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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 indicate an error occurring in a few articles. This error concerns a certain measure of noncompactness in the space BC.RC x RC/. We show how this error could be corrected. Moreover, we propose a more convenient approach using the more general Fréchet space C.RC x RC/in which we indicate a simpler measure of noncompactness. [ABSTRACT FROM AUTHOR]

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

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 are devoted to the study of a problem proposed in Aujol and Dossal (SIAM J Optim 29:3131–3153, 2019), which concerns the optimal convergence rates for damped inertial gradient dynamics with flat geometries. Fortunately, we find a solution to the problem. More specifically, we obtain an optimal decay rate for the system energy of the second-order differential equation with time-dependent damping in a Hilbert space H x ¨ (t) + γ (t) x ˙ (t) + ∇ F (x (t)) = 0 , where F : H → R is a differentiable convex function possessing at least one minimizer and γ (t) = α / t with α > 0. It is known that this system, with the time-dependent damping coefficient γ (t) , models a nonlinear oscillator with viscous damping, and is associated with the Nesterov acceleration scheme, FISTA, or the accelerated gradient method in the area of numerical optimization. [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]

An operator T acting on a Banach space X obeys property (gt) if the isolated points of the spectrum σ(T) of T which are eigenvalues are exactly those points λ of the spectrum for which T −λ is an upper semi-B-Fredholm with index less than or equal to 0. In this paper we study the stability of property (gt) under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic operators commuting with T . Moreover, we study the transfer of property (gt) from a bounded linear operator T acting on a Banach space X and a bounded linear operator S acting on a Banach space Y to their tensor product T ⊗S. [ABSTRACT FROM AUTHOR]

This paper is concerned with difference equation $ y(n+1) = A_ny_n+f(n, \, y_{n+\delta_n}, \, \lambda) $ with infinite delay in a Banach space $ X $, where $ \delta_n $ ($ n\in {\mathbb Z} $) is a given sequence taking values in $ \{0, 1\} $, and $ \lambda $ is a parameter. First, we construct a Lipschitz invariant manifold $ {\mathcal M}^{\lambda} $ of the equation which consists of all bounded forward solutions. Then we prove that $ {\mathcal M}^{\lambda} $ contains a unique bounded complete solution $ \gamma^{\lambda} = \gamma^{\lambda}(n) $ ($ n\in {\mathbb Z} $) which depends on $ \lambda $ continuously. To understand the dynamical behavior of the equation on the manifold $ {\mathcal M}^{\lambda} $, we establish a discrete inequality with infinite delay. By applying this inequality we finally show that $ \gamma^{\lambda} $ forward attracts (exponentially attracts) all solutions on $ {\mathcal M}^{\lambda} $ provided that the Lipschitz constant of $ f $ is sufficiently small. Smoothness of the manifold $ {\mathcal M}^\lambda $ is also addressed. [ABSTRACT FROM AUTHOR]

Let ℰ (X) denote the group of self-homotopy equivalences of an elliptic space X of formal dimension n. In this paper, we show that if ℰ (X) is finite, then ℰ (X) is isomorphic to a certain subgroup of ℰ (X n − 1). Moreover, we define the subgroup ℬ n (X) of aut (π n − 1 (X)) and we show that ℬ n (X) ≅ ℤ 2 provided that n is odd. Here X n − 1 denotes the (n − 1) -skeleton of X. [ABSTRACT FROM AUTHOR]

In the previous paper, we have defined together with I. Ionică the heights of a nontrivial triangle with respect to Birkhoff orthogonality in a real smooth space X, dimX = 2: In the present paper, we remark that, generally, the area of a nontrivial triangle in X has not the same value for different heights of the triangle. The purpose of this paper is to characterize the norm of X if this space has the property that the area of any triangle is well defined (independent of considered height). In this line we give five equivalent properties using the directional derivative of the norm. If X is strictly convex and dimX = 3, then each of these five properties characterizes the hilbertian norms (generated by inner products). [ABSTRACT FROM AUTHOR]

In this paper, we describe how intersections with a totality of some ideals affect the essentiality of an ideal. We mainly study intersections with every (a) annihilator ideal, (b) prime ideal (c) strongly irreducible ideal (d) irreducible ideal and every pure ideal. After some general results, the paper focuses on C (X) to characterize spaces X when every irreducible ideal of C (X) is pseudoprime. We also characterize the rings of continuous functions C (X) in which every pseudoprime ideal is strongly irreducible. We give a negative answer to a question raised by Gilmer and McAdam. [ABSTRACT FROM AUTHOR]

