On Orientations, Connectivity and Odd-Vertex-Pairings in Finite Graphs

1960 ◽  
Vol 12 ◽  
pp. 555-567 ◽  
Author(s):  
C. ST. J. A. Nash-Williams
Keyword(s):  
Real Number ◽  
Integer Part ◽  
Finite Graphs ◽  
Unordered Pair ◽  
Disjoint Sets ◽  
Negative Real Number

The integer part of a non-negative real number p will be denoted by [p]. For any integer n, n* will denote the greatest even integer less than or equal to n, that is, n* = n or n — 1 according as n is even or odd respectively.The order of a set A, denoted by |A|, is the number of elements in A. The set whose elements are a1, a2, … , an will be denoted by {a1, a2 … , an. The empty set will be denoted by Λ. A set will be said to include each of its elements. A set separates two elements if it includes one but not both of them.An unoriented graph U consists of two disjoint sets V(U), E(U), the elements of V(U) being called vertices of U and the elements of V(U) being called edges of U, together with a relationship whereby with each edge is associated an unordered pair of distinct vertices which the edge is said to join.

scholarly journals A result for semi-regular continued fractions

1969 ◽  
Vol 10 (1-2) ◽  
pp. 145-154 ◽  
Author(s):  
P. E. Blanksby
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Two Dimensional ◽  
Integer Part ◽  
Image Position

If Φ is a real number with |Φ| ≧ 1, then a semiregular continuet fraction development of Φ is denoted by where the ai are integers such that |ai| ≧ 2. The expansions arise geo-. metrically by considering the sequence of divided cells of two-dimensional grids (see [1]), and are described by the following algorithm: for all n ≧ 0, taking Φ = Φ.0 Hence where in this case the square brackets are used to signify the integer-part function. It follows that each irrational Φ has uncountably many such expansions, none of which has a constantly equal to 2 (or -2) for large n.

Numerical criteria for integral dependence

2011 ◽  
Vol 151 (1) ◽  
pp. 95-102 ◽  
Author(s):  
BERND ULRICH ◽  
JAVID VALIDASHTI
Keyword(s):  
Real Number ◽  
Direct Approach ◽  
Graded Algebras ◽  
Numerical Invariants ◽  
Negative Real Number

AbstractWe study multiplicity based criteria for integral dependence of modules or of standard graded algebras, known as ‘Rees criteria’. Rather than using the known numerical invariants, we achieve this goal with a more direct approach by introducing a multiplicity defined as a limit superior of a sequence of normalized lengths; this multiplicity is a non-negative real number that can be irrational.

LINEAR CENTERS WITH PERTURBATIONS OF DEGREE 2d + 5

2012 ◽  
Vol 22 (01) ◽  
pp. 1250007 ◽  
Author(s):  
WENTAO HUANG ◽  
XINGWU CHEN ◽  
VALERY G. ROMANOVSKI
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Differential Systems ◽  
Isochronous Centers ◽  
Negative Real Number

We describe a method for studying the center and isochronicity problems for a class of differential systems in the form of linear center perturbed by homogeneous series of degree 2d + m, where d is a non-negative real number and m is a positive integer. As an application, we classify centers and isochronous centers for a particular case when m = 5.

scholarly journals Decomposition of the n-Dimensional Lattice-Graph into Hamiltonian Lines

1961 ◽  
Vol 12 (3) ◽  
pp. 123-131 ◽  
Author(s):  
C. ST.J. A. Nash-Williams
Keyword(s):  
Euclidean Space ◽  
Unit Length ◽  
Lattice Points ◽  
Dimensional Euclidean Space ◽  
Dimensional Lattice ◽  
Edge Disjoint ◽  
Unordered Pair ◽  
Spanning Subgraph ◽  
Lattice Graph ◽  
Disjoint Sets

A graph G consists, for the purposes of this paper, of two disjoint sets V(G), E(G), whose elements are called vertices and edges respectively of G, together with a relationship whereby with each edge is associated an unordered pair of distinct vertices (called its end-vertices) which the edge is said to join, and whereby no two vertices are joined by more than one edge. An edge γ and vertex ξ are incident if ξ is an end-vertex of γ. A monomorphism [isomorphism] of a graph G into [onto] a graph H is a one-to-one function φ from V(G)∪E(G) into [onto] V(H)∪E(H) such that φ(V(G))⊂V(H), φ(E(G))⊂E(H) and an edge and vertex of G are incident in G if and only if their images under φ are incident in H. G and H are isomorphic (in symbols, G ≅ H) if there exists an isomorphism of G onto H. A subgraph of G is a graph H such that V(H) ⊂ V(G), E(H)⊂E(G) and an edge and vertex of H are incident in H if and only if they are incident in G; if V(H) = V(G), H is a spanning subgraph. A collection of graphs are edge-disjoint if no two of them have an edge in common. A decomposition of G is a set of edge-disjoint subgraphs of G which between them include all the edges and vertices of G. Ln is a graph whose vertices are the lattice points of n-dimensional Euclidean space, two vertices A and B being joined by an edge if and only if AB is of unit length (and therefore necessarily parallel to one of the co-ordinate axes). An endless Hamiltonian line of a graph G is a spanning subgraph of G which is isomorphic to L1. The object of this paper is to prove that Ln is decomposable into n endless Hamiltonian lines, a result previously established (1) for the case where n is a power of 2.

