An Eisenstein criterion for noncommutative polynomials

We consider the Fuchsian linear differential equation obtained (modulo a prime) for , the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination of and can be removed from and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth-order linear differential operator occurs as the left-most factor of the ‘depleted’ differential operator and it is shown to be equivalent to the symmetric fourth power of LE, the linear differential operator corresponding to the elliptic integral E. This result generalizes what we have found for the lower order terms and . We conjecture that a linear differential operator equivalent to a symmetric (n − 1) th power of LE occurs as a left-most factor in the minimal order linear differential operators for all 's.

On contact manifolds we describe a notion of (contact) finite-type for linear partial differential operators satisfying a natural condition on their leading terms. A large class of linear differential operators are of finite-type in this sense but are not well understood by currently available techniques. We resolve this in the following sense. For any such D we construct a partial connection r H on a (finite rank) vector bundle with the property that sections in the null space of D correspond bijectively, and via an explicit map, with sections parallel for the partial connection. It follows that the solution space of D is finite dimensional and bounded by the corank of the holonomy algebra of r H. The treatment is via a uniform procedure, even though in most cases no normal Cartan connection is available. The prolongations of a k th order linear differential operator between vector bundles arise by differentiating the given operator D : E ! F, and forming a new system comprising D along with auxiliary operators that capture some of this derived data. To exploit this effectively it is crucial to determine what part of this information should be retained, and then how best to manage it. With this understood, for many classes of operators the resulting prolonged operator can expose key properties of the original differential operator and its equation. Motivated by questions related to integrability and deformations of structure, a theory of overdetermined equations and prolonged systems was developed during the 1950s and 1960s by Goldschmidt, Spencer, and others (2, 17). Generally, results in these works are derived abstractly using jet bundle theory, and are severely restricted in the sense that they apply most readily to differential operators satisfying involutivity conditions. These features mean the theory can be difficult to apply. In the case that the given partial differential operator D : E ! F, has surjective symbol there is an effective algorithmic approach to this problem. The prolongations are constructed from the leading symbol �(D) : J k � 1 E ! F, where J k � 1 is the bundle of symmetric covariant tensors on M of rank k. At a point of M, denoting by K the kernel of �(D), the spaces K ` = ( J ` � 1 K) ( J k+` � 1 E), `� 0, capture spaces of new variables to be introduced, and the system closes up if K ` = 0 for

This article is a sequel to the earlier articles, which describe the invertible ordinary differential operators and their generalizations. The generalizations are invertible mappings of filtered modules generated by one differentiation, and are called invertible D-operators. In particular, invertible ordinary linear differential operators, invertible linear difference operators with periodic coefficients, maps defined by unimodular matrices, and C-transformations of control systems are invertible D-operators. C-Transformations are those invertible transformations for which the variables of one system are expressed in terms of the variables of the other system and their derivatives.In the article we consider the invertible D-operators whose inverses are D-operators of the same type. In previous papers, a classification of invertible D-operators was obtained. Namely, a table of integers was associated to each invertible D-operator. These tables were described in a clear elementary-geometric language. Thus, to each invertible D-operator one assigns an elementary-geometric model, which is called a d-scheme of squares. The class of invertible D-operators having the same d-scheme was also described. In this paper, the invertible D-operators whose d-schemes consist of a single square are called unicellular. It is proved that any unicellular operator in some bases is given by an upper triangular matrix that differs from the identity matrix only by the first row. The main result is representation of the arbitrary invertible D-operator as a composition of unicellular operators. The minimum number of unicellular operators in such a composition is equal to the number of squares of the d-scheme of the original D-operator. As in previous papers, the used method is based on the description of d-schemes in the language of spectral sequences of algebraic complexes.The results obtained can be useful in the transformation and classification of control systems, in particular to describe flat systems.

Euler type partial differential operators on real analytic functions

Holomorphic vector valued differential operators acting on Siegel modular forms and preserving automorphy under the restriction to diagonal blocks are important in many respects, including application to critical values of L functions. Such differential operators are associated with vectors of new special polynomials of several variables defined by certain harmonic conditions. They include the classical Gegenbauer polynomial as a prototype, and are interesting as themselves independently of Siegel modular forms. We will give formulas for all such polynomials in two different ways. One is to describe them using polynomials characterized by monomials in off-diagonal block variables. We will give an explicit and practical algorithm to give the vectors of polynomials through these. The other one is rather theoretical but seems much deeper. We construct an explicit generating series of polynomials mutually related under certain mixed Laplacians. Here substituting the variables of the polynomials to partial derivatives, we obtain the generic differential operator from which any other differential operators of this sort are obtained by certain projections. This process exhausts all the differential operators in question. This is also generic in the sense that for any number of variables and block partitions, it is given by a recursive unified expression. As an application, we prove that the Taylor coefficients of Siegel modular forms with respect to off-diagonal block variables, or of corresponding expansion of Jacobi forms, are essentially vector valued Siegel modular forms of lower degrees, which are obtained as images of the differential operators given above. We also show that the original forms are recovered by the images of our operators. This is an ultimate generalization of Eichler–Zagier’s results on Jacobi forms of degree one. Several more explicit results and practical construction are also given.

Linear perturbations of differential of difference operators with polynomials as eigenfunctions

Matrix elements of irreducible representations of SU(n + 1)×SU(n + 1) and multivariable matrix-valued orthogonal polynomials

A differential operator which acts on partial differentiation is defined as Partial Differential Operator (PDO). PDO works based on the order of the differential equation which then can solve the eigenvalues of the operator. On vector space of polynomials, PDO can be written in matrix representation. This can be helpful in finding the general form of eigenvalues of vector space polynomials. On this paper, a second order PDO: will be operated on two and three variable vector space polynomials.

