On certain properties of solutions of second-order linear differential equations

For a second-order linear differential equation with coefficients in a differential fieldL of meromorphic functions, a classical result of C. L. Siegel states that if the logarithmic derivative of every nontrivial solution is transcendental overL, then no nontrivial solution can satisfy a first-order algebraic differential equation overL. In Part 1, we consider a situation for the equation (*)w¨+A(z)w=0 where Siegel's condition is violated, namely where the logarithmic derivative of a nontrivial solution belongs toL. In this case, we obtain a representation for any solution of (*) which satisfies a first-order algebraic differential equation overL, and we illustrate our result with examples for various choices ofA(z) andL. In Part 2, we treat the question of determining when all solutions of (*) satisfy first-order algebraic differential equations overL, and in Part 3, we obtain a refinement of Siegel's classical theorem on Bessel's equation.This research was supported in part by the National Science Foundation (MCS-8002269).     

On differential equations satisfied by modular forms

We use the theory of modular functions to give a new proof of a result of P. F. Stiller, which asserts that, if t is a non-constant meromorphic modular function of weight 0 and F is a meromorphic modular form of weight k with respect to a discrete subgroup of SL ₂ () commensurable with SL ₂ (), then F, as a function of t, satisfies a (k+1)-st order linear differential equation with algebraic functions of t as coefficients. Furthermore, we show that the Schwarzian differential equation for the modular function t can be extracted from any given (k+1)-st order linear differential equation of this type. One advantage of our approach is that every coefficient in the differential equations can be relatively easily determined. Mathematics Subject Classification (2000):11F03, 11F11.     

Triviality of differential Galois cohomology of linear differential algebraic groups

For a partial differential field K, we show that the triviality of the first differential Galois cohomology of every linear differential algebraic group over K is equivalent to K being algebraically, Picard–Vessiot, and linearly differentially closed. This cohomological triviality condition is also known to be equivalent to the uniqueness up to an isomorphism of a Picard–Vessiot extension of a linear differential equation with parameters.     

Linear relations among algebraic solutions of differential equations

For any univariate polynomial with coefficients in a differential field of characteristic zero and any integer, q, there exists an associated nonzero linear ordinary differential equation (LODE) with the following two properties. Each term of the LODE lies in the differential field generated by the rational numbers and the coefficients of the polynomial, and the qth power of each root of the polynomial is a solution of this LODE. This LODE is called a qth power resolvent of the polynomial. We will show how one can get a resolvent for the logarithmic derivative of the roots of a polynomial from the αth power resolvent of the polynomial, where α is an indeterminate that takes the place of q. We will demonstrate some simple relations among the algebraic and differential equations for the roots and their logarithmic derivatives. We will also prove several theorems regarding linear relations of roots of a polynomial over constants or the coefficient field of the polynomial depending upon the (nondifferential) Galois group. Finally, we will use a differential resolvent to solve the Riccati equation.     

Tropical differential equations

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in the absolute values of its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed.We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is NP-hard, and this problem for arbitrary number of variables belongs to NP. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series.     

The nonlinear diffusion equation in cylindrical coordinates

Nonlinear corrections to some classical solutions of the linear diffusion equation in cylindrical coordinates are studied within quadratic approximation. When cylindrical coordinates are used, we try to find a nonlinear correction using quadratic polynomials of Bessel functions whose coefficients are Laurent polynomials of radius. This usual perturbation technique inevitably leads to a series of overdetermined systems of linear algebraic equations for the unknown coefficients (in contrast with the Cartesian coordinates). Using a computer algebra system, we show that all these overdetermined systems become compatible if we formally add one function on radius W(r). Solutions can be constructed as linear combinations of these quadratic polynomials of the Bessel functions and the functions W(r) and W′(r). This gives a series of solutions to the nonlinear diffusion equation; these are found with the same accuracy as the equation is derived. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 235–245, 2007.     

Approximation and Complexity: Liouvillean-Type Theorems for Linear Differential Equations on an Interval

Let u,v be solutions on an interval I of linear differential equations (LDEs) P=0 , Q=0 , respectively. We obtain a lower bound on the approximation of v by u in terms of bounds on the coefficients of LDE S _i =0 (for several i ), satisfied by the i th derivative of v and by the i th derivative of a basis of the LDE P=0 . One could view this result as a differential analog of the Liouville theorem which states that two different algebraic numbers are well separated if they satisfy algebraic equations with small enough integer coefficients. Unlike the algebraic situation, in the differential setting, in order to bound from below the difference |u-v| , we need to involve not only the coefficients of P,Q themselves, but also those of S _i . September 22, 2000. Final version received: March 11, 2001.     

On some new properties of the gamma function and the Riemann zeta function

In this paper, we have exhibited, by utilizing value distribution theory, some new properties of the Gamma function Γ(z) and the Riemann zeta function ζ(z). Specifically, we have proved that both of the two functions are prime and the Riemann zeta function, like Γ(z), does not satisfy any algebraic differential equation with coefficients in ??₀. Moreover, the two functions do not satisfy any functional equation of the form P(Γ, ζ, z) ≡ 0, where P(x, y, z) is a nonconstant polynomial in x, y and z.     

