The main purpose of this article is to extend some of the ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. Given a generic component Ψ of the moment map, which is a Morse function, we define a canonical class αp in the equivariant cohomology of the manifold M for each fixed point p ∈ M. When they ex...

A full-rank IC x n matrix G ( D ) over the rational functions F ( D ) generates a rate R = k / n convolutional code C. G ( D ) is minimal if it can be realized with as few memory elements as any encoder for C, and G ( D ) is canonical if it has a minimal realization in controller canonical form. We show that G ( D ) is minimal if and only if for all rational input sequences p1 ( D ) , the span ...

In this paper, we study Markov chains with innnite state block-structured transition matrix with repeated rows and rational structure. We provide an algorithmic approach to nd the stationary probability distribution based on the concepts of linear systems theory. These chains include well-known structured Markov chains such as canonical and non-canonical M/G/1 and G/M/1. We provide a truncation...

In this article, we prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps ft : P(C)→ P(C), parameterized by t in a quasi-projective complex variety. We use this to prove one implication in the if-and-only-if statement of [BD2, Conjecture 1.10] on unlikely intersections in the moduli space of rational maps; we present the co...

Canonical forms of Markovian distributions and processes provide an efficient way of describing these structures by eliminating the redundancy of general representations. Canonical forms of order 2 stationary Markov arrival processes (MAPs) have already been established for both continuous and discrete time. In this paper we present canonical form of continuous time non-stationary MAPs of order...

is the geometric multiplicity of λk which is also the number of Jordan blocks corresponding to λk . • The orders of the Jordan Blocks of λk must sum to the algebraic multiplicity of λk . • The number of Jordan blocks corresponding to an eigenvalue λk is its geometric multiplicity. • The matrix A is diagonalizable if and only if, for any eigenvalue λ of A , its geometric and algebraic multiplici...

Definition 2.1. Suppose p1, . . . , pk are polynomials in F[x] which are not all 0. Set I = 〈p1, . . . , pk〉. Let d denote the monic generator of I. We call d the greatest common divisor of the pi and write d = gcd(p1, . . . , pk). If d = 1, we say that the polynomials p1, . . . , pk is relatively prime. Note that, by definition, there exists polynomials q1, . . . , qk ∈ F[x] such that d = p1q1...

We consider the operation of intersecting with a locally principal Cartier divisor (i.e., a Cartier divisor which is principal on some neighborhood of its support). We describe this operation explicitly on the level of cycles and rational equivalences and as a corollary obtain a formula for rational equivalence between intersections of two locally principal Cartier divisors. Such canonical rati...

Let β1, . . . , βn be linearly independent vectors in a vector space. For all j with 0 ≤ j ≤ n and all vectors α1, . . . , αk, if β1, . . . , βn are in the span of β1, . . . , βj, α1, . . . , αk, then j + k ≥ n. The proof of the claim is by induction on k. For k = 0, the claim is obvious since β1, . . . , βn are linearly independent. Suppose the claim is true for k−1, and suppose that β1, . . ....