BACKGROUND: The acoustic environment in space flight vehicles and habitats remains a major operational concern. Risks associated with noise, radiation, and ototoxin exposure include reduced speech intelligibility and permanent shifts in hearing thresholds. This presentation assesses key elements of the NASA Hearing Conservation Program (HCP), reviews data availability in critical areas, and provides a prioritized list of recommendations for future study. OVERVIEW: Data collected in terrestrial environments primarily inform auditory damage risk criteria and recommendations on the use of hearing protective devices. A comprehensive understanding of short- and long-term changes in hearing sensitivity in spacecraft environments is required to mitigate unique habitability concerns and safety risks. Noise level monitoring and periodic audiometric testing are critical components of the HCP, but whether existing approaches can sufficiently characterize changes in hearing sensitivity and its impact on communication must be rigorously evaluated. This presentation discusses 1) the extent to which pure-tone audiometry can detect changes in hearing sensitivity affecting speech discrimination; 2) the limitations of existing retrospective studies investigating transient and permanent shifts in hearing thresholds; and 3) NASA Johnson Space Center (JSC) data availability and reporting. A workshop composed of relevant specialties across military and civilian spheres assembled primary peer-reviewed papers and JSC data in the following areas: pre- and post- flight audiograms, On-Orbit Hearing Assessments, Sound Level Meter measurements, and Acoustic Dosimeter measurements. Data quality and accessibility were evaluated, and reviewers generated a prioritized list of recommendations for further research. DISCUSSION: NASA's Hearing Conservation Program and effective programs in other professional disciplines use accurate, scientifically-based standards and countermeasures. Increasing interest in commercial and long-term space flight expands the variety of astronaut medical and physical conditions requiring accommodation, establishing the need to rigorously characterize the impact of the acoustic environment on auditory health and safety. This work is of broad interest to professionals who may be in a position to contribute to the scientific knowledge base, increase data accessibility, or implement additional testing and countermeasures in this area. Learning Objectives 1. Recognize the impact of acoustic conditions in space flight vehicles on auditory health and communication and review current approaches to hearing conservation. 2. Understand current methodologies and data utilized in monitoring auditory health in space environments, recognizing areas of potential improvement and further research needs. [ABSTRACT FROM AUTHOR]

In this paper we continue the work that we began in [ 6 ]. Given 1 < p < N 1 < p < N , two measurable functions V (r) ≥ 0 V (r) ≥ 0 and K (r) > 0 K (r) > 0 , and a continuous function A (r) > 0 A (r) > 0 (r > 0 r > 0), we consider the quasilinear elliptic equation − d i v (A (| x |) | ∇ u | p − 2 ∇ u) + V (| x |) | u | p − 2 u = K (| x |) f (u) in R N , − d i v (A (| x |) | ∇ u | p − 2 ∇ u) + V (| x |) | u | p − 2 u = K (| x |) f (u) in R N , where all the potentials A , V , K A , V , K may be singular or vanishing, at the origin or at infinity. We find existence of nonnegative solutions by the application of variational methods, for which we need to study the compactness of the embedding of a suitable function space X X into the sum of Lebesgue spaces L q 1 K + L q 2 K L K q 1 + L K q 2. The nonlinearity has a double-power super p p -linear behavior, as f (t) = min { t q 1 − 1 , t q 2 − 1 } f (t) = min { t q 1 − 1 , t q 2 − 1 } with q 1 , q 2 > p q 1 , q 2 > p (recovering the power case if q 1 = q 2 q 1 = q 2). With respect to [ 6 ], in the present paper we assume some more hypotheses on V V , and we are able to enlarge the set of values q 1 , q 2 q 1 , q 2 for which we get existence results. [ABSTRACT FROM AUTHOR]

This paper deals with the solvability in the semilinear abstract evolution equation with countable time delays, $ \begin{equation*} \begin{cases} \dfrac{du}{dt}(t)+Au(t) = F(u(t), (u(t-\tau_n))_{n\in\mathbb{N}}), & t>0, \\ u(t) = u_0(t), & t \in \bigcup\limits_{n \in \mathbb{N}}[-\tau_n,0], \end{cases} \end{equation*} $ in a Banach space $ X $ , where $ -A $ is a generator of a $ C_0 $ -semigroup with exponential decay and $ F: X \times X^\mathbb{N} \to X $ is Lipschitz continuous. Nicaise and Pignotti (J. Evol. Equ.; 2018;18;947–971) established global existence and exponential decay in time for solutions of the above equation with finite time delays in Hilbert spaces under global or local Lipschitz conditions. The purpose of the present paper is to generalize the result to the case of countable time delays in Banach spaces under a global Lipschitz condition. [ABSTRACT FROM AUTHOR]

