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The famous 1960s Lumer-Phillips theorem states that a closed and densely defined operator A:D(A) X → X on a Banach space X generates a strongly continuous contraction semigroup if and only if (AD(A)) is dissipative and the range of λ - A is surjective in X for some λ > 0. In this paper, we establish a version of this result for bi-continuous semigroups and apply the latter amongst other examples to the transport equation as well as to flows on infinite networks. [ABSTRACT FROM AUTHOR]

At the very beginning of the theory of finite dynamical systems, it was discovered that some relatively simple systems, even of ordinary differential equations, can generate very complicated (chaotic) behaviors. Furthermore these systems are extremely sensitive to perturbations, in the sense that trajectories with close but different initial data may diverge exponentially. Very often, the trajectories of such chaotic systems are localized, up to some transient process, in some subset of the phase space, the so-called strange attractors. Such subset have a very complicated geometric structure. They accumulate the nontrivial dynamics of the system. For a distributed system, whose time evolution is usually governed by partial differential equations (PDEs), the phase space X is (a subset of) an infinite dimensional function space. We will thus speak of infinite dimensional dynamical systems. Since the global existence and uniqueness of solutions has been proven for a large class of PDEs arising from different domains as Mechanics and Physics, it is therefore natural to investigate whether the features, in particular the notion of attractor, obtained for dynamical systems generated by systems of ODEs generalizes to systems of PDEs. In this paper we give a positive aftermath by proving the existence of pullback D D -attractor. The key point is to find a bounded family of pullback D D -absorbing sets then we apply the decomposition techniques and a method used in previous works to verify the pullback w w - D D -limit compactness. It is based on the concept of the Kuratowski measure of noncompactness of a bounded set as well as some new estimates of the equicontinuity of the solutions. [ABSTRACT FROM AUTHOR]

One of the most important results in domain theory is the Hofmann-Mislove Theorem, which reveals a very distinct characterization for the sober spaces via open filters. In this paper, we extend this result to the d -spaces and well-filtered spaces. We do this by introducing the notions of Hofmann-Mislove-system (HM-system for short) and $\Psi$ -well-filtered space, which provide a new unified approach to sober spaces, well-filtered spaces, and d -spaces. In addition, a characterization for $\Psi$ -well-filtered spaces is provided via $\Psi$ -sets. We also discuss the relationship between $\Psi$ -well-filtered spaces and H -sober spaces considered by Xu. We show that the category of complete $\Psi$ -well-filtered spaces is a full reflective subcategory of the category of $T_0$ spaces with continuous mappings. For each HM-system $\Psi$ that has a designated property, we show that a $T_0$ space X is $\Psi$ -well-filtered if and only if its Smyth power space $P_s(X)$ is $\Psi$ -well-filtered. [ABSTRACT FROM AUTHOR]

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

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

We are witnessing a new space race. A half century after the last Moon landing, and after a decade during which the United States could not launch its own astronauts to Earth orbit, there is new energy in the space activity. China has huge ambitions to rival or eclipse America as the major space power, and other countries are developing space programs. However, perhaps the greatest excitement attaches to the entrepreneurs who are trying to create a new business model for space travel based initially on tourism, and eventually, on colonizing the Moon and Mars and harvesting resources from asteroids. This paper presents a snapshot of the new space race and the rich men behind it, and it looks at some of the ethical and legal issues raised by this activity. The methodology is to consider the stated ambitions of the men leading private space companies, compare and contrast the space endeavor with earlier episodes of exploration and transportation innovation, review the regulatory environment for outer space, and consider two divergent scenarios for the future. Opinions are divided on whether commercial space flight is an expensive indulgence or potentially a way to find sustainability solutions for our life on Earth. It is concluded that the new space race can be characterized as unbounded: in ambition, in terms of laws and regulations, and in terms of ethical constraints on the activity. [ABSTRACT FROM AUTHOR]

