of UX, while in its second part focuses on cardinal characteristics of the lattice and how these relate to some cardinal functions of the space X. [ABSTRACT FROM AUTHOR]

According to intuition, two spaces X and Y are equivalent if they may be bent, shrunk, or expanded into one another. Spaces that are homotopy-equivalent to a point are called contractible. A vector space can be equipped with more than one norm. In this paper, the necessary and sufficient conditions for n-norms to be completely equivalent on linear n-Banach spaces are obtained. [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]

Self-commitments and increasing legal requirements lead to the compulsion to reduce carbon dioxide emissions in commercial aerospace. One viable approach is the reduction of structural mass, that would reduce emissions for every flight. This is especially true for high cadence singe aisle aircrafts. However, high quantities are required in this segment. To achieve these goals the newly developed low-melt Poly-Aryl-Ether-Ketone (LM-PAEK), a high-performance carbon fiber-reinforced thermoplastic composite was chosen to build a full-scale multi-functional fuselage demonstrator (MFFD) in order to delevelop an automated process that shows these savings. Single-step automated Fiber Placement (AFP) with in-situ consolidation offers distinct advantages in this field with its short curing times and especially by eliminating high amounts of waste otherwise caused by vacuum bagging and related tasks for post-consolidation in an autoclave. In order to ready this technology for future aircraft production this paper demonstrates how the processing window for LM-PAEK tape (TC1225) provided by TORAY was established. By closely linking robot and end-effector data with positionally accurate thermocouple measurements, the determined mechanical properties at coupon level and micro-sections of the manufactured specimens can be precisely correlated. This holistic approach is independent of the placement equipment and may enable global comparability within the community working in the field of thermoplastic AFP. In conclusion the procedure is evaluated and possible simplifications are discussed. [ABSTRACT FROM AUTHOR]

In this paper, the relationships between closure type properties of hyperspaces over a space X with that of covering properties of X are being investigated. Our main focus here is to study ai-properties of X and that of hyperspaces with hit-and-miss topology. Also some weak forms of Fréchet Urysohn property have been investigated in the hyperspace (Λ, τ+Δ). [ABSTRACT FROM AUTHOR]

The paper is motivated by E. Akin's book about dynamical systems and closed relations [ 1 ], and by J. Kennedy's and G. Erceg's recent paper about the entropy of closed relations on closed intervals [ 9 ]. In present paper, we introduce the entropy of a closed relation G on any compact metric space X and show its basic properties. We also introduce when such a relation G generates a periodic point or finitely generates a Cantor set. Then we show that periodic points, finitely generated Cantor sets, Mahavier products and the entropy of closed relations are preserved by topological conjugations. Among other things, this generalizes the well-known results about the topological conjugacy of continuous mappings. Finally, we prove a theorem, giving sufficient conditions for a closed relation G on [ 0 , 1 ] to have a non-zero entropy. Then we present various examples of closed relations G on [ 0 , 1 ] such that (1) the entropy of G is non-zero, (2) no periodic point or exactly one periodic point is generated by G , and (3) no Cantor set is finitely generated by G. [ABSTRACT FROM AUTHOR]

For two Banach spaces X and Y, we denote by K(X,Y) (resp. B(X,Y)) the space of all compact (resp. bounded) linear operators from X to Y. In this paper, we show that for 1\leq p,q<\infty, K(\ell _{p},\ell _{q}) is an 8-Lipschitz retract of B(\ell _{p},\ell _{q}). [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of finding the order 0<α<1$$ 0<\alpha <1 $$ in the fractional discrete system: C∇αun=Aun,n≥1,u0=x0,$$ \left\{\begin{array}{cll}{\kern0.1em }_C{\nabla}^{\alpha }{u}^n& =& A{u}^n,\kern0.30em n\ge 1,\\ {}{u}^0& =& {x}_0,\end{array}\right. $$where A$$ A $$ is a closed linear operator defined in a Banach space X,x0∈X$$ X,{x}_0\in X $$, and C∇αun$$ {\kern0.1em }_C{\nabla}^{\alpha }{u}^n $$ is the discrete Caputo fractional derivative of a given vector‐valued sequence (un)n∈ℕ0$$ {\left({u}^n\right)}_{n\in {\mathrm{\mathbb{N}}}_0} $$. [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 G = 핊d, d = 0, 1 or 3, act freely on a finitistic connected space X. This paper gives the cohomology classification of X if the mod 2 or rational cohomology of the orbit space X/G is isomorphic to the product of a projective space and sphere 픽ℙn קm, where 픽 = ℝ, ℂ or ℍ, respectively. For a free involution on X, a lower bound of covering dimension of the coincidence set of a continuous map f : X → ℝk is also determined. [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]