Inclusion Relations for General Riesz Typical Means

1974 ◽  
Vol 17 (1) ◽  
pp. 51-61 ◽  
Author(s):  
A. Jakimovski ◽  
J. Tzimbalario
Keyword(s):  
Real Number ◽  
Partial Sums ◽  
Unbounded Sequence ◽  
Riesz Method ◽  
Inclusion Relations ◽  
Image Position ◽  
Negative Real Number

Let α be a non-negative real number, λ≡{λ,n}(n≥0) a strictly increasing unbounded sequence with λ0≥0 and let be an arbitrary series with partial sums s≡{sn}. Writewhere s(t)=sn for λn<t≤λn+1, s(t)=0 for 0≤t≤λ0. The series ∑ an or the sequence of partial sums s={sn} is summable to ṡ by the Riesz method (R, λ, α) ifas ω→∞.

scholarly journals On non-negative spectrum in Banach algebras

1973 ◽  
Vol 18 (4) ◽  
pp. 295-298 ◽  
Author(s):  
Bertram Yood
Keyword(s):  
Banach Algebra ◽  
Real Number ◽  
Spectral Radius ◽  
Banach Algebras ◽  
Semi Group ◽  
Negative Spectrum ◽  
Complex Banach Algebra ◽  
Negative Real Number

Let A be a complex Banach algebra with an identity 1. In this note we study the subset Λ of A consisting of all g ∈ A such that the spectrum of g, sp(g), contains at least one non-negative real number. Clearly Λ is not, in general, a semi-group with respect to either addition or multiplication. However, Λ is an instance of a subset Q of A with the following properties, where ρ(f) denotes the spectral radius of f (4, p. 30).

scholarly journals On the Eventually Periodic Continued β-Fractions and Their Lévy Constants

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 127
Author(s):  
Qian Xiao ◽  
Chao Ma ◽  
Shuailing Wang
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Golden Ratio ◽  
Irreducible Polynomial ◽  
Negative Real Number ◽  
Lagrange's Theorem

In this paper, we consider continued β-fractions with golden ratio base β. We show that if the continued β-fraction expansion of a non-negative real number is eventually periodic, then it is the root of a quadratic irreducible polynomial with the coefficients in Z[β] and we conjecture the converse is false, which is different from Lagrange’s theorem for the regular continued fractions. We prove that the set of Lévy constants of the points with eventually periodic continued β-fraction expansion is dense in [c, +∞), where c=12logβ+2−5β+12.

scholarly journals ROTARU ALPHA - CONVEX FUNCTIONS

1992 ◽  
Vol 23 (4) ◽  
pp. 355-362
Author(s):  
SUBHAS S. TIHOOSNURMATH ◽  
S. R. SWAMY
Keyword(s):  
Real Number ◽  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
Representation Formula ◽  
Negative Real Number

Let $S^*(a,b)$ denote the class of analytic functions $f$ in the unit disc $E$, with $f(0) =f'(0) - 1 =0$, satisfying the condition $|(zf'(z)/f(z))- a|<b$, $a\in C$, $|a- 1|<b\le Re(a)$, $z\in E$. In this paper the class $S^*(\alpha, a, b)$ of functions $f$ analytic in $E$, with $f(0) = f'(0)- 1 =0$, $f(z)f'(z)/z\neq 0$ for $z$ in $E$ and satisfying in $E$ the condition $|J(\alpha,f)- a|<b$, $a \in C$, $|a-1|<b\le Re(a)$, where $J(\alpha, f) =(1- \alpha)(zf'(z)/f(z)) +\alpha((zf'(z))'/f'(z))$, $\alpha$ a non-negative real number is introduced. It is proved that $S^*(\alpha, a,b)\subset S^*(a,b)$, if $a> (4b/c)|Im(a)|$, $c=(b^2- |a- 1|^2)/b$. Further a representation formula for $f \in S^*(\alpha, a, b)$ and an inequality relating the coefficients of functions in $S^*(\alpha, a, b)$ are obtained.

scholarly journals On the Minimum Variable Connectivity Index of Unicyclic Graphs with a Given Order

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Shamaila Yousaf ◽  
Akhlaq Ahmad Bhatti ◽  
Akbar Ali
Keyword(s):  
Real Number ◽  
Connectivity Index ◽  
Star Graph ◽  
Randić Index ◽  
Unicyclic Graphs ◽  
Randic Index ◽  
Degree Of A Vertex ◽  
Fixed Order ◽  
Negative Real Number

The variable connectivity index, introduced by the chemist Milan Randić in the first quarter of 1990s, for a graph G is defined as ∑vw∈EGdv+γdw+γ−1/2, where γ is a non-negative real number and dw is the degree of a vertex w in G. We call this index as the variable Randić index and denote it by Rvγ. In this paper, we show that the graph created from the star graph of order n by adding an edge has the minimum Rvγ value among all unicyclic graphs of a fixed order n, for every n≥4 and γ≥0.

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