r-log-concavity of partition functions

The Ramanujan Journal - Let $$\hat{\mathscr {L}}$$ be the operator given by $$\hat{\mathscr {L}} \{a_n\}_{n \ge 0} = \{a_{n+1}^2 - a_{n} a_{n+2} \}_{n \ge 0}$$ . A sequence $$\{ a_n \}_{n \ge 0}$$...     

Fractional Gaussian estimates and holomorphy of semigroups

Let $$\Omega \subset {\mathbb {R}}^N$$ be an arbitrary open set, $$0<s<1$$ and denote by $$(e^{-t(-\Delta )_{{{\mathbb {R}}}^N}^s})_{t\ge 0}$$ the semigroup on $$L^2({{\mathbb {R}}}^N)$$ generated by the fractional Laplace operator. In the first part of the paper, we show that if T is a self-adjoint semigroup on $$L^2(\Omega )$$ satisfying a fractional Gaussian estimate in the sense that $$|T(t)f|\le Me^{-bt(-\Delta )_{{{\mathbb {R}}}^N}^s}|f|$$, $$0\le t \le 1$$, $$f\in L^2(\Omega )$$, for some constants $$M\ge 1$$ and $$b\ge 0$$, then T defines a bounded holomorphic semigroup of angle $$\frac{\pi }{2}$$ that interpolates on $$L^p(\Omega )$$, $$1\le p<\infty $$. Using a duality argument, we prove that the same result also holds on the space of continuous functions. In the second part, we apply the above results to the realization of fractional order operators with the exterior Dirichlet conditions.     

A note on $$AP_3$$ A P 3 -covering sequences

Periodica Mathematica Hungarica - For a given integer $$k\ge 3$$ , a sequence A of nonnegative integers is called an&nbsp; $$AP_k$$ -covering sequence if there exists an integer $$n_0$$ such...     

A combinatorial proof of the log-convexity of sequences in Riordan arrays

Journal of Algebraic Combinatorics - A Riordan array $$R=[r_{n,k}]_{n,k\ge 0}$$ can be characterized by two sequences $$A=(a_n)_{n\ge 0}$$ and $$Z=(z_n)_{n\ge 0}$$ such that $$r_{0,0}=1,...     

Menon-type identities with respect to sets of units

The Ramanujan Journal - Let $$n\ge 1$$ and let $$\mathbb {Z}_n^\star $$ be the group of units in the ring of residual classes modulo n, let $$m\ge 0$$ , $$k\ge 0$$ , $$m+k\ge 1$$ , $$u_1,\ldots...     

Korenblum maximum principle in mixed norm spaces

Archiv der Mathematik - It is well known that the Korenblum maximum principle holds in Bergman spaces $$\mathrm {A}^{p}$$ if and only if $$p\ge 1$$ . In this note, we improve this result by proving...     

OnQ-matrices

Recently, Jeter and Pye gave an example to show that Pang's conjecture, thatL ₁ Q , is false. We show in this article that the above conjecture is true for symmetric matrices. Specifically, we show that a symmetric copositive matrix is inQ if and only if it is strictly copositive.     

Reverse Inequalities for Subelliptic Functions

Mathematical Notes - We study a wedge $$\mathscr{K}(A)$$ of solutions of the inequality $$A(u) \ge 0$$ , where $$A$$ is a linear elliptic operator of order $$2m$$ . For the elements of the wedge,...     

Sufficient conditions for local equivalence of mappings

Let $$f,g:({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^m,0)$$ be $$C^{r+1}$$ mappings and let $$Z=\{x\in \mathbf {\mathbb {R}}^n:\nu (df (x))=0\}$$ , $$0\in Z$$ , $$m\le n$$ . We will show that if there exist a neighbourhood U of $$0\in {\mathbb {R}}^n$$ and constants $$C,C'>0$$ and $$k>1$$ such that for $$x\in U$$ $$\begin{aligned}&\nu (df(x))\ge C{\text {dist}}(x,Z)^{k-1}, \\&\left| \partial ^{s} (f_i-g_i)(x) \right| \le C'\nu (df(x))^{r+k-|s|}, \end{aligned}$$ for any $$i\in \{1,\dots , m\}$$ and for any $$s \in \mathbf {\mathbb {N}}^n_0$$ such that $$|s|\le r$$ , then there exists a $$C^r$$ diffeomorphism $$\varphi :({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^n,0)$$ such that $$f=g\circ \varphi $$ in a neighbourhood of $$0\in {\mathbb {R}}^n$$ . By $$\nu (df)$$ we denote the Rabier function.     

Improved SDP bounds for minimizing quadratic functions over the \ell^{1}-ball

Nesterov suggested an SDP-based bound for the problem to minimize a quadratic form over the -ball. In this note, we introduce a tighter SDP-based bound, the so-called copositive bound, and illustrate the improvement by simulation results. The copositive bound has the additional advantage that it can be easily extended to the inhomogeneous case of quadratic objectives including a linear term. We also indicate some improvements of the eigenvalue bound for the quadratic optimization over the -ball with 1 < p < 2, at least for p close to one.     