This paper shows that if f is a convex continuous function on a Gâteaux differentiability space X, then for any separable closed subspace E of X, there exists a sequence {fn}n=1∞${\big \lbrace f_{n}\big \rbrace} _{n=1}^{\infty }$ of continuous convex functions such that (1) fn(x)≤fn+1(x)≤f(x)$f_{n}(x)\le f_{n+1}(x)\le f(x)$ on X; (2) fn$f_{n}$ is Gâteaux differentiable at all points of a dense open subset of X; (3) fn(x)→f(x)${f_n}(x) \rightarrow f(x)$ on E. Moreover, if X is separable, then there exists a continuous convex function sequence {fn}n=1∞${\big \lbrace f_{n}\big \rbrace} _{n=1}^{\infty }$ such that (1) and (2) are true and fn(x)→f(x)${f_n}(x) \rightarrow f(x)$ on X. [ABSTRACT FROM AUTHOR]

Firstly, the new concepts of G − expansibility, G − almost periodic point, and G − limit shadowing property were introduced according to the concepts of expansibility, almost periodic point, and limit shadowing property in this paper. Secondly, we studied their dynamical relationship between the self-map f and the shift map σ in the inverse limit space under topological group action. The following new results are obtained. Let X , d be a metric G − space and X f , G ¯ , d ¯ , σ be the inverse limit space of X , G , d , f . (1) If the map f : X ⟶ X is an equivalent map, then we have A P G ¯ σ = Lim ← A p G f , f . (2) If the map f : X ⟶ X is an equivalent surjection, then the self-map f is G − expansive if and only if the shift map σ is G ¯ − expansive. (3) If the map f : X ⟶ X is an equivalent surjection, then the self-map f has G − limit shadowing property if and only if the shift map σ has G ¯ − limit shadowing property. The conclusions of this paper generalize the corresponding results given in the study by Li, Niu, and Liang and Li. Most importantly, it provided the theoretical basis and scientific foundation for the application of tracking property in computational mathematics and biological mathematics. [ABSTRACT FROM AUTHOR]

Let X be a uniformly smooth and 2-uniformly convex real Banach space with dual space X ∗ . In this paper, a Halpern-type subgradient extragradient algorithm is constructed. The sequence, generated by the algorithm, converges strongly to a common solution of variational inequality and two convex minimization problems. This result is obtained as an application of a Halpern-type subgradient extragradient algorithm, for approximating a common solution of variational inequality and J-fixed points of two continuous J-pseudocontractions. The theorem proved complements, improves and unifies many recent results in the literature. Finally, numerical experiments are given to illustrate the convergence of the sequence generated by the algorithm. [ABSTRACT FROM AUTHOR]

The existence of monads on products of projective spaces ... x ⋯ x ... is nontrivial. In this paper, we construct monads over the polycyclic variety P2n+1 x P2n+1, we prove that cohomology vector bundle associated to these monads is simple. We also construct a monad on P¹ x P¹ x P². We also study the vector bundles associated to monads and prove stability and simplicity. [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]

In this paper, we study the behavior of some topological and cardinal properties of topological spaces under the influence of the NτΦ-kernel of a space X. It has been proved that the NτΦ-kernel of a space X preserves the density and the network p - weight of normal spaces. Besides, shown that the N-compact kernel of a space X preserves the Souslin properties, the weight, the density, and the p-network weight of normal spaces. [ABSTRACT FROM AUTHOR]

Whenever the closure of an open set is also open, it is called e-open and if a space have a base consisting of e-open sets, it is called e-space. In this paper we first introduce and study e-spaces and e-continuous func-tions (we call a function f from a space X to a space Y an e-continuous at x 2 X if for each open set V containing f(x) there is an e-open set containing x with f(U) C V). We observe that the quasicomponent of each point in a space X is determined by e-continuous functions on X and it is characterized as the largest set containing the point on which every e-continuous function on X is constant. Next, we study the rings Ce(X) of all real valued e-continuous functions on a spaCe X. It turns out that Ce(X) coincides with the ring of real valued clopen continuous functions on X which is a C(Y) for a zero-dimensional spaCe Y whose elements are the quasicomponents of X. Using this fact we characterize the real maximal ideals of Ce(X) and also give a natural representation of its maximal ideals. Finally we have shown that Ce(X) determines the topology of X if and only if it is a zero-dimensional spaCe. [ABSTRACT FROM AUTHOR]

