Differential and Difference Algebra

Here we find the complete family of two degree of freedom classical Hamiltonians with invariant plane Γ = {q 2 = p 2 = 0} whose normal variational equation around integral curves in Γ is a generically a Hill-Schrödinger equation with cuartic polynomial potential. In particular, these Hamiltonian form a family of non-integrable Hamiltonians through rational first integrals.

Interpolative Kannan contractions are a refinement of Kannan contraction, which is considered as one of the significant notions in fixed point theory. Gb-metric spaces is considered as a generalized concept of both concepts b-metric and G-metric spaces therefore, the significant fixed and common fixed point results of the contraction based on this concept is generalized resultsfor both concepts. The purpose of this manuscript, is to take advantage to interpolative Kannan contraction together with the notion of Ωb which equipped with Gb-metric spaces and H simulation functions to formulate two new interpolative contractions namely, (H, Ωb)-interpolative contraction for self mapping f and generalized (H, Ωb)-interpolative contraction for pair of self mappings (f1, f2). We discuss new fixed and common fixed point theorems. Moreover, to demonstrate the applicability and novelty of our theorems, we formulate numerical examples and applications to illustrate the importance of fixed point ...

Here we find the complete family of two degree of freedom classical Hamiltonians with invariant plane Γ = {q 2 = p 2 = 0} whose normal variational equation around integral curves in Γ is a generically a Hill-Schrödinger equation with cuartic polynomial potential. In particular, these Hamiltonian form a family of non-integrable Hamiltonians through rational first integrals.

Recently, S. Meljanac proposed a construction of a class of examples of an algebraic structure with properties very close to the Hopf algebroids H over a noncommutative base A of other authors. His examples come along with a subalgebra B of H ⊗ H, here called the balancing subalgebra, which contains the image of the coproduct and such that the intersection of B with the kernel of the projection H ⊗ H → H ⊗ A H is a two-sided ideal in B which is moreover well behaved with respect to the antipode. We propose a set of abstract axioms covering this construction and make a detailed comparison to the Hopf algebroids of Lu. We prove that every scalar extension Hopf algebroid can be cast into this new set of axioms. We present an observation by G. Böhm that the Hopf algebroids constructed from weak Hopf algebras fit into our framework as well. At the end we discuss the change of balancing subalgebra under Drinfeld-Xu procedure of twisting of associative bialgebroids by invertible 2-cocycles.

In this paper we give a mechanism to compute the families of classical hamiltonians of two degrees of freedom with an invariant plane and normal variational equations of Hill-Schrödinger type selected in a suitable way. In particular we deeply study the case of these equations with polynomial or trigonometrical potentials, analyzing their integrability in the Picard-Vessiot sense using Kovacic's algorithm and introducing an algebraic method (algebrization) that transforms equations with transcendental coefficients in equations with rational coefficients without changing the Galoisian structure of the equation. We compute all Galois groups of Hill-Schrödinger type equations with polynomial and trigonometric (Mathieu equation) potentials, obtaining Galoisian obstructions to integrability of hamiltonian systems by means of meromorphic or rational first integrals via Morales-Ramis theory.

Here we find the complete family of two degree of freedom classical Hamiltonians with invariant plane Γ = {q 2 = p 2 = 0} whose normal variational equation around integral curves in Γ is a generically a Hill-Schrödinger equation with cuartic polynomial potential. In particular, these Hamiltonian form a family of non-integrable Hamiltonians through rational first integrals.

Let B → A be a homomorphism of Hopf algebras and let C be an algebra. We consider the induction from B to A of C in two cases: when C is a B-interior algebra and when C is a B-module algebra. Our main results establish the connection between the two inductions. The inspiration comes from finite group representation theory, and some constructions work in even more general contexts.

A fundamental tool of Differential Galois Theory is the assignment of an algebraic group to each finite-dimensional differential module over differential field in such a way that the category of differential modules it generates is equivalent, as a symmetric monoidal category, to the category of representations of the group. Its underlying set is then recognized as the group of differential automorphisms of the Picard-Vessiot field extension of the base field for this differential module. These results can be obtained by means of a Tannaka reconstruction process, applied to the abelian category of finite-dimensional differential modules. In this paper, we explore the possibility of extending this theory when the differential field is replaced by a more general differential ring $A$. In this case, it is reasonable to deal with differential modules which are finitely generated and projective over $A$. A major obstacle is that this category is not abelian, in contrast with the classica...

