In this article, we formalize a matrix of Z-module and its properties. Specially, we formalize a matrix of a linear transformation of Z-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free Z-module V , determinant of its Gramian matrix is constant regardless of selection of its basis. Z-module is necessary for lattice problems...

This paper gives a practical method of extending an n× r matrix P (z), r ≤ n, with Laurent polynomial entries in one complex variable z, to a square matrix also with Laurent polynomial entries. If P (z) has orthonormal columns when z is restricted to the torus T, it can be extended to a paraunitary matrix. If P (z) has rank r for each z ∈ T, it can be extended to a matrix with nonvanishing dete...

We investigate the existence and the nature of the solutions of the matrix equation Ax = b, where A is a Z-matrix and b is a nonnegative vector. When x is required to be nonnegative, then an existence theorem is due to Carlson and Victory and is re-proved in this paper. We apply our results to study nonnegative vectors in the range of Z-matrices.

This paper gives a practical method of extending an nr matrix P (z), r n, with Laurent polynomial entries in one complex variable z, to a square matrix also with Laurent polynomial entries. If P (z) has orthonormal columns when z is restricted to the torus T, it can be extended to a paraunitary matrix. If P (z) has rank r for each z 2 T, it can be extended to a matrix with nonvanishing determin...

The giant graviton in the AdS3×S×T 4 spacetime only exists when its angular momentum is a specific value and, moreover, it could have arbitrary size. In this paper we first apply the transformation of mixing azimuthal and internal coordinate or mixing time and internal coordinate to the 11D M-theory with a stack of M2-branes ⊥ M2-branes, then, through the mechanism of Kaluza-Klein reduction and...

I demonstrate the existence of quasi–realistic heterotic–string models in which all the untwisted Kähler and complex structure moduli are projected out by the generalized GSO projections. I discuss the conditions and characteristics of the models that produce this result. The existence of such models offers a novel perspective on the realization of extra dimensions in string theory. In this vie...

This paper aims to investigate and categorize all inequivalent irreducible Z+-modules of a commutative unit Z+-ring A, equipped with set {1, x, y, xy} satisfying x2=1,y2=1 as Z+-basis by using matrix equations, which was part call for Special Issue about inequalities equations Symmetry. If the rank Z+-module n≤2, we prove that there are finitely many Z+-modules, respectively, one three. However...

We find that the IIA Matrix models defined on the non-compact C/Z6, C /Z2 and C/Z4 orbifolds preserve supersymmetry where the fermions are on-massshell Majorana-Weyl fermions. In these examples supersymmetry is preserved both in the orbifolded space and in the non-orbifolded space at the same time. The Matrix model on C/Z6 orbifold has the same N = 2 supersymmetry as the case of C/Z3 orbifold, ...

Let P be an r×s matrix of Laurent polynomials with symmetry such that P(z)P∗(z) = Ir for all z ∈ C\{0} and the symmetry of P is compatible. The matrix extension problem with symmetry is to find an s × s square matrix Pe of Laurent polynomials with symmetry such that [Ir,0]Pe = P (that is, the submatrix of the first r rows of Pe is the given matrix P), Pe is paraunitary satisfying Pe(z)P∗e(z) = ...