As a continuation of the work on linear maps between operator algebras which preserve certain subsets of operators with finite rank, or finite corank, here we consider the problem inbetween, that is, we treat the question of preserving operators with infinite rank and infinite corank. Since, as it turns out, in this generality our preservers cannot be written in a nice form what we have got use...

This paper presents new results on the identities satisfied by hypoplactic monoid. We show how to embed monoid of any rank strictly greater than 2 (including infinite rank) into a direct product copies 2. confirms that all monoids or equal satisfy exactly same identities. then give complete characterization those identities, and prove variety generated has finite axiomatic rank, giving basis fo...

By introducing the concept of ǫ-convertibility, we extend Nielsen’s and Vidal’s theorems to the entanglement transformation of infinite-dimensional systems. Using an infinitedimensional version of Vidal’s theorem we derive a new stochastic-LOCC (SLOCC) monotone which can be considered as an extension of the Schmidt rank. We show that states with polynomially-damped Schmidt coefficients belong t...

By introducing the concept of -convertibility, we extend Nielsen’s and Vidal’s theorems to the entanglement transformation of infinite-dimensional systems. Using an infinitedimensional version of Vidal’s theorem we derive a new stochastic-LOCC (SLOCC) monotone which can be considered as an extension of the Schmidt rank. We show that states with polynomially-damped Schmidt coefficients belong to...

Infinite determinantal measures introduced in this note are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of a determinantal point process and a convergent, but not integrable, multiplicative functional. Theorem 2, the main result announced in this note, gives an ex...

In the higher rank paper [17], we reduced Dwork’s conjecture from higher rank case over any smooth affine variety X to the rank one case over the simplest affine space A. In the present paper, we finish our proof by proving the rank one case of Dwork’s conjecture over the affine space A, which is called the key lemma in [17]. The key lemma had already been proved in [16] in the special case whe...

Abstract. The construction of frames for a Hilbert space H can be equated to the decomposition of the frame operator as a sum of positive operators having rank one. This realization provides a different approach to questions regarding frames with particular properties and motivates our results. We find a necessary and sufficient condition under which any positive finite-rank operator B can be e...

Let K be a field of characteristic 6= 2 such that every finite separable extension of K is cyclic. Let A be an abelian variety over K. If K is infinite, then A(K) is Zariski-dense in A. If K is not locally finite, the rank of A over K is infinite.

The tensor rank of some Gabidulin codes small dimension is investigated. In particular, we determine the any metric code equivalent to an 8-dimensional Fq-linear generalized in Fq4×4. This shows that such a never minimum rank. this way, detect first infinite family which are not

Let K be a field of characteristic 6= 2 such that every finite separable extension of K is cyclic. Let A be an abelian variety over K. If K is infinite, then A(K) is Zariski-dense in A. Unless K ⊂ F̄p for some p, the rank of A over K is infinite.