In this work we compute the families of classical Hamiltonians in two degrees of freedom in which the Normal Variational Equation around an invariant plane falls in Schrödinger type with polynomial or trigonometrical potential. We analyze the integrability of Normal Variational Equation in Liouvillian sense using the Kovacic's algorithm. We compute all Galois groups of Schrödinger type equations with polynomial potential. We also introduce a method of algebrization that transforms equations with transcendental coefficients in equations with rational coefficients without changing essentially the Galoisian structure of the equation. We obtain Galoisian obstructions to existence of a rational first integral of the original Hamiltonian via Morales-Ramis theory.

Here we find the complete family of two degree of freedom classical Hamiltonians with invariant plane Γ = {q 2 = p 2 = 0} whose normal variational equation around integral curves in Γ is a generically a Hill-Schrödinger equation with cuartic polynomial potential. In particular, these Hamiltonian form a family of non-integrable Hamiltonians through rational first integrals.

A fundamental tool of Differential Galois Theory is the assignment of an algebraic group to each finite-dimensional differential module over differential field in such a way that the category of differential modules it generates is equivalent, as a symmetric monoidal category, to the category of representations of the group. Its underlying set is then recognized as the group of differential automorphisms of the Picard-Vessiot field extension of the base field for this differential module. These results can be obtained by means of a Tannaka reconstruction process, applied to the abelian category of finite-dimensional differential modules. In this paper, we explore the possibility of extending this theory when the differential field is replaced by a more general differential ring $A$. In this case, it is reasonable to deal with differential modules which are finitely generated and projective over $A$. A major obstacle is that this category is not abelian, in contrast with the classica...

Let H be a skew field of finite dimension over its center k. We solve the Inverse Galois Problem over the field of fractions H (X) of the ring of polynomial functions over H in the variable X , if k contains an ample field. Résumé. Soit H un corps gauche de dimension finie sur son centre k. Nous résolvons le Problème Inverse de Galois sur le corps des fractions H (X) de l'anneau des fonctions polynomiales en la variable X et à coefficients dans H , si k contient un corps ample.

This paper introduces the notion of a DG-Hopf algebra with Steenrod ∇-i (co)products, the idea being that the cohomology of such an object should be a Hopf algebra with an action of the Steenrod squares. After giving some explanation of the origin of the definition, the author shows that the singular cubical chain complex of a topological monoid satisfies the definition, but notes that the singular chain complex does not.

Linear finite transducers underlie a series of schemes for Public Key Criptography (PKC) proposed in the 90s of the last century. The uninspiring and arid language then used, condemned these works to oblivion. Although some of these schemes were after shown to be insecure, the promise of a new system of PKC relying in diferent complexity assumptions is still quite exciting. The algorithms there used depend heavilly on the results of invertibility of linear transducers. In this paper we introduce the notion of post-initial linear tranducer, which is an extension of the notion of linear finite tranducer with memory, and for which the previous fundamental results on invertibility hold. Using this notion, we give a necessary and sufficient condition for left invertibility with fixed delay of finite transducers, as well as a new explicit method to obtain a left inverse.

Let R be the associative k-algebra generated by two elements x and y with defining relation yx = 1. A complete description of simple modules over R is obtained by using the results of Irving and Gerritzen. We examine the short exact sequence 0 → U → E → V → 0, where U and V are simple R-modules. It shows that nonsplit extension only occurs when both U and V are one-dimensional, or, under certain condition, U is infinite-dimensional and V is one-dimensional.

