On an arithmetic convolution

The Cauchy-type product of two arithmetic functions $f$ and $g$ on nonnegative integers is defined as $(f\bullet g)(k):=\sum_{m=0}^{k} {k\choose m}f(m)g(k-m)$. We explore some algebraic properties of the aforementioned convolution, which is a fundamental-characteristic of the identities involving the Bernoulli numbers, the Bernoulli polynomials, the power sums, the sums of products, henceforth.Comment: 10 page

Rank Properties of the Semigroup of Endomorphisms over Brandt semigroup

Howie and Ribeiro \cite{a.Howie99,a.Howie00} introduced various ranks, viz. small rank, lower rank, intermediate rank, upper rank and the large rank of a finite semigroup. In this note, we investigate all these ranks of the semigroup of endomorphisms over Brandt semigroup.Comment: To Appear in Semigroup Forum, 201

Subgroups of the additive group of real line

Without assuming the field structure on the additive group of real numbers $\mathbb{R}$ with the usual order $<,$ we explore the fact that every proper subgroup of $\mathbb{R}$ is either closed or dense. This property of subgroups of the additive group of reals is special and well known (see Abels and Monoussos [4]). However, by revisiting it, we provide another direct proof. We also generalize this result to arbitrary topological groups in the sense that, any topological group having this property of the subgroups in a given topology is either connected or totally disconnected

Sums of products of power sums

For any two arithmetic functions $f,g$ let $\bullet$ be the commutative and associative arithmetic convolution $(f\bullet g)(k):=\sum_{m=0}^k \left( \begin{array}{c} k m \end{array} \right)f(m)g(k-m)$ and for any $n\in\mathbb{N},$ $f^n=f\bullet \cdots\bullet f$ be $n-$fold product of $f\in \mathcal{S}.$ For any $x\in\mathbb{C},$ let $\mathcal{S}_0=e$ be the multiplicative identity of the ring $(\mathcal{S},\bullet,+)$ and $\mathcal{S}_x(k):=\frac{\mathcal{B}_{x+1}(k+1)-\mathcal{B}_{1}(k+1)}{k+1},~x\neq 0$ denote the power sum defined by Bernoulli polynomials $\mathcal{B}_x(k)=B_k(x).$ We consider the sums of products $\mathcal{S}_x^N(k),~N\in\mathbb{N}_0.$ A closed form expression for $\mathcal{S}^N_x(k)(x)$ generalizing the classical Faulhaber formula, is derived. Furthermore, some properties of $\alpha-$Euler numbers \cite{JS9}(a variant of Apostol Bernoulli numbers) and their sums of products, are considered using which a closed form expression for the sums of products of infinite series of the form $\eta_\alpha(k):=\sum_{n=0}^{\infty}\alpha^n n^k,~0<|\alpha|<1,~k\in\mathbb{N}_0$ and the related Abel sums, is obtained which in particular, gives a closed form expression for well known Bernoulli numbers. A generalization of the sums of products of power sums to the sums of products of alternating power sums is also obtained. These considerations generalize in a unified way to define sums of products of power sums for all $k\in\mathbb{N}$ hence connecting them with zeta functions.Comment: This paper has been withdrawn by the author because of lot many error

Fractional Power-Law Spectral Response of CaCu3Ti4O12 Dielectric: Many-Body Effects

Spectral character of dielectric response in CaCu3Ti4O12 across 0.5Hz-4MHz over 45-200K corresponding to neither the Debyean nor the KWW relaxation patterns rather indicates a random-walk like diffusive dynamics of moments. Non-linear relaxation here is due to the many body dipole-interactions, as confirmed by spectral-fits of our measured permittivity to the Dissado-Hill behaviour. Fractional power-laws observed in {\epsilon}*({\omega}) macroscopically reflect the fractal microscopic configurations. Below ~100K, the power-law exponent m (n) steeply decreases (increases), indicating finite length-scale collective response of moment-bearing entities. At higher temperatures, m gradually approaches 1 and n falls to low values, reflecting tendency towards the single-particle/Debyean relaxation.Comment: 10 pages, 3 figures, 22 reference

