A Note on the Chevalley--Warning Theorems

Let f1, . . . , fr be polynomials in n variables, over the field Fq, and suppose that their degrees are d1, . . . , dr. It was shown by Warning in 1935 that if N is the number of common zeros of the polynomials fi, then N > qn?d. It is the main aim of the present paper to improve on this bound. When the set of common zeros does not form an affine linear subspace in Fnq , it is shown for example that N > 2qn?d if q ≥ 4, and that N ≥ qn+1?d/(n + 2 ? d) if the fi are all homogeneous. © 2011 RAS(DoM) and LMS

A fluid model for closed queueing networks with PS stations

This technical report introduces a closed multi-class queueing network (QN) model with class-switching, where the service rates are de ned to represent multi-processor stations with a processor-sharing (PS) allocation policy. These transition rates are also able to consider traditional delay nodes, and therefore a QN model with these transition rates is well-suited for multi-threaded software applications. In this report, we de ne the QN model and use the results in [1] to show that the transient sample paths of the QN model converge to the solution of a system of ordinary di erential equations (ODEs). As the size of the ODE system grows linearly with the number of stations and job classes in the QN model, solving the ODE system becomes a scalable alternative to Markov chain representations

On a function introduced by Erd\"{o}s and Nicolas

Erd\"os and Nicolas [erdos1976methodes] introduced an arithmetical function $F(n)$ related to divisors of $n$ in short intervals $\left] \frac{t}{2}, t\right]$. The aim of this note is to prove that $F(n)$ is the largest coefficient of polynomial $P_n(q)$ introduced by Kassel and Reutenauer [kassel2015counting]. We deduce that $P_n(q)$ has a coefficient larger than $1$ if and only if $2n$ is the perimeter of a Pythagorean triangle. We improve a result due to Vatne [vatne2017sequence] concerning the coefficients of $P_n(q)$

Universal partial sums of Taylor series as functions of the centre of expansion

V. Nestoridis conjectured that if $\Omega$ is a simply connected subset of $\mathbb{C}$ that does not contain $0$ and $S(\Omega)$ is the set of all functions $f\in \mathcal{H}(\Omega)$ with the property that the set $\left\{T_N(f)(z)\coloneqq\sum_{n=0}^N\dfrac{f^{(n)}(z)}{n!} (-z)^n : N = 0,1,2,\dots \right\}$ is dense in $\mathcal{H}(\Omega)$, then $S(\Omega)$ is a dense $G_\delta$ set in $\mathcal{H}(\Omega)$. We answer the conjecture in the affirmative in the special case where $\Omega$ is an open disc $D(z_0,r)$ that does not contain $0$.Comment: 9 page