On formal power series defined by infinite linear systems

SALOMAA1. Introiluction and preliminaries.The theory of formal power series in non- commuting variables was initiated around 1960 -apart from some scattered work done earlier in connection with free groups.Such power series are applicable in a number of areas but, in particular, they have turned out to be an indispensable

A connection between scalar conservation laws and infinite linear systems of PDE's

For a positive integer n and a subset S⊆[n−1], the descent polytope DP S is the set of points (x 1,…,x n ) in the n-dimensional unit cube [0,1] n such that x i ≥x i+1 if i∈S and x i ≤x i+1 otherwise. First, we express the f-vector as a sum over all subsets of [n−1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5,…}∩[n−1]. We derive a generating function for F S (t), written as a formal power series in two non-commuting variables with coefficients in ℤ[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.

Formal analysis is the study of formal power series, formal Laurent series, formal root series, and other formal series or formal functionals. This book is the first comprehensive presentation of the topic that systematically introduces formal analysis, including its algebraic, analytic, and topological structure, along with various applications.

This monograph covers the theory of finite and infinite matrices over the fields of real numbers, complex numbers and over quaternions. Emphasizing topics such as sections or truncations and their relationship to the linear operator theory on certain specific separable and sequence spaces, the authors explore techniques like conformal mapping, iterations and truncations that are used to derive precise estimates in some cases and explicit lower and upper bounds for solutions in the other cases. Most of the matrices considered in this monograph have typically special structures like being diagonally dominated or tridiagonal, possess certain sign distributions and are frequently nonsingular. Such matrices arise, for instance, from solution methods for elliptic partial differential equations. The authors focus on both theoretical and computational aspects concerning infinite linear algebraic equations, differential systems and infinite linear programming, among others. Additionally, the authors cover topics such as Bessel s and Mathieu s equations, viscous fluid flow in doubly connected regions, digital circuit dynamics and eigenvalues of the Laplacian

Helicon wave (whistler waves bounded transversely by a magnetic field surface) propagation is investigated for a cylindrically symmetric curved (flaring) boundary using a finite-element method where the propagation region is divided into a sequence of truncated cones. In each conical segment, a local spherical coordinate system is used with the origin at the apex of the cone. A vector wave equation for the fields is formulated for a cold plasma and reduced, in spherical coordinates, to a pair of coupled partial differential equations (PDEs) for two scalar functions. The PDEs are separable for the azimuthal eigenvalue m = 0. The -dependence is a Legendre function, with non-integer eigenvalues determined by the cone angle. The dependence on the radial coordinate, x, satisfies a fourth-order ordinary differential equation (ODE). A straightforward numerical integration of the equation from to fails for large values of , because of the existence of an exponentially growing solution to the equation. Consequently, a different approach is needed. Four independent solutions, valid for , in the form of power series in x (PS) are obtained which are each asymptotically proportional to as . For x large, four asymptotic expansions (AE) are obtained in the form of series in 1/x times or . The problem then is to find four linear combinations of the four PSs, each of which approaches one of the AEs in the limit by first deriving a double-integral representation of each PS, valid for all x, and then using its limit to match each AE to a linear combination of the PSs. For special values of or 4N, N an integer) closed-form solutions (polynomials times exponentials) result that are exact representations of the solution for all x. Solutions are computed and illustrated for an outgoing wave using the closed-form solution for special values of , and as a function of and x using the PS and AE, which have a large range of overlap. The propagating solutions of the ODE are also obtained using a WKB method and are used to calculate the propagation in a slowly diverging parabola of revolution.

