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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]

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]

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]

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]

Let B (X) be the algebra of all bounded linear operators on a complex or real Banach space X with dim ⁡ X ≥ 3. In this paper, we characterize the maps from B (X) into itself which preserve the ascent of product of operators or, they preserve the descent of product of operators. It turns out that both problems are connected with preservers of the rank-one nilpotency of the product. [ABSTRACT FROM AUTHOR]

With the major progress of the Artemis program, and the development of NASA's largest rocket since the Apollo era capable of interplanetary travel, Americans are closer than ever to reaching back for the Moon, but this time with the intent of establishing a constant human presence. This vision is further strengthened by the rise of the private rocket sector with SpaceX, RocketLab, Relativity Space, Blue Origin, etc. The establishment of the lunar infrastructure will be a stepping-stone towards reaching for a farther destination, Mars. This paper presents a novel idea of lunar infrastructure construction, with a focus on habitation modules, based on the recently published work by the authors, focusing on the preliminary trade study of various concepts of lunar habitats. The modular blocks are introduced, representing the main building components for any lunar infrastructure. The modular blocks for habitat structures are designed as hollow bricks, optimized in a way that requires the least amount of construction material while implementing 3D printing technologies and minimizing the on-site human construction and assembly. Furthermore, different constructability techniques are presented, together with the initial stages of their development and materials which can be used for their manufacturing. Future work will encompass material properties, a more detailed explanation of manufacturing, global design, and structural analyses. • Hollow modular blocks for lunar habitats allow practical use of lunar in-situ material. • Minimizing on-site human construction and mass during the space transportation. • Constructability with existing technologies, like 3D printing, regolith mobile rovers. • Applicable for various types of lunar infrastructure. [ABSTRACT FROM AUTHOR]

In 1972, Dana Scott proved a fundamental result on the connection between order and topology which says that injective T 0 spaces are precisely continuous lattices endowed with Scott topology. This paper investigates whether this is true in an enriched context, where the enrichment is the quantale obtained by equipping the interval [ 0 , 1 ] with a continuous t-norm. It is shown that for each continuous t-norm, the specialization [ 0 , 1 ] -order of a separated and injective [ 0 , 1 ] -approach space X is a continuous [ 0 , 1 ] -lattice and the [ 0 , 1 ] -approach structure of X coincides with the Scott [ 0 , 1 ] -approach structure of its specialization [ 0 , 1 ] -order; but, unlike in the classical situation, the converse fails in general. [ABSTRACT FROM AUTHOR]

In a recent paper, we defined generalized color orderings (GCO) and Feynman diagrams (GFD) to compute color-dressed generalized biadjoint amplitudes. In this work, we study the Cachazo-Early-Guevara-Mizera (CEGM) representation of generalized partial amplitudes and "decoupling" identities. This representation is a generalization of the Cachazo-He-Yuan (CHY) formulation as an integral over the configuration space X (k , n) of n points on CP k − 1 in generic position. Unlike the k = 2 case, Parke-Taylor-like integrands are not enough to compute all partial amplitudes for k > 2. Here we give a set of constraints that integrands associated with GCOs must satisfy and use them to construct all (3 , n < 9) integrands, all (3 , 9) integrands up to four undetermined constants, and 95% of (4 , 8) integrands up to 24 undetermined constants. k = 2 partial amplitudes are known to satisfy identities. Among them, the so-called U (1) decoupling identities are the simplest ones. These are characterized by a label i and a color ordering in X (2 , | [ n ] ∖ { i } |). Here we introduce decoupling identities for k > 2 determined combinatorially using GCOs. Moreover, we identify the natural analog of U (1) identities as those characterized by a pair of labels i ≠ j , and a pair of GCOs, one in X (k , | [ n ] ∖ { i } |) and the other in X (k − 1 , | [ n ] ∖ { j } |). We call them double extension identities. We also provide explicit connections among different ways of representing GCOs, such as configurations of lines, configurations of points, and reorientation classes of uniform oriented matroids (chirotopes). [ABSTRACT FROM AUTHOR]

We extend some of main results of Abasov et al., 2016, Fotiy et al., (2020), Mykhaylyuk et al., (2015), Pliev and Fang, (2017), Pliev and Popov, (2014) to the setting of orthogonally additive operators on lattice-normed spaces. We introduce a new class of C -complete lattice-normed spaces which strictly includes a class of Banach–Kantorovich spaces. The first main result of the paper asserts that every laterally-to-norm continuous C -compact orthogonally additive operator T : X → Y from an atomless C -complete lattice-normed space X to a Banach space Y is narrow. As a non expecting consequence of the first main result we obtain necessary and sufficient conditions for a nonlinear superposition operator T N : E (X) → E (X) to be C -compact. We also show that the sum of two orthogonally additive operators G and T , where G is narrow and T is laterally-to-norm continuous and C -compact, is a narrow operator. In the last part of the article we investigate dominated orthogonally additive operators. In particular we show the narrowness of dominated operators taking values in a Banach sequence space Y or in a separable symmetric operator space C E. [ABSTRACT FROM AUTHOR]

