Let w be a permutation of {1, 2, …, n}, and let D(w) the Rothe diagram w. The Schubert polynomial ${\mathfrak{S}_w}\left(x \right)$ can realized as dual character flagged Weyl module associated with D(w). This implies following coefficient-wise inequality: $${\rm{Mi}}{{\rm{n}}_w}\left(x \right) \le {\mathfrak{S}_w}\left(x {\rm{Ma}}{{\rm{x}}_w}\left(x \right),$$ where both Minw(x) Maxw(x) are po...

has d(k!) d((k I)!) solutions, where d(n) denotes the number of divisors of n. This follows from {x: S(x) = k} = {x: xlk!, x t (k I)!}. Thus, equation (1) always has at least a solution, if d(k!) > d((k I)!) for k ~ 2. In what follows, we shall prove this inequality, and in fact we will consider the arithmetical functions y, <7, d, w, 0 at factorials. Here y( n) = Euler's arithmetical function,...

The usual concept of shape invariance is discussed and one extension of this concept is suggested. Gedenshtein [1] defined the “shape invariant” potentials by the relationship V+(x; a0)− V−(x; a1) = W (x, a0) +W ′(x, a0)−W (x; a1) +W ′(x; a1) = R(a1) (1) where W (x; a) is the superpotential, a0 and a1 stand for parameters of the supersymmetric partner potentials V+ and V−, R(a) is a constant. T...

We study homogenization by Γ-convergence of functionals of type Z Ω W “ x ε ,∇φ(x) ” dx, where Ω ⊂ RN is a bounded open set, φ ∈ W 1,p(Ω; Rm) and p > 1, when the 1-periodic integrand W : RN ×Mm×N → [0, +∞] is not of p-polynomial growth. Our homogenization technique can be applied when m = N and W has a singular behavior of type W (x, ξ) → +∞ as detξ → 0. However, our technique is not consistent...

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In this paper, his only venture into control theory, Levinson solves the following basic problem of fundamental importance in the theory of linear control systems. Use IR m to denote real m-dimensional Euclidean space, thought of as a space of column vectors. Let P be a compact convex polyhedron in IR m , and let the set of vertices of P be denoted by V : Let a; b] be a given compact subinterva...

An important issue of computer network management is to distribute network resource fairly to its users. The first quantitative fairness score function   1 2 , , , n F x x x  was proposed in 1984 by Jain, Chiu, and Hawe [1] for evaluating network resource sharing fairness. Chen and Zhang [6] proposed another fairness score function   1 2 , , , n G x x x  which better fits the real world s...

The numerical range of an n× n matrix polynomial P (λ) = Amλ m + Am−1λ m−1 + . . . + A1λ + A0 is defined by W (P ) = {λ ∈ C : x∗P (λ)x = 0, x ∈ C, x 6= 0}. For the linear pencil P (λ) = Iλ− A, the range W (P ) coincides with the numerical range of matrix A, F (A) = {x∗Ax : x ∈ C, x∗x = 1}. In this paper, we obtain necessary conditions for the origin to be a boundary point of F (A). As a consequ...

Consider the following state space model (signal and observation model). Y t = H t X t + W t , W t ∼ N (0, R) (1) X t = F t X t−1 + U t , U t ∼ N (0, Q) (2)

Let X k be a smooth proper scheme over a perfect field of characteristic p and let n be a natural number. A fundamental theorem of Barry Mazur relates the Hodge numbers of X k to the action of Frobenius on the crystalline cohomology H cris (X W) of X over the Witt ring W of k, which can be viewed as a linear map 8: F*W H n cris (X W) H n cris (X W). If H n cris (X W) is torsion free, then since...