Formal power series and regular operations on fuzzy languages

We investigate the translation equation (T) , and the cocycle equations (Co1) , (Co2) in [ x ], the ring of formal power series over . Here we restrict ourselves to iteration groups of type I, i. e. to solutions of (T) of the form , where c 1 ? 1 is a generalized exponential function. It is easy to prove that the coefficient functions c n (s), a n (s), s n (s) of are polynomials, c 1(s), polynomials or rational functions in one more generalized exponential function and, in a special case, polynomials in an additive function. For example, we obtain . For this we do not need any detailed information about the polynomials P n . Under some conditions on the exponential and additive functions it is possible to replace the exponential and additive functions by independent indeterminates. In this way we obtain formal versions of the translation equation and the cocycle equations in rings of the form [ x ], [ x ], [ x ], [ x ], and [ x ]. We solve these equations in a completely algebraic way, by deriving formal differential equations or Aczel�Jabotinsky type equations for these problems. It is possible to get the structure of the coefficients in great detail which are now polynomials or rational functions. We prove the universal character (depending on certain parameters) of these polynomials or rational functions. And we deduce the canonical form S -1(yS(x)) for iteration groups of type I. This approach seems to be more clear and more general than the original one. Some simple substitutions allow us to solve these problems in rings of the form ( [ u ]) [ x ], i. e. where the coefficient functions are formal series.

A positive cosmological constant simplifies the asymptotics of forever expanding cosmological solutions of the Einstein equations. In this paper a general mathematical analysis on the level of formal power series is carried out for vacuum spacetimes of any dimension and perfect fluid spacetimes with linear equation of state in spacetime dimension four. For equations of state stiffer than radiation evidence for development of large gradients, analogous to spikes in Gowdy spacetimes, is found. It is shown that any vacuum solution satisfying minimal asymptotic conditions has a full asymptotic expansion given by the formal series. In four spacetime dimensions, and for spatially homogeneous spacetimes of any dimension, these minimal conditions can be derived for appropriate initial data. Using Fuchsian methods the existence of vacuum spacetimes with the given formal asymptotics depending on the maximal number of free functions is shown without symmetry assumptions.

The application of mathematics to natural phenomena often brings up the question : when a linear equation is changed slightly, how much of a change results in the solution? Here vector is used to include, for example, vectors in Hubert space and elements of (linear) function spaces; also it is assumedC) that the equation is uniquely solvable. We formulate and answer the question in §2 for a linear equation in a Banach space, then specialize to deduce perturbation limits for linear algebraic systems, infinite systems, and integral equations. Finally we obtain error limitations for certain approximate methods of solving infinite linear systems (method of segments) and integral equations (method of Goursat-Schmidt). It is hoped that these limitations may be useful to applied mathematicians. The methods and results of this paper unify and extend investigations concerning algebraic systems by F. R. Moulton [15](2), Etherington [6] and the author [13, 14]; and on integral equations of Fredholm type and second kind by Tricomi [21]. However, they do not cover perturbation questions associated with characteristic values and characteristic functions, which have been studied by Lord Rayleigh [16, p. 115], Courant [4, p. 296] and Mrs. Adams [l]. §1 is expository, containing as much about abstract spaces as is needed for §2. 1. Vector spaces. It will be useful to collect here some facts about normed linear spaces ispaces L), Banach spaces ispaces B), and linear transformations. A space L is linear: if xEL and a is any complex number, the product ax is defined and axEL; if also y EL, the sum x+y is defined and x+yEL. With each element x of L is associated a non-negative real number ||*||, its norm; ||*|| >0 unless x = @, the zero-element of L; |J@|| =0. The norm has properties of an absolute value: ||a*|| = \a -11*11, ||*+y|| a||*||+||y||(We have described a complex space L; in a real space, number a must be real.) A normed linear space is a Banach space 5 [2, p. 53] if it is furthermore complete: if {*„} is an infinite sequence of elements of 5, and if II*»—*»||—*0 as m, m—»oo, there exists a xo of 5 such that ||*„—*o||—»0 as n—+ oo (strong completeness). Examples of such spaces are listed in Banach's book [2, pp. 10-12, examples 3-10], and several occur in the remainder of this paper.

We use an elementary method to draw analytic conclusions from divergent formal power series solutions of systems of differential equations containing a parameter and give some applications to the theory of turning points. Our main result shows that a divergent formal series transformation of one system into another in which the coefficients satisfy certain estimates is necessarily the asymptotic expansion of an actual transformation. We use it to show the following. Given a two dimensional system ePdy/dx = A(x,e)y with A holomorphic at (x0,0), suppose that x0 is formally not a turning point in the sense that no singularities appear at x0 during the standard formal solution procedure with formal fractional power series in e. Then the formal solution is necessarily a uniform asymptotic representation of a fundamental matrix of the system on a full neighborhood of x0. (This conclusion is known to fail under weaker hypotheses on A). We also obtain similar but less complete results for higher order systems under more specialized hypotheses.