In this paper, we study strong coproximinality in Bochner Lp -spaces and in the Köthe Bochner function space E(X). We investigate some conditions to be imposed on the subspace G of the Banach space X such that Lp (µ, G) is strongly coproximinal in Lp (µ, X), 1 ≤ p < ∞. On the other hand, we prove that if G is a separable subspace of X then G is strongly coproximinal in X if and only if E(G) is strongly coproximinal in E(X), provided that E is a strictly monotone Köthe space. This generalizes some results in the literature. Some other results in this direction are also presented. [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 the first part of this paper, we present some investigations on the class of almost (L) limited operators. We show that an operator T : X → E , from a Banach space X to a Banach lattice E, is almost (L) limited iff its adjoint carries disjoint almost L-sequences to norm null ones. In addition, we improve several results obtained by Oughajji et al. In its second part, we study the relationship between the class of weakly precompact operators and that of order weakly compact (resp. b-weakly compact) operators. Among other things, we show that for a Banach lattice E and a Banach space X the following statements are equivalent: Every order weakly compact (resp. b-weakly compact) operator T : E → X is weakly precompact; The norm of E ′ is order continuous or X does not contain any isomorphic copy of ℓ 1 . [ABSTRACT FROM AUTHOR]

In this paper, we prove that if K is a nonempty weakly compact set in a Banach space X, T : K → K is a nonexpansive map satisfying x + T x 2 ∈ K for all x ∈ K , then T has a fixed point whenever X is uniformly rotund with respect to every k-dimensional subspace or X satisfies the property (P). These results improve the results of Veeramani and Radhakrishnan et al. [ABSTRACT FROM AUTHOR]

In this paper, we establish the global-in-time hydrodynamic limit of the Boltzmann equation to the planar rarefaction wave of a compressible Euler system in three dimensional space x ∈ ℝ³ for general collision kernels. Our approach is based on the Hilbert expansion and a recent L²-L∞ framework. In particular, we improve the L²-estimate to be a localized version because the planar rarefaction wave is indeed a one dimensional wave which makes the source terms not integrable in the L² energy estimate of the three dimensional problem. We also point out that the wave strength of rarefaction may be large. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some new multiplier spaces related to a series ∑ i x i in a normed space X through f – statistical summability and give some characterizations of completeness and barrelledness of the spaces through weakly unconditionally Cauchy series in X and X * , respectively. We also obtain a new version of the Orlicz-Pettis theorem within the frame of the f – statistical convergence. [ABSTRACT FROM AUTHOR]

In this paper, we give a number of characterizations for a Banach space X which is isometric to a subspace of c0, or, c0(Γ), successively, in terms of extreme points of its dual unit ball BX*, Fréchet and Gâteaux derivatives of its norm, or, in terms of w*-strongly exposed points and w*-exposed points of BX*. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some results on the relationships between different ideal convergence modes namely, IK, IK*, I, K, I [K and (I [K)*. We introduce a topological space namely IK-sequential space and show that the class of IK-sequential spaces contain the sequential spaces. Further IK-notions of cluster points and limit points of a function are also introduced here. For a given sequence in a topological space X, we characterize the set of IK-cluster points of the sequence as closed subsets of X. [ABSTRACT FROM AUTHOR]

