We shall develop here a galois theory for difference equations. In the PicardVessiot galois theory for differential equations, the basic objects of study are the Picard-Vessiot extension of a field and its associated galois group. Recall that a Picard-Vessiot extension of a differential field k is an extension K generated by a fundamental set of solutions of a linear differential equation and h...

We investigate the coradical filtration of pointed coalgebras. First, we generalize a theorem of Taft and Wilson using techniques developed by Radford in [Rad78] and [Rad82]. We then look at the coradical filtration of duals of inseparable field extensions L∗ upon extension of the base field K, where K ⊆ L is a field extension. We reduce the problem to the case that the field extension is purel...

We show finiteness results on torsion points of commutative algebraic groups over a $p$-adic field $K$ with values in various extensions $L/K$ infinite degree. mainly study the following cases: (1) $L$ is an abelian extension which splitting crystalline character (such as Lubin–Tate extension). (2) certain iterate associated formal groups.

We study the class of separable (real) Banach spaces X which can be embedded into a space C(K) (K compact metric) in only one way up to automorphism. We show that in addition to the known spaces c0 (and all it subspaces) and 1 (and all its weak∗-closed subspaces) the space c0( 1) has this property. We show on the other hand (answering a question of Castillo and Moreno) that p for 1< p <∞ fails ...

Clustering involves the partitioning of n objects into k clusters. Many clustering algorithms use hard-partitioning techniques where each object is assigned to one cluster. In this paper we propose an overlapping algorithm MCOKE which allows objects to belong to one or more clusters. The algorithm is different from fuzzy clustering techniques because objects that overlap are assigned a membersh...

K-Modes is an eminent algorithm for clustering data set with categorical attributes. This algorithm is famous for its simplicity and speed. The KModes is an extension of the K-Means algorithm for categorical data. Since K-Modes is used for categorical data so ‘Simple Matching Dissimilarity’ measure is used instead of Euclidean distance and the ‘Modes’ of clusters are used instead of ‘Means’. Ho...

We present a family of conformal field theories (or candidates for CFTs) that is build on extremal partition functions. Spectra of these theories can be decomposed into the irreducible representations of the Fischer-Griess Monster sporadic group. Interesting periodicities in the coefficients of extremal partition functions are observed and interpreted as a possible extension of Monster moonshin...

There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fitting into natural commutative diagrams and exact sequences involving K-theory...

This thesis is concerned with the definition and the study of properties of homotopic Hopf-Galois extensions in the category Ch 0 k of chain complexes over a field k, equipped with its projective model structure. Given a differential graded k-Hopf algebra H of finite type, we define a homotopic H-Hopf-Galois extension to be a morphism ' : B ! A of augmented H-comodule dg-k-algebras, where B is ...