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Let $ (X, T) $ (X , T) be a linear dynamical system defined on a Banach space X and $ {L_T}:{K_{{X^ * }}} \to {K_{{X^ * }}} $ L T : K X ∗ → K X ∗ be defined by $ {L_T}S = TS $ L T S = TS , where $ {K_{{X^ * }}} = \overline {span\{ \langle \cdot,{y^ * } \rangle x:{y^ * } \in {X^ * },x \in X\} } $ K X ∗ = span { 〈 ⋅ , y ∗ 〉 x : y ∗ ∈ X ∗ , x ∈ X } ¯ . This paper first obtains the equivalence of the (syndetically) weak multi-sensitive compactness between $ (X, T) $ (X , T) and $ ({K_{{X^ * }}},{L_T}) $ (K X ∗ , L T). Then, it is shown that (1) there exists a non-mean-sensitive linear dynamical system $ (X ,T) $ (X , T) satisfying that $ ({K_{{X^ * }}},{L_T}) $ (K X ∗ , L T) is mean sensitive; (2) There exists a non-topologically-transitive and weakly multi-sensitive compact system, which is not syndetically weakly multi-sensitive compact. Besides, it is proved that every weakly mixing linear dynamical system defined on a separable Banach space is weakly multi-sensitive compact. [ABSTRACT FROM AUTHOR]

Let L be a truncated Archimedean vector lattice whose truncation is denoted by *. In a recent paper, we proved that there exists a locally compact Hausdorff space X such that L is a lattice isomorphic with a truncated vector lattice of functions in C∞(X) whose truncation is provided by meet with some characteristic function on X. This representation, no matter how interesting it is, has a major drawback, namely, C∞(X) need not be a vector lattice, unless X is extremally disconnected. The main purpose of this paper is to remedy this shortcoming by proving that, indeed, an extremally disconnected locally compact space X can be found such that L can be seen as an order dense vector sublattice of C∞(X) whose truncation is provided, again, by meet with some characteristic function on X. [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]

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]

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 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]

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]

In this paper we give a quantitative analysis of an explicit iteration method due to C.E. Chidume for the approximation of a zero of an m-accretive operator A : X → 2 X in Banach spaces which does not involve the computation of the resolvent of A. [ABSTRACT FROM AUTHOR]

A CT-scanner was used to scan 25 mm thick radiata pine boards during drying at 90 °C dry-bulb and 60 °C wet-bulb temperatures. Twenty transverse sections spaced by 10 mm along the length of the boards were CT-scanned six times during drying, and an elastic image registration technique was used to transform the CT-scanned images into moisture content profiles. The resulting images suggested that moisture content distributions did not always resemble diffusion-like moisture transfer mechanism, so common assumptions based on moisture content gradients as the main driving force for wood drying may not always be enough to explain the development of wet and dry zones coexisting below and above fiber saturation point. This paper discusses these findings based on both experimental CT-scanner data and the theory of water potential, which it is argued provides a more complete description of driving forces in wood drying. [ABSTRACT FROM AUTHOR]

In this paper, Hyers–Ulam stability of the diffusion equation on the half space R + n + 1 with n ≥ 2 is investigated: ∂ u ∂ t (x , y , t) =△ u (x , y , t) where x ∈ R n , y>0, and t>0. Using integral transforms, precisely, Fourier transform with respect to the tangential space x and Laplace transform for time t, we obtain the generalized Hyers–Ulam stability of diffusion equation in the half space R + n + 1 involving special functions such as gamma and error functions. [ABSTRACT FROM AUTHOR]

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

In this paper a Banach function space (b.f.s. in short) X (Ω) on a n-dimensional bounded domain Ω with Lebesgue measure is considered. The Banach function space X (S) on the (n − 1) -dimensional surface S ⊂ Ω ¯ generated by the norm of the space X (Ω) is defined. The Sobolev function space W X m (Ω) is defined using the norm of the space X (Ω) , as well as the concept of the trace of an arbitrary function from this space on the surface S. Based on this concept, the space of traces W X m (S) is defined and a characterization of this space is given. It was proved that it is boundedly embedded in the space X (S). Particular cases of a functional space X (Ω) can be Lebesgue spaces, Lebesgue spaces with variable summability exponents, Morrey space, Orlicz space, grand Lebesgue spaces and etc. The obtained results allow us to consider boundary value problems for differential equations in b.f.s. W X m (Ω). [ABSTRACT FROM AUTHOR]