Tests for the convergence of continued fractions,based on the fundamental system of inequalities

We have proved that if the partial numerators of the continued fraction f(c)=1/1+c₂/l+c₃/l+... are all nonzero and for at least some number n?1 satisfy the inequalities $$p_n \left| {1 + c_n + c_{n + 1} } \right| \ge p_{n - 2} p_n \left| {c_n } \right| + \left| {c_{n + 1} } \right|(n \ge 1,p_{ - 1} = p_0 = c_1 = 0,p_n \ge 0),$$ then f(c) converges in the wide sense if and only if at least one of the series $$\begin{array}{l} \sum\nolimits_{n = 1}^\infty {\left| {c_3 c_5 \ldots c_{2n - 1} /(c_2 c_4 \ldots c_{2n} )} \right|} , \\ \sum\nolimits_{n = 1}^\infty {\left| {c_2 c_4 \ldots c_{2n} /(c_3 c_5 \ldots c_{2n + 1} )} \right|} \\ \end{array}$$      

Necessary and Sufficient Conditions for the Solvability of the L p Dirichlet Problem on Lipschitz Domains

We study the homogeneous elliptic systems of order $2\ellWe study the homogeneous elliptic systems of order with real constant coefficients on Lipschitz domains in, . For any fixed p > 2, we show that a reverse H?lder condition with exponent p is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in L ^p . We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the L ^p Dirichlet problem for and . The range of p is known to be sharp if and . For the polyharmonic equation, the sharp range of p is also found in the case n = 6, 7 if , and if .Research supported in part by the NSF.     

A Bilogarithmic Criterion for the Existence of a Regular Minorant that Does Not Satisfy the Bang Condition

Mathematical Notes - Problems of constructing regular majorants for sequences $$\mu=\{\mu_n\}_{n=0}^{\infty}$$ of numbers $$\mu_n\ge0$$ that are the Taylor coefficients of integer transcendental...     

Distance sets corresponding to convex bodies

Suppose that is a 0-symmetric convex body which denes the usual norm

Quasi-Cauchy quotients and means

Aequationes mathematicae - Let $$I\subset {\mathbb {R}}$$ be an interval that is closed under addition, and $$ k\in {\mathbb {N}}$$ , $$k\ge 2\,$$ . For a function $$f:I\rightarrow \left( 0,\infty...     

SECTIONAL CURVATURE OF KAEHLER SUBMANIFOLDS OF A COMPLEX PROJCETIVE SPACE

Let M~n (n≥2) be a complex Kaehler submanifold immersed in the complex projective space CP~m(1). Let K be the sectional curvature of M~n. Then K≥1/8 if and only if M~n is an imbedding submanifold congruent to the standard imbedding CP~n (1) or CP~n(1/2).     

A remark on the field of moduli of Riemann surfaces

Archiv der Mathematik - Let S be a closed Riemann surface of genus $$g\ge 2$$ and let $$\mathrm{Aut}(S)$$ be its group of conformal automorphisms. It is well known that if either: (i)...     

ASYMPTOTIC STABILITY FOR A CLASS OF NONAUTONOMOUS NEUTRAL DIFFERENTIAL EQUATIONS

ＡＳＹＭＰＴＯＴＩＣＳＴＡＢＩＬＩＴＹＦＯＲＡＣＬＡＳＳＯＦＮＯＮＡＵＴＯＮＯＭＯＵＳＮＥＵＴＲＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ＊＊ＹＵＪＩＡＮＳＨＥ＊ＭａｎｕｓｃｒｉｐｔｒｅｃｅｉｖｅｄＪｕｌｙ４，１９９５．ＲｅｖｉｓｅｄＭａｒｃｈ２...     

On sums of polynomial-type exceptional units in $$\varvec{\mathbb {Z}}/\varvec{n\mathbb {Z}}$$Z/nZ

A unit u in a commutative ring with unity R is called exceptional if $$u-1$$ is also a unit. We introduce the notion of a polynomial version of this (abbreviated as $$f\hbox {-exunits}$$) for any $$f(X) \in \mathbb {Z}[X]$$. We find the number of representations of a non-zero element of $$\mathbb {Z}/n\mathbb {Z}$$ as a sum of two f-exunits for an infinite family of polynomials f of each degree $$\ge 1$$. We also derive the exact formulae for certain infinite families of linear and quadratic polynomials. This generalizes a result proved by Sander (J Number Theory 159:1–6, 2016).     

An exponential Diophantine equation related to the sum of powers of two consecutive terms of a Lucas sequence and x-coordinates of Pell equations

Periodica Mathematica Hungarica - For the Fibonacci sequence the identity $$F_n^2+F_{n+1}^2 = F_{2n+1}$$ holds for all $$n \ge 0$$ . Let $${\mathcal {X}}:= (X_\ell )_{\ell \ge 1}$$ be the sequence...