scholarly journals Non-negative integral matrices with given spectral radius and controlled dimension

2021 ◽  
pp. 1-24
Author(s):  
MEHDI YAZDI
Keyword(s):  
Spectral Radius ◽  
Positive Real Number ◽  
Algebraic Integer ◽  
Log P ◽  
Irreducible Matrix ◽  
Positive Real ◽  
Shift Of Finite Type ◽  
Primitive Matrix ◽  
Integral Matrices ◽  
Perron Number

Abstract A celebrated theorem of Douglas Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number p, we prove that there is an integral irreducible matrix with spectral radius p, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number p, there is an irreducible shift of finite type with entropy $\log (p)$ defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

Some remarks on the solvability of non-local elliptic problems with the Hardy potential

2014 ◽  
Vol 16 (04) ◽  
pp. 1350046 ◽  
Author(s):  
B. Barrios ◽  
M. Medina ◽  
I. Peral
Keyword(s):  
Real Number ◽  
Sobolev Inequality ◽  
Positive Real Number ◽  
Fractional Power ◽  
Existence Of Solution ◽  
Sharp Constant ◽  
Hardy Potential ◽  
Positive Real ◽  
Existence And Multiplicity ◽  
Non Local

The aim of this paper is to study the solvability of the following problem, [Formula: see text] where (-Δ)s, with s ∈ (0, 1), is a fractional power of the positive operator -Δ, Ω ⊂ ℝN, N > 2s, is a Lipschitz bounded domain such that 0 ∈ Ω, μ is a positive real number, λ < ΛN,s, the sharp constant of the Hardy–Sobolev inequality, 0 < q < 1 and [Formula: see text], with αλ a parameter depending on λ and satisfying [Formula: see text]. We will discuss the existence and multiplicity of solutions depending on the value of p, proving in particular that p(λ, s) is the threshold for the existence of solution to problem (Pμ).

Ultradiversities and their spherical completeness

2020 ◽  
Vol 26 (2) ◽  
pp. 231-240
Author(s):  
Gholamreza H. Mehrabani ◽  
Kourosh Nourouzi
Keyword(s):  
Real Number ◽  
Finite Subset ◽  
Positive Real Number ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Ultrametric Spaces ◽  
Positive Real

AbstractDiversities are a generalization of metric spaces which associate a positive real number to every finite subset of the space. In this paper, we introduce ultradiversities which are themselves simultaneously diversities and a sort of generalization of ultrametric spaces. We also give the notion of spherical completeness for ultradiversities based on the balls defined in such spaces. In particular, with the help of nonexpansive mappings defined between ultradiversities, we show that an ultradiversity is spherically complete if and only if it is injective.

scholarly journals Exact solutions for unsteady axial Couette flow of a fractional Maxwell fluid due to an accelerated shear

2011 ◽  
Vol 16 (2) ◽  
pp. 135-151 ◽  
Author(s):  
Muhammad Athar ◽  
Corina Fetecau ◽  
Muhammad Kamran ◽  
Ahmad Sohail ◽  
Muhammad Imran
Keyword(s):  
Shear Stress ◽  
Positive Real Number ◽  
Fractional Integration ◽  
Maxwell Fluid ◽  
Unsteady Motion ◽  
Positive Real ◽  
Special Cases ◽  
Multiple Integrals ◽  
Similar Solutions ◽  
R Functions

The velocity field and the adequate shear stress corresponding to the flow of a fractional Maxwell fluid (FMF) between two infinite coaxial cylinders, are determined by means of the Laplace and finite Hankel transforms. The motion is produced by the inner cylinder that at time t = 0+ applies a shear stress fta (a ≥ 0) to the fluid. The solutions that have been obtained, presented under series form in terms of the generalized G and R functions, satisfy all imposed initial and boundary conditions. Similar solutions for ordinary Maxwell and Newtonian fluids are obtained as special cases of general solutions. The unsteady solutions corresponding to a = 1, 2, 3, ... can be written as simple or multiple integrals of similar solutions for a = 0 and we extend this for any positive real number a expressing in fractional integration. Furthermore, for a = 0, 1 and 2, the solutions corresponding to Maxwell fluid compared graphically with the solutions obtained in [1–3], earlier by a different technique. For a = 0 and 1 the unsteady motion of a Maxwell fluid, as well as that of a Newtonian fluid ultimately becomes steady and the required time to reach the steady-state is graphically established. Finally a comparison between the motions of FMF and Maxwell fluid is underlined by graphical illustrations.

scholarly journals Common Fixed Point Theorems in a New Fuzzy Metric Space

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Weiquan Zhang ◽  
Dong Qiu ◽  
Zhifeng Li ◽  
Gangqiang Xiong
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Positive Real Number ◽  
Fuzzy Metric Space ◽  
Positive Real ◽  
Fuzzy Metric ◽  
Contraction Condition ◽  
Common Fixed Point Theorems

We generalize the Hausdorff fuzzy metric in the sense of Rodríguez-López and Romaguera, and we introduce a newM∞-fuzzy metric, whereM∞-fuzzy metric can be thought of as the degree of nearness between two fuzzy sets with respect to any positive real number. Moreover, underϕ-contraction condition, in the fuzzy metric space, we give some common fixed point theorems for fuzzy mappings.

Ballots, queues and random graphs

1989 ◽  
Vol 26 (1) ◽  
pp. 103-112 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Real Number ◽  
Random Graph ◽  
Random Graphs ◽  
Limit Distribution ◽  
Positive Real Number ◽  
Positive Real ◽  
Ballot Theorem

This paper demonstrates how a simple ballot theorem leads, through the interjection of a queuing process, to the solution of a problem in the theory of random graphs connected with a study of polymers in chemistry. Let Γn(p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where 0 < p < 1. Denote by ρ n(s) the number of vertices in the union of all those components of Γn(p) which contain at least one vertex of a given set of s vertices. This paper is concerned with the determination of the distribution of ρ n(s) and the limit distribution of ρ n(s) as n → ∞and ρ → 0 in such a way that np → a where a is a positive real number.

AN EXACT FORMULA FOR THE HARMONIC CONTINUED FRACTION

2020 ◽  
pp. 1-11
Author(s):  
MARTIN BUNDER ◽  
PETER NICKOLAS ◽  
JOSEPH TONIEN
Keyword(s):  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Exact Formula ◽  
Positive Real ◽  
Image Position

For a positive real number $t$ , define the harmonic continued fraction $$\begin{eqnarray}\text{HCF}(t)=\biggl[\frac{t}{1},\frac{t}{2},\frac{t}{3},\ldots \biggr].\end{eqnarray}$$ We prove that $$\begin{eqnarray}\text{HCF}(t)=\frac{1}{1-2t(\frac{1}{t+2}-\frac{1}{t+4}+\frac{1}{t+6}-\cdots \,)}.\end{eqnarray}$$

scholarly journals Algebraic Numbers Satisfying Polynomials with Positive Rational Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Vichian Laohakosol ◽  
Suton Tadee
Keyword(s):  
Growth Factor ◽  
Real Number ◽  
Algebraic Number ◽  
Positive Real Number ◽  
Algebraic Numbers ◽  
Positive Real ◽  
Quantitative Version

A theorem of Dubickas, affirming a conjecture of Kuba, states that a nonzero algebraic number is a root of a polynomial f with positive rational coefficients if and only if none of its conjugates is a positive real number. A certain quantitative version of this result, yielding a growth factor for the coefficients of f similar to the condition of the classical Eneström-Kakeya theorem of such polynomial, is derived. The bound for the growth factor so obtained is shown to be sharp for some particular classes of algebraic numbers.

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