In this paper, we construct a family of Bernstein functions using class rational parametrization. The new basis on an index $\alpha \in {\left(-\infty \, , 0 \right)}\cup {\left(1 +\infty\right)}$, and for given degree $k\in \mathbb{N}^*$, these are with numerator denominator polynomials k. All the classical properties as positivity, partition unity hold they constitute approximation continuous...

This note is an attempt to attack a conjecture of Fraenkel and Simpson stated in 1998 concerning the number distinct squares finite word. By counting (right-)special factors, we give upper bound for k-powers word any integer k?3. k-power, mean form uu...u?ktimes.

We show that the operator \[ T_{K,s_1,s_2}f(z) := \int_{\mathbb{R}^n} A_{K,s_1,s_2}(z_1,z_2) f(z_2)\, dz_2 \] is a Calderon-Zygmund operator. Here for $K \in L^\infty(\mathbb{R}^n \times \mathbb{R}^n)$, and $s,s_1,s_2 (0,1)$ with $s_1+s_2 = 2s$ we have \frac{K(x,y) \left (|x-z_1|^{s_1-n} -|y-z_1|^{s_1-n} \right )\, (|x-z_2|^{s_2-n} -|y-z_2|^{s_2-n}\right )}{|x-y|^{n+2s}}\, dx\, dy. This motivat...

Let k be a field, G an abelian group with a bicharacter, A a colour algebra; i.e., an associative graded k-algebra with identity, C a graded A-coring that is projective as a right A-module, C∗ the graded dual ring of C and Λ a left graded C-comodule that is finitely generated as a graded right C∗-module. We give necessary and sufficient conditions for projectivity and flatness of a graded modul...

Let D be a list of single digits, and let k be a positive integer. We construct an infinite sequence of positive integers by repeatedly appending, in order, one at a time, the digits from the list D to the integer k, in one of four ways: always on the left, always on the right, alternating and starting on the left, or alternating and starting on the right. In each of these four situations, we i...

We consider the fractal percolation process on the unit square, with fixed decimation parameter N and level dependent retention parameters {pk}; that is, for all k ≥ 1, at the kth stage every retained square of side-length N1−k is partitioned into N2 congruent subsquares, and each of these is retained with probability pk, independently of all others. We show that if ∏ k pk = 0 (i.e., if the are...

Let Λ and Γ be artin algebras and ΛUΓ a faithfully balanced selforthogonal bimodule. In this paper, we first introduce the notion of k-Gorenstein modules with respect to ΛUΓ and then establish the left-right symmetry of the notion of k-Gorenstein modules, which develops a classical result of Auslander. As an application, we study the properties of dual modules relative to Gorenstein bimodules. ...

MAXIMAL QUOTIENT RINGS AND ESSENTIAL RIGHT IDEALS IN GROUP RINGS OF LOCALLY FINITE GROUPS Theorem . zero . FERRAN CEDÓ * AND BRIAN HARTLEY Dedicated to the memory of Pere Menal Let k be a commutative field . Let G be a locally finite group without elements of order p in case char k = p > 0 . In this paper it is proved that the type I. part of the maximal right quotient ring of the group algebgr...

We study the oscillatory and nonoscillatory behavior of solutions a class forced impulsive nonlinear neutral differential systems form $$ \left\{\begin{array}{c}{\left(r(t){\left(y(t)+p(t)y\left(t-\tau \right)\right)}^{\prime}\right)}^{\prime }+q(t)G\left(y\left(t-\sigma \right)\right)=f(t),\kern1.75em t\ne {\tau}_k,\kern0.75em k\ \epsilon\ \mathbb{N},\\ {}\Delta \left(r\left({\tau}_k\right){\l...

We propose SwiftAgg+ , a novel secure aggregation protocol for federated learning systems, where central server aggregates local models of $N \in \mathbb {N}$ distribut...