Related papers: Symbolic integration of hyperexponential 1-forms
It is well known that the real and imaginary parts of any holomorphic function are harmonic functions of two variables. In this paper we generalize this property to finite-dimensional commutative algebras. We prove that if some basis of a…
A general structure theorem on higher order invariants is proven. For an arithmetic group, the structure of the corresponding Hecke module is determined. It is shown that the module does not contain any irreducible submodule. This explains…
We consider a Hamiltonian system on the symplectic space $({\mathbb{R}}^{2n}, dy\wedge dx)$ with a real-analytic Hamiltonian $H : {\mathbb{R}}^{2n}\to {\mathbb{R}}$. We assume that the system has a non-degenerate equilibrium position at the…
Transcendental Liouvillian extensions are differential fields, in which one can model poly-logarithmic, hyperexponential, and trigonometric functions, logarithmic integrals, and their (nested) rational expressions. For such an extension…
Let $k[X] = k[x_{i,j}: i = 1,..., m; j = 1,..., n]$ be the polynomial ring in $m n$ variables $x_{i,j}$ over a field $k$ of arbitrary characteristic. Denote by $I_2(X)$ the ideal generated by the $2 \times 2$ minors of the generic $m \times…
Fox's H-function provide a unified and elegant framework to tackle several physical phenomena. We solve the space fractional diffusion equation on the real line equipped with a delta distribution initial condition and identify the…
The ring operations and the metric on $C(X)$ are extended to the set $\mathbb{H}_{nf}(X)$ of all nearly finite Hausdorff continuous interval valued functions and it is shown that $\mathbb{H}_{nf}(X)$ is both rationally and topologically…
In the paper we consider the one-frequency cohomological equation \begin{equation*} (\partial_x + \omega \partial_y) g(x,y) = a(x,y) \end{equation*} on the 2-torus with unknown $g$ and analytic initial data $a$. We identify all the…
Let $\mathcal F$ be either the set of all bounded holomorphic functions or the set of all $m$-homogeneous polynomials on the unit ball of $\ell\_r$. We give a systematic study of the sets of all $u\in\ell\_r$ for which the monomial…
A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for…
We describe the strong dual space $({\mathcal O} (D))^*$ for the space ${\mathcal O} (D)$ of holomorphic functions of several complex variables over a bounded Lipschitz domain $D$ with connected boundary $\partial D$ (as usual, ${\mathcal…
Let $I$ be a homogeneous ideal in the polynomial ring $R = k[z_1, \cdots, z_n]$ , where $k$ is an algebraically closed field of characteristic zero. Macaulay's Theorem provides constraints on the Hilbert function of $I$ or $R/I$ from one…
Using diagrammatic techniques, we provide explicit functional relations between the cumulant generating functions for the biunitarily invariant ensembles in the limit of large size of matrices. The formalism allows to map two distinct areas…
The article describes a purely topological counterpart of the $\epsilon$-factorization of constants in the functional equations (which is a key ingredient in the interplay between L-functions and classical automorphic forms). We consider…
New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…
This paper gives the pointwise H\"older (or multifractal) spectrum of continuous functions on the interval $[0,1]$ whose graph is the attractor of an iterated function system consisting of $r\geq 2$ affine maps on $\mathbb{R}^2$. These…
In this paper, we determine all irreducible spherical functions \Phi of any K -type associated to the pair (G,K)=(\SO(4),\SO(3)). This is accomplished by associating to \Phi a vector valued function H=H(u) of a real variable u, which is…
One computes the cohomology of the projective embedding of sl(m+1,R) acting on the differential operators on densities on R^m of various weights. This cohomology is non vanishing only for some special critical values of the weights. This…
The singularity structure of a second-order ordinary differential equation with polynomial coefficients often yields the type of solution. It is shown that the $\theta$-operator method can be used as a symbolic computational approach to…
In this article, I give an iterative closed form formula for the Hilbert-Kunz function for any binomial hypersurface in general, over any feild of arbitrary positive characteristic. I prove that the Hilbert-Kunz multiplicity associated to…