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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]

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]

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]

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]

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

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

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]

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 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]

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]

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]

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]

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]

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]

In this paper, we construct global solutions to the Cauchy problem on a nonlinear Fokker-Planck equation near Maxwellian with small-amplitude initial data in Sobolev space H x 2 L v 2 by a refined nonlinear energy method. Compared with the results of Liao et al. (Global existence and decay rates of the solutions near Maxwellian for non-linear Fokker-Planck equations, J. Stat. Phys. , 173 (2018), 222–241.), the regularity assumption on the initial data is much weaker. [ABSTRACT FROM AUTHOR]

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

The purpose of this paper is to study the existence of mild solutions to a class of second order nonlinear evolution equations of the form u"(t) + A(u'(t)) + B(u(t)) 3 f (t), t E (0, T), 1 u(0) = uo, u'(0) = g(u'), where A : D(A) Ç X 2X is an m-accretive operator on a Banach space X, B : X X is a Lipschitz mapping, g : C([0,T]; X) X is a function and f G L1 (0,T, X). We obtain sufficient conditions for this problem to have at least a mild solution. [ABSTRACT FROM AUTHOR]

This study contains an algorithmic solution of the Sine Gordon equation in three space and time dimensional problems. For discretization, the central difference formula is used for the time variable. In contrast, space variable x, y, and z are discretized using the non-polynominal cubic spline functions for each. The proposed scheme brings the accuracy of order O(h2 + k2 + σ2 + τ2h2 + τ2 k2 + τ2σ2) by electing suitable parametric values. The paper also discussed the truncation error of the proposed method and obtained the stability analysis. Numerical problems are elucidated by this method and compared to results taken from the literature. [ABSTRACT FROM AUTHOR]

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

In order to ensure reliable and secure image exchange, chaotic systems are often considered for their good performance in information security. In this work, we first propose an extended version of a chaotic tent map (TM)—the multiparametric 1D tent map (MTM). The latter contains six control parameters defined over an unlimited range. These parameters strongly influence the MTM output when they are slightly modified by ∓ 10 − 13 , which makes MTM stronger than the existing TM and other 1D chaotic maps in terms of security key space. Then, this paper proposes a simple, yet powerful method to make uniform the distribution of chaotic sequence values, making the latter suitable for use in cryptosystems. Next, a new image cryptosystem is introduced based on MTM and parallel computing. This computing mode is incorporated to boost the security level of our scheme and to speed up its runtime. Indeed, in only one running round, our encryption scheme generates a security key of space equal to 10 78 × n with n indicating the number of the available CPU cores. Therefore, the suggested scheme achieves a good trade-off between safety and efficiency. The results of the performed comparisons and numerical experiments indicate on the one hand that MTM exhibits good chaotic characteristics in comparison to its original version. On the other hand, the suggested cryptosystem demonstrates good strength against various kinds of attacks (brute force, statistical, classical, noise, differential, etc.). Furthermore, comparison with similar schemes indicate that the proposed scheme is competitive in terms of execution time and superior in terms of security level. [ABSTRACT FROM AUTHOR]

This paper deals with extremally disconnected spaces and extremally disconnected P-spaces. A space X is said to be extremally disconnected if, for every open subset V of X, the closure of V in X is also an open set. P-spaces are spaces in which the intersection of countably many open sets is an open set. The authors present a new characterization of extremally disconnected spaces, and the extremally disconnected P-spaces, by means of selection theory. If X is either an extremally disconnected space or an extremally disconnected P-space, then the usual theorems of extension of real-valued continuous functions for a dense subset S of X can be deduced from our results. A corollary of our outcomes is that every nondiscrete space X of nonmeasurable cardinality has a dense subset S such that S is not C-embedded in X. [ABSTRACT FROM AUTHOR]

In this paper, we provide several Arzelà–Ascoli-type results on the space of all continuous functions from a Tychonoff space X into the fuzzy sets of R n , (F U S C B (R n) , H e n d) , which are upper semi-continuous and have bounded support endowed with the endograph metric. Namely, we obtain positive results when X is considered to be a k r -space and C (X , (F U S C B (R n) , H e n d)) is endowed with the compact open topology, as well as when we assume that X is pseudocompact and C (X , (F U S C B (R n) , H e n d)) is equipped with the uniform topology. [ABSTRACT FROM AUTHOR]

