On a class of non-linear functional equations connected with modular functions

Schroder developed several notions which are central to the study of complex dynamics. Despite his failure to rigorously establish his fixed point theorem, it is a fundamental result, and his belief that iteration of an arbitrary function ∅(z) could be reduced to the solution of the so-called Abel and Schroder functional equations was prophetic. Not only was the next phase in the development of complex dynamics ushered in by an interest in the solution of the Schroder and Abel equations, but the study and solution of functional equations is fundamental in many contemporary studies of iteration.

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.

VECFEM is a black-box solver for the solution of a large class of nonlinear functional equations by finite element methods. It uses very robust solution methods for the linear FEM problem to compute reliably the Newton-Raphson correction and the error indicator. Kernel algorithms are conjugate gradient methods (CG) for the solution of the linear system. In this paper we present the optimal data structures on parallel computers for the matrix-vector multiplication, which is the key operation in the CG iteration, the principles of the element distribution onto the processors and the mounting of the global matrix over all processors as transformation of optimal data structures. VECFEM is portably implemented for message passing systems. Two examples with unstructured and structured grids will show the efficiency of the data structures.

It is shown that the set of integrodifferential and algebraic functional equations of the local energy transfer theory may be considerably reduced in dimension for the case of isotropic turbulence. This is achieved without restricting the solution space. The basis for this is a complete analytical solution to the functional equations Q(k;t,t('))=H(k;t,t('))Q(k;t('),t(')) and H(k;t,s)H(k;s,t('))=H(k;t,t(')). The solution is proved to depend only on a single function straight phi(k;t) solely determining Q and H. Hence the dimension of both the dependent and the independent variables is reduced by one. From the latter, the corresponding two integrodifferential equations are lowered to a single integrodifferential equation for straight phi(k;t), extended by an integral side condition on the k dependence of straight phi(k;t). In the limit nu-->0, a partial solution to the reduced set of equations is presented in the Appendix.

This paper discussed some existence theorems for nonlinear functional integral equations in the space L^1 of Lebesgue integrable functions,by using the Darbo fixed point theorem associated with the Hausdorff measure of noncompactness. Also, as an application, we discuss the existence of solutions for some nonlinear integral equations with fractional order.

A general method is given for defining architectural design complexity measures. Desired properties of a measure are described by functional equations. Two cases of descriptions are considered. Complexity measures are given as the solutions of functional equations. Other complexity measures can be regarded as special cases of the solutions. A new measure is also presented.

Approximation orders of formal Laurent series by β-expansions

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defined over the field of rational numbers Q is normal. They do this by noting that, over a field of characteristic 3, F (X) equals 1/ √ 1 + 1/X. The latter has a periodic continued fraction and, as a consequence, they are able to explicitly verify that F (X) is normal. They then use an “automatic” proof to show that the Laurent series of F (X), defined over Q, is normal. Here we shall give a different proof of the latter result (Theorem 4, below) and show, more generally, that if a formal Laurent series Γ (X) defined over Q is defined and normal (mod p) for any prime p, then it is also normal over Q. We shall also consider the continued fraction of the more general surd 1/ √ 1 + u/X + v/X2 over any field not of characteristic 2. Put δ = u/4−v and note that √ 1 + u/X + v/X2 = 1 + u/(2X) when δ = 0. Hence, when δ = 0, the continued fraction expansion is trivial and is not normal. Define

In this paper, the constrained dispersionless BKP hierarchy are given by restricting the formal Laurent series [Formula: see text] of the dispersionless BKP hierarchy to [Formula: see text]. Its additional symmetries are given by introducing vital formal Laurent series [Formula: see text]. The additional flows form the nonabelian [Formula: see text]-grading Lie algebra. Furthermore, the additional flows acting on [Formula: see text] and [Formula: see text] are presented.

A higher-dimensional Kurzweil theorem for formal Laurent series over finite fields

Consider a polynomial ring R with the ‐grading where the degree of each variable is a standard basis vector. In other words, R is the homogeneous coordinate ring of a product of n projective spaces. In this setting, we characterize the formal Laurent series which arise as Hilbert series of finitely generated R‐modules. We also provide necessary conditions for a formal Laurent series to be the Hilbert series of a finitely generated module with a given depth. In the bigraded case (corresponding to the product of two projective spaces), we completely classify the Hilbert series of finitely generated modules of positive depth.

In this article, we establish some results for the existence of solution of nonlinear functional integral equations by using Darbo’s fixed point theorem in Banach algebra. As an application, we discuss some examples of nonlinear functional integral equations and existence of solutions.

We study the regularity of solutions of functional equations of a generalized mean value type. In this paper we give sufficient conditions for the regularity by using hypoellipticity which is a concept of the theory of partial differential equations. We also give an affirmative answer to a conjecture of H. Światak. A part of the results was announced in the comprehensive paper [8] on our joint works. To prove the regularity of solutions of functional equations is one of the central problems in the theory of functional equations (see [1]).

We investigate the existence and Borel summability of formal power series in one variable with holomorphic coefficients, solutions of nonlinear partial differential equations with shrinkings having non-Kowalewski type in the complex domain. Sufficient conditions are given in terms of the shape of the functional partial differential equations and initial conditions. The existence and asymptotic behavior of local sectorial holomorphic solutions of these nonlinear functional partial differential equations are obtained as a by-product. We apply the results to the study of asymptotic the behavior of solutions of some advanced-argument partial differential equations in a neighborhood of infinity.