Differentially algebraic gaps

H-fields are ordered differential fields that capture some basic properties of Hardy fields and fields of transseries. Each H-field is equipped with a convex valuation, and solving first-order linear differential equations in H-field extensions is strongly affected by the presence of a “gap” in the value group. We construct a real closed H-field that solves every first-order linear differential equation, and that has a differentially algebraic H-field extension with a gap. This answers a question raised in [1]. The key is a combinatorial fact about the support of transseries obtained from iterated logarithms by algebraic operations, integration, and exponentiation.     

Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space C [0, T]. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding It? stochastic equation.     

On the Relation between Linear Difference and Differential Equations with Polynomial Coefficients

This paper represents the third part of a contribution to the “dictionary” of homogeneous linear differential equations with polynomial coefficients on one hand and corresponding difference equations on the other. In the first part (cf. [4]) we studied the case that the differential equation (D) has at most regular singularities at O and at ∞, and arbitrary singularities in the rest of the complex plane. We constructed fundamental systems of solutions of a corresponding difference equation (A), using integral transforms of microsolutions of (D) at its singular points in ?. In the second part ([5]) we considered differential equations having at most a regular singularity at O and an irregular one at O. We used integral transforms of asymptotically flat solutions of (D) to define it fundamental system of solutions of (Δ), holomorphic in a right half plane, and integral transforms of sections of the sheaf of solutions of (D) modulo solutions with moderate growth as t → 0 in some sector, to define a fundamental system of (Δ), holomorphic in a left half plane. In this final part we combine the techniques and results of the preceding papers to deal with the general case.     

The BS class of Hermite spline quasi-interpolants on nonuniform knot distributions

The BS Hermite spline quasi-interpolation scheme is presented. It is related to the continuous extension of the BS linear multistep methods, a class of Boundary Value Methods for the solution of Ordinary Differential Equations. In the ODE context, using the numerical solution and the associated numerical derivative produced by the BS methods, it is possible to compute, with a local approach, a suitable spline with knots at the mesh points collocating the differential equation at the knots and having the same convergence order as the numerical solution. Starting from this spline, here we derive a new quasi-interpolation scheme having the function and the derivative values at the knots as input data. When the knot distribution is uniform or the degree is low, explicit formulas can be given for the coefficients of the new quasi-interpolant in the B-spline basis. In the general case these coefficients are obtained as solution of suitable local linear systems of size 2d×2d, where d is the degree of the spline. The approximation order of the presented scheme is optimal and the numerical results prove that its performances can be very good, in particular when suitable knot distributions are used.     

Towards an Algebraic Proof of Deligne’s Regularity Criterion. An Informal Survey of Open Problems

We review the notion of regular singular point of a linear differential equation with meromorphic coefficients, from the viewpoint of algebraic geometry. We give several equivalent definitions of regularity along a divisor for a meromorphic connection on a complex algebraic manifold and discuss the global birational theory of fuchsian differential modules over a field of algebraic functions. We describe the generalized algebraic version of Deligne’s canonical extension, constructed in [1, I.4]. Our main interest lies in the algebraic form of Deligne’s regularity criterion [2, II.4.4 (iii)], asserting that, on a normal compactification, only one codimensional components of the locus at infinity need to be considered. If one considers the purely algebraic nature of the statement, it is surprising that the only existing proof of this criterion is the transcendental argument given by Deligne in his corrigendum to loc. cit. dated April 1971. The algebraic proof given in our book [1, I.5.4] is also incorrect, as J. Bernstein kindly indicated to us.We introduce some notions of logarithmic geometry to let the reader appreciate Bernstein’s (counter)examples to some statements in our book [1]. Standard methods of generic projection in projective spaces reduce the question to a two-dimensional puzzle. We report on ongoing correspondence with Y. André and N. Tsuzuki, leading to partial results and provide examples indicating the subtlety of the problem. Lecture held in the Seminario Matematico e Fisico on January 31, 2005 Received: June 2005     

General Linear Quadratic Optimal Stochastic Control Problem Driven by a Brownian Motion and a Poisson Random Martingale Measure with Random Coefficients

In this article, we consider a linear-quadratic optimal control problem (LQ problem) for a controlled linear stochastic differential equation driven by a multidimensional Browinan motion and a Poisson random martingale measure in the general case, where the coefficients are allowed to be predictable processes or random matrices. By the duality technique, the dual characterization of the optimal control is derived by the optimality system (so-called stochastic Hamilton system), which turns out to be a linear fully coupled forward-backward stochastic differential equation with jumps. Using a decoupling technique, the connection between the stochastic Hamilton system and the associated Riccati equation is established. As a result, the state feedback representation is obtained for the optimal control. As the coefficients for the LQ problem are random, here, the associated Riccati equation is a highly nonlinear backward stochastic differential equation (BSDE) with jumps, where the generator depends on the unknown variables K, L, and H in a quadratic way (see (5.9) herein). For the case where the generator is bounded and is linearly dependent on the unknown martingale terms L and H, the existence and uniqueness of the solution for the associated Riccati equation are established by Bellman's principle of quasi-linearization.     