Given a topological dynamical system (X,T), i.e., T is a continuous transformation on a compact metric space X), that admits mistakes and an invariant measure, this paper proves the Brin–Katok formula in this case. In particular, this paper finds that it suffices to study a mistake function which is linear in n and monotonic with respect to Δ. Consequently, this paper shows the Brin–Katok formula for the mean Bowen ball replacing the Bowen metrics with the mean Bowen metrics. The results presented in this paper may have some applications in the study of other properties of a dynamical system which admits small errors. [ABSTRACT FROM AUTHOR]

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

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

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

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

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]

In this paper we establish some properties of the multivalued mapping (x, d) → DC (x, d) that associates to every element x of a linear normed space X the set of linear continuous functionals of norm d ≥ 0 and which separates the closed ball B (x, d) from a closed convex set C ⊂ X. Using this mapping we give links with other important concepts in convex analysis (ε-approximation element, ε-subdifferential of distance function, duality mapping, polar cone). Thus, we establish a dual characterization of ε-approximation elements with respect to a nonvoid closed convex set as a generalization of a known result of Garkavi. Also, we give some properties of univocity and monotonicity of mapping DC. [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 stochastic Cahn–Hilliard–Navier–Stokes equations in two-dimensional spaces. Applying the Maslowski–Seidler method, we establish the existence of invariant measure in state space L x 2 × H 1 with the weak topology. We also prove the existence of global pathwise solutions using the stochastic compactness argument. [ABSTRACT FROM AUTHOR]

In this paper, we consider the spectral theory. More precisely, we study the spectral families and their corresponding operators on reflexive Banach spaces. Assume A is a well-bounded operator on reflexive Lebesgue spaces then the operator A is a scalar-type spectral operator. We establish that if a weak spectral family E (λ) is concentrated on [ a , b ] then there exists a linear well-bounded operator A ∈ L (X) on the reflexive Banach space X , such that 〈 A (x) , y * 〉 = b 〈 x , y * 〉 − ∫ [ a , b ] 〈 x , E (λ) y * 〉 d λ holds for all x ∈ X and y * ∈ X * . We prove that by assuming (Φ , X) is a functional calculus on the measurable space (Z , Σ) , then there are a semi-finite measure space (Ω , F , μ) and solitary operator U : X → L p (Ω , F , μ) , and an injective pointwise continuous *-homomorphism F : M (Z , Σ) → M (Ω , F) , such that Φ (f) = U M F f U − 1 holds for any operator of the multiplication M F f that multiplies by function f. [ABSTRACT FROM AUTHOR]

This paper is concerned with the propagation phenomena in a diffusion system with the Belousov–Zhabotinskii chemical reaction in ℝ N under the bistability assumption. We prove that there is a new type of entire solution originated from three moving planar traveling fronts, and evolved to a V-shaped traveling front as time changes, which means that the profile of this solution is not invariant at all. Here an entire solution is referred to a solution that is defined for all time t  ∈  ℝ and in the whole space x  ∈  ℝ N . Furthermore, we show that not only the entire solution but also all transition fronts share the same global mean speed by constructing suitable radially symmetric expanding and retracting super- and subsolutions. [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]

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

In this paper we show that the equivalence among the classical quasi-norms of the generalized Orlicz spaces XΦ — the Orlicz quasi-norm, the Luxemburg quasi-norm and the Amemiya quasi-norm — holds under some mild conditions on the underlying quasi-Banach function space X — mainly the weak Fatou property — improving previous results of [4] for which some lattice convexity requirements for the quasi-Banach function space X were needed. [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 note is of supplementary nature to our previous paper [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. Let S be the union of the nodal cubic curve and its three inflectional tangents in the complex projective plane ℙ 2 . Such S has appeared as the singular locus of certain hypergeometric system introduced in [J. Kaneko, K. Matsumoto and K. Ohara, A system of hypergeometric differential system in two variables of rank 9,Int. J. Math. 28 (2017) 1750015], and we have given generators and defining relations of the fundamental group of X = ℙ 2 ∖ S [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. X has a 9-fold Galois covering space X ̃ given by the complement of the Hesse configuration of 12 lines in ℙ 2 . Hence one can apply the method of Reidemeister–Schreier to derive a finite presentation of π 1 (X ̃) , which we carry out in this note. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a concept of norm-attainment in the projective symmetric tensor product of a Banach space X, which turns out to be naturally related to the classical norm-attainment of N -homogeneous polynomials on X. Due to this relation, we can prove that there exist symmetric tensors that do not attain their norms, which allows us to study the problem of when the set of norm-attaining elements in is dense. We show that the set of all normattaining symmetric tensors is dense in for a large set of Banach spaces such as Lp-spaces, isometric L1-predual spaces or Banach spaces with monotone Schauder basis, among others. Next, we prove that if X* satisfies the Radon-Nikodým and approximation properties, then the set of all norm-attaining symmetric tensors in is dense. From these techniques, we can present new examples of Banach spaces X and Y such that the set of all norm-attaining tensors in the projective tensor product is dense, answering positively an open question from the paper [10]. [ABSTRACT FROM AUTHOR]

