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

The aim of this paper is to investigate the existence of weak solutions for the coupled quasilinear elliptic system of gradient type$ \left\{ \begin{array}{ll} - {\rm div} (a(x, u, \nabla u)) + A_t (x, u,\nabla u) = g_1(x, u, v) &{\rm{ in}} \; \Omega ,\\ - {\rm div} (B(x, v, \nabla v)) + B_t (x, v,\nabla v) = g_2(x, u, v) &{\rm{ in}}\; \Omega ,\\ \quad u = v = 0 &{\rm{ on}}\;\partial\Omega , \end{array} \right. $where $ \Omega \subset \mathbb R^N $ is an open bounded domain, $ N \geq 2 $ and $ A(x,t,\xi) $, $ B(x,t, {\xi}) $ are $ \mathcal{C}^1 $–Carathéodory functions on $ \Omega \times \mathbb R \times { \mathbb R}^{N} $ with partial derivatives $ A_t = \frac{\partial A}{\partial t} $, $ a = {\nabla}_{\xi}A $, respectively $ B_t = \frac{\partial B}{\partial t} $, $ b = {\nabla}_{{\xi}}B $, while $ g_1(x,t,s) $, $ g_2(x,t,s) $ are given Carathéodory maps defined on $ \Omega \times \mathbb R\times \mathbb R $ which are partial derivatives with respect to $ t $ and $ s $ of a function $ G(x,t,s) $.We prove that, even if the general form of the terms $ A $ and $ B $ makes the variational approach more difficult, under suitable hypotheses, the functional related to the problem is bounded from below and attains its minimum in a 'right' Banach space $ X $.The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami–Palais–Smale condition and a suitable generalization of the Weierstrass Theorem. [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]

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 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]

A CT-scanner was used to scan 25 mm thick radiata pine boards during drying at 90 °C dry-bulb and 60 °C wet-bulb temperatures. Twenty transverse sections spaced by 10 mm along the length of the boards were CT-scanned six times during drying, and an elastic image registration technique was used to transform the CT-scanned images into moisture content profiles. The resulting images suggested that moisture content distributions did not always resemble diffusion-like moisture transfer mechanism, so common assumptions based on moisture content gradients as the main driving force for wood drying may not always be enough to explain the development of wet and dry zones coexisting below and above fiber saturation point. This paper discusses these findings based on both experimental CT-scanner data and the theory of water potential, which it is argued provides a more complete description of driving forces in wood drying. [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 this paper, Hyers–Ulam stability of the diffusion equation on the half space R + n + 1 with n ≥ 2 is investigated: ∂ u ∂ t (x , y , t) =△ u (x , y , t) where x ∈ R n , y>0, and t>0. Using integral transforms, precisely, Fourier transform with respect to the tangential space x and Laplace transform for time t, we obtain the generalized Hyers–Ulam stability of diffusion equation in the half space R + n + 1 involving special functions such as gamma and error functions. [ABSTRACT FROM AUTHOR]

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

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

In 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]

The aim of this paper is to introduce and solve the following p-radical functional equation related to Drygas mappings f (p√xp+yp) + f (p√xp−yp) = 2f(x) + f(y) + f(−y), x, y ∈ ℝ where f is a mapping from R into a vector space X and p ≥ 3 is an odd natural number. Using an analogue version of Brzdęk’s fixed point theorem [14], we establish some hyperstability results for the considered equation in ultrametric 2-Banach spaces. Also, we give some hyperstability results for the inhomogeneous p-radical functional equation related to Drygas mappings f (p√xp+yp) + f (p√xp−yp) = 2f(x) + f(y) + f(−y)+G(x,y) [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the long-time behavior of weak solutions for the three-dimensional primitive equations of large-scale ocean. Due to the low regularity of the triple nonlinearity term, it is very difficult to obtain the uniqueness of weak solutions such that we can not study the long-time behavior of weak solutions by using the classical theory of infinite-dimensional dynamical systems. Inspired by the idea of ℓ-trajectory in [1], we introduce a new phase space X ℓ (see Section 3.1 for the definition of X ℓ ) on which the translation semigroup { L (t) } t ≥ 0 can be imposed naturally. We first establish the finite-dimensional global attractor in X ℓ by using the classical theory of infinite-dimensional dynamical systems, via a Lipschitz continuous mapping from X ℓ into H(see Section 2 for the definition of H), we obtain the corresponding finite-dimensional global attractor in H. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and $T_{0}$ spaces X and Y , it is proved that the following three conditions are equivalent: (1) the Scott space $\Sigma \mathcal O(X)$ of the lattice of all open sets of X is H-sober; (2) for every H-sober space Y , the function space $\mathbb{C}(X, Y)$ of all continuous mappings from X to Y equipped with the Isbell topology is H-sober; (3) for every H-sober space Y , the Isbell topology on $\mathbb{C}(X, Y)$ has property S with respect to H. One immediate corollary is that for a $T_{0}$ space X , Y is a d -space (resp., well-filtered space) iff the function space $\mathbb{C}(X, Y)$ equipped with the Isbell topology is a d -space (resp., well-filtered space). It is shown that for any $T_0$ space X for which the Scott space $\Sigma \mathcal O(X)$ is non-sober, the function space $\mathbb{C}(X, \Sigma 2)$ equipped with the Isbell topology is not sober. The function spaces $\mathbb{C}(X, Y)$ equipped with the Scott topology, the compact-open topology and the pointwise convergence topology are also discussed. Our study also leads to a number of questions, whose answers will deepen our understanding of the function spaces related to H-sober spaces. [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]