A topological space X is called almost discrete, if it has precisely one nonisolated point. In this paper, we get that for a countable product X = ∏ X i of almost discrete spaces X i the space C p (X) of all continuous real-valued functions with the topology of pointwise convergence is a μ -space if, and only if, X is a weak q -space if, and only if, t (X) = ω if, and only if, X is functionally generated by the family of all its countable subspaces. This result makes it possible to solve Archangel'skii's problem on the product of Grothendieck spaces. It is proved that in the model of ZFC , obtained by adding one Cohen real, there are Grothendieck spaces X and Y such that X × Y is not weakly Grothendieck space. In (P F A) : the product of any countable family almost discrete Grothendieck spaces is a Grothendieck space. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel and rigorous approach to the Fokker–Planck equation, or Kolmogorov forward equation, for the Feynman–Kac transform of non-Markov anomalous processes. The equation describes the evolution of the density of the anomalous process Y t = X E t under the influence of potentials, where X is a strong Markov process on a Lusin space X that is in weak duality with another strong Markov process X ̂ on X and { E t , t ≥ 0 } is the inverse of a driftless subordinator S that is independent of X and has infinite Lévy measure. We derive a probabilistic representation of the density of the anomalous process under the Feynman–Kac transform by the dual Feynman–Kac transform in terms of the weak dual process X ̂ t and the inverse subordinator { E t ; t ≥ 0 }. We then establish the regularity of the density function, and show that it is the unique mild solution as well as the unique weak solution of a non-local Fokker–Planck equation that involves the dual generator of X and the potential measure of the subordinator S. During the course of the study, we are naturally led to extend the notation of Riemann–Liouville integral to measures that are locally finite on [ 0 , ∞). [ABSTRACT FROM AUTHOR]

Recent development of the various type of re-entry vehicles for tourism and, in near future, for frequent commercial flights to space as well as coming second generation supersonic transport (SST) rise important issue of flight safety of these vehicles, including sonic boom exposure on ground structures, people and environment. During the space plane descend the sonic boom carpet may cover thousands of kilometers wide territories on the earth or ocean and may cause some discomfort or even danger for people. Proper modelling of shock waves propagation in the non-homogeneous windy atmosphere is important task that should be solved based on the fast calculation methods to provide data to the flight deck in real time, while descending. In this paper the sonic boom modelling is based on augmented Burgers equation. A generalization of this equation is made for the case of a weakly inhomogeneous moving viscous heat-conducting medium with the conservation equations of the second order of approximation. The numerical method of sonic boom propagation and attenuation in real atmosphere and respective computer code "vBoom" have been developed and validated. The calculation data of sonic boom propagation and primary and secondary sonic boom exposure areas on the ground along with comparison with high speed aircraft data are presented. Measurement-based estimation for the loudness of sonic boom for re-entry vehicle at descending has been made. • Re-entry vehicle space safety. • Sonic boom modelling and carpets. • Augmented Burgers equation. [ABSTRACT FROM AUTHOR]

A commutative ring with identity is called a zip ring if each of its faithful ideals contains a finitely generated faithful ideal. A natural generalization of this notion is to require every faithful ideal to contain a countably generated faithful ideal. This is what we do in this paper. If a ring has this property, we call it a czip ring. Clearly, every zip ring is a czip ring. We give several characterizations of czip rings, including some in terms of subspaces of the space of minimal prime ideals with the Zariski topology. In rings of continuous functions we characterize the Tychonoff spaces X for which C (X) is a czip ring. This enables us to show that the class of czip rings properly contains that of zip rings. [ABSTRACT FROM AUTHOR]

Decomposition integrals provide a framework for non-linear integrals that include Choquet, Shilkret, the PAN, and the concave integrals. All of these integrals found their applications in mathematics, notably in decision-making and economy. An important class of decomposition integrals is the class of integrals extending Lebesgue integral in the sense that the decomposition integral with respect to classical measures coincides with Lebesgue integral. In this paper, we consider finite spaces X only and discuss some necessary and sufficient conditions for this property. Also, some construction methods are given and exemplified. [ABSTRACT FROM AUTHOR]

