Two patients developed corneal ectasia after photorefractive keratectomy (PRK). Case 1 had evidence of early keratoconus preoperatively, with manifest refractions of -4.00 +2.50 x 160 (20/20) in the right eye and -7.00 +3.00 x 180 (20/30) in the left eye; thin corneas (472 microm and 441 microm, respectively); and inferior paracentral steepening in the right eye and central steepening in the le...

The matrix functions appear in several applications in engineering and sciences. The computation of these functions almost involved complicated theory. Thus, improving the concept theoretically seems unavoidable to obtain some new relations and algorithms for evaluating these functions. The aim of this paper is proposing some new reciprocal for the function of block anti diagonal matrices. More...

We give a combinatorial formula for the non-symmetric Macdonald polynomials E µ (x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J µ (x; q, t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop and Sahi, that characterizes the non-symmetric Macdonald polynomials.

We give a combinatorial formula for the non-symmetric Macdonald polynomials Eμ(x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials Jμ(x; q, t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop and Sahi, that characterizes the non-symmetric Macdonald polynomials.

We study the L-spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces M = Γ\X with finite volume and arithmetic fundamental group Γ whose universal covering X is a symmetric space of non-compact type. We also show, how the obtained results for locally symmetric spaces can be generalized to manifolds with cusps of rank one.

Supplementary material A. Symmetric tensors A.1. Background Let Rd1⇥···⇥dm be the set of d1⇥ · · ·⇥dm real m-order tensors. In this paper, we focus on cubical tensors, i.e., d1 = · · · = dm = d. We denote the set of m-order cubical tensors by Rd m . We denote the elements of M 2 Rdm by Mj1,...,jm , where j1, . . . , jm 2 [d]. Let = [ 1, . . . , m] be a permutation of {1, . . . ,m}. Given M 2 Rd...

We give a combinatorial formula for the non-symmetric Macdonald polynomials E µ (x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J µ (x; q, t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop, that characterizes the non-symmetric Macdonald polynomials.

We give a combinatorial formula for the non-symmetric Macdonald polynomials E µ (x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J µ (x; q, t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop and Sahi, that characterizes the non-symmetric Macdonald polynomials.

A matrix A = (aij) ∈ Rn×n is said to be symmetric and antipersymmetric matrix if aij = aji = −an−j+1,n−i+1 for all 1 ≤ i, j ≤ n. Peng gave the bisymmetric solutions of the matrix equation A1X1B1+A2X2B2+. . .+AlXlBl = C, where [X1, X2, . . . , Xl] is a real matrices group. Based on this work, an adjusted iterative method is proposed to find the symmetric and antipersymmetric solutions of the abo...

There is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoidIX , that is, a monoid of partial one-to-one selfmaps of a set X . The present paper describes the structure of a categorical dual I Ł X to the symmetric inverse monoid and discusses representations of an inverse semigroup in this dual symme...