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

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

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

This paper studies the bi-compact- G δ -open topology on the family C (X) of all continuous functions from a completely regular space X to the space ℝ. We compare this topology with open-point topology, bi-point-open topology and bi-compact-open topology and mention an example of a space asked in [P. Garg and S. Kundu, The compact- G δ -open topology on C (X) , Topology Appl. 159(8) (2012) 2082–2089] which satisfies C p (X) < C k z (X) < C k (X). We also investigate submertrizability, mertrizability, separability, Č ech-completeness and Baire property of bi-compact- G δ -open topology on C (X). [ABSTRACT FROM AUTHOR]

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

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

We establish a sharp reciprocity inequality for modulus in compact metric spaces X with finite Hausdorff measure. In particular, when X is also homeomorphic to a planar rectangle, our result answers a question of K. Rajala and M. Romney [Ann. Acad. Sci. Fenn. Math. 44 (2019), pp. 681-692]. More specifically, we obtain a sharp inequality between the modulus of the family of curves connecting two disjoint continua E and F in X and the modulus of the family of surfaces of finite Hausdorff measure that separate E and F. The paper also develops approximation techniques, which may be of independent interest. [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 a Banach function space (b.f.s. in short) X (Ω) on a n-dimensional bounded domain Ω with Lebesgue measure is considered. The Banach function space X (S) on the (n − 1) -dimensional surface S ⊂ Ω ¯ generated by the norm of the space X (Ω) is defined. The Sobolev function space W X m (Ω) is defined using the norm of the space X (Ω) , as well as the concept of the trace of an arbitrary function from this space on the surface S. Based on this concept, the space of traces W X m (S) is defined and a characterization of this space is given. It was proved that it is boundedly embedded in the space X (S). Particular cases of a functional space X (Ω) can be Lebesgue spaces, Lebesgue spaces with variable summability exponents, Morrey space, Orlicz space, grand Lebesgue spaces and etc. The obtained results allow us to consider boundary value problems for differential equations in b.f.s. W X m (Ω). [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 consider the Schrödinger equation with nonlinear derivative term. Our goal is to initiate the study of this equation with non vanishing boundary conditions. We obtain the local well posedness for the Cauchy problem on Zhidkov spaces X k (R) X k (R) and in ϕ + H k (R) ϕ + H k (R). Moreover, we prove the existence of conservation laws by using localizing functions. Finally, we give explicit formulas for stationary solutions on Zhidkov spaces. [ABSTRACT FROM AUTHOR]

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

In this paper, we 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]

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

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]

Let A be a unital C⁎-algebra equipped with its natural order. As usual the effect algebra of A is the interval { a ∈ A : 0 ≤ a ≤ I A } , where I A denotes the unit of A. In this paper, we give a complete description of order isomorphisms between effect algebras of C⁎-algebras of type C (X) ⊗ B (H) , where C (X) stands for the algebra of all continuous complex valued functions on a (non pathological) Hausdorff compact space X and B (H) denotes the algebra of all bounded linear operators on a complex Hilbert space H. Our results generalize some works by L. Molnár and P. Šemrl. [ABSTRACT FROM AUTHOR]

