Tau Functions Associated to Pseudodifferential Operators of Several Variables

Let there be given a finite-gap operator L = d 2 /d x 2 + q and its Baker function ψ( x, p ), which is analytic for p on a certain hyperelliptic curve C . It is shown that a sequence of Bäcklund transformations maps C to a projective space. This embedding can be interpreted as a matrix representation of the Hill equation by the Neumann system of constrained harmonic oscillators. The image curve, C' , lies on a rational ruled surface; the structure of this surface is explained by use of ideas due to Burchnall & Chaundy ( Proc. R. Soc. Lond . A 118, 557-583 (1928)). Baker functions and Bäcklund transformations are then used to define a (many-to-many) correspondence between effective divisors on the curve C and points lying on a quadric, or in the intersection of two or more quadrics. This relates the theory of the Hill equation to earlier work of Knörrer, Moser and Reid. It is then shown that the Kummer image of the Jacobian of C can be realized as a hypersurface in the space of momentum variables of the Neumann system. Further projects, such as extensions to non-hyperelliptic curves, are outlined.

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.

Approximation orders of formal Laurent series by β-expansions

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.

: This research has led to new developments for solving nonlinear optimization problems involving functions that are not everywhere differentiable and/or are implicity defined. For the single variable case a method has been given which combined polyhedral and quadratic approximation, and automatic scale-free penalty technique and a safeguard that insures convergence to a stationary point, but does not detract from rapid convergence. Under relatively weak convergence rate assumptions the algorithm exhibits a new type of better than linear convergence. The safequard also has the practical advantage of keeping apart points that are used in denominators of difference quotients for approximating second derivatives. A practical single resource allocation problem with several bounded decision variables has been solved very efficiently via a dual technique that used the single variable method in a nested manner to solve both the outer dual problem and the inner Lagrangian subproblems. The new concept of better than linear convergence form the single variable case has been generalized to the multivariable case. Author supplied key words also include; constrained minimization, and line search.

We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain “motivic Galois group” U*, which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one-parameter subgroup of U*. The group U* arises through a Riemann-Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of the choice of a unit of mass. As an algebraic group scheme, U* is a semidirect product by the multiplicative group G_m of a prounipotent group scheme whose Lie algebra is freely generated by one generator in each positive integer degree. There is a universal singular frame in which all divergences disappear. When computed as iterated integrals, its coefficients are certain rational numbers that appear in the local index formula of Connes-Moscovici. When working with formal Laurent series over ℚ, the data of equisingular flat vector bundles define a Tannakian category whose properties are reminiscent of a category of mixed Tate motives.

We prove an explicit formula for infinitely many convergents of Hurwitzian continued fractions that repeat several copies of the same constant and elements of one arithmetic progression, in a quasi-periodic fashion. The proof involves combinatorics and formal Laurent series. Using very little analysis we can express their limits in terms of (modified) Bessel functions and Fibonacci polynomials. The limit formula is a generalization of Lehmer's theorem that implies the continuous fraction expansions of $e$ and $\tan(1)$, and it can also be derived from Lehmer's work using Fibonacci polynomial identities. We completely characterize those implementations of our limit formula for which the parameter of each Bessel function is the half of an odd integer, allowing them to be replaced with elementary functions.

Let p be a prime. This paper deals with solutions of functional equations in either formal Laurent series or in analytic functions. Examples connected to special modular functions are considered.

An extended integrable fractional-order KP soliton hierarchy

The Henon-Heiles system in the general form has been considered. In a nonintegrable case with the help of the Painleve test new solutions have been found as formal Laurent or Puiseux series, depending on three parameters. One of parameters determines a location of the singularity point, other parameters determine coefficients of series. It has been proved, that if absolute values of these two parameters are less or equal to unit, then obtained series converge in some ring. For some values of these parameters the obtained Laurent series coincide with the Laurent series of the known exact solutions.

Painlevé analysis of nonlinear evolution equations—an algorithmic method

The n-component Kadomtsev–Petviashvili (KP) hierarchy was introduced by Sato as the set of deformation equations, i.e., Lax equations, of n+1n×n-matrix pseudodifferential operators. On such a set of operators we define a Schlesinger–Bäcklund transformation, which produces from a set of solutions of the KP hierarchy a new set of solutions. Applying such transformations recursively we obtain a whole family of solutions, from which we can then derive the KP tau function.