The mathematical basis for the Gaussian entanglement is discussed in detail, as well as its implications in the internal space-time structure of relativistic extended particles. It is shown that the Gaussian entanglement shares the same set of mathematical formulas with the harmonic oscillator in the Lorentz-covariant world. It is thus possible to transfer the concept of entanglement to the Lor...

In a wide number of applications in computational science and engineering the solution of linear systems of equations with several right-hand sides given at once is required. Direct methods based on Gaussian elimination are known to be popular in that setting. Nevertheless if the dimension of the problem is very large or if matrix-free algorithms are preferred, preconditioned block Krylov space...

We consider the solution of a linear system Ax = b on a distributed memory machine when the matrix A has full rank and is large, sparse and nonsymmetric. We use our Cartesian nested dissection algorithm to compute a ll-reducingcolumn ordering of the matrix. We develop algorithms that use the associated separator tree to estimate the structure of the factor and to distribute and perform numeric ...

The usual procedure to compute the determinant is the so-called Gaussian elimination. We can view this as the transformation of the matrix into a lower triangular matrix with column operations. These transformations do not change the determinant but in the triangular matrix, the computation of the determinant is more convenient: we must only multiply the diagonal elements to obtain it. (It is a...

Tian and Styan have shown many rank equalities for the sum of two and three idempotent matrices and pointed out that rank equalities for the sum P₁ + ⋯+P k with P₁,…, P k be idempotent (k > 3) are still open. In this paper, by using block Gaussian elimination, we obtained rank equalities for the sum of finitely many idempotent matrices and then solved the open problem mentioned above. Extension...

We develop the Redfield equation for delta-correlated gaussian noise and apply it to the case of two neutrino flavor or spin precession in the presence of a noisy matter density or magnetic field, respectively. The criteria under which physical fluctuations can be well approximated by the delta-correlated gaussian noise for the above cases are examined. Current limits on the possible neutrino m...

One of the oldest problems in algebra is the solution of a system of linear equations over certain domains. The Gaussian elimination algorithm provides an effective solution over fields. The Smith normal form algorithm yields a method over the integers. Here we consider another variation of this theme. Let A be a finite (additive) abelian group, let αi,j ∈ End(A) for 1 ≤ i ≤ n and 1 ≤ j ≤ m and...

Observations where additive noise is present can for many models be grouped into a compound observation matrix, adhering to the same type of model. There are many ways the observations can be stacked, for instance vertically, horizontally, or quadratically. An estimator for the spectrum of the underlying model can be formulated for each stacking scenario in the case of Gaussian noise. We compar...

We prove a central limit theorem for the nite dimensional marginals of the Gibbs distribution of the macroscopicòverlap'-parameters in the Hoppeld model in the case where the number of random`patterns', M, as a function of the system size N satisses lim N"1 M(N)=N = 0, without any assumptions on the speed of convergence. The covariance matrix of the limiting gaussian distributions is diagonal a...

The solution of the fractional hyperbolic partial differential equation is obtained by means of the variational iteration method. Our numerical results are compared with those obtained by the modified Gauss elimination method. Our results reveal that the technique introduced here is very effective, convenient, and quite accurate to one-dimensional fractional hyperbolic partial differential equa...