In this paper, an alternative proof is presented of the following result on symbolic powers due to Ein, Lazarsfeld and Smith [3] (for the affine case over [Formula: see text]) and to Hochster and Huneke [4] (for the general case). Let A be a regular ring containing a field K. Let [Formula: see text] be a radical ideal of A and let h be the maximum of the heights of its minimal primes. Then for all n, we have an inclusion [Formula: see text], where the first ideal denotes the hnth symbolic power of [Formula: see text]. In prime characteristic, this result admits an easy tight closure proof due to Hochster and Huneke. In this paper, the characteristic zero version is obtained from this by an application of the Lefschetz Principle.

A differentially recursive sequence over a differential field is a sequence of elements satisfying a homogeneous differential equation with non-constant coefficients (namely, Taylor expansions of elements of the field) in the differential algebra of Hurwitz series. The main aim of this paper is to explore the space of all differentially recursive sequences over a given field with a non-zero differential. We show that these sequences form a two-sided vector space that admits, in a canonical way, a structure of Hopf algebroid over the subfield of constant elements. We prove that it is the direct limit, as a left comodule, of all spaces of formal solutions of linear differential equations and that it satisfies, as Hopf algebroid, an additional universal property. When the differential on the base field is zero, we recover the Hopf algebra structure of linearly recursive sequences.

We show that [Tre05] can be generalised to provide uniform ways of extending certain model complete theories of field expansions in characteristic 0 to model complete theories of differential field expansions with several commuting derivatives. Key to our method is a new notion of large sets we present here which generalises the notion of large fields to subsets of the field. Then, given n ∈ N and a class M of fields expanded to some fixed language whose definable sets reduce to a combination of affine varieties and large sets, we show that there exists a theory U N (which we call a uniform companion) such that given a model complete theory T whose models are in M then T ∪ U N is a model-complete theory of differential fields in N commuting derivatives. Similarly to [Tre05], U N also preserves quantifier elimination, model companionship, model completions, and completeness (upon adding certain constants to the language). We apply our results to classes of fields with finite number of orderings and valuations that satisfy local-global principles in the sense of [Pop96] as well as valued fields enriched with a subfield of the residue field. Note that in all expansions considered here, the additional structure is independent from the derivatives.

Since a new field of research in the theory of valuations has been opened by the notion of symmetric extensions of a valuation on a field K to K(X1,…, Xn), with respect to the indeterminates X1,…, Xn, it makes sense to look for applications of this theory in dealing with arbitrary valuations on rational function fields. This paper investigates the conditions and methods of transforming asymmetric valuations on rational function fields into symmetric ones-procedure that will be called valuation leveling-the motivation being the opportunity of applying the specific results from the theory of symmetric valuations in the general case. 1. Background Given K a field and v a (Krull) valuation on K, we will denote by vK the value group, by Kv the residue field of v on K, by va the value of an element a  K and by av  Kv its residue. We will use the classical additive notation for v, that is assuming the ultrametric triangle law as v(a + b) ≥ inf (a, b). Given v and u two valuations on K, we will say that v is equivalent to u and write v  u, if there exists an isomorphism of ordered groups j : vK  uK such that we get u = jv. Let L/K be an extension of fields. We will call a valuation u on L an extension of v if u(x) = v(x) for all x in K and, in this case, we will canonically identify Kv with a subfield of Lu and vK with a subgroup of uL. If we choose L

Our main result states that any causal discrete-time system can be realized with a reversible or invertible state variable representation. We thus bridge the gap with continuous time, where flows always satisfy this property, and settle an old debate on the relevance of invertibility in nonlinear discrete dynamics. As our result applies to the linear case, we show how the classic approach should slightly be modified and we give several examples of the benefits that can be obtained. Difference algebra is employed for nonlinear systems. It leads to a most elegant characterization of causality, which permits us to demonstrate the reversibility. Module theory suffices in the linear case, which is supplemented by elementary matrix manipulations.

The aim of this paper is to prove that there is an isomorphism of graded A-modules and of chain complexes between Vn(g), the Chevalley-Eilenberg complex for a Lie algebra g and Wn(g), the complex formed from of Dual numbers.
The paper is divided in two parts; in the rst part it is built Wn(g), for it, the steps necessary to reach the desired complex are described. Subsequently, the maps that allow to establish the isomorphism between the two structures are dened. This process will allow to establish an alternative description of the Chevalley-Eilenberg complex.