Power boundedness in the Fourier and Fourier–Stieltjes algebras on homogeneous spaces

We study power boundedness in the Fourier and Fourier–Stieltjes algebras, Open image in new window and Open image in new window of a homogeneous space Open image in new window The main results characterizes when all elements with spectral radius at most one, in any of these algebras, are power bounded.     

SINGULAR CONFORMALLY INVARIANT TRILINEAR FORMS,I THE MULTIPLICITY ONE THEOREM

A normalized holomorphic family (depending on Open image in new window ∈ ?³) of conformally invariant trilinear forms on the sphere is studied. Its zero set Z is described. For Open image in new window ? Z, the multiplicity of the space of conformally invariant trilinear forms is shown to be 1.     

Nonseparable closed vector subspaces of separable topological vector spaces

In 1983 P. Domański investigated the question: For which separable topological vector spaces E, does the separable space Open image in new window have a nonseparable closed vector subspace, where \(\hbox {c}\) is the cardinality of the continuum? He provided a partial answer, proving that every separable topological vector space whose completion is not q-minimal (in particular, every separable infinite-dimensional Banach space) E has this property. Using a result of S.A. Saxon, we show that for a separable locally convex space (lcs) E, the product space Open image in new window has a nonseparable closed vector subspace if and only if E does not have the weak topology. On the other hand, we prove that every metrizable vector subspace of the product of any number of separable Hausdorff lcs is separable. We show however that for the classical Michael line \(\mathbb M\) the space of all continuous real-valued functions on \(\mathbb M\) endowed with the pointwise convergence topology, \(C_p(\mathbb M)\) contains a nonseparable closed vector subspace while \(C_p(\mathbb M)\) is separable.     

Constructions of cyclic constant dimension codes

Subspace codes and particularly constant dimension codes have attracted much attention in recent years due to their applications in random network coding. As a particular subclass of subspace codes, cyclic subspace codes have additional properties that can be applied efficiently in encoding and decoding algorithms. It is desirable to find cyclic constant dimension codes such that both the code sizes and the minimum distances are as large as possible. In this paper, we explore the ideas of constructing cyclic constant dimension codes proposed in Ben-Sasson et al. (IEEE Trans Inf Theory 62(3):1157–1165, 2016) and Otal and Özbudak (Des Codes Cryptogr, doi: 10.1007/s10623-016-0297-1, 2016) to obtain further results. Consequently, new code constructions are provided and several previously known results in [2] and [17] are extended.     

The Nevo–Zimmer intermediate factor theorem over local fields

The Nevo–Zimmer theorem classifies the possible intermediate G-factors Y in Open image in new window , where G is a higher rank semisimple Lie group, P a minimal parabolic and X an irreducible G-space with an invariant probability measure. An important corollary is the Stuck–Zimmer theorem, which states that a faithful irreducible action of a higher rank Kazhdan semisimple Lie group with an invariant probability measure is either transitive or free, up to a null set. We present a different proof of the first theorem, that allows us to extend these two well-known theorems to linear groups over arbitrary local fields.     

Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into

Let Open image in new window denote a weight in Open image in new window which belongs to the Muckenhoupt class Open image in new window and let Open image in new window denote the uncentered Hardy–Littlewood maximal operator defined with respect to the measure Open image in new window . The sharp Tauberian constant of Open image in new window with respect to Open image in new window , denoted by Open image in new window , is defined by

Open image in new window

In this paper, we show that the Solyanik estimate

$$\begin{aligned} \lim _{\alpha \rightarrow 1^-}\mathsf{C}_{w}(\alpha ) = 1 \end{aligned}$$

holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy–Littlewood maximal operator Open image in new window and a weight Open image in new window :

Open image in new window

We show that we have Open image in new window if and only if Open image in new window . As a corollary of our methods we obtain a quantitative embedding of Open image in new window into Open image in new window .