Ripple Effect is a logic puzzle where the player has to fill numbers into empty cells in a rectangular grid. The grid is divided into rooms, and each room must contain consecutive integers starting from 1 to its size. Also, if two cells in the same row or column contain the same number x , there must be a space of at least x cells separating the two cells. In this paper, we develop a physical zero-knowledge proof for the Ripple Effect puzzle using a deck of cards, which allows a prover to convince a verifier that he/she knows a solution without revealing it. In particular, given a secret number x and a list of numbers, our protocol can physically verify that x does not appear among the first x numbers in the list without revealing x or any number in the list. [ABSTRACT FROM AUTHOR]

C ⁎ -algebras, group algebras, and the algebra A (X) of approximable operators on a Banach space X having the bounded approximation property are known to be zero product determined. In this paper we give a quantitative estimate of this property by showing that, for the Banach algebra A , there exists a constant α with the property that for every continuous bilinear functional φ : A × A → C there exists a continuous linear functional ξ on A such that sup ‖ a ‖ = ‖ b ‖ = 1 ⁡ | φ (a , b) − ξ (a b) | ≤ α sup ‖ a ‖ = ‖ b ‖ = 1 , a b = 0 ⁡ | φ (a , b) | in each of the following cases: (i) A is a C ⁎ -algebra, in which case α = 8 ; (ii) A = L 1 (G) for a locally compact group G , in which case α = 60 27 1 + sin ⁡ π 10 1 − 2 sin ⁡ π 10 ; (iii) A = A (X) for a Banach space X having property (A) (which is a rather strong approximation property for X), in which case α = 60 27 1 + sin ⁡ π 10 1 − 2 sin ⁡ π 10 C 2 , where C is a constant associated with the property (A) that we require for X. [ABSTRACT FROM AUTHOR]

A fuzzy compactum in a space X is an upper semi-continuous map f : A → [ 0 , 1 ] from a nonempty compact subspace A ⊂ X to the unit closed interval [ 0 , 1 ] such that { x ∈ A : f (x) > 0 } is dense in A. In this paper, we investigate the topological structure of the space K e (X) of fuzzy compacta of a separable metric space X with the Hausdorff metric. In particular, we prove that K e (X) ≈ ℓ 2 if X is an infinite, locally compact, locally connected, connected and separable metric space. [ABSTRACT FROM AUTHOR]

The paper presents a satisfactory version of the Lie–Trotter product formula in the context of the duality between a quasi-equicontinuos C 0 -semigroup T (t) t ≥ 0 on a locally convex vector space X endowed with some locally convex Hausdorff topology Υ and its adjoint T ∗ (t) t ≥ 0 on the dual space X ∗ endowed with the topology C (X ∗ , X) of uniform convergence on compact subsets of X Υ. An application is given for the transition semigroup of the diffusion associated to the generalized Schrödinger operator on L ∞ (R N , d x). [ABSTRACT FROM AUTHOR]

In this paper we prove existence and uniqueness of a mild solution to the Young equation d y (t) = A y (t) d t + σ (y (t)) d x (t) , t ∈ [ 0 , T ] , y (0) = ψ. Here, A is an unbounded operator which generates a semigroup of bounded linear operators (S (t)) t ≥ 0 on a Banach space X , x is a real-valued η -Hölder continuous. Our aim is to reduce, in comparison to Gubinelli et al. (2006) and Addona et al. (2022) (see also Deya et al. (2012) and Gubinelli and Tindel, (2010)), the regularity requirement on the initial datum ψ eventually dropping it. The main tool is the definition of a sewing map for a new class of increments which allows the construction of a Young convolution integral in a general interval [ a , b ] ⊂ R when the X α -norm of the function under the integral sign blows up approaching a and X α is an intermediate space between X and D (A). [ABSTRACT FROM AUTHOR]

This article is in line with earlier investigations done in [10–12,14,15,18] and several such works. Here our aim is to introduce and study two new completeness-like properties, namely, Bourbaki quasi-completeness and cofinally Bourbaki quasi-completeness (we use infinite chains instead of finite ones), which strictly lie between compactness and completeness, primarily in the setting of uniform spaces. We use the concept of finite-component covers [19] to define a new type of modification of a uniform space, which plays a crucial role throughout the paper. In Section 2, we relate cBq-completeness to the existing notion of superparacompactness [19]. Another significant and very natural problem we deal with is the topological problem of metrizability of a uniform space using a Bq-complete and a cBq-complete metric. We obtain results similar to the classical Čech theorem about the complete metrizability of a metric space X in terms of its Stone-Čech compactification βX , which are presented in sections 3 and 4 of the article. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Reznichenko property and the Pytkeev property in the hyperspace CL (X) of a space X with the hit-and-miss topology. [ABSTRACT FROM AUTHOR]

