Containing Symbolic Powers in Regular Rings

We extend some results of Ein–Lazarsfeld-Smith, Hochster–Huneke, and Takagi-Yoshida concerning the containments of symbolic powers of ideals in ordinary powers in regular rings.     

Non-linear behaviour of Castelnuovo–Mumford regularity

It is shown that the Castelnuovo–Mumford regularity of lex-segment ideals of ordinary powers as well as of Frobenius powers of a homogeneous polynomial ideal is a function of polynomial type of huge degree.     

POWERS OF REAL SYMMETRIC DIFFERENTIAL EXPRESSIONS WITHOUT SMOOTHNESS ASSUMPTIONS

Abstract

In recent years many authors have studied properties of powers of symmetric (formally self-adjoint) ordinary linear differential expressions. So that these powers can be formed in the classical way these authors have placed heavy smoothness assumptions on the coefficients. Here we show that no differentiability conditions whatsoever are needed on the coefficients in order to form powers of a given expression-provided these powers are formed in the quasi-differential sense rather than the classical one.     

Asymptotics of the Solution of a Singularly Perturbed System of Equations with a Multizone Internal Layer

Differential Equations - We consider a boundary value problem for a singularly perturbed system of two second-order ordinary differential equations with distinct powers of the small parameter...     

Asymptotic analysis of certain systems of linear differential equations with a large parameter

Two classes of systems of linear ordinary differential equations containing high-frequency terms proportional to certain positive powers of the oscillation frequency are considered. Complete asymptotic expansions of fundamental systems of solutions are constructed and substantiated.     

An elementary approach to containment relations between symbolic and ordinary powers of certain monomial ideals

We provide an elementary explanation of a surprising result of Ein–Lazarsfeld–Smith and Hochster–Huneke on the containment between symbolic and ordinary powers of ideals for a certain class of simple monomial ideals.     

Asymptotics of a solution to a second order linear differential equation with large summands

We consider the second order ordinary differential equations whose coefficients of the unknowns contain smooth and rapidly oscillating summands proportional to the positive powers of the oscillation frequency. The complete asymptotic expansions are constructed and justified of solutions to the Cauchy problem.     

A New Method of Summation of Fourier Series Converging in C-Norm

A new method is based on some properties of a semigroup of fractional powers of some positive operator in Banach space. This method uses an essentially smaller deformation of an ordinary partial sum of the Fourier series than in Cesaro's method or in its generalizations.     

Universal topological algebra needs closed topological categories

It is shown that a development of universal topological algebra, based in the obvious way on the category of topological spaces, leads in general to a pathological situation. The pathology disappears when the base category is changed to a cartesian closed topological category or to a topological category endowed with a compatible closed symmetric monoidal structure, provided that in the latter case, the algebraic operations are expressed in terms of monoidal powers rather than the usual cartesian powers. With such base categories, universal topological algebra becomes virtually as well-behaved as ordinary (setbased) universal algebra.     

Convex bodies and asymptotic invariants for powers of monomial ideals

Continuing a well established tradition of associating convex bodies to monomial ideals, we initiate a program to construct asymptotic Newton polyhedra from decompositions of monomial ideals. This is achieved by forming a graded family of ideals based on a given decomposition. We term these graded families powers since they generalize the notions of ordinary and symbolic powers. Asymptotic invariants for these graded families are expressed as solutions to linear optimization problems on the respective convex bodies. This allows to establish a lower bound on the Waldschmidt constant of a monomial ideal by means of a more easily computable invariant, which we introduce under the name of naive Waldschmidt constant.     

Geometric nullstellensatz and symbolic powers on arbitrary varieties

In recent years, a multiplier ideal defined on arbitrary varieties, so called Mather–Jacobian multiplier ideal, has been developed independently by Ein–Ishii–Mustata and de Fernex–Docampo. With this new tool, we have a chance of extending some classical results proved in nonsingular case to arbitrary varieties to establish their general forms. In this paper, we first extend a result of geometric nullstellensatz due to Ein–Lazarsfeld in nonsingular case to any projective varieties. Then we prove a result on comparison of symbolic powers with ordinary powers on any varieties, which extends results of Ein–Lazarsfeld–Smith and Hochster–Huneke.     

Asymptotic solvers for ordinary differential equations with multiple frequencies

We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms,focusing on the case of multiple,non-commensurate frequencies.We derive an asymptotic expansion in inverse powers of the oscillatory parameter and use its truncation as an exceedingly effective means to discretize the differential equation in question.Numerical examples illustrate the effectiveness of the method.     

Asymptotic expansion of a solution to a singular perturbation optimal control problem on an interval with integral constraint

A control problem for solutions of a boundary value problem for a second-order ordinary differential equation with a small parameter at the second derivative is considered on a closed interval. The control is scalar and subject to integral constraints. We construct a complete asymptotic expansion in powers of the small parameter in the Erdélyi sense.     

On Namias's Fractional Fourier Transforms

The objective of this paper is to make rigorous a formal study(Namias, 1980) of fractional powers for the Fourier transform.To make the theory unambiguous, it is found necessary to modifyNamias's fractional operators. Some theorems are then provedfor the modified operators and an operational calculus is developed.Finally, an application to an ordinary differential equationis considered.     

On Complicated Expansions of Solutions to ODES

Polynomial ordinary differential equations are studied by asymptotic methods. The truncated equation associated with a vertex or a nonhorizontal edge of their polygon of the initial equation is assumed to have a solution containing the logarithm of the independent variable. It is shown that, under very weak constraints, this nonpower asymptotic form of solutions to the original equation can be extended to an asymptotic expansion of these solutions. This is an expansion in powers of the independent variable with coefficients being Laurent series in decreasing powers of the logarithm. Such expansions are sometimes called psi-series. Algorithms for such computations are described. Six examples are given. Four of them are concern with Painlevé equations. An unexpected property of these expansions is revealed.     

Congruences between Siegel modular forms II

Using a p-adic monodromy theorem on the affine ordinary locus in the minimally compactified moduli scheme modulo powers of a prime p of abelian varieties, we extend Katz?s results on congruence and p-adic properties of elliptic modular forms to Siegel modular forms of higher degree.     

The general Jacobi matrix method for solving some nonlinear ordinary differential equations

In this paper, we obtain the approximate solutions for some nonlinear ordinary differential equations by using the general Jacobi matrix method. Explicit formulae which express the Jacobi expansion coefficients for the powers of derivatives and moments of any differentiable function in terms of the original expansion coefficients of the function itself are given in the matrix form. Three test problems are discussed to illustrate the efficiency of the proposed method.     

Analytic integration of a system of nonlinear ordinary differential equations with an irregular type singularity

Summary We shall discuss how to construct analytic expressions for bounded solutions of nonlinear ordinary differential equations of the form(A) which tend to0 with the order of certain positive powers of x as x approaches the origin through a sector. It will be shown that the opening angle of this sector is maximal. Entrata in Redazione il 1^o luglio 1971.     

Formal power series solutions in a parameter

This paper is a continuation of the previous paper (J. Differential Equations 165 (2000) 255). The main subject is the Gevrey property of formal solutions of an analytic ordinary differential equation in powers of a parameter. In one case, a given formal solution itself is of the Gevrey type, while, in another case, the existence of a formal solution implies the existence of formal solutions of the Gevrey types. These situations are explained systematically in this paper.