Topological spaces compact with respect to a set of filters

If is a family of filters over some set I, a topological space X is sequencewise -compact if for every I-indexed sequence of elements of X there is such that the sequence has an F-limit point. Countable compactness, sequential compactness, initial κ-compactness, [λ; µ]-compactness, the Menger and Rothberger properties can all be expressed in terms of sequencewise -compactness for appropriate choices of . We show that sequencewise -compactness is preserved under taking products if and only if there is a filter such that sequencewise -compactness is equivalent to F-compactness. If this is the case, and there exists a sequencewise -compact T ₁ topological space with more than one point, then F is necessarily an ultrafilter. The particular case of sequential compactness is analyzed in detail.     

Wavelet transform and radon transform on the quaternion Heisenberg group

Let Q be the quaternion Heisenberg group,and let P be the affine automorphism group of Q.We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on L2(Q).A class of radial wavelets is constructed.The inverse wavelet transform is simplified by using radial wavelets.Then we investigate the Radon transform on Q.A Semyanistyi–Lizorkin space is introduced,on which the Radon transform is a bijection.We deal with the Radon transform on Q both by the Euclidean Fourier transform and the group Fourier transform.These two treatments are essentially equivalent.We also give an inversion formula by using wavelets,which does not require the smoothness of functions if the wavelet is smooth.In addition,we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on Q.     

Anisotropic inverse harmonic mean curvature flow

We study the evolution of convex hypersurfaces H（., t） with initial H（., 0） = 0H0 at a rate equal to H - f along its outer normal, where H is the inverse of harmonic mean curvature of H（., t）, H0 is a smooth, closed, and uniformly convex hypersurface. We find a θ^＊ 〉 0 and a sufficient condition about the anisotropic function f, such that if θ 〉 θ^＊, then H（.,t） remains uniformly convex and expands to infinity as t →∞ and its scaling, H（-, t）e^-nt, converges to a sphere. In addition, the convergence result is generalized to the fully nonlinear case in which the evolution rate is log H - log f instead of H - f.     

Super -operators of Jordan superalgebras and super Jordan Yang-Baxter equations

Abstract In this paper, the super t~-operators of Jordan superalgebras are introduced and the solutions of super Jordan Yang-Baxter equation are discussed by super б-operators. Then pre-Jordan superalgebras are studied as the algebraic structure behind the super б-operators. Moreover, the relations among Jordan superalgebras, pre-Jordan superalgebras, and dendriform superalgebras are established. Keywords Super б-operator, dendriform superalgebra, pre-Jordan superalgebra     

Variations on ω-boundedness

Let be a property (or, equivalently, a class) of topological spaces. A space X is called -bounded if every subspace of X with (or in) has compact closure. Thus, countable-bounded has been known as ω-bounded and (σ-compact)-bounded as strongly ω-bounded. In this paper we present a systematic study of the interrelations of these two known “boundedness” concepts with -boundedness where is one of the further countability properties weakly Lindelöf, Lindelöf, hereditarily Lindelöf, and ccc.     

Some remarks on projective generators and injective cogenerators

In this paper,we give a relationship between projective generators(resp.,injective cogenerators) in the category of R-modules and the counterparts in the category of complexes of R-modules.As a consequence,we get that complexes of W-Gorenstein modules are actually W-Gorenstein complexes whenever W is a subcategory of R-modules satisfying W⊥W,where W is the subcategory of exact complexes with all cycles in W.We also study when all cycles of a W-Gorenstein complexes are W-Gorenstein modules.     

Characterization of compact quantum group

Given a C^＊-algebra A and a comultiplication Ф on A, we show that the pair （A, Ф） is a compact quantum group if and only if the associated multiplier Hopf ^＊-algebra （A, ФA） is a compact Hopf ^＊-algebra.     

On the asymptotic form of convex hulls of Gaussian random fields

We consider a centered Gaussian random field X = {X _t : t ∈ T} with values in a Banach space $\mathbb{B}$ defined on a parametric set T equal to ? ^m or ? ^m . It is supposed that the distribution of X _t is independent of t. We consider the asymptotic behavior of closed convex hulls W _n = conv{X _t : t ∈ T _n}, where (T _n ) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b _n ) _n≥1 with probability 1, (in the sense of Hausdorff distance), where the limit set is the concentration ellipsoid of . The asymptotic behavior of the mathematical expectations Ef(W _n ), where f is some function, is also studied.     

Convolution functions of several variables with generalized bounded variation

Let L ¹ be the class of all complex-valued functions, with period 2π in each variable, in the space , where $\mathbb{T} = [0,2\pi )$ is the one-dimensional torus. Here, it is observed that L ¹ * E ? E for E = Lip(p; α ₁, α ₂, ..., α _N ) over , for , for , and for in the sense of Vitali as well as Hardy.     

On minimal non- -groups

A finite group G is called an J N J-group if every proper subgroup H of G is either subnormal in G or self-normalizing. We determinate the structure of non-J N J-groups in which all proper subgroups are J N J- groups.     

Fixed points of <Emphasis Type="Italic">n</Emphasis>-valued maps on surfaces and the Wecken property—a configuration space approach

In this paper, we explore the fixed point theory of n-valued maps using configuration spaces and braid groups, focusing on two fundamental problems, the Wecken property, and the computation of the Nielsen number. We show that the projective plane (resp. the 2-sphere S²) has the Wecken property for n-valued maps for all n ∈ ? (resp. all n ≥ 3). In the case n = 2 and S², we prove a partial result about the Wecken property. We then describe the Nielsen number of a non-split n-valued map Open image in new window of an orientable, compact manifold without boundary in terms of the Nielsen coincidence numbers of a certain finite covering q: X? → X with a subset of the coordinate maps of a lift of the n-valued split map Open image in new window .     