In this paper, we show that the density in energy of Lipschitz functions in a Sobolev space N 1 , p (X) holds for all p ∈ [ 1 , ∞) whenever the space X is complete and separable and the measure is Radon and positive and finite on balls. Emphatically, p = 1 is allowed. We also give a few corollaries and pose questions for future work. The proof is direct and does not involve the usual flow techniques from prior work. It also yields a new approximation technique, which has not appeared in prior work. Notable with all of this work is that we do not use any form of Poincaré inequality or doubling assumption. The techniques are flexible and suggest a unification of a variety of approaches that have appeared in the literature on the topic. [ABSTRACT FROM AUTHOR]

For bounded linear operators A, B, C and D on a Banach space X, we show that if BAC = BDB and CDB = CAC then I — AC is generalized Drazin—Riesz invertible if and only if I — BD is generalized Drazin—Riesz invertible, which gives a positive answer to Question 4.9 in Yan, Zeng and Zhu [Complex Anal. Oper. Theory 14, Paper No. 12 (2020)]. In particular, we show that Jacobson's lemma holds for generalized Drazin—Riesz inverses. [ABSTRACT FROM AUTHOR]

This paper deals with the well‐posedness for the fully nonlinear functional differential equation u′(t)∈A(t)ut in a general Banach space X, where {A(t);t∈[a,b)} is a family of operators whose domains are subsets of the so‐called initial‐history space X and whose ranges are subsets of the space X. The special case where A(t)ϕ=B(t)ϕ(0)+G(t,ϕ) for t∈[a,b) and ϕ∈X with ϕ(0)∈D(B(t)) has been extensively studied so far, but there has not been a satisfactory solution to the flow invariance problem. This paper establishes the well‐posedness for the fully nonlinear functional differential equations and solves the above‐mentioned problem on flow invariance. [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]

Let P be a topological property. A space X is said to be star P if whenever is an open cover of X, there exists a subspace A ⊆ X with property P such that , where. In this paper, we investigate the relationships among star Lindelöf spaces, star almost Lindelöf spaces and star weakly Lindelöf spaces, and also study topological properties of star weakly Lindelöf spaces. [ABSTRACT FROM AUTHOR]

Let X be a K3 surface, and g be a primitive non-symplectic automorphism of X of order 3, 4, or 6. We write the quotient space as X / 〈 g 〉 and the quotient morphism as p : X → X / 〈 g 〉 . In this paper, we study curves on X obtained by pulling back curves on X / 〈 g 〉 by p. As a result, we show that if X / 〈 g 〉 is smooth, then X / 〈 g 〉 is the projective plane ℙ 2 or Hirzebruch surfaces F l of degree l = 0 , 1 , 2 , 3 , 4 , 6 , or 12. In addition, we show that if X / 〈 g 〉 is smooth, then X / 〈 g 〉 is determined by the set of fixed points of g and the action of g on rational curves on X. [ABSTRACT FROM AUTHOR]

In this paper, we recognize the finite propagation speed of the solution for the relativistic Bhatnagar-Gross-Krook model of the Marle-type near equilibrium regime in the whole space R x 3 . The global solutions vanish outside a modified line cone ⟨x⟩ = aMt with a, M > 1. Moreover, one can see that the slope of the modified line cone aM can be as close as to the maximum speed of the relativistic transport part. This means that the propagation speed is optimal. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new properties of (ω , c) -periodic and asymptotically (ω , c) -periodic functions, then we apply them to study the existence and uniqueness of mild solutions of these types to the following semilinear fractional differential equations: (1) { c D t α u (t) = A u (t) + c D t α − 1 f (t , u (t)) , 1 < α < 2 , t ∈ R , u (0) = 0 and (2) { c D t α u (t) = A u (t) + c D t α − 1 f (t , u (t − h)) , 1 < α < 2 , t , h ∈ R + , u (0) = 0 where c D t α (⋅) (1 < α < 2) stands for the Caputo derivative and A is a linear densely defined operator of sectorial type on a complex Banach space X and the function f (t , x) is (ω , c) -periodic or asymptotically (ω , c) -periodic with respect to the first variable. Our results are obtained using the Leray–Schauder alternative theorem, the Banach fixed point principle and the Schauder theorem. Then we illustrate our main results with an application to fractional diffusion-wave equations. [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]