     

Global existence of small solutions for the fourth-order nonlinear Schrödinger equation

We consider the Cauchy problem for the fourth-order nonlinear Schrödinger equation

Open image in new window

where \(n=1,2\). We prove global existence of small solutions under the growth condition of \(f\left( u\right) \) satisfying \(\left| \partial _{u}^{j}f\left( u\right) \right| \le C\left| u\right| ^{p-j},\) where \(p>1+\frac{4}{n},0\le j\le 3\).

     

Unbounded norm convergence in Banach lattices

A net \((x_\alpha )\) in a vector lattice X is unbounded order convergent to \(x \in X\) if \(|x_\alpha - x| \wedge u\) converges to 0 in order for all \(u\in X_+\). This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net \((x_\alpha )\) in a Banach lattice X is unbounded norm convergent to x if Open image in new window for all \(u\in X_+\). We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.     

Positive definite and Gram tensor complementarity problems

Given a continuous function Open image in new window and Open image in new window , the non-linear complementarity problem \(\text{ NCP }(g,q)\) is to find a vector Open image in new window such that

$$\begin{aligned} x \ge 0,~~y:=g(x) +q\ge 0~~\text{ and }~~x^Ty=0. \end{aligned}$$

We say that g has the Globally Uniquely Solvable (\(\text{ GUS }\))-property if \(\text{ NCP }(g,q)\) has a unique solution for all Open image in new window and C-property if \(\mathrm{NCP}(g,q)\) has a convex solution set for all Open image in new window . In this paper, we find a class of non-linear functions that have the \(\text{ GUS }\)-property and C-property. These functions are constructed by some special tensors which are positive semidefinite. We call these tensors as Gram tensors.

     

Asymptotic Normality of Some Graph Sequences

For a simple finite graph G denote by Open image in new window the number of ways of partitioning the vertex set of G into k non-empty independent sets (that is, into classes that span no edges of G). If \(E_n\) is the graph on n vertices with no edges then Open image in new window coincides with Open image in new window , the ordinary Stirling number of the second kind, and so we refer to Open image in new window as a graph Stirling number. Harper showed that the sequence of Stirling numbers of the second kind, and thus the graph Stirling sequence of \(E_n\), is asymptotically normal—essentially, as n grows, the histogram of Open image in new window , suitably normalized, approaches the density function of the standard normal distribution. In light of Harper’s result, it is natural to ask for which sequences \((G_n)_{n \ge 0}\) of graphs is there asymptotic normality of Open image in new window . Thanh and Galvin conjectured that if for each n, \(G_n\) is acyclic and has n vertices, then asymptotic normality occurs, and they gave a proof under the added condition that \(G_n\) has no more than \(o(\sqrt{n/\log n})\) components. Here we settle Thanh and Galvin’s conjecture in the affirmative, and significantly extend it, replacing “acyclic” in their conjecture with “co-chromatic with a quasi-threshold graph, and with negligible chromatic number”. Our proof combines old work of Navon and recent work of Engbers, Galvin and Hilyard on the normal order problem in the Weyl algebra, and work of Kahn on the matching polynomial of a graph.     

A weird relation between two cardinals

For a set M, let \({\text {seq}}(M)\) denote the set of all finite sequences which can be formed with elements of M, and let \([M]^2\) denote the set of all 2-element subsets of M. Furthermore, for a set A, let Open image in new window denote the cardinality of A. It will be shown that the following statement is consistent with Zermelo–Fraenkel Set Theory \(\textsf {ZF}\): There exists a set M such that Open image in new window and no function Open image in new window is finite-to-one.     

Dualities and algebras with a near-unanimity term

Let P be a finite relational structure that admits a (k +  1)-ary nearunanimity polymorphism. Then the NU Duality Theorem tells us that the algebra, whose operations are the polymorphisms of P, is dualisable with a dualising alter ego given by. We show that a more efficient alter ego can be obtained by using obstructions, as introduced by Zádori. We show that in the case that P is an ordered set (and therefore is an order-primal algebra), the duality that we obtain is strong. We close the paper by showing that if P is a finite fence, then our duality is optimal.     

