in this paper‎, ‎we study the extremal‎ ‎ranks and inertias of the hermitian matrix expression $$‎ ‎f(x,y)=c_{4}-b_{4}y-(b_{4}y)^{*}-a_{4}xa_{4}^{*},$$ where $c_{4}$ is‎ ‎hermitian‎, ‎$*$ denotes the conjugate transpose‎, ‎$x$ and $y$ satisfy‎ ‎the following consistent system of matrix equations $a_{3}y=c_{3}‎, ‎a_{1}x=c_{1},xb_{1}=d_{1},a_{2}xa_{2}^{*}=c_{2},x=x^{*}.$ as‎ ‎consequences‎, ‎we g...

Let M be an n × n matrix with i.i.d. CN(0, 1) entries (Complex Normal random variables with mean 0 and variance 1), and define A = M +M ∗ √ 2 , where M∗ denotes the conjugate transpose of M . The resulting random matrix is a member of the Gaussian Unitary Ensemble (GUE), and we would like to classify the distribution of its eigenvalues. In a previous section, the measure of the Gaussian Unitary...

Once ReΨ and ImΨ are found from these two coupled di erential equations, the time-dependent probability density |Ψ| for a particle represented by the wave function Ψ can be calculated from |Ψ| = Ψ∗Ψ = (ReΨ) + (ImΨ), where the * denotes the complex conjugate. Our task, then, is to solve Equations (2) and (3) numerically, and this computational approach requires that the partial derivatives be ap...

We present a method to produce examples of non-homeomorphic conjugate complex varieties based on the genus theory of lattices. As an application, we give examples of arithmetic Zariski pairs.

Here λ denotes a partition, λ its conjugate, and the condition “λ even” (or “λ even”) implies that all parts of λ (or all parts of λ) must be even. Furthermore, sλ(x) = sλ(x1, x2, . . . ) is a Schur function of a finite or infinite number of variables. When x = (x1, . . . , xn) the identities (1.1a)–(1.1c) may be viewed as reciprocals of Weyl denominator formulas; the latter expressing the prod...

We show that for each element g of a Garside group, there exists a positive integer m such that gm is conjugate to a periodically geodesic element h, an element with |hn|D = |n| · |h|D for all integers n, where |g|D denotes the shortest word length of g with respect to the set D of simple elements. We also show that there is a finite-time algorithm that computes, given an element of a Garside g...

Throughout this paper, we always make use of the following notation: N = {1, 2, 3, . . .} denotes the set of natural numbers, N0 = {0, 1, 2, 3, . . .} denotes the set of nonnegative integers, Z−0 = {0,−1,−2,−3, . . .} denotes the set of nonpositive integers, Z denotes the set of integers, R denotes the set of real numbers, C denotes the set of complex numbers. The generalized Bernoulli polynomi...

It is well-known, that the ring C[X1, . . . , Xn]n of polynomial invariants of the alternating group An has no finite SAGBI basis with respect to the lexicographical order for any number of variables n ≥ 3. This note proves the existence of a nonsingular matrix δn ∈ GL(n,C) such that the ring of polynomial invariants C[X1, . . . ,Xn] δn n , where An n denotes the conjugate of An with respect to...