Let X be a linear space over K ∈ { R , C } , Y be a real or complex Banach space and f : X n → Y . With some fixed a j i , C i 1 ... i n ∈ K ( j ∈ { 1 , ... , n } , i , i k ∈ { 1 , 2 } , k ∈ { 1 , ... , n } ), we study, using the direct and the fixed point methods, the stability and the general stability of the equation f (a 11 x 11 + a 12 x 12 , ... , a n 1 x n 1 + a n 2 x n 2) = ∑ 1 ≤ i 1 , ... , i n ≤ 2 C i 1 ... i n f (x 1 i 1 , ... , x n i n ) , for all x j i j ∈ X ( j ∈ { 1 , ... , n } , i j ∈ { 1 , 2 } ). Our paper generalizes several known results, among others concerning equations with symmetric coefficients, such as the multi-Cauchy equation or the multi-Jensen equation as well as the multi-Cauchy–Jensen equation. We also prove the hyperstability of the above equation in m-normed spaces with m ≥ 2 . [ABSTRACT FROM AUTHOR]

At present, the research on the prediction of the remaining useful life (RUL) of machinery mainly focuses on multi-sensor feature extraction and then uses the features to predict RUL. In complex operations and multiple abnormal environments, the impact of noise may result in increased model complexity and decreased accuracy of RUL predictions. At the same time, how to use the sensor characteristics of time is also a problem. To overcome these issues, this paper proposes a dual-channel long short-term memory (LSTM) neural network model. Compared with the existing methods, the advantage of this method is to adaptively select the time feature and then perform first-order processing on the time feature value and use LSTM to extract the time feature and first-order time feature information. As the RUL curve predicted by the neural network is zigzag, we creatively designed a momentum-smoothing module to smooth the predicted RUL curve and improve the prediction accuracy. Experimental verification on the commercial modular aerospace propulsion system simulation (C-MAPSS) dataset proves the effectiveness and stability of the proposed method. [ABSTRACT FROM AUTHOR]

Let Baut1X and Baut1p be the Dold-Lashof classifying spaces of a space X and a fibration p : X → Y, respectively. In this paper, we give an example that there exists a fibration ξ : S7 X S11 X S15 X S19 → x p→ S5 such that Baut1X and Baut1p are not coformal and are rational H(2)-spaces. [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]

LetMD.M; be a metric space and let X be a Banach space. Let F be a set-valued mapping fromM into the family Km. of all compact convex subsets of X of dimension at most m. The main result in our recent joint paper with Charles Fefferman (which is referred to as a "finiteness principle for Lipschitz selections") provides efficient conditions for the existence of a Lipschitz selection of F, i.e., a Lipschitz mapping f WM ! X such that f .x/2 F.x/for every x 2 M. We give new alternative proofs of this result in two special cases. When m D 2, we prove it for X D R2, and when m D 1 we prove it for all choices of X. Both of these proofs make use of a simple reiteration formula for the "core" of a set-valued mapping F, i.e., for a mapping GWM ! Km.X/which is Lipschitz with respect to the Hausdorff distance, and such that G.x/- F.x/for all x 2 M. [ABSTRACT FROM AUTHOR]

Introduction: Since Gagarin became the first human to travel into space and complete one orbit around the Earth, on 12 April 1961, the number of manned spaceflights has increased significantly. Spaceflight is still complex and has potential risk for incidents and accidents. The aim of this study was to analyze how safe it is for humans to travel in space. Objectives: This paper, therefore, summarizes incidents and accidents covering the six decades of manned spaceflight (1961–2020). Material and methods: Extensive PubMed, Cochrane, and Google Scholar searches were made with search strings of "incidents", "accident", "spaceflight", and "orbit", and including all vehicles so far. Search terms were combined by AND or OR in search strings. Of the results obtained, studies which evaluated manned spaceflight were included in the study. Data from the National Aeronautics Space Agency (NASA), the Russian Space Agency (ROSCOSMOS), the European Space Agency (ESA), and the Chinese Space Agency (CNSA), as well as from the Virgin Galactic and the SpaceX databases, were searched to complete data and to identify all the accomplished manned spaceflights, as well as all incidents and accidents that have occurred in the specific period. Search results were compared to findings on Wikipedia, Encyclopedia Astronautica, and other public webpages. Reference lists of included articles/homepages were also included for further potential data. Results: From 1961–2020, our data revealed an increasing number of manned space flights, n = 327. The number of times an astronaut has been sent to space, n = 1294, resulted in an accumulated n = 19,414 days spent in space. The number of days spent in orbit has constantly increased from 1961 until today. The number of incidents (altogether n = 36) and accidents (altogether n = 5) has constantly decreased. The number of astronauts who have died during spaceflight is represented by n = 19. The current statistical fatality rate is 5.8% (deaths per spaceflight) with the highest fatality rate in the 1960s (0.013 deaths/day spent in space), and the lowest rates in the 1990s and the period from 2010 until the present (no deaths). The most dangerous phases of spaceflight are launch, landing and staying in orbit. Altogether, n = 12 incidents (incident rate per spaceflight: 0.04) and one accident (accident rate: 0.003) during launch have been reported, n = 9 incidents (incident rate: 0.03) and two accidents (accident rate: 0.006) have been reported during landing and n = 10 incidents (incident rate: 0.03) have been reported in orbit. Discussion: Manned spaceflight over the last six decades has become significantly safer. Since 2003, no astronaut fatality has been reported. With greater international cooperation and maintaining of the International Space Station (ISS), the number of manned spaceflights and days spent in space has constantly increased, with constantly lower rates of incidents and accidents. [ABSTRACT FROM AUTHOR]