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

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]

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]

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]

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]

In this paper we introduce and study a class of ideals between z-ideals and z°-ideals (=d-ideals) namely zr-ideals. A zr-ideal is a z-ideal which is at the same time an r-ideal (an ideal I in a ring R is called an r-ideal if for each non-zerodivisor r ∈ R and each a ∈ R, ra ∈ I implies a ∈ I). In contrast to the sum of z-ideals in C(X) which is a z-ideal, the sum of zr-ideals need not be a zr-ideal. We prove that the sum of every two zr-ideals of C(X) is a zr-ideal if and only if X is a quasi F -space. In C(X) every z°-ideal is a zr-ideal and we characterize the spaces X for which the converse is also true. We observe that X is a cozero complemented space if and only if every (prime) r-ideal in C(X) is a z-ideal and whenever every (prime) z-ideal of C(X) is an r-ideal it is equivalent to X being an almost P-space. Using these facts it turns out that the set of all r-ideals and the set of all z-ideals of C(X) coincide if and only if X is a P-space. [ABSTRACT FROM AUTHOR]

With the latest published annual climate-based daylighting metrics and performance criteria, annual daylighting simulation has become more important to the design professions, but interpretation and application of these data are still relatively novel. This paper documents an eight-year human factors daylighting field research project using students' qualitative assessments of daylight sufficiency and corresponding point-in-time and annual climate-based daylighting simulation in a variety of commercial spaces (n = 22) in order to provide insight to the building performance simulation community about application of these new annual daylighting metrics. The sensitivity analysis on horizontal illuminance thresholds consistently represents high correlation with students' subjective responses and annual simulation results at 250 lx, whereas point-in-time analyses produces a range of best-fitting thresholds from 200 to 300 lx for daylight sufficiency. Additionally, the study reveals that ASE 1000 lx 250 hours at 10% threshold warrants refinement, and perhaps temporal or space type sensitivity should be applied to these criteria. [ABSTRACT FROM AUTHOR]

In this paper, we show the following statements: (1) There exists a pseudocompact star Lindelöf Tychonoff space which is not star σ-compact. (2) There exists a Tychonoff pseudocompact star countable (hence, star Lindelöf) space having a pseudocompact, Gδ regular closed subspace which is not star Lindelöf. (3) Assuming , there exists a normal star countable (hence, star Lindelöf) space having a Gδ regular closed subspace which is not star Lindelöf. (4) Let X be a space, then A(X) is star Lindelöf if and only if e(X) ≤ ω. The statement (1) gives a negative answer to Song [13, Remark 2.3]. [ABSTRACT FROM AUTHOR]

In this paper, we investigate conditions on a pair of Banach lattices E and F that tells us when every positive almost L-weakly compact (resp. almost M- weakly compact) operator T : E → F is weakly compact. Also, we present some necessary conditions that tells us when every weakly compact operator T : E → F is almost M-weakly compact (resp. almost L-weakly compact). In particular, we will prove that if every weakly compact operator from a Banach lattice E into a Banach space X is almost L-weakly compact, then E is a KB-space or X has the Dunford-Pettis property and the norm of E is order continuous. [ABSTRACT FROM AUTHOR]

Medieval prohibitions abound against artisans selling stolen or bloodied goods, and refabricating used clothing to deceive customers. Such prohibitions have led scholars to suggest the secondhand trade in the Middle Ages was one of poverty or marginality. Yet, if we read trade regulations carefully and alongside other types of sources as well as consider the context of a late medieval economy undergirded by credit, a complex image of fripperers materialises. This was a trade populated by both men and women of various means vying with each other and other merchants and artisans for control of retail space. This paper works to uncover the economic and cultural standing of fripperers in thirteenth- and fourteenth-century Paris and the commercial space of Les Halles. [ABSTRACT FROM PUBLISHER]

