then σ is a singular value of A and u and v are corresponding left and right singular vectors, respectively. (For generality it is assumed that the matrices here are complex, although given these results, the analogs for real matrices are obvious.) If, for a given positive singular value, there are exactly t linearly independent corresponding right singular vectors and t linearly independent co...

Abstract. Let Mn;m be the set of n-by-m matrices with entries inthe field of real numbers. A matrix R in Mn = Mn;n is a generalizedrow substochastic matrix (g-row substochastic, for short) if Re e, where e = (1; 1; : : : ; 1)t. For X; Y 2 Mn;m, X is said to besgut-majorized by Y (denoted by X sgut Y ) if there exists ann-by-n upper triangular g-row substochastic matrix R such thatX = RY . This ...

The application of strain into GeSn alloys can effectively modulate the band structures, thus creating novel electronic and photonic devices. Raman spectroscopy is a powerful tool for characterizing strain; however, lack coefficient makes it difficult accurate determination in alloys. Here, we have investigated Raman-strain function Ge1-xSnxalong ?1 0 0? 1 directions. nanomembranes (NMs) with d...

<abstract><p>Let $ \tau be the substitution 1\to 101 and 0\to 1 on alphabet \{0, 1\} $. The fixed point of obtained starting from 1, denoted by {\bf{s}} $, is a Sturmian sequence. We first give characterization using f $-representation. Then we show that distribution zeros in determinants induces partition integer lattices quadrant. Combining those properties, explicit values Hankel...

Usually we shall just call A an algebra if the field k is clear from the context. The algebra A is associative if multiplication is associative i.e. for all a, b, c ∈ A, (ab)c = a(bc), and unital if there is a multiplicative identity, i.e. an element usually denoted by 1 such that, for all a ∈ A, 1a = a1 = a. Note that, in this case, 1 = 0 ⇐⇒ A = {0}. Otherwise, the map k → A defined by t 7→ t·...

Throughout the paper, Γ = (V,E) denotes a simple, finite and connected graph without loops, with vertex set V and edge set E. Two different vertices, x, y ∈ V , are called adjacent, which will be represented by x ∼ y, if {x, y} ∈ E. Given a vertex subset F ⊂ V , we denote by F c its complementary in V and we call boundary and closure of F , the sets δ(F ) = {x ∈ V : x ∼ y for some y ∈ F} and F̄ ...

1 Parameter estimation in ODE systems with inputs Let ẏ = f(y, [x], p) (1) be an ODE for components y ∈ RN with continuous input function [x] ∈ C0(R,RM ). The system depends on dynamic parameters p ∈ RK and on initial conditions y(t = 0) which are combined to P = (p, y(t = 0)). The solution to eq. (1) is denoted by yP [x] and is a differentiable function of time t. Let us assume that the compon...