c-realcompact spaces are introduced by Karamzadeh and Keshtkar in Quaest. Math. 41, no. 8 (2018), 1135-1167. We offer a characterization of these spaces X via c-stable family of closed sets in X by showing that X is c-realcompact if and only if each c-stable family of closed sets in X with finite intersection property has nonempty intersection. This last condition which makes sense for an arbitrary topological space can be taken as an alternative definition of a c-realcompact space. We show that each topological space can be extended as a dense subspace to a c-realcompact space with some desired extension properties. An allied class of spaces viz CP-compact spaces akin to that of c-realcompact spaces are introduced. The paper ends after examining how far a known class of c-realcompact spaces could be realized as CP-compact for appropriately chosen ideal P of closed sets in X. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a τ$$ \tau $$‐discrete semigroup {Tτn}n∈ℕ0$$ {\left\{{T}_{\tau}&amp;#x0005E;n\right\}}_{n\in {\mathbb{N}}_0} $$ generated by a closed linear operator A$$ A $$ in a Banach space X$$ X $$. We show that {Tτn}n∈ℕ0$$ {\left\{{T}_{\tau}&amp;#x0005E;n\right\}}_{n\in {\mathbb{N}}_0} $$ allows us to write the solution to an abstract discrete difference equation of first order as a discrete variation of parameters formula. Moreover, we study the main properties of {Tτn}n∈ℕ0$$ {\left\{{T}_{\tau}&amp;#x0005E;n\right\}}_{n\in {\mathbb{N}}_0} $$ and its relation with the well‐known notion of discrete semigroups. Finally, we characterize uniform exponential stability of a C0$$ {C}_0 $$‐semigroup {T(t)}t≥0$$ {\left\{T(t)\right\}}_{t\ge 0} $$ in terms of the τ$$ \tau $$‐discrete semigroup {Tτn}n∈ℕ0$$ {\left\{{T}_{\tau}&amp;#x0005E;n\right\}}_{n\in {\mathbb{N}}_0} $$. [ABSTRACT FROM AUTHOR]

We consider an abstract problem P in a metric space X which has a unique solution u ∈ X. Our aim in this current paper is two folds: first, to provide a convergence criterion to the solution of Problem P, that is, to give necessary and sufficient conditions on a sequence {un} ⊂ X which guarantee the convergence un → u in the space X; second, to find a Tyknonov triple T such that a sequence {un} ⊂ X is a T-approximating sequence if and only if it converges to u. The two problems stated above, associated to the original Problem P, are closely related. We illustrate how they can be solved in three particular cases of Problem P: a variational inequality in a Hilbert space, a fixed point problem in a metric space and a minimization problem in a reflexive Banach space. For each of these problems we state and prove a convergence criterion that we use to define a convenient Tykhonov triple T which requires the condition stated above. We also show how the convergence criterion and the corresponding T -well posedness concept can be used to deduce convergence and classical well-posedness results, respectively. [ABSTRACT FROM AUTHOR]

The unstable Adams spectral sequence is a spectral sequence that starts fromalgebraic information about the mod 2 cohomology H* (X) of a space X as an unstable algebra over the Steenrod algebra A, and converges, in good cases, to the 2-localized homotopy groups of X. Bousfield and Don Davis looked at the case when X was either of the infinite matrix groups SO or U. Bousfield and Davis created algebraic spectral sequences and conjectured that they agreed with the unstable Adams spectral sequences for SO and U. To this end the following algebraic decomposition must hold ... where M1 ∈ M2 ∈ ⋯ is the well known dyadic filtration of the A-module H* (ℝP∞, ℤ/2) ≄ F2 [u] given by the dyadic expansion of the powers of u. This paper aims at showing that this decomposition holds for numerous values of s and t. [ABSTRACT FROM AUTHOR]

One of the fundamental problems in rings of continuous functions is to extract those spaces X for which C(X) determines X, that is to investigate X and Y such that C(X) isomorphic with C(Y ) implies X homeomorphic with Y . Later S. Banach and M. Stone proved independently with slight variance, that if X is a compact Hausdorff space, C(X) also determine X. Their works were maximally extended by E. Hewitt who introduced realcompact spaces and later Melvin Henriksen and Biswajit Mitra solved the problem for locally compact and nearly realcompact spaces. In this paper we tried to develop a unified theory of this problem to cover up all the works in the literature introducing the notion P-compact spaces. [ABSTRACT FROM AUTHOR]

