An operator on formal power series of the form SμS , where μ is an invertible power series, and σ is a series of the form t+（t^{2}） is called a unipotent substitution with pre-function. Such operators, denoted by a pair (μ ，σ ） , form a group. The objective of this contribution is to show that it is possible to define a generalized powers for such operators, as for instance fractional powers σ for every .

Abstract:
A locally finite category is defined as a category in which every arrow admits only finitely many different ways to be factorized by composable arrows. The large algebra of such categories over some fields may be defined, and with it a group of invertible series (under multiplication). For certain particular locally finite categories, a substitution operation, generalizing the usual substitution of formal power series, may be defined, and with it a group of reversible series (invertible under substitution). Moreover, both groups are actually affine groups. In this contribution, we introduce their coordinate Hopf algebras which are both free as commutative algebras. The semidirect product structure obtained from the action of reversible series on invertible series by anti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebras under the form of a smash coproduct. 1. Introduction The set of formal power series in one variable , such as, , where , forms a group under the usual multiplication of series (whenever is a commutative ring with a unit). Moreover, the set of series, such as, , , forms a group under another operation, namely, the substitution. For any , and , the substitution of by is defined as the series (the fact that begins with implies that is summable in the usual topology of series). This actually gives rise to a semidirect product of groups (where is the opposite group of ). Actually, this situation may be generalized in the following way. Let be a category in which any arrow admits only finitely many factorizations by composable arrows. Such a category is referred to as a locally finite category. A locally finite category admits a large algebra, that is, the set of all set-theoretic maps from the arrows of the category to some base (commutative) ring may be multiplied by a Cauchy-kind product inherited from the composition of arrows in the category. Now, the set of all series in this large algebra with a coefficient at each identity arrow in the category forms a group under multiplication. Moreover, given a finite semicategory , roughly speaking a category without identities, we may construct the free category over the underlying graph structure of , which is a locally finite category. According to a universal property, we may define in a unique way an evaluation functor that maps formal nonvoid paths in (nonvoid sequences of composable arrows in ) to the result in of their compositions. This gives rise to an operation of substitution on the large algebra of similar to the substitution of formal power

Abstract:
The famous Poincaré-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds into its associative envelope—its universal enveloping algebra—as a sub-Lie algebra for the usual commutator Lie bracket. However, there is another functorial way—less known—to associate a Lie algebra to an associative algebra and inversely. Any commutative algebra equipped with a derivation , that is, a commutative differential algebra, admits a Wronskian bracket under which it becomes a Lie algebra. Conversely, to any Lie algebra a commutative differential algebra is universally associated, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property. 1. Introduction Any (associative) algebra (and more generally a Lie-admissible algebra), say , admits a derived structure of Lie algebra under the commutator bracket , . This actually describes a forgetful functor, more precisely an algebraic functor, from associative to Lie algebras. This functor admits a left adjoint that enables to associate to any Lie algebra its universal associative envelope. In this way the theory of Lie algebras may be explored through (but not reduced to) that of associative algebras. A Lie algebra which embeds into its universal enveloping algebra is referred to as special. The famous Poincaré-Birkhoff-Witt theorem states that any Lie algebra which is free as a module (and therefore, any Lie algebra over a field) is special. When the Lie algebra is Abelian, then its universal enveloping algebra reduces to the symmetric algebra of its underlying module structure, and thus any commutative Lie algebra is trivially special. However, there is another way to associate an associative algebra to a Lie algebra, and reciprocally, in a functorial way. The idea does not consist anymore to consider noncommutative algebras under commutators but differential commutative algebras together with the so-called Wronskian determinant. A derivation of an algebra is a linear map that satisfies the usual Leibniz rule. An algebra with a distinguished derivation is said to be a differential algebra. For any such pair may be defined a bilinear map by . When the algebra is commutative, then is alternating and satisfies the Jacobi identity so that it