Substituting non-unit formal power series into formal power series is a well-understood concept. This article describes a sense in which one can define substitution of unit formal power series, implemented in Maple’s MultivariatePowerSeries package.

Let S n be the symmetric group, T d be the subspace of C[x 1 , x 2 ,…,x n ] which contains all of the homogeneous polynomials of degree d. The representation of S n determined by T d is denoted by λ d , and the irreducible representation characterized by partition a is denoted by π(a). We denote the multiplicity of π(a) entries into λ d by N a d . As an extension of [4], this note simplifies the recursion formula of the formal power series (?)N a d t d , and obtains the explicity expressions of generating function...

A new method for the solution of the Boltzmann equation for time-dependent or stationary neutron transport is presented. The essentials of the method are a Fourier transform of the transport integral equation and the decomposition of the new kernel into a bilinear form. In the monoenergetic case, the transport problem is reduced to the solution of an (infinite) linear system of equations. It is important that the matrix elements in this linear system be determined only once for any geometry and that this be done in an analytical way. In the present paper, the matrix elements for plane, spherical, and cylindrical geometries are constructed explicitly and are presented in a form suitable for numerical applications. To illustrate the efficiency of the method, the critical dimensions for these geometries have been determined in various integral transform ITN approximations, N being the order of the linear system after truncation. For samples of a thickness up to ∼10 mean-free-paths, the results for the critical dimension in stationary problems, or for the fundamental time-decay constants in nonstationary problems, have in the IT4 approximation, at least, the accuracy of the corresponding S16 results.The convergence of the ITN results, with respect to the order N, is very fast. The results for N = 4 can be considered as exact within one or two units in the fifth digit. The convergence behavior of the SN and the PN method is decidedly slower. In contrast to the SN, PN or Case’s method, the smaller the system, the better the accuracy of the ITN calculations for fixed N is. The increase in accuracy by increasing N requires less computational effort for the ITN approximation than for SN or PN approximation. Apart from providing an analytical standard for the check of numerical approximations, the present method is a simple and rigorous tool for the calculation of neutron distributions, particularly in small systems.

In this paper, we develop a theory of free holomorphic functions on noncommutative Reinhardt domains , generated by positive regular free holomorphic functions f in n noncommuting variables and by positive integers m ≥ 1, where is the algebra of all bounded linear operators on a Hilbert space . Noncommutative Berezin transforms are used to study Hardy algebras H ∞ (D f, rad m ) and domain algebras A(D f, rad m ) associated with and compositions of free holomorphic functions. We obtain noncommutative Cartan type results for formal power series, in several noncommuting indeterminates, which leave invariant the nilpotent parts of the corresponding domains. As a consequence, we characterize the set of all free biholomorphic functions with F(0) = 0. We show that the free biholomorphic classification of the domains is the same as the classification, up to unital completely isometric isomorphisms having completely contractive hereditary extension, of the corresponding noncommutative domain algebras A(D f, rad m ). In particular, we prove that Ψ : A(D f, rad 1 ) → A(D g, rad 1 ) is a unital completely isometric isomorphism if and only if there is a free biholomorphic map φ ∈ Bih(D g 1 , D f 1 ) such that This implies that the noncommutative domains and are free biholomorphic equivalent if and only if the domain algebras A(D f, rad 1 ) and A(D g, rad 1 ) are completely isometrically isomorphic. Using the interaction between the theory of functions in several complex variables and our noncommutative theory, we provide several results concerning the free biholomorphic classification of the noncommutative domains and the classification, up to completely isometric isomorphisms, of the associated noncommutative domain (resp. Hardy) algebras. In particular, we characterize the unit ball of among the noncommutative domains , up to free biholomorphisms. We also obtain characterizations for the unitarily implemented isomorphisms of noncommutative Hardy (resp. domain) algebras in terms of free biholomorphic functions between the corresponding noncommutative domains.