The representation type of rational normal scrolls

The goal of this paper is to demonstrate that all non-singular rational normal scrolls \(S(a_0,\ldots ,a_k)\subseteq \mathbb P ^N\), \(N =\sum _{i=0}^k(a_i)+k\), (unless \(\mathbb P ^{k+1}=S(0,\ldots ,0,1)\), the rational normal curve \(S(a)\) in \(\mathbb P ^a\), the quadric surface \(S(1,1)\) in \(\mathbb P ^3\) and the cubic scroll \(S(1,2)\) in \(\mathbb P ^4\)) support families of arbitrarily large rank and dimension of simple Ulrich (and hence indecomposable ACM) vector bundles. Therefore, they are all of wild representation type unless \(\mathbb P ^{k+1}\), \(S(a)\), \(S(1,1)\) and \(S(1,2)\) which are of finite representation type.     

Optimal bounds for convergence of expected spectral distributions to the semi-circular law

Let \(\mathbf {X}=(X_{jk})_{j,k=1}^n\) denote a Hermitian random matrix with entries \(X_{jk}\), which are independent for \(1\le j\le k\le n\). We consider the rate of convergence of the empirical spectral distribution function of the matrix \(\mathbf {X}\) to the semi-circular law assuming that \(\mathbf{E}X_{jk}=0\), \(\mathbf{E}X_{jk}^2=1\) and that

$$\begin{aligned} \sup _{n\ge 1}\sup _{1\le j,k\le n}\mathbf{E}|X_{jk}|^4=:\mu _4<\infty , \end{aligned}$$

and

$$\begin{aligned} \sup _{1\le j,k\le n}|X_{jk}|\le D_0n^{\frac{1}{4}}. \end{aligned}$$

By means of a recursion argument it is shown that the Kolmogorov distance between the expected spectral distribution of the Wigner matrix \(\mathbf {W}=\frac{1}{\sqrt{n}}\mathbf {X}\) and the semicircular law is of order \(O(n^{-1})\).

     

Diophantine approximation with four squares and one <Emphasis Type="Italic">k</Emphasis>th power of primes

Let k be an integer with \(k\ge 3\) and \(\eta \) be any real number. Suppose that \(\lambda _1, \lambda _2, \lambda _3, \lambda _4, \mu \) are non-zero real numbers, not all of the same sign and \(\lambda _1/\lambda _2\) is irrational. It is proved that the inequality \(|\lambda _1p_1^2+\lambda _2p_2^2+\lambda _3p_3^2+\lambda _4p_4^2+\mu p_5^k+\eta |<(\max \ p_j)^{-\sigma }\) has infinitely many solutions in prime variables \(p_1, p_2, \ldots , p_5\), where \(0<\sigma <\frac{1}{16}\) for \(k=3,\ 0<\sigma <\frac{5}{3k2^k}\) for \(4\le k\le 5\) and \(0<\sigma <\frac{40}{21k2^k}\) for \(k\ge 6\). This gives an improvement of an earlier result.     

Upper triangular matrix operators with diagonal (T_1,T_2), T_2 k-nilpotent

If \(T=\left(\begin{array}{clcr}T_1&\quad C\\ 0&\quad T_2\end{array}\right) \in B(\mathcal{X }_1\oplus \mathcal{X }_2)\) is a Banach space upper triangular operator matrix with diagonal \((T_1, T_2)\) such that \(T_2\) is \(k\)-nilpotent for some integer \(k\ge 1\), then \(T\) inherits a number of its spectral properties, such as SVEP, Bishop’s property \((\beta )\) and the equality of Browder and Weyl spectrum, from those of \(T_1\). This paper studies such operators. The conclusions are then applied to provide a general framework for results pertaining (for example) to Browder, Weyl type theorems and supercyclicity for classes of Hilbert space operators, such as \(k\)-quasi hyponormal, \(k\)-quasi isometric and \(k\)-quasi paranormal operators, defined by a positivity condition.     