Given a norm ν on ℝ², the set of ν-Dirichlet improvable numbers DIν was defined and studied in the papers (Andersen and Duke, Acta Arith. 198 (2021) 37-75 and Kleinbock and Rao, Internat. Math. Res. Notices 2022 (2022) 5617-5657). When ν is the supremumnorm, DIν = BA ∪ ℚ, where BA is the set of badly approximable numbers. Each of the sets DIν, like BA, is of measure zero and satisfies the winning property of Schmidt. Hence for every norm ν, BA ∩ DIν is winning and thus has full Hausdorff dimension. In this article, we prove the following dichotomy phenomenon: either BA ⊂ DIν or else BA \ DIν has full Hausdorff dimension. We give several examples for each of the two cases. The dichotomy is based on whether the critical locus of ν intersects a precompact gt-orbit, where {gt} is the one-parameter diagonal subgroup of SL2(ℝ) acting on the space X of unimodular lattices in ℝ². Thus, the aforementioned dichotomy follows from the following dynamical statement: for a lattice Λ ∈ X, either gℝΛ is unbounded (and then any precompact gℝ>0-orbit must eventually avoid a neighborhood of Λ), or not, in which case the set of lattices in X whose gℝ>0 -trajectories are precompact and contain Λ in their closure has full Hausdorff dimension. [ABSTRACT FROM AUTHOR]

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

For a $T_0$ space X , let $\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets of X endowed with the Smyth order $\sqsubseteq$. $(\mathsf{K}(X), \sqsubseteq)$ (shortly $\mathsf{K}(X)$) is called the Smyth power poset of X. In this paper, we mainly discuss some basic properties of the Scott topology on Smyth power posets. It is proved that for a well-filtered space X , its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is still well-filtered, and a $T_0$ space Y is well-filtered iff the Smyth power poset $\mathsf{K}(Y)$ with the Scott topology is well-filtered and the upper Vietoris topology is coarser than the Scott topology on $\mathsf{K}(Y)$. A sober space Z is constructed for which the Smyth power poset $\mathsf{K}(Z)$ with the Scott topology is not sober. A few sufficient conditions are given for a $T_0$ space X under which its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is sober. Some other properties, such as local compactness, first-countability, Rudin property and well-filtered determinedness, of Smyth power spaces, and the Scott topology on Smyth power posets, are also investigated. [ABSTRACT FROM AUTHOR]

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

In this paper, we are concerned with a 3 × 3 block operator matrices acting in a Banach or Hilbert space X 1 × X 2 × X 3 given by A 1 B 1 C 1 A 2 B 2 C 2 A 3 B 3 C 3 , where the linear entries are assumed to be unbounded. We study the closure as well as the self-adjointness in the case where the linear operators A 2 and A 3 are A 1 -bounded, B 1 and B 3 are B 2 -bounded and C 1 and C 2 are C 3 -bounded. These results are of importance to a non-relativistic three-channel potential scattering model. [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]

The paper studies separation properties for subsets of the space N of normlinear functions on the Banach space X, i.e. the sum of a linear function and a multiple of the norm. We use as separating functionals the ones that are generated by the duality pairing [ x , p ] = p (x) = 〈 x , a 〉 + δ ‖ x ‖. We show that normlinear functions can be used to separate points from radiant or coradiant sets in X. Then we pass to separation defined by means of the pairing [ x , ⋅ ] , seen as a function defined on N . Since in this case the evaluation functionals are linear, separation describes a special subclass of convex sets in N . We characterize X-convex sets by exploiting the isomorphism between N and the space Y = X ∗ × R . We also study polarity relations defined on N . Polar sets are X-convex. And we describe what further properties are needed in order to make a subset C ⊂ Y the polar, or the reverse polar, of some set in X. For polar sets C ⊂ Y , it is possible to deduce the unique closed, radiant prepolar set U ⊂ X. To conclude, we emphasize some connections between upward (downward) X-convex sets in Y and sets of upper (resp. lower) bounds in Y. [ABSTRACT FROM AUTHOR]