We study equivariant sheaves over profinite spaces, where the group is also taken to be profinite. We resolve a serious deficit in the existing theory by constructing a good notion of equivariant presheaves, with a suitable equivariant sheafification functor. Using equivariant sheafification, we develop the general theory of equivariant sheaves of modules over a ring, give explicit constructions of infinite products and introduce an equivariant analogue of skyscraper sheaves. These results underlie recent work by the authors which proves that there is an algebraic model for rational G -spectra in terms of equivariant sheaves over profinite spaces. That model is constructed in terms of Weyl- G -sheaves over the space of closed subgroups of G , where the term Weyl indicates that the stalk over H is H -fixed. In this paper, we prove that Weyl- G -sheaves of R -modules form an abelian category with enough injectives and is a coreflective subcategory of equivariant sheaves of R -modules. We end the paper with a structural result that provides another way to conveniently build equivariant sheaves from simpler data. We prove that a G -equivariant sheaf over a profinite base space X is a colimit of equivariant sheaves over finite discrete spaces X i with actions of finite groups G i , where X is the limit of the X i and G is the limit of the G i. [ABSTRACT FROM AUTHOR]

Let X be a compact metric space and f : X → X be a continuous map having the shadowing property. In this paper, first, we prove that if X is connected, then ω f is continuous if, and only if, f is minimal. Second, we prove that the continuity of ω f yield C R (f) = R (f) = A P (f). Third, it is shown that if any point is recurrent then the collection of minimal sets is closed in (2 X , d H). Finally, we show that the existence of a Li-Yorke pair is sufficient to have positive topological sequence entropy and if furthermore, f has s -shadowing property then f is Li-Yorke chaotic. [ABSTRACT FROM AUTHOR]

We investigate computable metrizability of Polish spaces up to homeomorphism. In this paper we focus on Stone spaces. We use Stone duality to construct the first known example of a computable topological Polish space not homeomorphic to any computably metrized space. In fact, in our proof we construct a right-c.e. metrized Stone space which is not homeomorphic to any computably metrized space. Then we introduce a new notion of effective categoricity for effectively compact spaces and prove that effectively categorical Stone spaces are exactly the duals of computably categorical Boolean algebras. Finally, we prove that, for a Stone space X , the Banach space C (X ; R) has a computable presentation if, and only if, X is homeomorphic to a computably metrized space. This gives an unexpected positive partial answer to a question recently posed by McNicholl. [ABSTRACT FROM AUTHOR]

The space of non-resultant systems of bounded multiplicity for a toric variety X is a generalization of the space of rational curves on it. In our earlier work [24] we determined explicitly the homotopy type of this space for the case X = C P m. In this paper we consider this space for general non-singular toric varieties and prove that the homotopy stability holds for them. [ABSTRACT FROM AUTHOR]

In this paper, using the concept of the real-boards of ideals in C (X) , the ring of all real-valued continuous functions on a completely regular Hausdorff space X , some characterizations of nearly pseudocompact and nearly realcompact spaces are given. For an ideal I ⊆ C (X) , the real-board of I is the largest subspace of βX , the Stone-Čech compactification of X , on which every member of I can be extended continuously. Using this concept, the subrings C K I (X) and C ∞ I (X) of C (X) are introduced and studied, and I -pseudocompact spaces which are generalizations of pseudocompact spaces are presented as well as some of their properties. [ABSTRACT FROM AUTHOR]

For each separable space X , we consider the family of all countable dense sets of X modulo an equivalence relation defined as follows: we say that M ∼ N if and only if there is a homeomorphism h : X → X such that h [ M ] = N. The countable dense homogeneity degree of X is defined as the cardinality of the set of the equivalence classes under the relation ∼. In case such degree equals 1, the space is said to be countable dense homogeneous. Countable dense homogeneity has become a classical concept and has been studied with interest in the last decades. In this paper we present a full characterization of the countable dense homogeneity degree for the class of dendrites. [ABSTRACT FROM AUTHOR]