We use transfinite induction to show that if L is an epimorphism of a vector space V and maps a vector subspace W of V into a proper subspace of itself, then there is a smallest subspace E of V containing W such that L(E) E (or a minimal space of surjectivity or solvability) and we give examples where there are infinitely many distinct minimal spaces of solvability. We produce an example showing that if L 1 and L 2 are two epimorphisms of a vector space V which are endomorphisms of a proper subspace W of V such that LI(W) n L2(W is a proper subspace of W, then there may not exist a smallest subspace E of V containing W such that LI(E) E L2(E). While no nonconstant linear partial differential operator maps the field of meromorphic functions onto itself, we construct a locally convex topological vector space of formal power series containing the meromorphic functions such that every linear partial differential operator with constant coefficients maps this space linearly and continuously onto itself. Furthermore, we show that algebraically there is for every linear partial differential operator P(D) with constant coefficients a smallest extension E of the meromorphic functions in n complex variables, where

Unified canonical forms for matrices over a differential ring

This is the first of two papers on the ALPAK system for nonnumerical algebra on a digital computer. This paper is concerned with polynomials in several variables and truncated power series with polynomial coefficients. The second paper will discuss rational functions of several variables, truncated power series with rational-function coefficients, and systems of linear equations with rational-function coefficients. The ALPAK system has been programmed within the BE-SYS-4 monitor system on the IBM 7090 computer, but the language and concepts are machine independent. The available polynomial arithmetic operations are add, subtract, multiply, divide (if divisible), substitute, differentiate, zero test, nonzero test, and equality test. The speed of the system is indicated by the rule of thumb that one man-hour equals one 7090-second. The available space in core is usually sufficient for approximately 8000 polynomial terms. Section I of this paper consists of a nontechnical description of the system and a brief glimpse into the future. Section II discusses several specific problems to which the ALPAK system has been applied. These two parts do not presuppose any knowledge of computers or computer programming. Section III describes the use and the implementation of the algebraic operations relating to polynomials in several variables and truncated power series with polynomial coefficients. The reader of Section III is assumed to be acquainted with the elements of FAP (FORTRAN Assembly Program) programming, including the use of macros, as described in a series of IBM publications, the latest of which is IBM 7090–7094 Programming Systems MAP (Macro Assembly Program) Language (Form Number C28–6311).

It has been shown that, under suitable hypotheses, boundary value problems of the form, $Ly+\lambda y=f,$ $BC y =0$ where $L$ is a linear ordinary or partial differential operator and $BC$ denotes a linear boundary operator, then there exists $\Lambda >0$ such that $f\ge 0$ implies $\lambda y \ge 0$ for $\lambda\in [-\Lambda ,\Lambda ]\setminus\{0\},$ where $y$ is the unique solution of $Ly+\lambda y=f,$ $BC y =0$. So, the boundary value problem satisfies a maximum principle for $\lambda\in [-\Lambda ,0)$ and the boundary value problem satisfies an anti-maximum principle for $\lambda\in (0, \Lambda ]$. In an abstract result, we shall provide suitable hypotheses such that boundary value problems of the form, $D_{0}^{\alpha}y+\beta D_{0}^{\alpha -1}y=f,$ $BC y =0$ where $D_{0}^{\alpha}$ is a Riemann-Liouville fractional differentiable operator of order $\alpha$, $1<\alpha \le 2$, and $BC$ denotes a linear boundary operator, then there exists $\mathcal{B} >0$ such that $f\ge 0$ implies $\beta D_{0}^{\alpha -1}y \ge 0$ for $\beta \in [-\mathcal{B} ,\mathcal{B} ]\setminus\{0\},$ where $y$ is the unique solution of $D_{0}^{\alpha}y+\beta D_{0}^{\alpha -1}y =f,$ $BC y =0$. Two examples are provided in which the hypotheses of the abstract theorem are satisfied to obtain the sign property of $\beta D_{0}^{\alpha -1}y.$ The boundary conditions are chosen so that with further analysis a sign property of $\beta y$ is also obtained. One application of monotone methods is developed to illustrate the utility of the abstract result.

We describe the spectrum of linear difference and differential operators with unbounded operator coefficients in weight spaces.

In this paper we construct local elementary solutions for linear differential operators P(x9 Dx) whose principal symbols are real and of simple characteristics and investigate their regularity properties using Sato's theory of the sheaf ^ (Sato DC^CAl). Throughout this paper we assume that the coefficients of differential operators are analytic. In §1 we prepare some theorems which extend the classical existence theorem of Cauchy-Kowalevsky in complex domain to cases of singular initial data. (Cf. Hamada [1].) In §2 we employ the results of §1 to construct local elementary solutions for Cauchy problems for (/-) hyperbolic operators. As an application of the method employed there, we construct a singular solution u(x) of P(x, Dx)u = Q whose singular support is very small. (See Theorem 2.8 for the precise meaning of this statement.) In §3 we construct a local elementary solution for a linear partial differential operator of real principal symbols with simple characteristics and decide where its singularities locate to conclude that the result of §2 on the existence of singular solutions with small singularities is the best possible one. Throughout this paper we denote by M an ra-dimensional real analytic manifold which we identify with a domain in R containing the origin.

We consider nonlinear elliptic problems driven by a nonhomogeneous nonlinear differential operator. Using variational methods combined with Morse theory (critical groups), we prove two multiplicity results establishing three nontrivial smooth solutions. For the semilinear problem (linear differential operator), we produce four nontrivial smooth solutions. In the special case of the p-Laplacian differential operator, our framework of analysis incorporates equations which are resonant at infinity with respect to the principal eigenvalue.