A space X is said to be star-σ-countably compact if for every open cover U of X, there exists a σ-countably compact subset C of X such that St(C, U) = X. In this paper, we investigate the relationship between star-σ-countably compact spaces and other related spaces and also study the topological properties of star-_-countably compact spaces. [ABSTRACT FROM AUTHOR]

Given a closed convex set A in a Banach space X, motivated by the continuity and Fréchet differentiability of A introduced, respectively, in [D. Gale and V. Klee, Math. Scand., 7 (1959), pp. 379-391] and [X. Y. Zheng, SIAM J. Optim., 30 (2020), pp. 490-512], this paper considers the Cp-differentiability, subdifferentiability, and G\^ateaux differentiability of A. Using the technique of variational analysis, it is proved that A is Cp-differentiable (resp., subdifferentiable or G\^ateaux differentiable) if and only if for every continuous convex function f : X → R with infx∈A f(x) > infx∈X f(x) the corresponding constrained optimization problem PA(f) is 2/p-order-well-posed solvable (resp., generalized well-posed solvable or weak well-posed solvable). It is also proved that if the conjugate function f* of a continuous convex function f on X is Cp-differentiable on dom(f*), then for every closed convex set A in X with infx∈A f(x) > -∞ the corresponding optimization problem PA(f) is 2/p-order-well-posed solvable. As a byproduct, every constrained convex optimization problem with a strongly convex quadratic objective function is proved to be globally second-order-well-posed solvable. Our main results are new even in the case of finite dimensional spaces. [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]

Every (full) finite Gabor system generated by a unit-norm vector g ∈ Cd is a finite unit-norm tight frame (FUNTF) and can thus be associated with a (Gabor) positive operator valued measure (POVM). Such a POVM is informationally complete if the d² corresponding rank 1 matrices form a basis for the space of d x d matrices. A sufficient condition for this to happen is that the POVM is symmetric, which is equivalent to the fact that the associated Gabor frame is an equiangular tight frame (ETF). The existence of Gabor ETF is an important special case of the Zauner conjecture. It is known that generically all Gabor FUNTFs lead to informationally complete POVMs. In this paper, we initiate a classification of non-complete Gabor POVMs. In the process, we establish some seemingly simple facts about the eigenvalues of the Gram matrix of the rank 1 matrices generated by a finite Gabor frame. We also use these results to construct some sets of d² unit vectors in Cd with a relatively smaller number of distinct inner products. [ABSTRACT FROM AUTHOR]

A bounded linear operator T on a complex Banach space X is superconvex-cyclic, if for some vector x in the space, C. c o (o r b (T , x)) is dense. Here, co(orb(T, x)) means the convex hull of { T n x : n ≥ 0 }. In this paper, we give necessary conditions for the superconvex-cyclicity of an operator in terms of the point spectrum of its adjoint. Also, we prove that positive multiples of superconvex-cyclic operators are superconvex-cyclic. However, this concept is not necessarily transferred to positive integer powers of an operator. Moreover, we completely characterize superconvex-cyclic matrices whose spectra contain no real number or contain a positive real number. [ABSTRACT FROM AUTHOR]

In this paper, we study two properties viz., property-U and property-SU of a subspace Y of a Banach space X, which correspond to the uniqueness of the Hahn–Banach extension of each linear functional in Y ∗ and when this association forms a linear operator of norm-1 from Y ∗ to X ∗ . It is proved that, under certain geometric assumptions on X, Y, Z, these properties are stable with respect to the injective tensor product; Y has property-U (SU) in Z if and only if X ⊗ ε ∨ Y has property-U (SU) in X ⊗ ε ∨ Z. We prove that when X ∗ has the Radon–Nikodým Property for 1 < p < ∞, Lp(μ, Y) has property-U (property-SU) in Lp(μ, X) if and only if Y is so in X. We show that if Z⊆ Y⊆ X and Y has property-U (SU) in X then Y/Z has property-U (SU) in X/Z. On the other hand, Y has property-SU in X if Y/Z has property-SU in X/Z and Z (⊆ Y) is an M-ideal in X. This partly solves the 3-space problem for property-SU. We characterize all hyperplanes in c0 which have property-SU. We derive necessary and sufficient conditions for all finite codimensional proximinal subspaces of c0 which have property-U (SU). [ABSTRACT FROM AUTHOR]

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