Approximation complexity of tensor product-type random fields with heavy spectrum

We consider a sequence of Gaussian tensor product-type random fields , where and are all positive eigenvalues and eigenfunctions of the covariance operator of the process X ₁, are standard Gaussian random variables, and is a subset of positive integers. For each d ∈ ?, the sample paths of X _d almost surely belong to L ₂([0, 1] ^d ) with norm ∥·∥_2,d . The tuples , are the eigenpairs of the covariance operator of X _d . We approximate the random fields X _d , d ∈ , by the finite sums X _d ⁽ⁿ⁾ corresponding to the n maximal eigenvalues λ _k , . We investigate the logarithmic asymptotics of the average approximation complexity $n_d^{pr} (\varepsilon ,\delta ): = \min \left\{ {n \in \mathbb{N}:\mathbb{P}(\left\| {X_d - X_d^{(n)} } \right\|_{2,d}^2 > \varepsilon ^2 \mathbb{E}\left\| {X_d } \right\|_{2,d}^2 ) \leqslant \delta } \right\},$ and the probabilistic approximation complexity $n_d^{avg} (\varepsilon ): = \min \left\{ {n \in \mathbb{N}:\mathbb{E}\left\| {X_d - X_d^{(n)} } \right\|_{2,d}^2 \leqslant \varepsilon ^2 \mathbb{E}\left\| {X_d } \right\|_{2,d}^2 } \right\}$ , as the parametric dimension d → ∞ the error threshold ? ∈ (0, 1) is fixed, and the confidence level δ = δ(d, ?) is allowed to approach zero. Supplementing recent results of M.A. Lifshits and E.V. Tulyakova, we consider the case where the sequence decreases regularly and sufficiently slowly to zero, which has not been previously studied.     

Martindale rings and H-module algebras with invariant characteristic polynomials

Under study is the category of the possibly noncommutative H-module algebras that are mapped homomorphically onto commutative algebras. The H-equivariant Martindale ring of quotients Q _H (A) is shown to be a finite-dimensional Frobenius algebra over the subfield of invariant elements Q _H (A) ^H and also the classical ring of quotients for A. We introduce a full subcategory of such that the algebras in are integral over its subalgebras of invariants and construct a functor ?? , which is left adjoined to the inclusion ?? .     

Associative n-Tuple algebras

In the paper,we study algebras having n bilinearmultiplication operations : A×A → A, s = 1, …, n, such that (a b) c = a (b c), s, r = 1,..., n, a, b, c ∈ A. The radical of such an algebra is defined as the intersection of the annihilators of irreducible A-modules, and it is proved that the radical coincides with the intersection of the maximal right ideals each of which is s-regular for some operation . This implies that the quotient algebra by the radical is semisimple. If an n-tuple algebra is Artinian, then the radical is nilpotent, and the semisimple Artinian n-tuple algebra is the direct sum of two-sided ideals each of which is a simple algebra. Moreover, in terms of sandwich algebras, we describe a finite-dimensional n-tuple algebra A, over an algebraically closed field, which is a simple A-module.     

Coefficient characterizations and sections for some univalent functions

Let (α) denote the class of locally univalent normalized analytic functions f in the unit disk |z| < 1 satisfying the condition $Re\left( {1 + \frac{{zf''(z)}} {{f'(z)}}} \right) < 1 + \frac{\alpha } {2}for|z| < 1 $ and for some 0 < α ≤ 1. We firstly prove sharp coefficient bounds for the moduli of the Taylor coefficients a _n of f ∈ (α). Secondly, we determine the sharp bound for the Fekete-Szegö functional for functions in (α) with complex parameter λ. Thirdly, we present a convolution characterization for functions f belonging to (α) and as a consequence we obtain a number of sufficient coefficient conditions for f to belong to (α). Finally, we discuss the close-to-convexity and starlikeness of partial sums of f ∈ (α). In particular, each partial sum s _n (z) of f ∈ (1) is starlike in the disk |z| ≤ 1/2 for n ≥ 11. Moreover, for f ∈ (1), we also have Re(s′ _n (z)) > 0 in |z| ≤ 1/2 for n ≥ 11.     

Continuous embeddings of Besov-Morrey function spaces

We study embeddings of spaces of Besov-Morrey type, M Bp1,q1s1,r1(Rd ) → M Bp2 ,q2s2 ,r2 (R d ), and obtain necessary and sufficient conditions for this. Moreover, we can also characterise the special weighted situation Bp1 ,r1s1 (R d , w) → M Bp2 ,q2s2 ,r2 (Rd ) for a Muckenhoupt A ∞ weight w, with wα(x) = |x|α , α -d1, as a typical example.     

On the class of groups with pronormal hall π-subgroups

Given a set π of prime numbers, we define the class of all finite groups in which Hall π-subgroups exist and are pronormal by analogy with the Hall classes , , and . We study whether is closed under the main class-theoretic closure operations. In particular, we establish that is a saturated formation.     

Hausdorff dimensions of the Julia sets of reluctantly recurrent rational maps

In this paper, we consider a rational map f of degree at least two acting on Riemman sphere that is expanding away from critical points. Assuming that all critical points of f in the Julia set J(f) are reluctantly recurrent, we prove that the Hausdorff dimension of the Julia set J(f) is equal to the hyperbolic dimension, and the Lebesgue measure of Julia set is zero when the Julia set J(f) ≠ .