In this paper we continue to research relationships between closure-type properties of hyperspaces over a space X and covering properties of X. For a Hausdorff space X we denote by 2X the family of all closed subsets of X. We investigate selection properties of the bitopological space (2X, Δ1+, Δ2+) where Δi+ is the upper Δi-topology for each i=1,2. [ABSTRACT FROM AUTHOR]

This paper present results of ω-order preserving partial contraction mapping generating a general class of semilinear initial value problems. We consider the use of fractional powers of unbounded linear operators for its application by starting with some results concerning such fractional powers. We assume A to be the infinitesimal generator of an analytic semigroup in a Banach space X, 0 ∈ ρ(A) and defined the fractional powers of A for 0 < α ≤ 1. We also show that Aα is a closed linear operator whose domain D(Aα) ⊃ D(A) is dense in X. Finally we established that the operator is bounded, continuous and Holder continuous. [ABSTRACT FROM AUTHOR]

The Hofmann-Mislove theorem says that the ordered set of open filters of the open-set lattice of a sober topological space is isomorphic to the ordered set of compact saturated sets (ordered by reverse inclusion) of that space. This paper concerns a metric analogy of this result. To this end, the notion of compact functions of approach spaces is introduced. Such functions are an analog of compact subsets in the enriched context. It is shown that for a sober approach space X, the metric space of proper open [0;∞]-filters of the metric space of upper regular functions of X is isomorphic to the opposite of the metric space of inhabited and saturated compact functions of X, establishing a Hofmann-Mislove theorem for approach spaces. [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]

This paper considers von Neumann-Schatten p-Bessel sequences, von NeumannSchatten p-frames and von Neumann-Schatten q-Riesz bases. Sequences, consisting of operators from a separable Banach space X into the von NeumannSchatten p-class Cp, which are von Neumann-Schatten p-Bessel sequences are characterized. It is shown that the set of all von Neumann-Schatten p-Bessel sequences can be considered as a Banach algebra. Also, some necessary and sufficient conditions for a reflexive Banach space to possess von NeumannSchatten p-frames or von Neumann-Schatten q-Riesz bases are presented. Moreover, the stability of von Neumann-Schatten p-frames and von NeumannSchatten q-Riesz bases under different kinds of perturbations is studied, especially some sufficient conditions for the preservation of the frame property under perturbations are derived. [ABSTRACT FROM AUTHOR]

In this paper, we introduced the mildly version of the Hurewicz basis covering property, studied by Babinkostova, Kočinac, and Scheepers. A space X is said to have mildly-Hurewicz property if for each sequence ⟨ Un : n ∈ ω ⟩ of clopen covers of X there is a sequence ⟨ Vn : n ∈ ω ⟩ such that for each n, Vn is a finite subset of Un and for each x ∈ X, x belongs to ⋃ Vn for all but finitely many n. Then we characterized mildly-Hurewicz property by mildly-Hurewicz Basis property and mildly-Hurewicz measure zero property for metrizable spaces. [ABSTRACT FROM AUTHOR]

The time-series data generated by turbofan engines has a great degree of complexity and dynamics. At present, recurrent neural networks are commonly used to model and forecast the remaining useful life (RUL). The relationship of the sample data is not taken into account, and there are issues such as gradient explosion. In view of this, a spatio-temporal attention model is proposed, which comprehensively relates to the temporal association of data features and the hidden state of data features in space. At the same time, position coding is performed on the temporal relationship, avoiding the use of recurrent neural networks. Experimental results show that by combining the two dimensions, the predictive performance of the model is significantly improved. Compared with different methods on the four data sets of the commercial modular aerospace propulsion system simulation (C-MAPSS), the stability and prediction accuracy of the spatio-temporal attention model are better than that of alternative methods. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce and study new topological properties called C-almost normality and L-almost normality. A space X is called a C-almost normal (resp. L-almost normal) space if there exist an almost normal space Y and a bijective function f : X → Y such that the restriction function f|A : A → f(A) is a homeomorphism for each compact (resp. Lindel¨of) subspace A ⊆ X. We investigate these properties and present some examples to illustrate the relationships among them with other kinds of topological properties. [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]

