Band 32, Heft 1

For the completed Riemann zeta function ξ⁢(s){\xi(s)}, it is known that the Riemann hypothesis for ξ⁢(s){\xi(s)} implies the Riemann hypothesis for ξ(m)⁢(s){\xi^{(m)}(s)}, where m is any positive integer. In this paper, we investigate the distribution of the fractional parts of the sequence (α⁢γm){(\alpha\gamma_{m})}, where α is any fixed non-zero real number and γm{\gamma_{m}} runs over the imaginary parts of the zeros of ξ(m)⁢(s){\xi^{(m)}(s)}. We also obtain a zero density estimate and an explicit formula for the zeros of ξ(m)⁢(s){\xi^{(m)}(s)}. In particular, all our results hold uniformly for 0≤m≤g⁢(T){0\leq m\leq g(T)}, where the function g⁢(T){g(T)} tends to infinity with T and g⁢(T)=o⁢(log⁡log⁡T){g(T)=o(\log\log T)}.

Inspired by recent work of Carlson, Friedlander and Pevtsova concerning modules for p -elementary abelian groups Er{E_{r}} of rank r over a field of characteristic p>0{p>0}, we introduce the notions of modules with constant d -radical rank and modules with constant d -socle rank for the generalized Kronecker algebra 𝒦r=k⁢Γr{\mathcal{K}_{r}=k\Gamma_{r}} with r≥2{r\geq 2} arrows and 1≤d≤r-1{1\leq d\leq r-1}. We study subcategories given by modules with the equal d -radical property and the equal d -socle property. Utilizing the simplification method due to Ringel, we prove that these subcategories in mod⁡𝒦r{\operatorname{mod}\mathcal{K}_{r}} are of wild type. Then we use a natural functor 𝔉:mod⁡𝒦r→mod⁡k⁢Er{\operatorname{\mathfrak{F}}\colon{\operatorname{mod}\mathcal{K}_{r}}\to% \operatorname{mod}kE_{r}} to transfer our results to mod⁡k⁢Er{\operatorname{mod}kE_{r}}.

We previously showed that abstract Cuntz semigroups form a closed symmetric monoidal category. This automatically provides additional structure in the category, such as a composition and an external tensor product, for which we give concrete constructions in order to be used in applications. We further analyze the structure of not necessarily commutative Cu{\mathrm{Cu}}-semirings, and we obtain, under mild conditions, a new characterization of solid Cu{\mathrm{Cu}}-semirings R by the condition that R≅〚R,R〛{R\cong\llbracket R,R\rrbracket}.

We study locally Cohen–Macaulay curves of low degree in the Segre threefold ℙ1×ℙ1×ℙ1{\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}} and investigate the irreducible and connected components, respectively, of the Hilbert scheme of them. We also apply the similar argument to the Segre threefold ℙ2×ℙ1{\mathbb{P}^{2}\times\mathbb{P}^{1}}.

The purpose of this paper is to study a new class of function spaces, called 𝒩⁢(p,q,s){\mathcal{N}(p,q,s)}-type spaces, in the unit ball 𝔹{{\mathbb{B}}} of ℂn{{\mathbb{C}}^{n}}. The Carleson measure on such spaces is investigated. Some embedding theorems among 𝒩⁢(p,q,s){\mathcal{N}(p,q,s)}-type spaces, weighted Bergman spaces and weighted Hardy spaces are established. As for applications, the Hadamard products and random power series on 𝒩⁢(p,q,s){\mathcal{N}(p,q,s)}-type spaces are also studied.

We consider a nonlinear parametric Robin problem. In the reaction, there are two terms, one critical and the other locally defined. Using cut-off techniques, together with variational tools and critical groups, we show that, for all small values of the parameter, the problem has at least three nontrivial smooth solutions all with sign information, which converge to zero in C1⁢(Ω¯){C^{1}(\bar{\Omega})} as the parameter λ→0+{\lambda\to 0^{+}}.

Hausdorff operators originated from some classical summation methods. Now this is an active research field. In the present article, a spectral representation for multidimensional normal Hausdorff operator is given. We show that normal Hausdorff operator in L2⁢(ℝn){L^{2}(\mathbb{R}^{n})} is unitary equivalent to the operator of multiplication by some matrix-valued function (its matrix symbol) in the space L2⁢(ℝn;ℂ2n){L^{2}(\mathbb{R}^{n};\mathbb{C}^{2^{n}})}. Several corollaries that show that properties of a Hausdorff operator are closely related to the properties of its symbol are considered. In particular, the norm and the spectrum of such operators are described in terms of the symbol.