Bavrin’s Type Factorization of the Temljakov Operator for Holomorphic Functions in Circular Domains of $$\mathbb {C}^{n}$$

The paper concerns investigations of holomorphic functions of several complex variables with a factorization of their Temljakov transform. Firstly, there were considered some inclusions between the families \(\mathcal {C}_{\mathcal {G}},\mathcal {M}_{\mathcal {G}},\mathcal {N}_{\mathcal {G}},\mathcal {R}_{\mathcal {G}},\mathcal {V}_{\mathcal {G}}\) of such holomorphic functions on complete n-circular domain \(\mathcal {G}\) of \(\mathbb {C}^{n}\) in some papers of Bavrin, Fukui, Higuchi, Michiwaki. A motivation of our investigations is a condensation of the mentioned inclusions by some new families of Bavrin’s type. Hence we consider some families \(\mathcal {K}_{ \mathcal {G}}^{k},k\ge 2,\) of holomorphic functions f :  \(\mathcal {G}\rightarrow \mathbb {C},f(0)=1,\) defined also by a factorization of \( \mathcal {L}f\) onto factors from \(\mathcal {C}_{\mathcal {G}}\) and \(\mathcal {M} _{\mathcal {G}}.\) We present some interesting properties and extremal problems on \(\mathcal {K}_{\mathcal {G}}^{k}\).     

Finite Gap Jacobi Matrices,III. Beyond the Szegő Class

Let \({\frak {e}}\subset {\mathbb {R}}\) be a finite union of ?+1 disjoint closed intervals, and denote by ω _j the harmonic measure of the j left-most bands. The frequency module for \({\frak {e}}\) is the set of all integral combinations of ω ₁,…,ω _? . Let \(\{\tilde{a}_{n}, \tilde{b}_{n}\}_{n=-\infty}^{\infty}\) be a point in the isospectral torus for \({\frak {e}}\) and \(\tilde{p}_{n}\) its orthogonal polynomials. Let \(\{a_{n},b_{n}\}_{n=1}^{\infty}\) be a half-line Jacobi matrix with \(a_{n} = \tilde{a}_{n} + \delta a_{n}\), \(b_{n} = \tilde{b}_{n} +\delta b_{n}\). Suppose

$\sum_{n=1}^\infty \lvert \delta a_n\rvert ^2 + \lvert \delta b_n\rvert ^2 <\infty $

and \(\sum_{n=1}^{N} e^{2\pi i\omega n} \delta a_{n}\), \(\sum_{n=1}^{N} e^{2\pi i\omega n} \delta b_{n}\) have finite limits as N→∞ for all ω in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to ω, then for z∈???, \(p_{n}(z)/\tilde{p}_{n}(z)\) has a limit as n→∞. Moreover, we show that there are non-Szeg? class J’s for which this holds.

     

Sharp Moser–Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition

Let \(\Phi _{n}(x)=e^x-\sum _{j=0}^{n-2}\frac{x^j}{j!}\) and \(\alpha _{n} =n\omega _{n-1}^{\frac{1}{n-1}}\) be the sharp constant in Moser’s inequality (where \(\omega _{n-1}\) is the area of the surface of the unit \(n\)-ball in \(\mathbb {R}^n\)), and \(dV\) be the volume element on the \(n\)-dimensional hyperbolic space \((\mathbb {H}^n, g)\) (\(n\ge {2}\)). In this paper, we establish the following sharp Moser–Trudinger type inequalities with the exact growth condition on \(\mathbb {H}^n\):

For any \(u\in {W^{1,n}(\mathbb {H}^n)}\) satisfying \(\Vert \nabla _{g}u\Vert _{n}\le {1}\), there exists a constant \(C(n)>0\) such that

$$\begin{aligned} \int _{\mathbb {H}^n}\frac{\Phi _{n}(\alpha _{n}|u|^{\frac{n}{n-1}})}{(1+|u|)^{\frac{n}{n-1}}}dV \le {C(n)\Vert u\Vert _{L^n}^{n}}. \end{aligned}$$

The power \(\frac{n}{n-1}\) and the constant \(\alpha _{n}\) are optimal in the following senses:

1. (i)
   If the power \(\frac{n}{n-1}\) in the denominator is replaced by any \(p<\frac{n}{n-1}\), then there exists a sequence of functions \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha _{n}(|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV \rightarrow {\infty }. \end{aligned}$$
    
2. (ii)
   If \(\alpha >\alpha _{n}\), then there exists a sequence of function \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha (|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV\rightarrow {\infty }, \end{aligned}$$

   for any \(p\ge {0}\).
    