In many research works Bouaouid et al. have proved the existence of mild solutions of an abstract class of nonlocal conformable fractional Cauchy problem of the form: d α x t / d t α = A x t + f t , x t , x 0 = x 0 + g x , t ∈ 0 , τ. The present paper is a continuation of these works in order to study the controllability of mild solution of the above Cauchy problem. Precisely, we shall be concerned with the controllability of mild solution of the following Cauchy problem d α x t / d t α = A x t + f t , x t + B u t , x 0 = x 0 + g x , t ∈ 0 , τ , where d α . / d t α is the vectorial conformable fractional derivative of order α ∈ 0 , 1 in a Banach space X and A is the infinitesimal generator of a semigroup T t t ≥ 0 on X. The element x 0 is a fixed vector in X and f , g are given functions. The control function u is an element of L 2 0 , τ , U with U is a Banach space and B is a bounded linear operator from U into X. [ABSTRACT FROM AUTHOR]

In this paper, we mainly discuss the angle modulus of convexity δ X a (ϵ) and the angle modulus of smoothness ρ X a (ϵ) in a real normed linear space X, which are closely related to the classical modulus of convexity δ X (ϵ) and the modulus of smoothness ρ X (ϵ) . Some geometric properties of the two moduli were investigated. In particular, we obtained a characterization of uniform non-squareness in terms of ρ X a (1) . Meanwhile, we studied the relationships between δ X a (ϵ) , ρ X a (ϵ) and other geometric constants of real normed linear spaces through some equalities and inequalities. Moreover, these two coefficients were computed for some concrete spaces. [ABSTRACT FROM AUTHOR]

A space X is said to be set star-Lindelöf if for each nonempty subset A of X and each collection U of open sets in X such that Ā⊑⋃u, there is a countable subset V of U such that A ⊑ St(⋃V; u). The class of set star-Lindelöf spaces lie between the class of Lindelöf spaces and the class of star-Lindelöf spaces. In this paper, we investigate the relationship between set star-Lindelöf spaces and other related spaces by providing some suitable examples and study the topological properties of set star-Lindelöf spaces. [ABSTRACT FROM AUTHOR]

In this paper, we construct a nonhomogeneous geometric quadratic stochastic operator generated by 2-partition ξ on countable state space X = Z*. The limiting behavior of such operator is studied. We have proved that such operator possesses the regular property. [ABSTRACT FROM AUTHOR]

Continuing investigations initiated by the first author, we associate relational structures for metric spaces and investigate their model theoretic properties. In this paper, we consider ultrametric spaces. Among others, we show that any elementary substructure of the relational structure associated with a totally bounded ultrametric space X is dense in X. Further, we provide an explicit upper bound for a splitting chain of atomic types in ultrametric spaces of a finite spectrum. For ultrametric spaces, these results improve previous ones of the present authors and may have further practical applications in designing similarity detecting algorithms. [ABSTRACT FROM AUTHOR]

