Uniform Convergence of Translation Operators

We denote N, R, C the sets of natural, real and complex numbers respectively. Let (λn), n ∈ N be an unbounded sequence of complex numbers. Costakis has proved the following result. There exists an entire function f with the following property: for every x, y ∈ R with 0 , every θ ∈(0,1) and every a ∈ C there is a subsequence of natural numbers (mn), n ∈ N such that, for every compact subset L ⊆C , In the present paper we show that the constant function a cannot be replaced by any non-constant entire function G. This is so even if one demands the convergence in (*) only for a single radius r and a single positive number θ. This result is related with the problem of existence of common universal vectors for an uncountable family of sequences of translation operators.

Introducing the general condition for an operator in curved space to be unitary

We investigate the general condition for an operator to be unitary.This condition is introduced according to the definition of the position operator in curved space.In a particular case,we discuss the concept of translation operator in curved space followed by its relation with an anti-Hermitian generator.Also we introduce a universal formula for adjoint of an arbitrary linear operator.Our procedure in this paper is totally different from others,as we explore a general approach based only on the algebra of the operators.Our approach is only discussed for the translation operators in one-dimensional space and not for general operators.

Harmonic Analysis Associated with the Heckman-Opdam-Jacobi Operator on R^(d+1)

In this paper we consider the Heckman-Opdam-Jacobi operatorΔ_(H J)on R^(d+1).We define the Heckman-Opdam-Jacobi intertwining operator V_(H J),which turns out to be the transmutation operator betweenΔ_(H J)and the LaplacianΔ_(d+1).Next we construct^(t)V_(H J)the dual of this intertwining operator.We exploit these operators to develop a new harmonic analysis corresponding toΔ_(H J).

Entity-related paths modeling for knowledge base completion

Knowledge bases(KBs)are far from complete,necessitating a demand for KB completion.Among various methods,embedding has received increasing attention in recent years.PTransE,an important approach using embedding method in KB completion,considers multiple-step relation paths based on TransE,but ignores the association between entity and their related entities with the same direct relationships.In this paper,we propose an approach called EP-TransE,which considers this kind of association.As a matter of fact,the dissimilarity of these related entities should be taken into consideration and it should not exceed a certain threshold.EPTransE adjusts the embedding vector of an entity by comparing it with its related entities which are connected by the same direct relationship.EPTransE further makes the euclidean distance between them less than a certain threshold.Therefore,the embedding vectors of entities are able to contain rich semantic information,which is valuable for KB completion.In experiments,we evaluated our approach on two tasks,including entity prediction and relation prediction.Experimental results show that our idea of considering the dissimilarity of related entities with the same direct relationships is effective.