Assume that P is a topological property of a space X , then we say that X is dense- P if each dense subset of X has the property P. In this paper, we mainly discuss dense subsets of a space X , and we prove that: (1) if X is Tychonoff space, then X is dense-pseudocompact if and only if the range of each continuous real-valued function f on X is finite, if and only if X is finite, if and only if X is hereditarily pseudocompact; (2) X is dense-connected if and only if U ‾ = X for any non-empty open subset U of X ; (3) X is dense-ultraconnected if and only if for point x ∈ X , we have { x } ‾ = X or { x } ∪ (X ∖ { x } ‾) is the unique open neighborhood of x in { x } ∪ (X ∖ { x } ‾) , if and only if for any two points x and y in X , we have x ∈ { y } ‾ or y ∈ { x } ‾. Moreover, we give a characterization of a topological group (resp., paratopological group, quasi-topological group) G such that G is dense-connected. [ABSTRACT FROM AUTHOR]

The article studies topological games that arise in the study of the continuity of operations in groups with topology, such as paratopological and semitopological groups. These games are modifications of the Banach–Mazur game. Given a two-player game G (X) of the Banach–Mazur type, we define Γ G -Baire, Γ G -nonmeager and Γ G -spaces. A space X is a Γ G -Baire if the second player does not have a winning strategy in G (X). The classes of Γ G -nonmeager spaces and Γ G -spaces are defined similarly, with the help of modifications of the game G (X). For the games under consideration, equivalent games are found, which facilitates studying the relationship between the resulting classes of spaces and determining which spaces belong to these classes. For this purpose, we introduce a modification of the Banach–Mazur game with four players. Results of this paper find application in the study the continuity of operations in groups with topology. [ABSTRACT FROM AUTHOR]

A space X is called weakly normal if for every two disjoint closed subsets A and B of X there exists a continuous mapping f of X into R ω such that images of A and B are disjoint. A space X is called weakly normal off of the diagonal if the space X 2 ∖ Δ is weakly normal. The aim of this paper is to prove that it is consistent that every weakly normal off of the diagonal, countably compact space is first-countable and compact. [ABSTRACT FROM AUTHOR]

In the paper we applied the method of Containing Spaces to categories, whose objects are homotopies φ : X × I → Y , connected two mappings of spaces f : X → Y and g : X → Y (that is, f (x) = φ (x , 0) and g (x) = φ (x , 1) , x ∈ X) and morphisms are proper embeddings, for the construction of universal objects. [ABSTRACT FROM AUTHOR]

For a regular space X , the hyperspace (C L (X) , τ F) (resp., (C L (X) , τ V)) is the space of all nonempty closed subsets of X with the Fell topology (resp., Vietoris topology). In this paper, we give the characterization of the space X such that the hyperspace (C L (X) , τ F) (resp., (C L (X) , τ V)) with a countable character of closed subsets. We mainly prove that (C L (X) , τ F) has a countable character on each closed subset if and only if X is compact metrizable, and (C L (X) , τ F) has a countable character on each compact subset if and only if X is locally compact and separable metrizable. Moreover, we prove that (K (X) , τ V) have the compact- G δ -property if and only if X have the compact- G δ -property and every compact subset of X is metrizable. [ABSTRACT FROM AUTHOR]

In this paper, we study variational inequalities of the form 〈 − Δ N u , v − u 〉 + 〈 F (u) , v − u 〉 ≥ 0 , ∀ v ∈ K u ∈ K , where Δ N u = div | ∇ u | N − 2 ∇ u is the N -Laplacian, K is a closed convex set in the Beppo–Levi space X = D 0 1 , N (Ω) , where Ω is the exterior of the closed unit ball B (0 , 1) ¯ in R N (N ≥ 2). We are interested here in the case where F is given by a multivalued function f = f (x , u , ∇ u) that depends also on the gradient ∇ u of the unknown function. We consider a functional analytic framework of the above problem, including conditions on the lower order term f such that the problem can be appropriately formulated in X and a related weighted Orlicz space, and the involved mappings have certain useful monotonicity-continuity properties. Existence of solutions in coercive and noncoercive cases are studied. In the noncoercive case, we follow a sub-supersolution approach and prove the existence of solutions of the above inequality under a multivalued local growth condition of Bernstein–Nagumo type. [ABSTRACT FROM AUTHOR]

For a metrizable space X , we denote by Met (X) the space of all metric that generate the same topology of X. The space Met (X) is equipped with the supremum distance. In this paper, for every strongly zero-dimensional metrizable space X , we prove that the set of all metrics whose ranges are closed totally disconnected subsets of the line is a dense G δ subspace in Met (X). As its application, we show that some sets of universal metrics are meager in spaces of metrics. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study I s n -sequential spaces and I - snf -countable spaces. The following main results are obtained: let I be an ideal on N and X be a topological space, then (1) X is an I s n -sequential space if and only if each I s n -continuous (resp. quotient) mapping on the space X is continuous (resp. I s n -quotient). (2) X is an I -neighborhood space if and only if each I s n -quotient mapping on the space X is I -quotient, which answers [10, Question 4.12(2)] in the affirmative. (3) X is an I - snf -countable space if and only if X is a continuous and 1- I -covering image of a metric space. (4) If I is a K -uniform ideal on N , then each topological space is an I -neighborhood space. [ABSTRACT FROM AUTHOR]