This result sharpens the earlier work of the authors Lu and Tang (Adv Nonlinear Stud 13(4):1035–1052, 2013) on best constants for the Moser–Trudinger inequalities on hyperbolic spaces.

     

Martin’s Axiom and separated mad families

Two families \(\mathcal{A}, \mathcal{B}\) of subsets of ω are said to be separated if there is a subset of ω which mod finite contains every member of \(\mathcal{A}\) and is almost disjoint from every member of \(\mathcal{B}\). If \(\mathcal{A}\) and \(\mathcal{B}\) are countable disjoint subsets of an almost disjoint family, then they are separated. Luzin gaps are well-known examples of ω ₁-sized subfamilies of an almost disjoint family which can not be separated. An almost disjoint family will be said to be ω ₁-separated if any disjoint pair of ≤ω ₁-sized subsets are separated. It is known that the proper forcing axiom (PFA) implies that no maximal almost disjoint family is ≤ω ₁-separated. We prove that this does not follow from Martin’s Axiom.     

Some remarks on energy inequalities for harmonic maps with potential

We call the \({\delta}\)-vector of an integral convex polytope of dimension d flat if the \({\delta}\)-vector is of the form \({(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)}\), where \({a \geq 1}\). In this paper, we give the complete characterization of possible flat \({\delta}\)-vectors. Moreover, for an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^N}\) of dimension d, we let \({i(\mathcal{P},n)=|n\mathcal{P}\cap \mathbb{Z}^N|}\) and \({i^*(\mathcal{P},n)=|n(\mathcal{P} {\setminus}\partial \mathcal{P})\cap \mathbb{Z}^N|}\). By this characterization, we show that for any \({d \geq 1}\) and for any \({k,\ell \geq 0}\) with \({k+\ell \leq d-1}\), there exist integral convex polytopes \({\mathcal{P}}\) and \({\mathcal{Q}}\) of dimension d such that (i) For \({t=1,\ldots,k}\), we have \({i(\mathcal{P},t)=i(\mathcal{Q},t),}\) (ii) For \({t=1,\ldots,\ell}\), we have \({i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)}\), and (iii) \({i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)}\) and \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

Orthogonal Polynomials for Modified Chebyshev Measure of the First Kind

Given numbers \({n,s \in \mathbb{N}}\), \({n \geq 2}\), and the \({n}\)th-degree monic Chebyshev polynomial of the first kind \({\widehat T_n(x)}\), the polynomial system “induced” by \({\widehat T_n(x)}\) is the system of orthogonal polynomials \({\{p_{k}^{n,s} \}}\) corresponding to the modified measure \({d \sigma^{n,s}(x)=\widehat T^{2s}_n(x) d\sigma(x)}\), where \({d\sigma(x)=1/\sqrt{1-x^{2}}dx}\) is the Chebyshev measure of the first kind. Here we are concerned with the problem of determining the coefficients in the three-term recurrence relation for the polynomials \({p^{n,s}_{k}}\). The desired coefficients are obtained analytically in a closed form.     

On geometric properties of the generating function for the Ramanujan sequence

The Ramanujan sequence \(\{\theta _{n}\}_{n \ge 0}\), defined as \(\theta _{0}= {1}/{2}\), \({n^{n}} \theta _{n}/{n !} = {e^{n}}/{2} - \sum _{k=0}^{n-1} {n^{k}}/{k !}\, \), \(n \ge 1\), has been studied on many occasions and in many different contexts. Adell and Jodrá (Ramanujan J 16:1–5, 2008) and Koumandos (Ramanujan J 30:447–459, 2013) showed, respectively, that the sequences \(\{\theta _{n}\}_{n \ge 0}\) and \(\{4/135 - n \cdot (\theta _{n}- 1/3 )\}_{n \ge 0}\) are completely monotone. In the present paper, we establish that the sequence \(\{(n+1) (\theta _{n}- 1/3 )\}_{n \ge 0}\) is also completely monotone. Furthermore, we prove that the analytic function \((\theta _{1}- 1/3 )^{-1}\sum _{n=1}^{\infty } (\theta _{n}- 1/3 ) z^{n} / n^{\alpha }\) is universally starlike for every \(\alpha \ge 1\) in the slit domain \(\mathbb {C}\setminus [1,\infty )\). This seems to be the first result putting the Ramanujan sequence into the context of analytic univalent functions and is a step towards a previous stronger conjecture, proposed by Ruscheweyh et al. (Israel J Math 171:285–304, 2009), namely that the function \((\theta _{1}- 1/3 )^{-1}\sum _{n=1}^{\infty } (\theta _{n}- 1/3 ) z^{n} \) is universally convex.     