Abstract:
The two operations, deletion and contraction of an edge, on multigraphs directly lead to the Tutte polynomial which satisfies a universal problem. As observed by Brylawski (1972) in terms of order relations, these operations may be interpreted as a particular instance of a general theory which involves universal invariants like the Tutte polynomial and a universal group, called the Tutte-Grothendieck group. In this contribution, Brylawski’s theory is extended in two ways: first of all, the order relation is replaced by a string rewriting system, and secondly, commutativity by partial commutations (that permits a kind of interpolation between noncommutativity and full commutativity). This allows us to clarify the relations between the semigroup subject to rewriting and the Tutte-Grothendieck group: the latter is actually the Grothendieck group completion of the former, up to the free adjunction of a unit (this was not even mentioned by Brylawski), and normal forms may be seen as universal invariants. Moreover we prove that such universal constructions are also possible in case of a nonconvergent rewriting system, outside the scope of Brylawski’s work. 1. Introduction In his paper [1], Tutte took advantage of two natural operations on (finite multi) graphs (actually on isomorphism classes of multigraphs), deletion and contraction of an edge, in order to introduce the ring and a polynomial in two commuting variables , also known by Whitney [2], unique up to isomorphism since solutions of a universal problem. This polynomial, since called the Tutte polynomial, is a graph invariant in at least two different meanings: first of all, it is defined on isomorphism classes, rather than on actual graphs, in such a way that two graphs with distinct Tutte polynomials are not isomorphic (a well-known functorial point of view), and, secondly, it is invariant with respect to a graph decomposition. Indeed, let be a graph, and let be an edge of , which is not a loop (an edge with the same vertex as source and target) nor a bridge (an edge that connects two connected components of a graph). The edge contraction of is the graph obtained by identifying the vertices source and target of , and removing the edge . We write for the graph where the edge is merely removed; this operation is the edge deletion. Let us consider the graph (well-defined as isomorphic classes) which can be interpreted as a decomposition of . Then, the Tutte polynomial is invariant with respect to this decomposition in the sense that . Moreover this decomposition eventually terminates with graphs with

Abstract:
Even in spaces of formal power series is required a topology in order to legitimate some operations, in particular to compute infinite summations. Many topologies can be exploited for different purposes. Combinatorists and algebraists may think to usual order topologies, or the product topology induced by a discrete coefficient field, or some inverse limit topologies. Analysists will take into account the valued field structure of real or complex numbers. As the main result of this paper we prove that the topological dual spaces of formal power series, relative to the class of product topologies with respect to Hausdorff field topologies on the coefficient field, are all the same, namely the space of polynomials. As a consequence, this kind of rigidity forces linear maps, continuous for any (and then for all) of those topologies, to be defined by very particular infinite matrices similar to row-finite matrices.

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The set of natural integers is fundamental for at least two reasons: it is the free induction algebra over the empty set (and at such allows definitions of maps by primitive recursion) and it is the free monoid over a one-element set, the latter structure being a consequence of the former. In this contribution, we study the corresponding structure in the linear setting, i.e., in the category of modules over a commutative ring rather than in the category of sets, namely the free module generated by the integers. It also provides free structures of induction algebra and of monoid (in the category of modules). Moreover we prove that each of its linear endomorphisms admits a unique normal form, explicitly constructed, as a non-commutative formal power series.

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Perfect nonlinear functions from a finite group $G$ to another one $H$ are those functions $f: G \rightarrow H$ such that for all nonzero $\alpha \in G$, the derivative $d_{\alpha}f: x \mapsto f(\alpha x) f(x)^{-1}$ is balanced. In the case where both $G$ and $H$ are Abelian groups, $f: G \rightarrow H$ is perfect nonlinear if and only if $f$ is bent i.e for all nonprincipal character $\chi$ of $H$, the (discrete) Fourier transform of $\chi \circ f$ has a constant magnitude equals to $|G|$. In this paper, using the theory of linear representations, we exhibit similar bentness-like characterizations in the cases where $G$ and/or $H$ are (finite) non Abelian groups. Thus we extend the concept of bent functions to the framework of non Abelian groups.

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It is well-known that degree two finite field extensions can be equipped with a Hermitian-like structure similar to the extension of the complex field over the reals. In this contribution, using this structure, we develop a modular character theory and the appropriate Fourier transform for some particular kind of finite Abelian groups. Moreover we introduce the notion of bent functions for finite field valued functions rather than usual complex-valued functions, and we study several of their properties. In particular we prove that this bentness notion is a consequence of that of Logachev, Salnikov and Yashchenko, introduced in "Bent functions on a finite Abelian group" (1997). In addition this new bentness notion is also generalized to a vectorial setting.

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Due to implementation constraints the XOR operation is widely used in order to combine plaintext and key bit-strings in secret-key block ciphers. This choice directly induces the classical version of the differential attack by the use of XOR-kind differences. While very natural, there are many alternatives to the XOR. Each of them inducing a new form for its corresponding differential attack (using the appropriate notion of difference) and therefore block-ciphers need to use S-boxes that are resistant against these nonstandard differential cryptanalysis. In this contribution we study the functions that offer the best resistance against a differential attack based on a finite field multiplication. We also show that in some particular cases, there are robust permutations which offers the best resistant against both multiplication and exponentiation base differential attacks. We call them doubly perfect nonlinear permutations.

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The purpose of this paper is to present the extended definitions and characterizations of the classical notions of APN and maximum nonlinear Boolean functions to deal with the case of mappings from a finite group K to another one N with the possibility that one or both groups are non-Abelian.