We study the limiting behavior of the Dirichlet and Neumann eigenvalue counting function of generalized second-order differential operators dd⁢μ⁢dd⁢x{\frac{\mathop{}\!d}{\mathop{}\!d\mu}\frac{\mathop{}\!d}{\mathop{}\!dx}}, where μ is a finite atomless Borel measure on some compact interval [a,b]{[a,b]}. Therefore, we firstly recall the results of the spectral asymptotics for these operators received so far. Afterwards, we make a proposition about the convergence behavior for so-called random V -variable Cantor measures.

Let j⁢(z){j(z)} be the modular j -invariant function. Let τ be an algebraic number in the complex upper half plane ℍ{\mathbb{H}}. It was proved by Schneider and Siegel that if τ is not a CM point, i.e., [ℚ(τ):ℚ]≠2{[\mathbb{Q}(\tau):\mathbb{Q}]\neq 2}, then j⁢(τ){j(\tau)} is transcendental. Let f be a harmonic weak Maass form of weight 0 on Γ0⁢(N){\Gamma_{0}(N)}. In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of f on Hecke orbits of τ. For a positive integer m , let Tm{T_{m}} denote the m -th Hecke operator. Suppose that the coefficients of the principal part of f at the cusp i⁢∞{i\infty} are algebraic, and that f has its poles only at cusps equivalent to i⁢∞{i\infty}. We prove, under a mild assumption on f , that, for any fixed τ, if N is a prime such that N≥23{N\geq 23} and N∉{23,29,31,41,47,59,71}{N\notin\{23,29,31,41,47,59,71\}}, then f(Tm.τ){f(T_{m}.\tau)} are transcendental for infinitely many positive integers m prime to N .

In this paper, the well-posedness of the higher-order Benjamin–Ono equation ut+ℋ⁢(ux⁢x)+ux⁢x⁢x=u⁢ux-∂x⁡(u⁢ℋ⁢∂x⁡u+ℋ⁢(u⁢∂x⁡u))u_{t}+\mathcal{H}(u_{xx})+u_{xxx}=uu_{x}-\partial_{x}(u\mathcal{H}\partial_{x}% u+\mathcal{H}(u\partial_{x}u)) is considered. The modified energy method is introduced to consider the equation. It is shown that the Cauchy problem of the higher-order Benjamin–Ono equation is locally well-posed in H3/4{H^{3/4}} without using the gauge transformation. Moreover, the well-posedness of the higher-order intermediate long wave equation ut+𝒢δ⁢(ux⁢x)+ux⁢x⁢x=u⁢ux-∂x⁡(u⁢𝒢δ⁢∂x⁡u+𝒢δ⁢(u⁢∂x⁡u)),𝒢δ=ℱx-1⁢i⁢(coth⁡(δ⁢ξ))⁢ℱx,u_{t}+\mathcal{G}_{\delta}(u_{xx})+u_{xxx}=uu_{x}-\partial_{x}(u\mathcal{G}_{% \delta}\partial_{x}u+\mathcal{G}_{\delta}(u\partial_{x}u)),\quad\mathcal{G}_{% \delta}=\mathcal{F}_{x}^{-1}i(\coth(\delta\xi))\mathcal{F}_{x}, is considered. It is shown that the Cauchy problem of the higher-order intermediate long wave equation is locally well-posed in H3/4{H^{3/4}}.

We obtain some new results about products of large and small sets in the Heisenberg group as well as in the affine group over the prime field. We apply these growth results to Freiman’s isomorphism in nonabelian groups.

Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero and let I be the T-ideal of polynomial identities of the adjoint representation of L . We prove that the number of multilinear central polynomials in n variables, linearly independent modulo I , grows exponentially like (dim⁡L)n{(\dim L)^{n}}.

For a Brylinski–Deligne covering group of a general linear group, we calculate some values of unramified Whittaker functions for certain representations that are analogous to the theta representations.

We classify all tilting classes over an arbitrary commutative ring via certain sequences of Thomason subsets of the spectrum, generalizing the classification for noetherian commutative rings by Angeleri, Pospíšil, ŠÅ¥ovíček and Trlifaj (2014). We show that the n -tilting classes can equivalently be expressed as classes of all modules vanishing in the first n degrees of one of the following homology theories arising from a finitely generated ideal: Tor*⁡(R/I,-){\operatorname{Tor}_{*}(R/I,-)}, Koszul homology, Čech homology, or local homology (even though in general none of those theories coincide). Cofinite-type n -cotilting classes are described by vanishing of the corresponding cohomology theories. For any cotilting class of cofinite type, we also construct a corresponding cotilting module, generalizing the construction of Šťovíček, Trlifaj and Herbera (2014). Finally, we characterize cotilting classes of cofinite type amongst the general ones, and construct new examples of n -cotilting classes not of cofinite type, which are in a sense hard to tell apart from those of cofinite type.