Probabilistic hesitant fuzzy set (PHFS) is a fruitful concept that adds to hesitant fuzzy set (HFS) the term of probability which is able to retain more information than the usual HFS. Here, we demonstrate that the existing definitions of PHFS are not still reasonable, and therefore, we first improve the PHFS definition. By endowing the set and algebraic operations with a new re-definition of P...

We prove additivity of the minimal conditional entropy associated with a quantum channel Φ, represented by a completely positive (CP), trace-preserving map, when the infimum of S(γ12)−S(γ1) is restricted to states of the form (I⊗Φ) ( |ψ〉〈ψ| ) . We show that this follows from multiplicativity of the completely bounded norm of Φ considered as a map from L1 → Lp for Lp spaces defined by the Schatt...

In this paper, the connection between Menger probabilistic norms and H"{o}hle probabilistic norms is discussed. In addition, the correspondence between probabilistic norms and Wu-Fang fuzzy (semi-) norms is established. It is shown that a probabilistic norm (with triangular norm $min$) can generate a Wu-Fang fuzzy semi-norm and conversely, a Wu-Fang fuzzy norm can generate a probabilistic norm.

First we introduce some of the basic notations. For any vector u = (u1, . . . , up) T ∈ R, denote by |u|q the vector `q-norm defined by |u|q = (∑p k=1 |uk| )1/q for q ≥ 1 and write |u|0 = ∑p k=1 I(uk 6= 0). For any set S, denote by S its complement. For a matrix A = (ak`) ∈ Rp×p, we denote by ‖A‖2 the spectral norm, ‖A‖F the Frobenius norm, and ‖A‖1 = ∑p k,`=1 |ak`| the elementwise `1-norm. Rec...

We denote by Bp,T (Rd) the Banach space which consists of continuous paths starting at 0 with finite p-variation norm ‖ ‖p. The 1-variation norm of x is the same as the total variation of x. The p-variation norm defines a weaker topology of the C1-path space with x0 = 0. Actually it is proved that I0,T (x) is a continuous functional of x in the p-variation topology for any 1 ≤ p < 2. However it...

T he properties of large conductance C a2+-dependent K + channels in smooth muscle cells (smc) isolated from norm al and atherosclerotic hum an aorta were studied using the patch-clam p technique. It was shown tha t smc from norm al hum an aorta possess a homogeneous population of norm al C a2+-dependent K + channels. In atherosclerotic aorta two kinetically different types of these channels co...

Abstract. We develop conditions under which a product ∏∞ i=0 Ti of matrices chosen from a possibly infinite set of matrices S = {Tj |j ∈ J} converges. We obtain the following conditions which are sufficient for the convergence of the product: There exists a vector norm such that all matrices in S are nonexpansive with respect to this norm and there exists a subsequence {ik}k=0 of the sequence o...

this paper uses the l1−norm and the concept of the non-dominated vector, topropose a method to find a well-dispersed subset of non-dominated (wdsnd) vectorsof a multi-objective mixed integer linear programming (momilp) problem.the proposed method generalizes the proposed approach by tohidi and razavyan[tohidi g., s. razavyan (2014), determining a well-dispersed subset of non-dominatedvectors of...

We study affine invariant 2D triangulation methods. That is, methods that produce the same for a point set S any (unknown) transformation of S. Our work is based on method by Nielson (1993) uses inverse covariance matrix to define an norm, denoted AS, and triangulation, DTAS[S]. revisit AS-norm from geometric perspective, show DTAS[S] can be seen as standard Delaunay transformed prove it retain...

This paper uses the L1−norm and the concept of the non-dominated vector, topropose a method to find a well-dispersed subset of non-dominated (WDSND) vectorsof a multi-objective mixed integer linear programming (MOMILP) problem.The proposed method generalizes the proposed approach by Tohidi and Razavyan[Tohidi G., S. Razavyan (2014), determining a well-dispersed subset of non-dominatedvectors of...