In this paper we study the behavior of some properties of topological spaces under the influence of the N τ φ -kernel of a space X. We prove that the N τ φ -kernel of a space X preserves the density, network and π -weight of normal spaces. Besides, it is shown that the N -compact kernel of a space X preserves the Souslin number, weight, density, and π -network weight of normal spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the topological structure of a universal construction related to quasitopological groups: the free quasitopological group F q (X) on a space X. We show that free quasitopological groups may be constructed directly as quotient spaces of free semitopological monoids, which are themselves constructed by iterating product spaces equipped with the "cross topology." Using this explicit description of F q (X) , we show that for any T 1 space X , F q (X) is the direct limit of closed subspaces F q (X) n of words of length at most n. We also prove that the natural map i n : ∐ i = 0 n (X ⊔ X − 1) ⊗ i → F q (X) n is quotient for all n ⩾ 0. Equipped with this convenient characterization of the topology of free quasitopological groups, we show, among other things, that a subspace Y ⊆ X is closed if and only if the inclusion Y → X induces a closed embedding F q (Y) → F q (X) of free quasitopological groups. [ABSTRACT FROM AUTHOR]

For any complex Banach space X and each p ∈ [ 1 , ∞) , we introduce the p -Bohr radius of order N (∈ N) is R ˜ p , N (X) defined by R ˜ p , N (X) = sup ⁡ { r ≥ 0 : ∑ k = 0 N ‖ x k ‖ p r p k ≤ ‖ f ‖ H ∞ (D , X) p } , where f (z) = ∑ k = 0 ∞ x k z k ∈ H ∞ (D , X). Here D = { z ∈ C : | z | < 1 } denotes the unit disk. We also introduce the following geometric notion of p -uniformly C -convexity of order N for a complex Banach space X for some N ∈ N. For p ∈ [ 2 , ∞) , a complex Banach space X is called p -uniformly C -convex of order N if there exists a constant λ > 0 such that (0.1) (‖ x 0 ‖ p + λ ‖ x 1 ‖ p + λ 2 ‖ x 2 ‖ p + ⋯ + λ N ‖ x N ‖ p) 1 / p ≤ max θ ∈ [ 0 , 2 π) ⁡ ‖ x 0 + ∑ k = 1 N e i θ x k ‖ for all x 0 , x 1 , ..., x N ∈ X. We denote A p , N (X) , the supremum of all such constants λ satisfying (0.1). We obtain the lower and upper bounds of R ˜ p , N (X) in terms of A p , N (X). In this paper, for p ∈ [ 2 , ∞) and each N ∈ N , we prove that a complex Banach space X is p -uniformly C -convex of order N if, and only if, the p -Bohr radius of order N R ˜ p , N (X) > 0. We also study the p -Bohr radius of order N for the Lebesgue spaces L q (μ) for 1 ≤ p < q < ∞ or 1 ≤ q ≤ p < 2. Finally, we prove an operator valued analogue of a refined version of Bohr and Rogosinski inequality for bounded holomorphic functions from the unit disk D into B (H) , where B (H) denotes the space of all bounded linear operator on a complex Hilbert space H. [ABSTRACT FROM AUTHOR]

• An autonomous multi-robot planning method is proposed for the flexible assembly of large-scale lunar facilities. • A distributed multi-agent planning frame with multi-replica public agents is introduced. • The hierarchical task graph (HTG) is proposed to help with suspending the mission at a relatively safe state when the environment is not suitable to work. • The Hierarchical Multi-Agent Planning method based on Two-stage Two-side Matching (HMAP-TTM) is proposed to optimize the time of the flexible assembly mission. • Compared to the baseline methods, HMAP-TTM can generate plans with shorter mission time and require smaller communication costs. Several national space agencies and commercial aerospace companies plan to set up lunar bases with large-scale facilities that rely on multiple lunar robots' assembly. Mission planning is necessary to achieve efficient multi-robot cooperation. This paper aims at autonomous multi-robot planning for the flexible assembly of the large-scale lunar facility, considering the harsh lunar environment, mission time optimization, and joint actions. The lunar robots and modules are scattered around the mission area without fixed assembly lines. Thus, the traditional assembly planning methods ignoring the optimal selection of modules are unable to handle this problem. We propose a hierarchical multi-agent planning method based on two-stage two-sided matching (HMAP-TTM) to solve this critical problem. First, the distributed planning framework with multi-replica public agents is introduced, ensuring robot plan knowledge consistency through public agents' communication. Second, the hierarchical task graph (HTG) divides the mission into task layers based on task dependency knowledge. Third, we develop a novel two-stage two-sided matching algorithm. Time-optimal plans emerge from the matching games among public and private agents in each layer of HTG. Agents make decisions in the game based on action knowledge updated during planning. Finally, an assembly mission is presented to prove the method's effectiveness. The simulation results show that the HMAP-TTM can generate plans with shorter mission time and require smaller communication costs than the baseline methods. [ABSTRACT FROM AUTHOR]

