Space Xn Research Articles

Relatively functionally countable subsets of products

A subset A of a topological space X is called relatively functionally countable (RFC) in X, if for each continuous function f:X→R the set f[A] is countable. We prove that all RFC subsets of a product ∏n∈ωXn are countable, assuming that spaces Xn are Tychonoff and all RFC subsets of every Xn are countable. In particular, in a metrizable space every RFC subset is countable.The main tool in the proof is the following result: for every Tychonoff space X and any countable set Q⊆X there is a continuous function f:Xω→R2 such that the restriction of f to Qω is injective.

Framed Matrices and A∞-Bialgebras *

We complete the construction of the biassociahedra KK, construct the free matrad H∞, realize H∞ as the cellular chains of KK, and define an A∞-bialgebra as an algebra over H∞. We construct the bimultiplihedra JJ, construct the relative free matrad rH∞ as a H∞-bimodule, realize rH∞ as the cellular chains of JJ, and define a morphism of A∞-bialgebras as a bimodule over H∞. We prove that the homology of every A∞-bialgebra over a commutative ring with unity admits an induced A∞-bialgebra structure. We extend the Bott-Samelson isomorphism to an isomorphism of A∞-bialgebras and determine the A∞-bialgebra structure of H*ΩΣX;ℚ. For each n≥2, we construct a space Xn and identify an induced nontrivial A∞-bialgebra operation ω2n:H*ΩXn;ℤ2⊗2→H*ΩXn;ℤ2⊗n.

On the distribution of angles between increasingly many short lattice vectors

Following Södergren, we consider a collection of random variables on the space Xn of unimodular lattices in dimension n: normalizations of the angles between the N=N(n) shortest vectors in a random unimodular lattice, and the volumes of spheres with radii equal to the lengths of these vectors. We investigate the expected values of certain functions (whose support depends on a parameter K=K(n)) evaluated at these random variables in the regime where K and N are allowed to tend to infinity with n at the rate KN=o(n1/6). Our main result is that as n⟶∞, these random variables exhibit a joint Poissonian and Gaussian behavior.

Canonical reduction of stabilizers for Artin stacks with good moduli spaces

We present a complete generalization of Kirwan’s partial desingularization theorem on quotients of smooth varieties. Precisely, we prove that if X is an irreducible Artin stack with stable good moduli space X→πX, then there is a canonical sequence of birational morphisms of Artin stacks Xn→Xn−1→⋯→X0=X with the following properties: (1) the maximum dimension of a stabilizer of a point of Xk+1 is strictly smaller than the maximum dimension of a stabilizer of Xk and the final stack Xn has constant stabilizer dimension; (2) the morphisms Xk+1→Xk induce projective and birational morphisms of good moduli spaces Xk+1→Xk. If in addition the stack X is smooth, then each of the intermediate stacks Xk is smooth and the final stack Xn is a gerbe over a tame stack. In this case the algebraic space Xn has tame quotient singularities and is a partial desingularization of the good moduli space X. When X is smooth our result can be combined with D. Bergh’s recent destackification theorem for tame stacks to obtain a full desingularization of the algebraic space X.

Uniform and Lq-ensemble reachability of parameter-dependent linear systems

In this paper, we consider families of linear systems (linear ensembles) defined by matrix pairs (A(θ),B(θ)) depending on a parameter θ∈P that is varying over a compact subset P of the complex plane. In particular, we investigate the existence of open-loop controls which are independent of the parameter θ∈P and steer a given family of initial states x0(θ) arbitrarily close to a desired family of terminal states f(θ) in finite time. Here, the maps θ↦x0(θ) and θ↦f(θ) are assumed to lie in a common appropriately chosen Banach space Xn(P) of Cn-valued functions. If this task is solvable for all initial and terminal states, the pair (A(θ),B(θ)) is called (completely) ensemble controllable with respect to Xn(P).Using a well-known infinite-dimensional version of the Kalman rank condition for systems on Banach spaces, we derive sufficient conditions for cascade and parallel connections of linear ensembles. Moreover, we prove an abstract decomposition theorem which results from a spectral splitting of the matrix family A(θ). Based on these findings as well as on cyclicity conditions for multiplication operators and approximation theory, we obtain necessary and sufficient conditions for ensemble controllability (reachability) with respect to the space of continuous functions and the space of Lq-functions. In the last section, results on averaged controllability (reachability) for linear families (A(θ),B(θ),C(θ)) are presented.

