The smallest and largest elements of L are denoted 0 and 1 respectively, so PL(0) is the empty set and PL(1) is the spectrum of L. As a ranges over L, PL(a) form a basis of closed sets for the so-called Zariski’s topology on Spec(L) which turns Spec(L) into a spectral space, that is a topological space homeomorphic to the spectrum of a ring. The following definitions for an element a and a prim...

Let R = K[y] = K[y1, . . . , yn] be a polynomial ring over an arbitrary field K and let F be a finite set {f1/g1, . . . , fs/gs} of rational functions in K(y), the quotient field of R, where fi (resp. gi) is in R (resp. R \ {0}) for all i. As usual we denote the affine and projective spaces over the field K by As and Ps−1, respectively. Points of the projective space Ps−1 are denoted by [α], wh...

In this paper we will consider hamiltonian properties of so-called Toeplitz graphs, which are defined as follows: Definition 1 Let n,m, a1, . . . , am ∈ N, 0 < a1 < · · · < am < n and V := {0, . . . , n− 1}. Define E := {[i, j] ∈ V 2 : ∃k ∈ {1, . . . , m} : |j − i| = ak}. Then the graph (V,E) with set of vertices V and set of edges E is called an undirected Toeplitz graph with entries (or strip...

Definition. A group G consists of a set G together with a binary operation ∗ for which the following properties are satisfied: • (x ∗ y) ∗ z = x ∗ (y ∗ z) for all elements x, y, and z of G (the Associative Law); • there exists an element e of G (known as the identity element of G) such that e ∗ x = x = x ∗ e, for all elements x of G; • for each element x of G there exists an element x′ of G (kn...

Edo liberty, adapted from class notes by Yoel Shkolnisky We will see that any matrix A ∈ R m×n can be written as A = U ΣV T such that U ∈ R m×m is unitary, V ∈ R n×n is unitary, and Σ ∈ R m×n is a non-negative real diagonal matrix. Σ(i, i), denoted σ i , are unique. If A the singular values are distinct, then the singular vectors are unique up to a multiplication by z ∈ C with |z| = 1. Remark 0...

A ring is a set R with two binary operations usually called addition, denoted as +, and multiplication, denoted as ·, such that (R, +) satisfies the five axioms of closure, associativity, commutativity, identity element (called zero, 0), and inverse element; and (R, ·) satisfies the three axioms of closure, associativity, and identity element (called one, 1). Furthermore, multiplication (·) dis...

For integers n ≥ 1, k ≥ 0, and k ≤ n, the graph Γn has vertices the 2 vectors of F n 2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular Γn is the n-cube, usually denoted by Qn. We examine the binary codes obtained from the adjacency matrices of these graphs when k = 1, 2, 3, following results obtained for the binary codes of the n-cube ...

The parameters of a linear block code over the finite field Fq of length n, dimension k and minimum distance d will be denoted by [n, k, d]q or [n, k, d]. The quotient k/n is called the information rate and denoted by R = k/n and the relative minimum distance d/n is denoted by δ. A sequence of codes (Cm|m ∈ N) with parameters [nm, km, dm] over a fixed finite field Fq is called asymptotically go...

The L1 performance analysis is studied for continuous-time interval positive systems. In this paper, necessary and sufficient conditions for stability and L1 performance are established for positive linear systems through linear inequalities. Introduction In real world, positive systems exist in many branches of technology and science such as industrial engineering [1], ecology [2]. For non-neg...