A consequence of a proof of the one-way function existence for the problem of macroscopic superpositions

One-way functions are functions that are easy to compute but hard to invert. Their existence is an open conjecture; it would imply the existence of intractable problems (i.e. NP-problems which are not in the P complexity class).If true, the existence of one-way functions would have an impact on the theoretical framework of physics, in particularly, quantum mechanics. Such aspect of one-way functions has never been shown before.In the present work, we put forward the following.We can calculate the microscopic state (say, the particle spin in the z direction) of a macroscopic system (a measuring apparatus registering the particle z-spin) by the system macroscopic state (the apparatus output); let us call this association the function F. The question is whether we can compute the function F in the inverse direction. In other words, can we compute the macroscopic state of the system through its microscopic state (the preimage F⁻¹)?In the paper, we assume that the function F is a one-way function. The assumption implies that at the macroscopic level the Schrödinger equation becomes unfeasible to compute. This unfeasibility plays a role of limit of the validity of the linear Schrödinger equation.     

<Literal>smt</Literal>: a Matlab toolbox for structured matrices

The full exploitation of the structure of large scale algebraic problems is often crucial for their numerical solution. Matlab is a computational environment which supports sparse matrices, besides full ones, and allows one to add new types of variables (classes) and define the action of arithmetic operators and functions on them. The smt toolbox for Matlab introduces two new classes for circulant and Toeplitz matrices, and implements optimized storage and fast computational routines for them, transparently to the user. The toolbox, available in Netlib, is intended to be easily extensible, and provides a collection of test matrices and a function to compute three circulant preconditioners, to speed up iterative methods for linear systems. Moreover, it incorporates a simple device to add to the toolbox new routines for solving Toeplitz linear systems.     

Coercive boundary value problems for regular degenerate differential-operator equations

This study focuses on nonlocal boundary value problems (BVP) for degenerate elliptic differential-operator equations (DOE), that are defined in Banach-valued function spaces, where boundary conditions contain a degenerate function and a principal part of the equation possess varying coefficients. Several conditions obtained, that guarantee the maximal L_p regularity and Fredholmness. These results are also applied to nonlocal BVP for regular degenerate partial differential equations on cylindrical domain to obtain the algebraic conditions that ensure the same properties.     

First Order Differential Operators Associated to the Cauchy—Riemann Operator in the Plane

The article deals with initial value problems of type δw/δt = Fw, w(0, ·) = φ where t is the time and F is a linear first order operator acting in the z = x ? iy-plane. In view of the classical Cauchy-Kovalevkaya Theorem, the initial value problem is solvable provided F has holomorphic coefficients and the initial function is holomorphic. On the other hand, the Lewy example [H. Lewy (1957). An example of a smooth linear partial differential equation without solution. Ann. of Math., 66, 155–158.] shows that there are equations of the above form with infinitely differentiable coefficients not having any solutions. The article in hand constructs, conversely, all linear operators F for which the initial value problem with an arbitrary holomorphic initial function is always solvable. In particular, we shall see that there are equations of that type whose coefficients are only continuous.     

Multiblock matrix methods for solution of integrodifferential equations of (0, k) type

Easily implementable numerical-analytical (so-called matrix) methods are developed and analyzed for approximate solution of one-dimensional linear integrodifferential equations with constant integration limits containing a differential expression only in the integrand. Zero-rank matrix algorithms of numerical differentiation and integration are used for construction. The original equation is reduced to a linear algebraic vector equation for the values of the sought solution at the grid points and to an interpolation formula.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 69–75, 1985.     

Locally Optimal Block Preconditioned Conjugate Gradient Method for Hierarchical Matrices

We present a method of almost linear complexity to approximate some (inner) eigenvalues of symmetric self-adjoint integral or differential operators. Using ℋ-arithmetic the discretisation of the operator leads to a large hierarchical (ℋ-) matrix M. We assume that M is symmetric, positive definite. Then we compute the smallest eigenvalues by the locally optimal block preconditioned conjugate gradient method (LOBPCG), which has been extensively investigated by Knyazev and Neymeyr. Hierarchical matrices were introduced by W. Hackbusch in 1998. They are data-sparse and require only O(nlog₂ n) storage. There is an approximative inverse, besides other matrix operations, within the set of ℋ-matrices, which can be computed in linear-polylogarithmic complexity. We will use the approximative inverse as preconditioner in the LOBPCG method. Further we combine the LOBPCG method with the folded spectrum method to compute inner eigenvalues of M. This is equivalent to the application of LOBPCG to the matrix M_μ = (M − μI)² . The matrix M_μ is symmetric, positive definite, too. Numerical experiments illustrate the behavior of the suggested approach. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)