The constrained dispersionless BKP hierarchy and its additional symmetries

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.

In this paper, the additional symmetries of the dispersionless cKP hierarchy are given by introducing vital formal Laurent series Yl. The additional flows form a subalgebra of the Virasoro algebra. Furthermore, the additional flows acting on μ(t,z) and ρ(t,z) of the dispersionless cKP hierarchy are presented.

This paper aims at the auto-Bäcklund transformations for the dispersionless mKP hierarchy, dispersionless Miura map between the constrained dispersionless mKP hierarchy and the constrained dispersionless Harry Dym hierarchy and Virasoro symmetries of the constrained dispersionless mKP hierarchy. The additional symmetries of the constrained dispersionless mKP hierarchy are constructed in a manner similar to that of the dispersionless mKP hierarchy. By comparison, a new Laurent series Yl is introduced. The additional flows form a subalgebra of the Virasoro algebra.

Approximation orders of formal Laurent series by β-expansions

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.

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.

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

Pseudodifferential operators of several variables are formal Laurent series in the formal inverses of ∂ 1 , …, ∂ n with ∂ i=d/dx i for 1≤i ≤ n. As in the single variable case, Lax equations can be constructed using such pseudodifferential operators, whose solutions can be provided by Baker functions. We extend the usual notion of tau functions to the case of pseudodifferential operators of several variables so that each Baker function can be expressed in terms of the corresponding tau function.

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.

In the paper, we investigate integrability characteristics for the dispersionless Kadomtsev-Petviashvili hierarchy. These characteristics include symmetries, Hamiltonian structures, and conserved quantities. We give a Lax triad to construct a master symmetry and a hierarchy of non-isospectral dispersionless Kadomtsev-Petviashvili flows. These non-isospectral flows, together with the known isospectral dispersionless Kadomtsev-Petviashvili flows, form a Lie algebra, which is used to derive two sets of symmetries for the isospectral dispersionless Kadomtsev-Petviashvili hierarchy. By means of the master symmetry, symmetries, Noether operator, and conserved covariants, Hamiltonian structures are constructed for both isospectral and non-isospectral dispersionless Kadomtsev-Petviashvili hierarchies. Finally, two sets of conserved quantities and their Lie algebra are derived for the isospectral dispersionless Kadomtsev-Petviashvili 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.

Factorials and stirling numbers in the algebra of formal Laurent series

It is proved that a commutative domain $R$ is $\omega$-Euclidean if and only if the ring of formal Laurent series over $R$ is $\omega$-Euclidean domain. It is also proved that every singular matrice over ring of formal Laurent series $R_{X}$ are products of idempotent matrices if $R$ is $\omega$-Euclidean domain.