Strongly Gauduchon Spaces

The authors define strongly Gauduchon spaces and the class■■, which are generalization of strongly Gauduchon manifolds in complex spaces. Comparing with the case of Kahlerian, the strongly Gauduchon space and the class■are similar to the Kahler space and the Fujiki class■■ respectively. Some properties about these complex spaces are obtained. Moreover, the relations between the strongly Gauduchon spaces and the class■■ are studied.     

Some new families of partial difference sets in finite fields

In this paper we present some new cyclotomic families of partial difference sets. The argument rests on a general procedure for constructing cyclotomic difference sets or partial difference sets in Galois domains due to Ott (Des Codes Cryptogr, doi:10.1007/s10623-015-0082-6, 2015). Definitions and various properties of partial difference sets can be found for instance in Ma (Des Codes Cryptogr 4:221–261, 1994).     

Commuting variety of Witt algebra

Let \(\mathfrak{g} = W_1 \) be the Witt algebra over an algebraically closed field k of characteristic p > 3; and let Open image in new window be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250: 473–484], we show that the variety Open image in new window is reducible, and not equidimensional. Irreducible components of Open image in new window and their dimensions are precisely given. As a consequence, the variety Open image in new window is not normal.     

Symmetric quiver Hecke algebras and <Emphasis Type="Italic">R</Emphasis>-matrices of quantum affine algebras IV

Let \(U'_q(\mathfrak {g})\) be a twisted affine quantum group of type \(A_{N}^{(2)}\) or \(D_{N}^{(2)}\) and let \(\mathfrak {g}_{0}\) be the finite-dimensional simple Lie algebra of type \(A_{N}\) or \(D_{N}\). For a Dynkin quiver of type \(\mathfrak {g}_{0}\), we define a full subcategory \({\mathcal C}_{Q}^{(2)}\) of the category of finite-dimensional integrable \(U'_q(\mathfrak {g})\)-modules, a twisted version of the category \({\mathcal C}^{(1)}_{Q}\) introduced by Hernandez and Leclerc. Applying the general scheme of affine Schur–Weyl duality, we construct an exact faithful KLR-type duality functor \({\mathcal F}_{Q}^{(2)}:\mathrm{Rep}(R) \rightarrow {\mathcal C}_{Q}^{(2)}\), where \(\mathrm{Rep}(R)\) is the category of finite-dimensional modules over the quiver Hecke algebra R of type \(\mathfrak {g}_{0}\) with nilpotent actions of the generators \(x_k\). We show that \({\mathcal F}_{Q}^{(2)}\) sends any simple object to a simple object and induces a ring isomorphism Open image in new window .     

A note “On <Emphasis Type="Italic">ss</Emphasis>-quasinormal or weakly <Emphasis Type="Italic">s</Emphasis>-permutably embedded subgroups of finite groups”

We point out some gaps in the paper (Monatsh Math doi: 10.1007/s00605-016-0877-1, 2016) by Kong and Guo, and give new brief proofs of the main results.     

Invariant and stationary measures for the action on Moduli space

We prove some ergodic-theoretic rigidity properties of the action of Open image in new window on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of Open image in new window is supported on an invariant affine submanifold.The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work.     

Geodesic mapping onto Kählerian spaces of the first kind

In the present paper a generalized Kählerian space Open image in new window of the first kind is considered as a generalized Riemannian space \(\mathbb{G}\mathbb{R}_N \) with almost complex structure F _i ^h that is covariantly constant with respect to the first kind of covariant derivative.Using a non-symmetric metric tensor we find necessary and sufficient conditions for geodesic mappings f: Open image in new window with respect to the four kinds of covariant derivatives. These conditions have the form of a closed system of partial differential equations in covariant derivatives with respect to unknown components of the metric tensor and the complex structure of the Kählerian space Open image in new window .     

On a characterization of finite dimensional real linear spaces

LetX be a real linear space; if is a linear topology onX, let denote the class of all bounded with respect to subsets ofX. In this paper it is shown that the spaceX is algebraically finite dimensional if and only if the class of all convex, absorbing and radially bounded subsets ofX is included in the intersection of all , where runs the set of all linear topologies onX.