The number of occupants in a space influences the risk of far-field airborne transmission of SARS-CoV-2 because the likelihood of having infectious and susceptible people both correlate with the number of occupants. This paper explores the relationship between occupancy and the probability of infection, and how this affects an individual person and a population of people. Mass-balance and dose–response models determine far-field transmission risks for an individual person and a population of people after sub-dividing a large reference space into 10 identical comparator spaces. For a single infected person, the dose received by an individual person in the comparator space is 10 times higher because the equivalent ventilation rate per infected person is lower when the per capita ventilation rate is preserved. However, accounting for population dispersion, such as the community prevalence of the virus, the probability of an infected person being present and uncertainty in their viral load, shows the transmission probability increases with occupancy and the reference space has a higher transmission risk. Also, far-field transmission is likely to be a rare event that requires a high emission rate, and there are a set of Goldilocks conditions that are just right when equivalent ventilation is effective at mitigating against transmission. These conditions depend on the viral load, because when they are very high or low, equivalent ventilation has little effect on transmission risk. Nevertheless, resilient buildings should deliver the equivalent ventilation rate required by standards as minimum. • Accounts for population dispersion when considering the risks of far-field exposure to a virus. • Far-field transmission is likely to be a rare event that requires a high emission rate. • Ventilation cannot is not an effective mitigation method when viral load is very high or low. • Larger spaces can pose a greater transmission risk than smaller spaces for a population of people. [ABSTRACT FROM AUTHOR]

Kočinac and Singh [4] introduced and studied the class of set selectively k-star-ccc spaces. Let k ∈ N , a space X is said to be set selectively k-star-ccc if for each nonempty subset B of X and for every collection U of open sets in X such that B ‾ ⊂ ∪ U and for every sequence (A n : n ∈ N) of maximal cellular open families in X , there is a sequence (A n : n ∈ N) such that for every n ∈ N , A n ∈ A n and B ⊆ S t k (⋃ n ∈ N A n , U). In this paper, we show that for k ∈ N , there exists a Tychonoff pseudocompact selectively 2-star-ccc space X which is not set selectively k-star-ccc, which gives an answer to the Problem 5.1 of [4] , and construct an example of a Tychonoff set selectively star-ccc space having a regular closed G δ -subspace which is not set selectively star-ccc, which gives an answer to the Problem 5.3 of [4]. We also prove that every open F σ -subset of a set selectively star-ccc space is set selectively star-ccc, which gives a positive answer to the Problem 5.6 of [4]. Finally, we obtain a normal example of a selectively 2-star-ccc space X which is neither set strongly star-Lindelöf nor set selectively star-ccc under some set-theoretic assumption, which gives a consistent positive answer to Problem 5.4 of [4]. [ABSTRACT FROM AUTHOR]

The defect levels in semiconductor are crucial to the performance of related devices. However, the positions of the defect levels in monoclinic β -Ga 2 O 3 , such as that for oxygen vacancy (V O), Ga–O divacancy, and self-trapped hole (STH), are still controversial. In this paper, we systematically investigated the photoluminescence and photoluminescence excitation spectra of undoped, Si-doped, and Mg-doped β -Ga 2 O 3 single crystals grown by the edge-defined film-fed growth method. It is found that the oxygen vacancies are deep donors and their levels are related to the coordinations of the vacancies. For each crystal, the recombination between V O and Ga–O divacancy levels is the origin of the blue emission, and the red emission originates from the transition from V O to STH levels. The location of STH level is also related to the geometrical position of oxygen. For the Si-doped crystal, no red emission is observed due to the annihilation of STHs. While a Mg impurity level is produced at ∼1 eV above the valence band maximum, which causes an extra red emission in the spectra of Mg-doped crystal. Our results strongly support the recent theoretical results obtained via first-principle calculations [Q. D. Ho et al., Phys. Rev. B 97 (2018) 115163] and could be benefit for designing β -Ga 2 O 3 related devices. • It is found that the oxygen vacancies in β -Ga 2 O 3 crystal are deep donors. • The location of oxygen vacancy level depends on the oxygen coordination. • The locations of Ga–O divacancy and self-trapped hole levels are also determined. [ABSTRACT FROM AUTHOR]

