Recall that if (K, v) is a valued field, one may consider its completion (K̂, v̂). The extension K̂/K is immediate, i.e. kK = kK̂ and ΓK = ΓK̂ . Definition. • A valued difference field is a structure (K, v, σ), where (K, v) is a valued field and σ is a field endomorphism satisfying σ(x) ∈ OK iff x ∈ OK . It induces a field endomorphism σk on kK and an endomorphism (of ordered abelian groups) σΓ on ΓK .

We prove that a C orientation-preserving circle endomorphism which is Hölder conjugate to a C circle expanding endomorphism is itself expanding.

Infinite words over a finite special confluent rewriting system R are considered and endowed with natural algebraic and topological structures. Their geometric significance is explored in the context of Gromov hyperbolic spaces. Given an endomorphism φ of the monoid generated by R, existence and uniqueness of several types of extensions of φ to infinite words (endomorphism extensions, weak endo...

SECTION 1 introduces a method for constructing a measure by inner approximation, starting from a set function defined on a lattice of sets. SECTION 2 defines a “tightness” property, which ensures that a set function has an extension to a finitely additive measure on a field determined by the class of approximating sets. SECTION 3 defines a “sigma-smoothness” property, which ensures that a tight...

This paper considers the following question: Given an Anosov endomorphism f on T, whether f is topologically conjugate to some hyperbolic toral endomorphism? It is well known that the answer for Anosov diffeomorphisms and expanding endomorphisms is affirmative. However, for the remainder Anosov endomorphisms, a quite different answer is obtained in this paper, i.e., for generic Anosov endomorph...

Hydrogen bonding in methanol clusters has been investigated by using inner-shell photoabsorption spectroscopy and density functional theory (DFT) calculations in the carbon and oxygen K-edge regions. The partial-ion-yield (PIY) curves of H(CH(3)OH)(n)(+) were measured as the soft x-ray absorption spectra of methanol clusters. The first resonance peak in the PIY curves, which is assigned to the ...

in this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $z_{2}$. we also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.

Let $A$ be a Noetherian ring, $I$ be an ideal of $A$ and $sigma$ be a semi-prime operation, different from the identity map on the set of all ideals of $A$. Results of Essan proved that the sets of associated prime ideals of $sigma(I^n)$, which denoted by $Ass(A/sigma(I^n))$, stabilize to $A_{sigma}(I)$. We give some properties of the sets $S^{sigma}_{n}(I)=Ass(A/sigma(I^n))setminus A_{sigma}(I...

let $g$ be a simple graph‎, ‎and $g^{sigma}$‎ ‎be an oriented graph of $g$ with the orientation ‎$sigma$ and skew-adjacency matrix $s(g^{sigma})$‎. ‎the $k-$th skew spectral‎ ‎moment of $g^{sigma}$‎, ‎denoted by‎ ‎$t_k(g^{sigma})$‎, ‎is defined as $sum_{i=1}^{n}( ‎‎‎lambda_{i})^{k}$‎, ‎where $lambda_{1}‎, ‎lambda_{2},cdots‎, ‎lambda_{n}$ are the eigenvalues of $g^{sigma}$‎. ‎suppose‎ ‎$g^{sigma...