ОБ АДДИТИВНОЙ МОДИФИКАЦИИ γ-СВОЙСТВА

The aim of this article is to modify the notion of the γ-property so that the modified property is preserved under taking a direct topological sum. To this end, we use a procedure of “saturation” of ω-covers of topological spaces. Namely, we add to each ω-cover the family of all unions of its subfamilies consisting of at most k elements (“k-saturation”), or the family of all unions of its finite subfamilies (“saturation”). Thus, we obtain for Tychonoff spaces a sequence ( k )k<ω γ′ of covering properties such that k k +1 ′ ′ γ ⇒ γ for any integer k, and 1 ′γ is the well-known γ-property. Also, the property ωγ′ such that k ω γ′ ⇒ γ′ for any integer k is obtained. It is proved that each kγ′ -property is preserved under usual topological operations, namely, taking closed subspaces, continuous images, and finite powers. It is known that the classical γ-property can be failed under passing to topological sum. Our main result means that it is impossible with respect to whole sequence ( k )k<ω γ′ . More precisely, if a space X satisfies kγ′ , Y satisfies mγ′ , then the sum X ⊕Y satisfies k +m γ′ . Nevertheless, X ⊕ X satisfies kγ′ as X itself. As another main result, we establish that the ωγ′ -property and the Lindelöf property are equivalent. It follows that any countable union of spaces Xn with the k (n) γ′ -property is a Lindelöf space.

On proximinality of subspaces and the lineability of the set of norm-attaining functionals of Banach spaces

We show that for every 1<n<∞, there exists a Banach space Xn containing proximinal subspaces of codimension n but no proximinal finite codimensional subspaces of higher codimension. Moreover, the set of norm-attaining functionals of Xn contains n-dimensional subspaces, but no subspace of higher dimension. This gives an n-by-n version of the solutions given by Read and Rmoutil to problems of Singer and Godefroy. We also study the existence of strongly proximinal subspaces of finite codimension, showing that for every 1<n<∞ and 1⩽k<n, there is a Banach space Xn,k containing proximinal subspaces of finite codimension up to n but not higher, and containing strongly proximinal subspaces of finite codimension up to k but not higher. Finally, we deal with possible infinite-dimensional versions of the previous results showing, among other results, that there are non-separable Banach spaces whose set of norm-attaining functionals contains infinite-dimensional separable linear subspaces but no non-separable subspaces.

On the Gromov-Hausdorff limit of metric spaces

Abstract In this paper, we show that a class of metric spaces determined by a continuous function f, which defines on the metric space of all real, n × n-matrices m is closed under the Gromov-Hausdorff convergence. This conclusion can be used to prove some metric properties of metric space is stable under the Gromov-Hausdorff convergence. Secondly, we consider the stability problem in Gromov hyperbolic space and show that if a sequence of Gromov hyperbolic spaces (Xn , dn ) is said to converge to (X, d) in the sense of Gromov-Hausdorff convergence, then the Gromov hyperbolicity δ(Xn ) of (Xn , dn ) tends to the Gromov hyperbolicity δ(X) of (X, d).

Snowflake universality of Wasserstein spaces

For p ∈ (1,∞) let Pp(R) denote the metric space of all p-integrable Borel probability measures on R, equipped with the Wasserstein p metric Wp. We prove that for every e > 0, every θ ∈ (0, 1/p] and every finite metric space (X, dX), the metric space (X, dX) embeds into Pp(R) with distortion at most 1 + e. We show that this is sharp when p ∈ (1, 2] in the sense that the exponent 1/p cannot be replaced by any larger number. In fact, for arbitrarily large n ∈ N there exists an n-point metric space (Xn, dn) such that for every α ∈ (1/p, 1] any embedding of the metric space (Xn, d α n) into Pp(R) incurs distortion that is at least a constant multiple of (logn)α−1/p. These statements establish that there exists an Alexandrov space of nonnegative curvature, namely P2(R), with respect to which there does not exist a sequence of bounded degree expander graphs. It also follows that P2(R) does not admit a uniform, coarse, or quasisymmetric embedding into any Banach space of nontrivial type. Links to several longstanding open questions in metric geometry are discussed, including the characterization of subsets of Alexandrov spaces, existence of expanders, the universality problem for P2(R), and the metric cotype dichotomy problem.