The Large Rank of a Finite Semigroup using Prime Subsets

The \emph{large rank} of a finite semigroup $\Gamma$, denoted by $r_5(\Gamma)$, is the least number $n$ such that every subset of $\Gamma$ with $n$ elements generates $\Gamma$. Howie and Ribeiro showed that $r_5(\Gamma) = |V| + 1$, where $V$ is a largest proper subsemigroup of $\Gamma$. This work considers the complementary concept of subsemigroups, called \emph{prime subsets}, and gives an alternative approach to find the large rank of a finite semigroup. In this connection, the paper provides a shorter proof of Howie and Ribeiro's result about the large rank of Brandt semigroups. Further, this work obtains the large rank of the semigroup of order-preserving singular selfmaps.Comment: Semigroup Forum, To appea

Glassy Domain Wall Matter in KH2PO4 Crystal: Field-Induced Transition

We have investigated the domain wall (DW) dielectric response of potassium dihydrogen phosphate (KH2PO4) crystal under 0-500V dc-bias electric field. Activated DW-contribution onsets freezing at Tf({\omega}, V), some 27K below the ferroelectric TC; timescale {\tau}f(T, V) exhibiting Vogel-Fulcher (VFT) divergence. Sharply distinct low- and high-field behaviors of TC(V), DW-Tg(V), VFT-T0(V), barrier energy Ua(V), and DW glass-fragility m(V) signify a field-induced transition from randomly-pinned/vitreous to clustered/glass-ceramic phases of domain wall matter. Field-hysteresis ({\epsilon}'poled > {\epsilon}'unpoled) observed at high dc-bias indicates coexistent unclustered DW phase, quenched-in during the field-cooling. We construct a paradigm T-E phase diagram depicting the complex glassy patterns of domain wall matter.Comment: 16 pages, 5 figures, 28 reference

The Ranks of the Additive Semigroup Reduct of Affine Near-Semiring over Brandt Semigroup

This work investigates the rank properties of $A^+(B_n)$, the additive semigroup reduct of affine near-semiring over Brandt semigroup $B_n$. In this connection, this work reports the ranks $r_1$, $r_2$, $r_3$ and $r_5$ of $A^+(B_n)$ and identifies a lower bound for the upper rank $r_4(A^+(B_n))$. While this lower bound is found to be the $r_4(A^+(B_n))$ for $n \ge 6$, in other cases where $2 \le n \le 5$, the upper rank of $A^+(B_n)$ is still open for investigation.Comment: Contributed talk titled "On the Ranks of Additive Semigroup Reducts of Affine Near-Semirings over Brandt Semigroups" the Conference General Algebra and Its Applications: GAIA 2013, Melbourne, Australia, July 15-19, 201

Absence of a multiglass state in some transition metal doped quantum paraelectrics

We critically investigate the purported existence of a multiglass state in the quantum paraelectrics SrTiO${_3}$ and KTaO${_3}$ doped with magnetic 3d transition metals. We observe that the transition metals have limited solubility in these hosts, and that traces of impurity magnetic oxides persist even in the most well processed specimens. Our dielectric measurements indicate that the polar nano-regions formed as a consequence of doping appear to lack co-operativity, and the associated relaxation process exhibits a thermally activated Arrhenius form. At lower temperatures, the dielectric susceptibility could be fit using the Barrett's formalism, indicating that the quantum-paraelectric nature of the host lattices are unaltered by the doping of magnetic transition metal oxides. All these doped quantum paraelectrics exhibit a crossover from the high temperature Curie-Weiss regime to one dominated by quantum fluctuations, as evidenced by a $T{^2}$ dependence of the temperature dependent dielectric susceptibility. The temperature dependence of the magnetic susceptibility indicate that magnetic signatures observed in some of the specimens could be solely ascribed to the presence of impurity oxides corresponding to the magnetic dopants used. Hence, the doped quantum paraelectrics appear to remain intrinsically paramagnetic down to the lowest measured temperatures, ruling out the presence of a multiglass state

Syntactic semigroup problem for the semigroup reducts of Affine Near-semirings over Brandt Semigroups

The syntactic semigroup problem is to decide whether a given finite semigroup is syntactic or not. This work investigates the syntactic semigroup problem for both the semigroup reducts of $A^+(B_n)$, the affine near-semiring over a Brandt semigroup $B_n$. It is ascertained that both the semigroup reducts of $A^+(B_n)$ are syntactic semigroups