Abelian covers of alternating groups

Let \({C={\rm inf} (k/n)\sum_{i=1}^n x_i(x_{i+1}+\cdots+x_{i+k})^{-1}}\), where the infimum is taken over all pairs of integers \({n\geq k\geq 1}\) and all positive \({x_1,\ldots,x_n}\), \({x_{n+i}=x_i}\). We prove that \({\ln 2 \leq C < 0.9305}\). In the definition of the constant C, the operation \({{\rm inf}_{k}\, {\rm inf}_{n}\, {\rm inf}_{x}}\) can be replaced by \({{\rm lim}_{k \to \infty}\, {\rm lim}_{n \to \infty} {\rm inf}_{x}}\).     

Lefschetz coincidence class for several maps

The aim of this paper is to define a Lefschetz coincidence class for several maps. More specifically, for maps \({f_{1}, \ldots , f_{k} : X \rightarrow N}\) from a topological space X into a connected closed n-manifold (even nonorientable) N, a cohomological class

$$L(f_{1}, \ldots , f_{k}) \in H^{n(k-1)}(X; (f_{1}, \ldots , f_{k}) ^{\ast} (R \times \Gamma^{\ast}_{N} \times \ldots \times \Gamma^{\ast} _{N}))$$

is defined in such a way that \({L(f_{1}, \ldots , f_{k}) \neq 0}\) implies that the set of coincidences

$${\rm Coin}(f_{1}, \ldots , f_{k}) = \{x \in X\,|\,f_{1}(x) = \ldots = f_{k}(x)\}$$

is nonempty.

     

A saturation phenomenon for a nonlinear nonlocal eigenvalue problem

Given \(1\le q \le 2\) and \(\alpha \in \mathbb {R}\), we study the properties of the solutions of the minimum problem

$$\begin{aligned} \lambda (\alpha ,q)=\min \left\{ \dfrac{\displaystyle \int _{-1}^{1}|u'|^{2}dx+\alpha \left| \int _{-1}^{1}|u|^{q-1}u\, dx\right| ^{\frac{2}{q}}}{\displaystyle \int _{-1}^{1}|u|^{2}dx}, u\in H_{0}^{1}(-1,1),\,u\not \equiv 0\right\} . \end{aligned}$$

In particular, depending on \(\alpha \) and q, we show that the minimizers have constant sign up to a critical value of \(\alpha =\alpha _{q}\), and when \(\alpha >\alpha _{q}\) the minimizers are odd.

     

Universal Cycle Packings and Coverings for <Emphasis Type="Italic">k</Emphasis>-Subsets of an <Emphasis Type="Italic">n</Emphasis>-Set

A cyclic sequence of elements of [n] is an (n, k)-Ucycle packing (respectively, (n, k)-Ucycle covering) if every k-subset of [n] appears in this sequence at most once (resp. at least once) as a subsequence of consecutive terms. Let \(p_{n,k}\) be the length of a longest (n, k)-Ucycle packing and \(c_{n,k}\) the length of a shortest (n, k)-Ucycle covering. We show that, for a fixed \(k,p_{n,k}={n\atopwithdelims ()k}-O(n^{\lfloor k/2\rfloor })\). Moreover, when k is not fixed, we prove that if \(k=k(n)\le n^{\alpha }\), where \(0<\alpha <1/3\), then \(p_{n,k}={n\atopwithdelims ()k}-o({n\atopwithdelims ()k}^\beta )\) and \(c_{n,k}={n\atopwithdelims ()k}+o({n\atopwithdelims ()k}^\beta )\), for some \(\beta <1\). Finally, we show that if \(k=o(n)\), then \(p_{n,k}={n\atopwithdelims ()k}(1-o(1))\).     

