On the complexity of skew arithmetic

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.

Introduction. In the investigation of a non-degenerate quadratic form f (x) with coefficients in a local or global field K, one associates two groups with f (x), the orthogonal and metaplectic groups. If K is, e.g., the field R of real numbers, the second group is the unitary group generated by Ut = exp(it -f(x)) and its conjugate V, under the Fourier transformation for all t in R; and it is locally isomorphic to SL2(R). By using a similar definition one can associate with any polynomial f (x) of higher degree a group. Such groups have appeared in connection with the Siegel-Weil formulas for the norm forms of certain simple Jordan algebras. On the other hand Kubota [4], [5] and Yamazaki [8] have associated for K = C and R metaplectic groups with where n need not be 2, in fact n 2 and n = 2k 2, respectively. In both cases they have used certain Bessel transformations instead of the Fourier transformation to get the metaplectic groups. In a recent paper [3] we have shown that the Lie algebra of a hypermetaplectic group over R is infinite dimensional (at least if the Fourier transformation is defined relative to the bilinear form associated with a definite quadratic form). In this paper we have restricted ourselves to the one-variable case and tried to obtain all metaplectic groups over K = R and C. More precisely we have started from an arbitrary polynomial f (x) of one variable x with coefficients in K = R or C, put U, = exp (it .f (x)) or exp(2iRe(tf(x)) and determined all cases where U, and another oneparameter subgroup V, of the unitary group of a certain L2-space generate a Lie group G locally isomorphic to SL2(K). In doing so we have assumed that the tangent vector or vectors of V, at t = 0 is a linear differential operator with polynomial coefficients, a condition satisfied in the known cases. The result is that f (x) simply becomes x, x2 under a suitable normalization and, accordingly, the differential operator becomes i-times

Euler type partial differential operators on real analytic functions

We apply Rossi’s half-plane version of Borel’s theorem to study the zero distribution of linear combinations of A \mathcal {A} -entire functions (Theorem 1.2). This provides a unified way to study linear q q -difference, difference and differential operators (with entire coefficients) preserving subsets of A \mathcal {A} -entire functions, and hence obtain several analogous results for the Hermite-Poulain theorem to linear finite ( q q -)difference operators with polynomial coefficients. The method also produces a result on the existence of infinitely many non-real zeros of some differential polynomials of functions in certain sub-classes of A \mathcal {A} -entire functions.

An analogue of the irreducibility is developed for linear differential operators, or, more generally, noncommutative polynomials, and is applied to a few simple examples. Introduction. The question of the irreducibility of an ordinary linear differential operator is of interest in the Picard-Vessiot theory (see Kolchin [1, ?22]). Indeed, the operator is irreducible if and only if the Galois group is linearly irreducible. In this note we develop an Eisenstein criterion to help answer this question. Because a linear differential operator is a special type of noncommutative polynomial, it is more natural to phrase the result in this more general context. The result, and indeed the proof with only slight changes in notation, is valid for partial differential operators, or more generally, noncommutative polynomials in several variables. Because the hypothesis that the ring be principal is unnatural in this setting, and because the reader may make the necessary changes if he so desires, we confine our attention to the ordinary case. 1. The irreducibility criterion. By a differential ring M we shall mean a ring (associative, commutative, with unit) together with two commuting operators a and 3 such that a is an automorphism of R, 3 is additive, and 6(ab)=aa3b+6a.b for every a, b in S. 1[6] will denote the noncommutative ring of differential operators with coefficients in R, so that 1[6] is the set of 0 ai6i, with ai E $, n E N, and right multiplication is defined by 6 * a-=oa * + 3a. Evidently if or=id, then W is an ordinary differential ring with derivation operator 6. By a differential ideal of R we mean an ideal which is closed under the two operators a and 6. The smallest differential ideal which contains the set Ec$S is denoted by [2]. Received by the editors October 7, 1971. AMS 1970 subject classifications. Primary 12H05, 12E05; Secondary 13B25, 16A02.

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

The regularized trace of an ordinary linear differential operator with polynomial coefficients in L2(−∞,+∞)

Some Applications of a Closed-form Solution for Compound Options of Order N.- Surjective Linear Partial Differential Operators with Variable Coefficients on Non-quasianalytic Classes of Roumieu Type.- The Fundamental Solution for a Second Order Weakly Hyperbolic Cauchy problem.- Pseudoholomorphic Discs Attached to Pseudoconcave Domains.- Vorticity and Regularity for Solutions of Initial-boundary Value Problems for the Navier-Stokes Equations.- Exponential Decay and Regularity for SG-elliptic Operators with Polynomial Coefficients.- A Short Description of Kinetic Models for Chemotaxis.- Eigenvalues, Eigenfunctions in Domains Becoming Unbounded.- Loss of Derivatives for t?? in Strictly Hyperbolic Cauchy Problems.- On the Operator Splitting Method: Nonlinear Balance Laws and a Generalization of Trotter-Kato Formulas.- Subelliptic Estimates for some Systems of Complex Vector Fields.- Approximate Solutions to the 2-D Unsteady Navier-Stokes System with Free Surface.- Time Decay Estimates of Solutions for Wave Equations with Variable Coefficients.- On Weakly Pseudoconcave CR Manifolds.- A Note on Kohn's and Christ's Examples.- On the Nonstationary Two-dimensional Navier-Stokes Problem in Domains with Strip-like Outlets to Infinity.- A Link between Local Solvability and Partial Analyticity of Several Classes of Degenerate Parabolic Operators.- The Solution of the Equation with .- On Schauder Estimates for the Evolution Generalized Stokes Problem.- Local Analyticity and Nonlinear Vector Fields.- Strongly Hyperbolic Complex Systems Reduced Dimension, Hermitian Systems.

Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients

We introduce a supersymmetric analog of the classical Coxeter frieze patterns. Our approach is based on the relation with linear difference operators. We define supersymmetric analogs of linear difference operators called Hill’s operators. The space of these “superfriezes” is an algebraic supervariety, isomorphic to the space of supersymmetric second order difference equations, called Hill’s equations.

Linear perturbations of differential of difference operators with polynomials as eigenfunctions

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.