The solar cells that currently dominate the industry require a high-purity silicon produced through a costly, energy-intensive process, yet these cells currently convert less than 25 percent of the sunlight they capture into electricity. Tandem cells, for instance, pair two materials such as silicon and halide perovskite, each of which absorbs and converts a different part of the light spectrum to produce more electricity in combination than either layer on its own. Silicon cells also provide power to satellites and the International Space Station, but the radiation levels in higher orbits, outside Earth's protective atmosphere, would shorten their life to weeks or days without a protective layer of glass. For now, that's still the benchmark for power-conversion efficiency outside the lab, but researchers are testing new designs, alternate materials, and combinations of all these approaches to push past the theoretical efficiency limit of roughly 33 percent for a standard single-junction silicon cell. [Extracted from the article]

We return to the work of Banaschewski and extract from it a theorem of Fuchs, Heinzer, and Olberding. As an application of Fuchs-Heinzer-Olberding's theorem, we generalize a result of Gillman and Kohls. We study pseudo-irreducible ideals and show that every ideal of a pm-ring is the (not necessarily finite) intersection of pairwise comaximal pseudo-irreducible ideals. After some general results, the article focuses on primal and pseudo-irreducible ideals in rings of continuous functions. We determine when every pseudo-irreducible ideal of C(X) is primal. We give a characterization of spaces X for which every Op is a primal ideal of C(X). [ABSTRACT FROM AUTHOR]

We study BDF theory in the context of (not necessarily simple) purely infinite corona algebras. Let B be a nonunital separable simple C*-algebra with standard regularity properties for which C (B) is purely infinite (though not necessarily simple). Let A be a separable nuclear C*-algebra. We prove a BDF Voiculescu decomposition theorem for maps from A to C (B) , and use this to prove that E x t σ (A , B) is a group. Then, restricting to the case A = C (X) , for some compact metric space X, and B has continuous scale, we provide characterizations of the neutral element for E x t (C (X) , B). [ABSTRACT FROM AUTHOR]

Formerly a solely government sphere, space is now an attractive frontier for private companies. As research and production costs were unreachable for private actors in the past, only government agencies such as NASA, could afford the development of their space program, with participation from private companies through contracting for design and manufacturing. Today, companies, such as Blue Origin and SpaceX are privately developing different aspects of the space industry and making the topic of cooperation between government agencies and private companies for commercial services relevant. Despite the new programs that private companies are bringing to life, such as space tourism and space launch, a major part of their activity is related to cooperation with NASA. The structure of this article consists of several parts, beginning with a brief overview of the types of space companies. This is followed by a history of private space companies and the economic aspect of public-private cooperation, and finally the military-diplomatic effects of this cooperation. After decades of coexisting and collaborating, the new model whereby NASA only places orders and private companies provide commercial services is the one that is the most cost-effective. [ABSTRACT FROM AUTHOR]

Asmar et al. [Note on norm convergence in the space of weak type multipliers. J Operator Theory. 1998;39(1):139–149] proved that the space of weak-type Fourier multipliers acting from L p into L p , ∞ is continuously embedded into L ∞ . We obtain a sharper result in the setting of abstract Lorentz spaces Λ q (X) with 0 < q ≤ ∞ built upon a Banach function space X on R n . We consider a source space S and a target space T in the class of admissible spaces A := { X , Λ q (X) : 0 < q ≤ ∞ }. Let M S , T 0 denote the space of Fourier multipliers acting from S to T . We show that if the space X satisfies the weak doubling property, then the space M Λ q (X) , Λ ∞ (X) 0 is continuously embedded into L ∞ for every 0 < q ≤ ∞. This implies that M S , T 0 is a quasi-Banach space for all choices of source and target spaces S , T ∈ A . [ABSTRACT FROM AUTHOR]

