Related papers: The Gauss-Dirichlet Orbit Number

We study a simple analytic solution to Einstein's field equations describing a thin spherical shell consisting of collisionless particles in circular orbit. We then apply two independent criteria for the identification of circular orbits,…

We consider the numerical solution of the equation - \Delta u - f(u) = g, for the unknown u satisfying Dirichlet conditions in a bounded domain. The nonlinearity f has bounded, continuous derivative. The algorithm uses the finite element…

In these lectures, we discuss two approaches to studying orbit spaces of algebraic Lie groups. Due to algebraic approach orbit space, or quotient, is an algebraic manifold, while from the differential viewpoint a quotient is a differential…

In this paper, we study the Dirichlet problem for the implicit degen- erate nonlinear elliptic equation with variable exponent in a bounded domain. We obtain sufficient conditions for the existence of a solution with- out regularization and…

The problem of the classification of the indefinite binary quadratic forms with integer coefficients is solved introducing a special partition of the de Sitter world, where the coefficients of the forms lie, into separate domains. Every…

There is developed a differential-algebraic approach to studying the representations of commuting differentiations in functional differential rings under nonlinear differential constraints. An example of the differential ideal with the only…

We construct a class of representations of the Heisenberg algebra in terms of the complex shift operators subject to the proper continuous limit imposed by the correspondence principle. We find a suitable Hilbert space formulation of our…

We obtain a description of the irreducible representation algebra of the alternating group of degree four over the ring of 2-adic integers.

We study the class numbers of integral binary cubic forms. For each $SL_2(Z)$ invariant lattice $L$, Shintani introduced Dirichlet series whose coefficients are the class numbers of binary cubic forms in $L$. We classify the invariant…

Over 200 years ago, Gauss discovered a composition law on the $SL_2({\mathbb Z})$-equivalence classes of primitive binary quadratic forms. Since then, bijections of classes of binary forms have been found with ideal class groups of…

The infinitesimal space of a quasiregular mapping was introduced by Gutlyanskii et al and generalized the idea of a derivative for this class of mappings which is only differentiable almost everywhere. In this paper, we show that the…

In this note we complete the calculation of the number of $GL(\mathbb R^n)$-orbits on $\Lambda^k(\mathbb R^n)^*$, by treating the cases $(n,k)= (7,4)$ and $(8,5)$ not covered in the literature. We also calculate the number of of…

Comparison results for solutions to the Dirichlet problems for a class of nonlinear, anisotropic parabolic equations are established. These results are obtained through a semi-discretization method in time after providing estimates for…

Let $p$ be an odd prime number. In this article, we study the number of quadratic residues and non-residues modulo $p$ which are multiples of $2$ or $3$ or $4$ and lying in the interval $[1, p-1]$, by applying the Dirichlet's class number…

In this paper we will discuss the Dirichlet problem of nonlinear second order partial differential equations resolved with any derivatives. First, we transform it into generalized integral equations. Next, we discuss the existence of the…

A classical result of Dirichlet shows that certain elementary character sums compute class numbers of quadratic imaginary number fields. We obtain analogous relations between class numbers and a weighted character sum associated to a…

We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations,…

We develop an algorithm for computing the closure of a given nilpotent $G_0$-orbit in $\g_1$, where $\g_1$ and $G_0$ are coming from a $\Z$ or a $\Z/m\Z$-grading $\g= \bigoplus \g_i$ of a simple complex Lie algebra $\g$.

In 2001, Bhargava proved a composition law for $2 \times 2 \times 2$ integer cubes, which generalized Gauss composition of integral binary quadratic forms. Furthermore, he derived four new composition laws defined on the following spaces:…

We consider quantum mechanical systems of spin chain type, with finite-dimensional Hilbert spaces and $\mathcal{N}=2$ or $\mathcal{N}=4$ supersymmetry, described in $\mathcal{N}=2$ superspace in terms of nonlinear chiral multiplets. We…