In this paper, we study (ω , T) (ω , T) -periodic impulsive evolution equations via the operator semigroups theory in Banach spaces X X , where T : X → X T : X → X is a linear isomorphism. Existence and uniqueness of (ω , T) (ω , T) -periodic solutions results for linear and semilinear problems are obtained by Fredholm alternative theorem and fixed point theorems, which extend the related results for periodic impulsive differential equations. [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 A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and ρ , τ : I ⟶ A , S , T : I ⟶ B be maps on a non-empty set I whose ranges are closed under multiplication and contain exponential functions. In this paper, we first show that if ‖ S (p) ‖ Y = ‖ ρ (p) ‖ X , ‖ T (p) ‖ Y = ‖ τ (p) ‖ X and ‖ S (p) T (q) ‖ Y = ‖ ρ (p) τ (q) ‖ X , for all p , q ∈ I , then there exists a homeomorphism φ from Šilov boundary ∂ A of A onto ∂ B such that for each x ∈ ∂ A and p ∈ I , | S (p) (φ (x)) | = | ρ (p) (x) | and | T (p) (φ (x)) | = | τ (p) (x) | . Then we prove that, if for some ε ≥ 0 , Ran π (T (p) S (q)) is contained in an ε ‖ τ (p) ρ (q) ‖ -neighborhood of Ran π (τ (p) ρ (q)) , for all p , q ∈ I , then, under a certain condition, there exist continuous functions α , β ∈ B such that | α (φ (x)) T (p) (φ (x)) - τ (p) (x) | ⩽ 4 ε | τ (p) (x) | and | β (φ (x)) S (p) (φ (x)) - ρ (p) (x) | ⩽ 4 ε | ρ (p) (x) | , for all p ∈ I and x ∈ ∂ (A) . Our results can be applied for Banach algebras of Lipschitz functions. [ABSTRACT FROM AUTHOR]

In the present paper we identify those filtered probability spaces (Ω , F , (F n) , P) that determine already the martingale type of a Banach space X. We isolate intrinsic conditions on the filtration (F n) of purely atomic σ -algebras which determine that the upper ℓ p estimates ‖ f ‖ L p (Ω , X) p ⩽ C p (‖ E (f | F 0) ‖ L p (Ω , X) p + ∑ n = 1 ∞ ‖ E (f | F n) − E (f | F n − 1) ‖ L p (Ω , X) p) , f ∈ L p (Ω , X) imply that the Banach space X is of martingale type p. Our paper complements G. Pisier's investigation [12] and continues the work by S. Geiss and second named author in [3]. [ABSTRACT FROM AUTHOR]

Within the next decade it is likely that the space tourism industry will grow and the number of humans travelling into space via commercial entities such as Virgin Galactic and Blue Origin, will increase significantly. Current space tourism ventures focus on short duration sub-orbital flights and visits to Low Earth Orbit (LEO). In the next few decades, a journey into space could become as normal as a transatlantic flight. During these new commercial ventures, the effects of cosmic radiation exposure, especially during sudden changes in space weather, such as ground level enhancement (GLE) or solar particle events (SPEs), could have significant health implications for crew and passengers. Such changes in space weather could expose space tourists to radiation doses in excess of the recommended maximum 1 mSv yearly effective dose uptake for a member of the public and 20 mSv yearly effective dose limit for those working with radiation (ICRP Publication 103, 2007 [1]). Domestic legislation and regulation focussing on potential radiation exposure for space tourists is limited and largely untested; there is heavy focus on conventional risk and wider safety, with guidance stemming from regulation of commercial high-level flights, which are significantly different to space tourism enterprises. In this paper we consider the current domestic legislation and regulations adopted by the USA and the UK, as two examples of launch nations with legislation and regulation relating to space tourism activities. We acknowledge and consider feedback we have received from the UK Civil Aviation Authority (CAA) on current regulations and topics outlined in this paper. We discuss whether current legislation and regulation offers sufficient protection for space flight participants (space tourists), and whether risk is balanced appropriately between the operators who provide space tourism services and those taking part. Finally, we discuss the routes to acceptance of the radiation risks and make recommendations for legislators, regulators and operators to support them in ensuring that the risks are managed appropriately while also supporting the development of the industry. [ABSTRACT FROM AUTHOR]

Suppose that K is a compact Hausdorff space, S is a locally compact Hausdorff space and X is a Banach space with Schäffer constant \mathrm {S}(X). In this paper, we prove that if there is a map T from C(K) to C_{0}(S, X) satisfying \begin{equation*} \frac {1}{M} \|f-g\| \leq \|T(f)-T(g)\|\leq M \|f-g\|,\ \forall f, g \in C(K), \end{equation*} with 1\leq M^{2}<\mathrm {S}(X), then there exists a compact subset S_{0} of S and a continuous function \varphi from S_{0} onto K. This theorem on Lipschitz embeddings of C(K) into C_{0}(S, X) is the first nonlinear vector extension of the classical 1966 Holsztyński Theorem. Our result is optimal for many Banach spaces X including the spaces \ell _p^n, \ell _p and L_{p}([0,1]), 1 < p <\infty, n \geq 2, even when T is linear. The motivation to prove this result comes from the fact that it immediately yields a nontrivial lower bound for the C_{0}(S, X)-distortion of the class of all separable Banach spaces whenever S is a scattered space and \mathrm {S}(X)>1, namely \mathrm {S}(X) itself. [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]