The strong Pytkeev property in topological spaces

A topological space X has the strong Pytkeev property at a point x∈X if there exists a countable family N of subsets of X such that for each neighborhood Ox⊂X and subset A⊂X accumulating at x, there is a set N∈N such that N⊂Ox and N∩A is infinite. We prove that for any ℵ0-space X and any space Y with the strong Pytkeev property at a point y∈Y the function space Ck(X,Y) has the strong Pytkeev property at the constant function X→{y}⊂Y. If the space Y is rectifiable, then the function space Ck(X,Y) is rectifiable and has the strong Pytkeev property at each point. We also prove that for any pointed spaces (Xn,⁎n), n∈ω, with the strong Pytkeev property their Tychonoff product ∏n∈ωXn and their small box-product ⊡n∈ωXn both have the strong Pytkeev property at the distinguished point (⁎n)n∈ω. We prove that a sequential rectifiable space X has the strong Pytkeev property if and only if X is metrizable or contains a clopen submetrizable kω-subspace. A locally precompact topological group is metrizable if and only if it contains a dense subgroup with the strong Pytkeev property.

Estimate of observable diameter of l p -product spaces

We estimate the observable diameter of the lp-product space Xn of an mm-space X by using the limit formula in our previous paper (Ozawa and Shioya, Math. Zeit. (to appear)). The idea of our proof is based on Gromov’s book (In: Metric structures for Riemannian and non-Riemannian spaces, 2007). As a corollary we obtain the phase transition property of \({\{X^n\}_{n=1}^\infty}\) under a uniform discreteness condition.

Green’s function of a chemotaxis model in the half space and its applications

In this paper, we are interested in a degenerate Keller–Segel model in the half space xn>lt in high dimensions, with a constant l. Under the conservative boundary condition, we study the global solvability, regularity and long time behavior of solutions. We first construct Green’s function of the half space problem and study its properties. Then by Green’s function, we give the implicit formula of solutions. We prove that when the initial data is small enough, the half space problem is always globally and classically solvable. We also obtain decay estimates of solutions when the parameter l have different signs. For l<0, the time decay of solutions is (1+t)−n/2. While for l>0, the time decay is (1+t)−(n−1)/2, and in this case, we also obtain an exponential decay in the spatial space in the xn-direction.

Algebraic vector bundles on punctured affine spaces and smooth quadrics

There is a canonical An-fibration from a (2n+1)-dimensional smooth affine quadric Yn to an (n+1)-dimensional punctured affine space Xn. For each rank r at least n, we construct a nontrivial algebraic vector bundle of rank r on Xn with trivial pullback to Yn. Moreover, we construct continuous families of arbitrarily large dimensional pairwise non-isomorphic rank n bundles on Xn with trivial pullbacks to Yn.

The local multiplier algebra of a [formula omitted]-algebra with finite dimensional irreducible representations

We describe, up to isomorphism, the local multiplier algebra Mloc(A) of a C∗-algebra A which admits only finite dimensional irreducible representations: Mloc(A) in this case is a finite or countable direct product of C∗-algebras of the form C(Xn)⊗Mn, where each space Xn is Stonean. In particular, Mloc(A) is an AW∗-algebra of type I, so it admits only inner derivations and it coincides with the injective envelope I(A) of A.