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

In this paper, we present some unexpected answers to two problems with longstanding interest. More specifically, we show that a bounded primitive function of an asymptotically almost periodic function from R to a Banach space X is remotely almost periodic if and only if c 0 ⊄ X. This gives a natural analogy of the classical Bohl-Bohr-Amerio-Kadets theorem for asymptotically almost periodic functions. Based on this result, we succeed in extending the classical Loomis theorem in [13] to asymptotically almost periodic case without any ergodic conditions, and apply our Loomis type theorems to abstract Cauchy problems. Moreover, we give an example of inhomogeneous Schrödinger equation with asymptotically almost periodic coefficient, whose solution is not asymptotically almost periodic but remotely almost periodic. This demonstrates that remotely almost periodic functions is the natural class for solutions to some partial differential equations, which is similar to a remarkable phenomenon revealed by Shen and Yi (1998) [20] using dynamical system approach. [ABSTRACT FROM AUTHOR]

Let L be a non-negative self-adjoint operator acting on L 2 (X) where X is a space of homogeneous type with a dimension n. Suppose that the heat kernel of L satisfies the Davies-Gaffney estimates of order m ≥ 2. Let H L 1 (X) be the Hardy space associated with L. In this paper we obtain the sharp endpoint estimate for the Schrödinger group e i t L associated with L such that ‖ (I + L) − n / 2 e i t L f ‖ L 1 (X) + ‖ (I + L) − n / 2 e i t L f ‖ H L 1 (X) ≤ C (1 + | t |) n / 2 ‖ f ‖ H L 1 (X) , t ∈ R for some constant C = C (n , m) > 0 independent of t. We further apply our result to provide the sharp estimate for Schrödinger group of the Kohn Laplacian □ b on polynomial model domains treated by Nagel–Stein [41] , where e − t □ b satisfies only the second order Davies-Gaffney estimates. Moreover, when the heat kernel of L satisfies a Gaussian upper bound, by a duality and interpolation argument, it gives a new proof of a recent result of [13] for sharp endpoint L p -Sobolev bound for e i t L : ‖ (I + L) − s e i t L f ‖ L p (X) ≤ C (1 + | t |) s ‖ f ‖ L p (X) , t ∈ R , s ≥ n | 1 2 − 1 p | for every 1 < p < ∞ , which extends the classical results due to Miyachi ([39,40]) for the Laplacian on the Euclidean space R n. [ABSTRACT FROM AUTHOR]

. In the present paper proved that if for a given compact Hausdorff space X the hyperspace exp X is a contractible compact space then the space OSf (X) is also a contractible compact space. As a consequence it is established that the space OSf (X) of semi-additive functionals is absolute (neighbourhood) retract if and only if the hyperspace exp X is so. [ABSTRACT FROM AUTHOR]

Assume that (X , d , μ) is a metric measure space that satisfies a Q-doubling condition with Q > 1 and supports an L 2 -Poincaré inequality. Let be a nonnegative operator generalized by a Dirichlet form E and V be a Muckenhoupt weight belonging to a reverse Hölder class R H q (X) for some q ≥ (Q + 1) / 2 . In this paper, we consider the Dirichlet problem for the Schrödinger equation − ∂ t 2 u + u + V u = 0 on the upper half-space X × R + , which has f as its the boundary value on X. We show that a solution u of the Schrödinger equation satisfies the Carleson type condition if and only if there exists a square Morrey function f such that u can be expressed by the Poisson integral of f. This extends the results of Song-Tian-Yan [Acta Math. Sin. (Engl. Ser.) 34 (2018), 787-800] from the Euclidean space R Q to the metric measure space X and improves the reverse Hölder index from q ≥ Q to q ≥ (Q + 1) / 2 . [ABSTRACT FROM AUTHOR]

Let C a non-empty, bounded, closed and convex subset of a Banach space X, and denote by l∞ (C) the l∞-sum of C. In the present paper, by using the degree of nondensifiability (DND), we introduce the class of r-Δ-DND-contraction maps f : l∞(C) → X and prove that if f(l∞(C)) ⊂ C then there is some x* ∈ C with f(x*, x*, ?..., x*, ...) = x* Our result, in the specified framework, generalizes other fixed point results for the so called generalized r-contraction and even other existing fixed point result based on the DND. Also, we derive a new Krasnosel'skῐῐ-type fixed point result. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the notion of ij-continuity. We show that a function mapping one bitopological space X to another bitopological space Y is continuous if for every open subset of Y and the inverse of that subset is in open set X. Moreover, a function χ : X → Y is πλ-continuous if and only if for every open subset H of Y, χ −1 (H) is πλ-open in X. Furthermore, we give other new notions of continuity in bitopological spaces. [ABSTRACT FROM AUTHOR]

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