Volume 31, Issue 1

In this paper, we aim to establish the sharp maximal pointwise estimates for the multilinear commutators generated by multilinear strongly singular Calderón–Zygmund operators and BMO functions or Lipschitz functions, respectively. As applications, the boundedness of these multilinear commutators on product of weighted Lebesgue spaces are obtained. It is interesting to note that there is no size condition assumption for the kernel of the multilinear strongly singular Calderón–Zygmund operator. Due to the stronger singularity for the kernel of the multilinear strongly singular Calderón–Zygmund operator, we need to be more careful in estimating the mean oscillation over the small balls to get the sharp maximal function estimates.

Let Endk⁢(V){\mathrm{End}_{k}(V)} denote the ring of all linear transformations of an arbitrary k -vector space V over a field k . We define X⊆Endk⁢(V){X\subseteq\mathrm{End}_{k}(V)} to be triangularizable if V has a well-ordered basis such that X sends each vector in that basis to the subspace spanned by basis vectors no greater than it. We then show that an arbitrary subset of Endk⁢(V){\mathrm{End}_{k}(V)} is strictly triangularizable (defined in the obvious way) if and only if it is topologically nilpotent. This generalizes the theorem of Levitzki that every nilpotent semigroup of matrices is triangularizable. We also give a description of the triangularizable subalgebras of Endk⁢(V){\mathrm{End}_{k}(V)}, which generalizes a theorem of McCoy classifying triangularizable algebras of matrices over algebraically closed fields.

In this paper, we give strong lower bounds on the size of the sets of products of matrices in some certain groups. More precisely, we prove an analogue of a result due to Chapman and Iosevich for matrices in SL2⁢(𝔽p){\mathrm{SL}_{2}(\mathbb{F}_{p})} with restricted entries on a small set. We also provide extensions of some recent results on expansion for cubes in Heisenberg group due to Hegyvári and Hennecart.

Excluding some exceptional cases, we determine the asphericity of the relative presentation 𝒫=〈G,x∣a⁢xm⁢b⁢xn〉{\mathcal{P}=\langle G,x\mid ax^{m}bx^{n}\rangle}, where a,b∈G∖{1}{a,b\in G\setminus\{1\}} and 1≤m≤n{1\leq m\leq n}. If H=〈a,b〉≤G{H=\langle a,b\rangle\leq G}, the exceptional cases occur when a=b2{a=b^{2}} or when H is isomorphic to C6{C_{6}}.

This paper classifies the equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology for complex E6{E_{6}}. This is achieved by using our finiteness result, and by improving the computing method. According to a conjecture of Barbasch and Pandžić, our classification should also be helpful for understanding the entire unitary dual of complex E6{E_{6}}.

We introduce a certain Artin–Schreier scheme over a finite field associated to a pair of coprime integers (m,n){(m,n)} with n≥3{n\geq 3} divisible by the characteristic of the base field, and study the middle étale cohomology group of it. If m is even, the variety admits actions of some finite Heisenberg groups. We study the middle cohomology as representations of the Heisenberg groups. If m is odd, we compute the Frobenius eigenvalues of it concretely. This affine scheme comes from the reduction of a certain affinoid in a Lubin–Tate space.

Let p be a prime integer and let ℤp{\mathbb{Z}_{p}} be the ring of p -adic integers. By a purely computational approach we prove that each nonzero normal element of a noncommutative Iwasawa algebra over the special linear group SL3⁢(ℤp){\mathrm{SL}_{3}(\mathbb{Z}_{p})} is a unit. This gives a positive answer to an open question in [F. Wei and D. Bian, Erratum: Normal elements of completed group algebras over SLn⁢(ℤp)\mathrm{SL}_{n}(\mathbb{Z}_{p}) [mr2747414], Internat. J. Algebra Comput. 23 2013, 1, 215] and makes up for an earlier mistake in [F. Wei and D. Bian, Normal elements of completed group algebras over SLn⁢(ℤp)\mathrm{SL}_{n}(\mathbb{Z}_{p}), Internat. J. Algebra Comput. 20 2010, 8, 1021–1039] simultaneously.