We demonstrate that if J is an M-ideal in a Lindenstrauss space X, then X satisfies the famous Smith and Ward identity, that is, rad J (F) = rad X (F) + d ( cent X (F) , J) for finite subsets F of X. We introduce strong property- (R 1) for a triplet (X , V , F) , where X is a Banach space, V is a closed convex subset of X, and F is a subfamily of closed, bounded subsets of X. We show that for a subspace V, the restricted Chebyshev centre for F ∈ F with respect to the unit ball of V is non-empty if the triplet (X , V , F) has the strong property- (R 1). We demonstrate that for an M-ideal J in a Lindenstrauss space X, the triplet (X , J , K (X)) has the strong property- (R 1) , where K (X) is the family of compact subsets of X. Some characterizations of the strong property- (R 1) are given. Similar to the strong 1 1 2 -ball property, we show that for a subspace V of X, a triplet (X , V , F) has the strong property- (R 1) if and only if (X , V , F) has the property- (R 1) and the set of restricted Chebyshev centres with respect to the unit ball of V is non-empty for F ∈ F. [ABSTRACT FROM AUTHOR]

Let L (X , Y) be the space of bounded linear operators between real Banach spaces X, Y. For a closed subspace Z ⊂ Y , we partially solve the operator version of Birkhoff–James orthogonality problem, if T ∈ L (X , Y) is orthogonal to L (X , Z) , when does there exist a unit vector x0 such that | | T (x 0) | | = | | T | | and T (x 0) is orthogonal to Z? In order to achieve this we first develop a compact optimization for a Y-valued compact operator T, via a minimax formula for d (T , L (X , Z)) in terms of point-wise best approximations, that links local optimization and global optimization. Our result gives an operator version of a classical minimax formula of Light and Cheney, proved for continuous vector-valued functions. For any separable reflexive Banach space X and for Z ⊂ Y is a L1-predual space as well as a M-ideal in Y, we show that if T ∈ K (X , Y) is orthogonal to L (X , Z) , then there is a unit vector x0 with | | T (x 0) | | = | | T | | and T (x 0) is orthogonal to Z. In the general case we also give some local conditions on T, when this can be achieved. [ABSTRACT FROM AUTHOR]

Launching spacecraft into orbit is one of the most challenging and critical aspects of spaceflight, and thus, is a key element of any national space strategy. Over the last several decades, the United States, seeing the high cost of space access as a primary impediment to greater government and commercial utilization of space, has made several attempts to lower the cost of space access, but without success. This analysis challenges whether cost should be the primary criterion for any national space launch strategy. A review of six different civil, commercial, and national security space launch customers shows that for the four major existing customer segments, cost is a lower priority than launch reliability, crew safety, and schedule assurance. Only for two emerging customer segments, Operationally Responsive Space and entrepreneurial space ventures, is cost a primary factor. Given this analysis, existing launch systems actually meet most customers' needs, creating equilibrium in the market. Several events, though, could disrupt that equilibrium in the near future, including the development of new reusable suborbital vehicles and a shift to commercial providers for crew transportation to low Earth orbit. [ABSTRACT FROM AUTHOR]

The threats to space systems posed by hostile states and non-state actors are evolving. Vulnerabilities span the national security, civil, and commercial space communities. The weaknesses have been studied, and adversaries to the United States and its allies have noted the asymmetric advantages space capabilities provide. They see the tremendous leverage that can be obtained by disrupting them. These adversaries are becoming much more diverse, sophisticated, and technologically competent; they are equipped and able to disrupt access to space capabilities. Defending space systems demands new tools as deterring or eliminating these evolving threats will be difficult. Risk posed by an increasingly dangerous space environment must also be addressed. As it has done for decades, the United States enjoys a unique position in which it can shape the direction of global space activities. With this position comes great responsibility-to forge behaviors to mitigate space debris, deter armed conflict, and enhance the peace, security, and prosperity of spacefaring nations and the rest of the world. A space assurance strategy involving deterrence and defense, global engagement, situational awareness, and responsive infrastructure presents the best opportunity to collectively defend, protect, and secure the high frontier. [ABSTRACT FROM AUTHOR]