The paper continues the study on the distinguished property of the space Cp (X) (of real-valued continuous functions over a Tychonoff space X in the pointwise topology) under the formation of tensor products towards the following research directions: (i) the injective tensor product Cp(X) ⊗ε E of Cp (X) and a real locally convex space E (and its completion Cp (X) ⊗bε E), and (ii) the space Cp (X, E) of all E-valued continuous functions (with a normed space E) endowed with the pointwise topology. This work leads also to a new characterization of distinguished Fréchet locally convex spaces E. We show, e. g., that if Cp(X) is metrizable, then E is distinguished if and only if metrizable Cp (X) ⊗ε E is distinguished. [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]

Given a metric pair (X,A), i.e. a metric space X and a distinguished closed set A\subset X, one may construct in a functorial way a pointed pseudometric space \mathcal {D}_\infty (X,A) of persistence diagrams equipped with the bottleneck distance. We investigate the basic metric properties of the spaces \mathcal {D}_\infty (X,A) and obtain characterizations of their metrizability, completeness, separability, and geodesicity. [ABSTRACT FROM AUTHOR]

We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if (X,d,\mu) is a locally complete and separable metric measure space, then continuous functions are dense in the Newtonian space N^{1,p}(X). Here the measure \mu is Borel and is finite and positive on all metric balls. In particular, we don't assume properness of X, doubling of \mu or any Poincaré inequalities. These resolve, partially or fully, questions posed by a number of authors, including J. Heinonen, A. Björn and J. Björn. In contrast to much of the past work, our results apply to locally complete spaces X and dispenses with the frequently used regularity assumptions: doubling, properness, Poincaré inequality, Loewner property or quasiconvexity. [ABSTRACT FROM AUTHOR]

In the complex landscape of global security, operating in the gray zone-employing strategies that fall below the threshold of armed conflict-has become increasingly vital. This article explores adapting gray-zone strategies to the unique operational dynamics of space. It emphasizes synchronizing diplomatic, informational, and economic instruments to dissuade and deter adversaries while avoiding kinetic conflict. Focusing on the Indo-Pacific region's strategic significance, particularly China's activities, it delves into nonkinetic actions such as jamming and cyber operations aimed at degrading satellite capabilities, which can impact homeland defense operations. The article addresses legal frameworks governing space activities, the rise of commercial space ventures, and the need for collaboration between military and commercial entities to mitigate space threats. It advocates for transparency, international cooperation, and response strategy development to uphold space sustainability and stability. Ultimately, the article stresses collective responsibility in navigating the gray zone to preserve space accessibility for future generations. [ABSTRACT FROM AUTHOR]

Let BX be the unit ball of a complex Banach space X, which may be infinite dimensional. The authors characterize the boundedness of the product of the radial derivative operator and the weighted composition operator from the natural Bloch space (or the little Bloch-type space) into the weighted-type space (or the little weighted-type space) on BX. [ABSTRACT FROM AUTHOR]

We show that if 퐺 is an admissible group acting geometrically on a CAT ⁡ (0) space 푋, then 퐺 is a hierarchically hyperbolic space and its 휅-Morse boundary (∂ κ G , ν) is a model for the Poisson boundary of (G , μ) , where 휈 is the hitting measure associated to the random walk driven by 휇. [ABSTRACT FROM AUTHOR]