Absolute continuity and singularity of Palm measures of the Ginibre point process

We prove a dichotomy between absolute continuity and singularity of the Ginibre point process \(\mathsf {G}\) and its reduced Palm measures \(\{\mathsf {G}_{\mathbf {x}}, \mathbf {x} \in \mathbb {C}^{\ell }, \ell = 0,1,2\ldots \}\), namely, reduced Palm measures \(\mathsf {G}_{\mathbf {x}}\) and \(\mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x} \in \mathbb {C}^{\ell }\) and \(\mathbf {y} \in \mathbb {C}^{n}\) are mutually absolutely continuous if and only if \(\ell = n\); they are singular each other if and only if \(\ell \not = n\). Furthermore, we give an explicit expression of the Radon–Nikodym density \(d\mathsf {G}_{\mathbf {x}}/d \mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x}, \mathbf {y} \in \mathbb {C}^{\ell }\).     

Smoothness and Asymptotic Properties of Functions with General Monotone Fourier Coefficients

In this paper we study trigonometric series with general monotone coefficients, i.e., satisfying

$$\begin{aligned} \sum \limits _{k=n}^{2n} |a_k - a_{k+1}| \le C \sum \limits _{k=[{n}/{\gamma }]}^{[\gamma n]} \frac{|a_k|}{k}, \quad n\in \mathbb {N}, \end{aligned}$$

for some \(C \ge 1\) and \(\gamma >1\). We first prove the Lebesgue-type inequalities for such series

$$\begin{aligned} n|a_n|\le C \omega (f,1/n). \end{aligned}$$

Moreover, we obtain necessary and sufficient conditions for the sum of such series to belong to the generalized Lipschitz, Nikolskii, and Zygmund spaces. We also prove similar results for trigonometric series with weak monotone coefficients, i.e., satisfying

$$\begin{aligned} |a_n | \le C \sum \limits _{k=[{n}/{\gamma }]}^{\infty } \frac{|a_k|}{k}, \quad n\in \mathbb {N}, \end{aligned}$$

for some \(C \ge 1\) and \(\gamma >1\). Sharpness of the obtained results is given. Finally, we study the asymptotic results of Salem–Hardy type.

     

On the estimates of the upper and lower bounds of Ramanujan primes

For \(n\ge 1\), the nth Ramanujan prime is defined as the least positive integer \(R_{n}\) such that for all \(x\ge R_{n}\), the interval \((\frac{x}{2}, x]\) has at least n primes. Let \(p_{i}\) be the ith prime and \(R_{n}=p_{s}\). Sondow, Laishram, and other scholars gave a series of upper bounds of s. In this paper we establish several results giving estimates of upper and lower bounds of Ramanujan primes. Using these estimates, we discuss a conjecture on Ramanujan primes of Sondow–Nicholson–Noe and prove that if \(n>10^{300}\), then \(\pi (R_{mn})\le m\pi (R_{n})\) for \(m\ge 1\).     

Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight

In this article, using the heat kernel approach from Bouche (Asymptotic results for Hermitian line bundles over complex manifolds: The heat kernel approach, Higher-dimensional complex varieties, pp 67–81, de Gruyter, Berlin, 1996), we derive sup-norm bounds for cusp forms of integral and half-integral weight. Let \({\Gamma\subset \mathrm{PSL}_{2}(\mathbb{R})}\) be a cocompact Fuchsian subgroup of first kind. For \({k \in \frac{1}{2} \mathbb{Z}}\) (or \({k \in 2\mathbb{Z}}\)), let \({S^{k}_{\nu}(\Gamma)}\) denote the complex vector space of cusp forms of weight-k and nebentypus \({\nu^{2k}}\) (\({\nu^{k\slash 2}}\), if \({k \in 2\mathbb{Z}}\)) with respect to \({\Gamma}\), where \({\nu}\) is a unitary character. Let \({\lbrace f_{1},\ldots,f_{j_{k}} \rbrace}\) denote an orthonormal basis of \({S^{k}_{\nu}(\Gamma)}\). In this article, we show that as \({k \rightarrow \infty,}\) the sup-norm for \({\sum_{i=1}^{j_{k}}y^{k}|f_{i}(z)|^{2}}\) is bounded by O(k), where the implied constant is independent of \({\Gamma}\). Furthermore, using results from Berman (Math. Z. 248:325–344, 2004), we extend these results to the case when \({\Gamma}\) is cofinite.