Inequalities for the overpartition function

The Ramanujan Journal - Let $$\overline{p}(n)$$ denote the overpartition function. Engel showed that for $$n\ge 2$$ , $$\overline{p}(n)$$ satisfy the Turán inequalities, that is,...     

A generalized variety problem

Semigroup Forum - Let $${\mathscr {C}}\! om $$ denote the variety of all commutative semigroups. For $$n\ge 1$$ let $${{\mathscr {N}}\! il }_n$$ (respectively $${\mathscr {N}}_n$$ ) denote the...     

Proof of a conjecture on a congruence modulo 243 for overpartitions

Periodica Mathematica Hungarica - Let $$\bar{p}(n)$$ denote the number of overpartitions of n. Recently, numerous congruences modulo powers of 2, 3 and 5 were established regarding $$\bar{p}(n)$$ ....     

Roots of crosscap slides and crosscap transpositions

Periodica Mathematica Hungarica - Let $$N_{g}$$ denote a closed nonorientable surface of genus g. For $$g \ge 2$$ the mapping class group $$\mathcal {M}(N_{g})$$ is generated by Dehn twists and one...     

The Iwasawa invariant $$\mu$$ μ vanishes for $$\mathbb{Z}_{2}$$ Z 2 -extensions of certain real biquadratic fields

Acta Mathematica Hungarica - For a real biquadratic field, we denote by $$\lambda$$ , $$\mu$$ and $$\nu$$ the Iwasawa invariants of cyclotomic $$\mathbb{Z}_{2}$$ -extension of $$k$$ . We give...     

Quantum observable generalized orthoalgebras

Positivity - Let $${{{\mathcal {S}}}}({\mathcal {H}})$$ denote the set of all self-adjoint operators (not necessarily bounded) on a Hilbert space $${\mathcal {H}}$$, which is the set of all...     

On the Diophantine equation $$Cx^{2}+D=2y^{q}$$ C x 2 + D = 2 y q

The Ramanujan Journal - Let C and D denote positive integers such that $$CD>1$$ . In this paper we investigate the solvability of the Diophantine equation $$Cx^{2}+D=2y^{q}$$ , in positive...     

A note on the Lebesgue–Ljunggren–Nagell equation $$ax^2+b^{2m}=4y^n$$ a x 2 + b 2 m = 4 y n

Periodica Mathematica Hungarica - Let a, b be fixed positive integers such that $$2\not \mid {ab}$$ , $$\gcd (a,b)=1$$ and a is square-free, and let $$h(-a)$$ denote the class number of...     

On the exponent of geometric unipotent radicals of pseudo-reductive groups

Archiv der Mathematik - Let $$k'/k$$ be a finite purely inseparable field extension and let $$G'$$ be a reductive $$k'$$ -group. We denote by $$G=\mathrm {R}_{k'/k}(G')$$ , the...     

Sign changes in the prime number theorem

The Ramanujan Journal - Let V(T) denote the number of sign changes in $$\psi (x) - x$$ for $$x\in [1, T]$$ . We show that $$\liminf _{T\rightarrow \infty } V(T)/\log T\ge \gamma _{1}/\pi +...     

Generalized Sidon sets of perfect powers

The Ramanujan Journal - For $$h \ge 2$$ and an infinite set of positive integers A, let $$R_{A,h}(n)$$ denote the number of representations of the positive integer n as the sum of h distinct terms...     

Characterization of numerical radius parallelism in $$C^*$$C∗-algebras

Positivity - Let v(x) denote the numerical radius of an element x in a $$C^*$$-algebra $$\mathfrak {A}$$. First, we prove several numerical radius inequalities in $$\mathfrak {A}$$. Particularly,...     

ON THE ERROR FUNCTION OF THE SQUARE-FULL INTEGERS

Let L(x) denote the number of square-full integers not exceeding x. It is proved in [1] thatL(x)～(ζ(3/2)/ζ(3))x~(1/2) (ζ(2/3)/ζ(2))x~(1/3) as x→∞,where ζ(s) denotes the Riemann zeta function. Let △(x) denote the error function in the asymptotic formula for L(x). It was shown by D. Suryanaryana~([2]) on the Riemann hypothesis (RH) that1/x integral from n=1 to x |△(t)|dt=O(x~(1/10 s))for every ε>0. In this paper the author proves the following asymptotic formula for the mean-value of △(x) under the assumption of R. H.integral from n=1 to T (△~2(t/t~(6/5))) dt～c log T,where c>0 is a constant.     

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.     

Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group

Periodica Mathematica Hungarica - Let G be an additive finite abelian group. For a sequence T over G and $$g\in G$$ , let $$\mathrm {v}_{g}(T)$$ denote the multiplicity of g in T. Let $$\mathcal...     

On the sum of divisors of mixed powers in short intervals

The Ramanujan Journal - Let d(n) denote the Dirichlet divisor function. Define $$\begin{aligned} \mathcal {S}_k(x,y):= \sum _{\begin{array}{c} x-y     

On the average number of subgroups of the group $${\mathbb {Z}}_m \times {\mathbb {Z}}_n$$ Z m × Z n with k-th power sequence

The Ramanujan Journal - Let $$\mathbb {Z}_{n}$$ be the additive group of residue classes modulo n. Let s(m,&nbsp;n) denote the total number of subgroups of the group $$\mathbb {Z}_{m} \times...     

Explicit congruences modulo 2048 for overpartitions

The Ramanujan Journal - Let $${\bar{p}}(n)$$ denote the number of overpartitions of n. Recently, a number of congruences modulo powers of 2, 3 and 5 have been discovered. The moduli for these...     

Asymptotic Behavior of the Fundamental Solutions for Critical Schrödinger Forms

Journal of Theoretical Probability - Let $$\{X_t\}_{t \ge 0}$$ be a transient $$\alpha $$ -stable process on $${\mathbb {R}}^d$$ and denote by H its generator. We consider the perturbation of the...     

Lower Bound for the Maximal Number of Facets of a 0/1 Polytope

Let $f_{n-1}(P)$ denote the number of facets of a polytope $P$ in ${\Bbb R}^n$. We show that there exist 0/1 polytopes $P$ with $$f_{n-1}(P)\geq\left (\frac{cn}{\log^2 n}\right )^{n/2},$$ where $c>0$ is an absolute constant. This improves earlier work of Barany and Por on a question of Fukuda and Ziegler.