Incidences between points and generalized spheres over finite fields and related problems

Let $\mathbb{F}_q$ be a finite field of $q$ elements where $q$ is a large odd prime power and $Q =a_1 x_1^{c_1}+...+a_dx_d^{c_d}\in \mathbb{F}_q[x_1,...,x_d]$, where $2\le c_i\le N$, $\gcd(c_i,q)=1$, and $a_i\in \mathbb{F}_q$ for all $1\le i\le d$. A $Q$-sphere is a set of the form $\lbrace x\in \mathbb{F}_q^d | Q(x-b)=r\rbrace$, where $b\in \mathbb{F}_q^d, r\in \mathbb{F}_q$. We prove bounds on the number of incidences between a point set $\mathcal{P}$ and a $Q$-sphere set $\mathcal{S}$, denoted by $I(\mathcal{P},\mathcal{S})$, as the following. $$| I(\mathcal{P},\mathcal{S})-\frac{|\mathcal{P}||\mathcal{S}|}{q}|\le q^{d/2}\sqrt{|\mathcal{P}||\mathcal{S}|}.$$ We prove this estimate by studying the spectra of directed graphs. We also give a version of this estimate over finite rings $\mathbb{Z}_q$ where $q$ is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem. In Sections $4$ and $5$, we prove a bound on the number of incidences between a random point set and a random $Q$-sphere set in $\mathbb{F}_q^d$. We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances.Comment: to appear in Forum Mat

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電気通信大学201

Architecture Parallel for the Renewable Energy System

This chapter present one possible evolution is the parallel topology on the high-voltage bus for the renewable energy system. The system is not connected to a chain of photovoltaic (PV) modules and the different sources renewable. This evolution retains all the advantages of this system, while increasing the level of discretization of the Maximum Power Point Tracker (MPPT). So it is no longer a chain of PV modules that works at its MPPT but each PV module. In addition, this greater discretization allows a finer control and monitoring of operation and a faster detection of defects. The main interest of parallel step-up voltage systems, in this case, lies in the fact that the use of relatively high DC voltages is possible in these architectures distributed

Analyses of Thin-Walled Sections under Localised Loading for General End Boundary Conditions – Part II: Buckling

Thin-walled sections under localised loading may lead to buckling of the sections. This paper briefly introduces the development of the Semi-Analytical Finite Strip Method (SAFSM) for buckling analyses of thin-walled sections under localised loading for general end boundary conditions. This method is benchmarked against the Finite Element Method (FEM). For different support and loading conditions, different functions are required for flexural and membrane displacements. In Part 1- Pre-buckling described in a companion paper at this conference, the analysis provides the computation of the stresses for use in the buckling analyses in this paper. Numerical examples of buckling analyses of thin-walled sections under localised loading with different end boundary conditions are also given in the paper in comparison with the FEM

Analyses of Thin-Walled Sections under Localised Loading for General End Boundary Conditions – Part I: Pre-Buckling

The Semi-Analytical Finite Strip Method (SAFSM) for pre-buckling analysis of thin-walled sections under localised loading has been developed for general end boundary conditions. For different boundary conditions at supports and loading point, different displacement functions are required for both flexural and membrane displacements. As the stresses are not uniform along the member due to localised loading, the pre-buckling analysis also requires multiple series terms with orthogonal functions. This paper briefly summaries the displacement functions used for different boundary conditions. In addition, the theory of the SAFSM for pre-buckling analysis of thin-walled sections under localised loading with general end boundary conditions is developed. The analysis is benchmarked against the Finite Element Method (FEM) using software package ABAQUS/Standard. The results from this pre-buckling analysis are deflections (pre-buckling modes) and membrane stresses which are used for the buckling analysis described in Part 2 - Buckling in the companion paper