A map f : X → Y from a Baire space X to a topological space (Y , T 2) is said to be generically continuous on X if the set G (f , T 2) : = { x ∈ X : f : X → (Y , T 2) is continuous at x } includes a dense G δ -set in X. In the paper, we investigate the following question: What conditions on spaces X , (Y , T 1) and (Y , T 2) do imply that f : X → (Y , T 2) is generically continuous on X for every continuous map f : X → (Y , T 1) ? We answer this question for Y being a poset with a topology T 2 : = T d induced by a metric d on Y. It is proved that if Y is a poset with a metric d satisfying some natural conditions, then f : X → (Y , T d) is generically continuous for every Baire metric space (or Baire space) X and every continuous map f : X → (Y , ↑ T d) , where ↑ T d : = { ↑ U : U ∈ T d }. Based on this result, we extend some known results for Y being hyperspaces and give an application to dynamical systems. Some related counterexamples are presented. [ABSTRACT FROM AUTHOR]

In this paper we introduce a light Dirac particle ψ as thermal dark matter candidate in a U (1) L μ − L τ model. Together with the new gauge boson X , we find a possible parameter space with m X ≃ 20 MeV, U (1) L μ − L τ coupling g X ≃ 5 ⋅ 10 − 4 and m ψ ≳ m X / 2 where the (g − 2) μ anomaly, dark matter, the Hubble tension, and (part of) the excess of 511 keV photons from the region near the galactic center can be explained simultaneously. This model is safe from current experimental and astrophysical constraints, but can be probed by the next generation of neutrino experiments as well as low–energy e + e − colliders. [ABSTRACT FROM AUTHOR]

Let f : X → X be a continuous map on a compact metric space X and let α f , ω f and I C T f denote the set of α -limit sets, ω -limit sets and nonempty closed internally chain transitive sets respectively. In this paper we characterise, by introducing novel variants of shadowing, maps for which every element of I C T f is equal to (resp. may be approximated by) the α -limit set and the ω -limit set of the same full trajectory. We construct examples highlighting the difference between these properties. [ABSTRACT FROM AUTHOR]

The concept of index and co-index of a paracompact Hausdorff space X equipped with free involutions relative to the antipodal action on spheres were introduced by Conner and Floyd [2]. In this paper, we extend the notion of index and co-index for free G -spaces X , where X is a finitistic space and G = S 1 (with complex multiplication) and G = S 3 (with quaternionic multiplication). We prove that the index of X is less than or equal to the mod 2 cohomology index of X. We also compute the ring cohomology of the orbit space X / G , where G = S 1 or G = S 3 and X is a finitistic space whose cohomology ring is the same as the product of spheres S n × S m , 1 ≤ n ≤ m. Using these cohomological calculations, we obtain some Borsuk-Ulam type results. [ABSTRACT FROM AUTHOR]

Results in this paper concern finding a cardinal number κ , as small as possible, such that every uniformly or proximally continuous mapping from a given subspace X of a product ∏ I X i into some A factorizes via a subspace of ∏ J X i with | J | ≤ κ. In some cases we substantially improve Vidossich theorem from [12]. Influenced by investigating locally co-presentable categories in [8] , also cardinals κ are found such that the above factorization hold for all products ∏ I X i from an epireflective subclass C of Unif 2 , their closed subspaces X and spaces A generating C. That situation is close to factorizing maps on limits of inverse systems via limits of smaller inverse subsystems. We define and investigate a new cardinal function on uniform spaces helping to find a convenient factorization. [ABSTRACT FROM AUTHOR]

This paper aims at developing a functional approach to investigate the Morse structures of attractors for infinite-dimensional dynamical systems generated by PDEs. Let S (t) be a semiflow on a complete metric space X , and A an attractor of S (t) with a Morse decomposition M. We first construct some good Morse–Lyapunov functions for M which seem to be of independent interest in their own right. Then we establish some fundamental deformation lemmas. Based on these results, we finally introduce the concept of critical groups for Morse sets and prove Morse inequalities and equations. The problem of how to calculate Morse equations of attractors by using a natural Morse–Lyapunov function which may be defined only on a dense subspace of X is also addressed, and an illustrating example of parabolic equation is presented. [ABSTRACT FROM AUTHOR]