Let X be a class of metric spaces and let Px be the set of all f: [0, oo) [0, oo) preserving X, i.e., (/, f o p) e X whenever (/, p) e X. For arbitrary subset A of the set of all metric preserving functions, we show that the equality Px = A has a solution if A is a monoid with respect to the operation of function composition. In particular, for the set SI of all amenable subadditive increasing functions, there is a class X of metric spaces such that Px = SI holds. 2020 Mathematics Subject Classification: Primary 26A30, Secondary 54E35, 20M20 [ABSTRACT FROM AUTHOR]

This perspective article reviews how the chemometrics community approached non-linear methods in the early years. In addition to the basic chemometric methods, some methods that fall under the term "machine learning" are also mentioned. Thereafter, types of non-linearity are briefly presented, followed by discussions on important aspects of modeling related to non-linear data. Lastly, a simulated data set with non-linear properties is analyzed for quantitative prediction and batch monitoring. The conclusion is that the latent variable methods to a large extent handle non-linearities by adding more linear combinations of the original variables. Nevertheless, with strong nonlinearities between the X and Y space, non-linear methods such as Support Vector Machines might improve prediction performance at the cost of interpretability into both the sample and variable space. Applying multiple local models can improve performance compared to a single global model, of both linear and non-linear nature. When non-linear methods are applied, the need for conservative model validation is even more important. Another approach is pre-processing of the data which can make the data more linear before the actual modeling and prediction phase. [ABSTRACT FROM AUTHOR]

We consider an elliptic variational inequality with unilateral constraints in a Hilbert space X which, under appropriate assumptions on the data, has a unique solution u. We formulate a convergence criterion to the solution u, i.e. we provide necessary and sufficient conditions on a sequence $ \{u_n\}\subset X $ { u n } ⊂ X which guarantee the convergence $ u_n\to u $ u n → u in the space X. Then we illustrate the use of this criterion to recover well-known convergence results and well-posedness results in the sense of Tykhonov and Levitin–Polyak. We also provide two applications of our results, in the study of a heat transfer problem and an elastic frictionless contact problem, respectively. [ABSTRACT FROM AUTHOR]

Japanese art tradition, contrary to the case of China, is characterized by an efficient and continued, although mostly undocumented, use of smalt from the late 16th century onward. Recent studies have successfully identified this pigment, the cobalt-colored glass that spread throughout the Old Continent during the Renaissance period, as the coloring agent employed for overglaze-blue enameling on Japanese porcelains produced at the kilns of Arita (the porcelain production center of Japan) from the early 1640s until the 20th century. Fragmentary evidence of the use of smalt in Japanese sacred art has also been reported, yet its earliest incorporation into such a type of traditional art form could not be identified. In order to resolve this crucial issue, portable EDXRF was employed for the non-destructive analyses of Japanese porcelains and sacred images bearing blue decoration. Scientific analysis allowed, for the first time ever, to establish a clear timeline of smalt use. Furthermore, this evidence and the literature data both agree, leading to the identification of the origin of the blue material used on both art productions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the complexity of random dynamical systems. Assume that ϕ is a continuous random dynamical system on Polish space X over an ergodic Polish system (Ω , F , P , θ). Let K be a ϕ -invariant random set with compact ω -section K (ω) such that ϕ (ω) (K (ω)) = K (θ ω). We proved that if h top (ϕ , K) > 0 , then there exists chaotic phenomenon (strong second type of distributional chaos) in this random dynamical system. [ABSTRACT FROM AUTHOR]

Given a functionally countable space X, if K is a scattered Corson compact space, then the product X × K is functionally countable. If the space X is, additionally, weakly ω 1 -Lindelöf, then X × L is functionally countable for any scattered Lindelöf space L. In particular, if X is a weakly Lindelöf functionally countable space, then the product X × L is functionally countable whenever L is a scattered Lindelöf space. We also establish that any finite product of countably compact functionally countable spaces is functionally countable and exponential separability of countably compact spaces X and Y implies that X × Y is exponentially separable if one of them is sequential. If the product X = X 1 × ⋯ × X n of functionally countable spaces X 1 ,... , X n is pseudocompact, then X is functionally countable. [ABSTRACT FROM AUTHOR]