In this paper, regularity properties and Strichartz type estimates for solutions of the Cauchy problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator A defined in a Banach space E , in which by choosing E and A , we can obtain numerous classes of initial value problems for Schrödinger equations, which occur in a wide variety of physical systems.

A two-dimensional central limit theorem for the eigenvalues of GL⁢(n){\mathrm{GL}(n)} Hecke–Maass cusp forms is newly derived. The covariance matrix is diagonal and hence verifies the statistical independence between the real and imaginary parts of the eigenvalues. We also prove a central limit theorem for the number of weighted eigenvalues in a compact region of the complex plane, and evaluate some moments of eigenvalues for the Hecke operator Tp{T_{p}} which reveal interesting interferences.

In this paper, it is shown that bilinear Calderón–Zygmund operators are bounded from products of variable Hardy spaces into variable Lebesgue or Hardy spaces.

The Reidemeister number of an automorphism ϕ of an Abelian group G is calculated by determining the cardinality of the quotient group G/(ϕ-1G)⁢(G){G/(\phi-1_{G})(G)}, and the Reidemeister spectrum of G is precisely the set of Reidemeister numbers of the automorphisms of G . In this work we determine the full spectrum of several types of group, paying particular attention to groups of torsion-free rank 1 and to direct sums and products. We show how to make use of strong realization results for Abelian groups to exhibit many groups where the Reidemeister number is infinite for all automorphisms; such groups then possess the so-called R∞{R_{\infty}}-property. We also answer a query of Dekimpe and Gonçalves by exhibiting an Abelian 2-group which has the R∞{R_{\infty}}-property.

A countable poset is ultrahomogeneous if every isomorphism between its finite subposets can be extended to an automorphism. If A is such a poset, then the group Aut⁡(A){\operatorname{Aut}(A)} has a natural topology in which Aut⁡(A){\operatorname{Aut}(A)} is a Polish topological group. We consider the problem of whether Aut⁡(A){\operatorname{Aut}(A)} contains a dense free subgroup of two generators. We show that if A is ultrahomogeneous, then Aut⁡(A){\operatorname{Aut}(A)} contains such a subgroup. Moreover, we characterize those countable ultrahomogeneous posets A such that for each natural number m the set of all cyclically dense elements g¯∈Aut(A)m{\overline{g}\in\operatorname{Aut}(A)^{m}} for the diagonal action is comeager in Aut(A)m{\operatorname{Aut}(A)^{m}}. In our considerations we strongly use the result of Schmerl which says that there are essentially four types of countably infinite ultrahomogeneous posets.

Let r:X2→X2{r\colon X^{2}\rightarrow X^{2}} be a set-theoretic solution of the Yang–Baxter equation on a finite set X . It was proven by Gateva-Ivanova and Van den Bergh that if r is non-degenerate and involutive, then the algebra K⁢〈x∈X∣x⁢y=u⁢v⁢ if ⁢r⁢(x,y)=(u,v)〉{K\langle x\in X\mid xy=uv\text{ if }r(x,y)=(u,v)\rangle} shares many properties with commutative polynomial algebras in finitely many variables; in particular, this algebra is Noetherian, satisfies a polynomial identity and has Gelfand–Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions rB{r_{B}} that are associated to a left semi-brace B ; such solutions can be degenerate or can even be idempotent. In order to do so, we first describe such semi-braces and then prove some decompositions results extending those of Catino, Colazzo and Stefanelli.

We prove that the automorphism group of a compact 6-manifold M endowed with a symplectic half-flat SU⁢(3){\mathrm{SU}(3)}-structure has Abelian Lie algebra with dimension bounded by min⁡{5,b1⁢(M)}{\min\{5,b_{1}(M)\}}. Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new complete examples on T⁢𝕊3{T\mathbb{S}^{3}} which are invariant under a cohomogeneity one action of SO⁢(4){\mathrm{SO}(4)}.