On the fixed point property for inverse limits of fans

In 1984, M. M. Marsh gave conditions under which the inverse limit of an inverse sequence of fans has the fixed point property. In this paper we give an extension of that result. 1. Definitions and notation By a space we mean a topological space. A continuum is a nonempty compact connected metric space. By a map we mean a continuous function. A fixed point of a map f : X → X is a point x ∈ X such that f(x) = x. A space X has the fixed point property if every map f : X → X has a fixed point. The symbols N and R denote the set of the positive integers and the set of the real numbers, respectively. The diameter of a set A in a metric space will be denoted by diam(A). Given a point x in a product ∏ {Xn : n ∈ N} we will write x = (xn)n∈N. An inverse sequence is a double sequence {Xn, fn} of spaces Xn and maps fn : Xn+1 → Xn. The inverse limit of the inverse sequence {Xn, fn}, which is denoted by lim ←− {Xn, fn} or by X∞, is the space of all points x ∈ ∏ {Xn : n ∈ N} such that xn = fn(xn+1). If Xn = X for each n ∈ N, then we write {X, fn} instead of {Xn, fn}. The projection map from X∞ into Xn will be denoted by f n . 2000 Mathematics Subject Classification. 54H25.

Left–right symmetric gauge model in generalized differential geometry on M4 × X4

The left–right symmetric gauge model with the symmetry of SU(3)c × SU(2)L × SU(2)R × U(1) is reconstructed in the generalized differential geometry (GDG) on the product space of Minkowski space M4 and four-point discrete space XN with N = 4. Ordinary gauge fields are regarded as the connection on M4; on the other hand, Higgs boson fields are regarded on the discrete space XN. This is a common view in the non-commutative geometry initiated by Connes. Gauge and Higgs fields are described in this paper as connections with the 1-form base on M4 × XN, a situation which is more similar to the ordinary differential geometry. We explain this formulation based on the matrix representation by use of the N × N matrix form. In addition to the generalized field strength obtained from the generalized gauge field, the anomalous field strength is introduced to yield sufficient Higgs potential and interacting terms by combining with corresponding terms in the generalized field strength. The resultant Higgs potential terms deduce the particular mass pattern such that among the Higgs doublet bosons, one CP-even scalar boson survives in the weak energy scale and the other four scalar bosons acquire the mass of the SU(2)R × U(1) breaking scale.

Type and Cotype in Vector-Valued Nakano Sequence Spaces

Given a sequence of Banach spaces {Xn}n and a sequence of real numbers {pn}n in [1,∞), the vector-valued Nakano sequence spaces ℓ({pn},{Xn}) consist of elements {xn}n in ∏nXn for which there is a constant λ>0 such that ∑n(‖xn‖/λ)pn<∞. In this paper we find the conditions on the Banach spaces Xn and on the sequence {pn}n for the spaces ℓ({pn},{Xn}) to have cotype q or type p.

Linear Chaos and Approximation

Let {Tn(t)}∞n=1 be a sequence of strongly continuous linear semigroups on Banach spaces Xn converging in the sense of Kato to a semigroup T(t) on the Banach space X. We discuss under what conditions the chaoticity of Tn(t) is inherited by T(t). We apply our results to a discrete parabolic equation.

The rationality of configuration spaces of lines in ℙ3

Let k be an algebraically closed field of arbitrary characteristic. Lines in ℙ3 are parametrized by the Grassmannian G(2, 4), which is isomorphic to a smooth quadric in ℙ5. We can consider the configuration space Xn = G(2, 4)n / PGL4(k) parametrizing ordered n-tuples of lines in ℙ3 up to projective equivalence. dim PGL4(k) = 15 and for n [ges ] 5, the stabilizer of a general n-tuple of lines is trivial, so for n [ges ] 5, Xn has the expected dimension 4n − 15.The question of rationality of Xn was posed by Dolgachev. The space Xn is clearly unirational, since there is a dominant rational map to it from the rational variety G(2, 4)n. The following results are known in characteristic 0: it is a special case of a theorem by Dolgachev and Boden [1] for configuration spaces in greater generality that if Xn is rational for some n [ges ] 5 then so is XN for any N [ges ] n. They also proved that the configuration space of lines in ℙm is rational if m is odd and recently Zaitsev [2] proved this for all m.Our proof uses different methods and it also has the advantage that it is valid in any characteristic. The main result is the following:

Projection constants of symmetric spaces and variants of Khintchine's inequality

Abstract The projection constants of the lpn-spaces for 1 ≦ p ≦ 2 satisfy with in the real case and in the complex case. Further, there is c &lt; 1 such that the projection constant of any n-dimensional space Xn with 1-symmetric basis can be estimated by . The proofs of the results are based on averaging techniques over permutations and a variant of Khintchine's inequality which states that