Related papers: Separation of variables in the semistable range
We consider a (pseudo)Riemannian manifold of arbitrary dimension. The Hamilton-Jacobi equation for geodesic Hamiltonian admits complete separation of variables for some (separable) metrics in some (separable) coordinate systems. Separable…
In this paper, we solve a separation of singularities problem in the Bergman space. More precisely, we show that if $P\subset \mathbb{C}$ is a convex polygon which is the intersection of $n$ half planes, then the Bergman space on $P$…
A polynomial f (multivariate over a field) is decomposable if f = g(h) with g univariate of degree at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and an approximation to their number…
The second order partial difference equation of two variables $ \CD u:= A_{1,1}(x) \Delta_1 \nabla_1 u + A_{1,2}(x) \Delta_1 \nabla_2 u + A_{2,1}(x) \Delta_2 \nabla_1 u + A_{2,2}(x) \Delta_2 \nabla_2 u & \qquad \qquad \qquad \qquad + B_1(x)…
We study the stability of the mesoscopic fluctuations of certain orthogonal polynomial ensembles on the real line utilizing the recurrence relation of the associated orthogonal polynomials. We prove that under a sparse enough decaying…
In this paper we address the following question arising from the work of P. Etingof, D. Kazhdan and A. Polishchuk (math.AG/0003009): given a homogeneous complex polynomial, when the rational map defined by its partials is of degree 1? We…
We study a new kind of symmetric polynomials P_n(x_1,...,x_m) of degree n in m real variables, which have arisen in the theory of numerical semigroups. We establish their basic properties and find their representation through the power sums…
The wave equation for vectors and symmetric tensors in spherical coordinates is studied under the divergence-free constraint. We describe a numerical method, based on the spectral decomposition of vector/tensor components onto spherical…
We consider a semilinear elliptic problem in an annulus of R^N, with N>1. Recent results ensure that there exists a sequence p_k of exponents of the nonlinear term at which a nonradial bifurcation from the radial solution occurs. Exploiting…
The necessary and sufficient conditions for a function to be totally or partially separable are derived. It is shown that a function is totally separable if and only if each component of the gradient vector of depends only on the…
It is known that random monic integral polynomials of bounded degree $d$ and integral coefficients distributed uniformly and independently in $[-H,H]$ are irreducible over $\mathbb{Z}$ with probability tending to $1$ as $H\to \infty$. In…
We prove an analogue of Miller's stable splitting of the unitary group $U(m)$ for spaces of commuting elements in $U(m)$. After inverting $m!$, the space $\text{Hom}(\mathbb{Z}^n,U(m))$ splits stably as a wedge of Thom-like spaces of…
We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is…
Found all equivalence classes for electromagnetic potentials and space-time metrics of Stackel spaces, provided that the equations of motion of the classical charged test particles are integrated by the method of complete separation of…
We prove that the scalar and $2\times 2$ matrix differential operators which preserve the simplest scalar and vector-valued polynomial modules in two variables have a fundamental Lie algebraic structure. Our approach is based on a general…
For finite complex reflexion groups, we consider the graded $W$-modules of diagonally harmonic polynomials in $r$ sets of variables, and show that associated Hilbert series may be described in a global manner, independent of the value of…
We show that for any semilinear partial differential equation of order m, the infinitesimals of the independent variables depend only on the independent variables and, if m>1 and the equation is also linear in its derivatives of order m-1…
Found all equivalence classes for electromagnetic potentials and space-time metrics of Stackel spaces, provided that the equations of motion of the classical charged test particles are integrated by the method of complete separation of…
Let $f: \CN \rightarrow \C $ be a reduced polynomial map, with $D=f^{-1}(0)$, $\U=\CN \setminus D$ and boundary manifold $M=\partial \U$. Assume that $f$ is transversal at infinity and $D$ has only isolated singularities. Then the only…
In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector…