Nonnegative Real Numbers Research Articles

A measure of ambiguity (Knightian uncertainty)

Uncertain or ambiguous events cannot be objectively measured by probabilities, i.e. different decision-makers may disagree about their likelihood of occurrence. This paper proposes a new decision-theoretical approach on how to measure ambiguity (Knightian uncertainty) that is analogous to axiomatic risk measurement in finance. A decision-theoretical measure of ambiguity is a function from choice alternatives (acts) to non-negative real numbers. Our proposed measure of ambiguity is derived from a novel assumption that ambiguity of any choice alternative can be decomposed into a left-tail ambiguity (uncertainty in the realization of relatively undesirable outcomes) and a right-tail ambiguity (uncertainty in the realization of relatively desirable outcomes). This decomposability assumption is combined with two standard assumptions: ambiguity sources (events) are independent (separable) from outcomes (consequences) and any elementary increase in uncertainty (increasing a more desirable outcome in a binary act) necessarily increases ambiguity.

Innovative q-rung orthopair fuzzy prioritized aggregation operators based on priority degrees with application to sustainable energy planning: A case study of Gwadar

<abstract><p>Clean energy potential can be used on a large scale in order to achieve cost competitiveness and market effectiveness. This paper offers sufficient information to choose renewable technology for improving the living conditions of the local community while meeting energy requirements by employing the notion of q-rung orthopair fuzzy numbers (q-ROFNs). In real-world situations, a q-ROFN is exceptionally useful for representing ambiguous/vague data. A multi-criteria decision-making (MCDM) is proposed in which the parameters have a prioritization relationship and the idea of a priority degree is employed. The aggregation operators (AOs) are formed by awarding non-negative real numbers known as priority degrees among strict priority levels. Consequently, some prioritized operators with q-ROFNs are proposed named as "q-rung orthopair fuzzy prioritized averaging (q-ROFPA<sub><italic>d</italic></sub>) operator with priority degrees and q-rung orthopair fuzzy prioritized geometric (q-ROFPG<sub><italic>d</italic></sub>) operator with priority degrees". The results of the proposed approach are compared with several other related studies. The comparative analysis results indicate that the proposed approach is valid and accurate which provides feasible results. The characteristics of the existing method are often compared to other current methods, emphasizing the superiority of the presented work over currently used operators. Additionally, the effect of priority degrees is analyzed for information fusion and feasible ranking of objects.</p></abstract>

Circular Distance-Two Labelling of Book Graphs Related to Code Assignment in Computer Wireless Networks

Let d be a positive real number. An L(1,d)-labeling of a graph G is an assignment of nonnegative real numbers to the vertices of G such that the adjacent vertices are assigned two different numbers (labels) whose difference is at least one, and the difference between numbers (labels) for any two distance-two vertices is at least d. The minimum range of labels over all L(1,d)-labelings of a graph G is called the L(1,d)-labeling number of G, denoted by λ_(1,d)(G). The L(1,d)-labeling with d≥1 of graph arose from the code assignment problem of computer wireless network and the L(1,d)-labeling with 0<d≤1 of graph is motivated by channel assignment problem. Let σ and d be two positive real numbers, a circular σ-L(1,d)-labeling of a graph G is a function f: V(G)→[0,σ) satisfying |f(u)-f(v)|σ≥1 if u v∈E(G), and |f(u)-f(v)|σ≥d if u and v are distance two apart, where |f(u)-f(v)|σ=min{|f(u)-f(v)|, σ-|f(u)-f(v)|}. The circular L(1,d)-labeling number of a graph G, denoted by σ_(1,d) (G), is the minimum σ such that there exists a circular σ-L(1,d)-labeling of G. In this paper, the code assignment of 3-D computer wireless network is abstracted as the circular L(1,d)-labeling of book graph, and the authors determined the circular L(1,d)-labeling numbers of book graph for any positive real number d≥2 basing on the properties and constructions of book graphs.

The Best Possible Constants of the Inequalities with Power Exponential Functions

The author in [7] conjectured the following inequality; If a and b are nonnegative real numbers with a + b = 1/2, then the inequality 1/2 ≤ a(2b)k+ b(2a)k ≤ 1 holds for 0 ≤ k ≤ 1. In this paper, we shall prove the conjecture affirmatively and give the upper and lower estimation of the power exponential functions ab + ba for the nonnegative real numbers a and b with a + b = 2. Moreover, we pose some inequalities with power exponential functions.

On a System of k-Difference Equations of Order Three

In this paper, we deal with the global behavior of the positive solutions of the system of k -difference equations u n + 1 1 = α 1 u n − 1 1 / β 1 + α 1 u n − 2 2 r 1 , u n + 1 2 = α 2 u n − 1 2 / β 2 + α 2 u n − 2 3 r 2 , … , u n + 1 k = α k u n − 1 k / β k + α k u n − 2 1 r k , n ∈ ℕ 0 , where the initial conditions u − l i l = 0,1,2 are nonnegative real numbers and the parameters α i , β i , γ i , and r i are positive real numbers for i = 1,2 , … , k , by extending some results in the literature. By the end of the paper, we give three numerical examples to support our theoretical results related to the system with some restrictions on the parameters.

Non-negative differential evolution for particle sizing from ultrasonic attenuation spectroscopy

Ultrasonic attenuation spectroscopy (UAS) is a potentially characterising technique for nano-meter size materials. It is the inversion of the ultrasonic attenuation spectrum to particle size distribution (PSD) by using an optimisation technique. Because the particle size is non-negative number, the inversion has to be optimised in the non-negative real number domain. However, many optimisation techniques have traditionally added the non-negative constrain into a cost function as a result the function becoming complex. In this article, the modification of the differential evolution (DE) technique has been proposed to narrow the searching solution domain bounded in the non-negative real number domain. An absolute function has been added to a mutant vector of the DE technique to guarantee the solution is non-negative real number. This technique is called DE-Nonneg technique. The technique has been demonstrated by characterising size of silica suspension. The optimisation results show that the PSD predictions are comparatively to simulated annealing (SA) technique. Moreover, the performance of DE-Nonneg could be improved via adjusting the parameters (expanding factor, cross over, number of parameters and mutation type) following the guideline addressed here. In addition, it is also possible to predict the PSD without using the structural PSD model. With this advantage, DE-Nonneg could possibly be applied for other non-negative optimisation problems.

Generalização de uma sequência infinita definida por radicais encaixados

Neste trabalho, realizamos um estudo de uma classe de sequências de números reais definidas por radicais encaixados, a qual generaliza sequências bem conhecidas na teoria. Mais precisamente, estaremos interessadas em analisar dois tipos de sequências, dadas pelas seguintes recorrências:para x e y números reais não negativos.

Isomorphisms of Semirings of Continuous Nonnegative Functions and the Lattices of Their Subalgebras

Let $$S$$ be the semifield $$\mathbb{R}_{+}$$ with zero of nonnegative real numbers or the semifield $$\mathbb{P}$$ of positive real numbers. The set of all continuous functions $$f\colon X\to S$$ with pointwise operations of addition and multiplication of functions defined on an arbitrary topological space $$X$$ forms the semiring $$C(X,S).$$ By a subalgebra we mean a nonempty subset $$A$$ of $$C(X,S)$$ such that $$f+g,fg,rf\in A$$ for any $$f,g\in A$$ and any $$r\in S.$$ For arbitrary topological spaces $$X$$ and $$Y,$$ we describe isomorphisms of the semirings $$C(X,S)$$ and $$C(Y,S)$$ and isomorphisms of the lattices of their subalgebras (subalgebras with unity).

Quasilinear elliptic equations with a source reaction term involving the function and its gradient and measure data

We study the equation $$-\text {div}(A(x,\nabla u))=|u|^{q_1-1}u|\nabla u|^{q_2}+\mu $$ where $$A(x,\nabla u)\sim |\nabla u|^{p-2}\nabla u$$ in some suitable sense, $$\mu $$ is a measure and $$q_1$$ , $$q_2$$ are nonnegative real numbers and satisfy $$q_1+q_2>p-1$$ . We give sufficient conditions for existence of solutions expressed in terms of the Wolff potential or the Riesz potentials of the measure. Finally we connect the potential estimates on the measure with Lipchitz estimates with respect to some Bessel or Riesz capacity.

Minimum Variable Connectivity Index of Trees of a Fixed Order

The connectivity index, introduced by the chemist Milan Randić in 1975, is one of the topological indices with many applications. In the first quarter of 1990s, Randić proposed the variable connectivity index by extending the definition of the connectivity index. The variable connectivity index for graph G is defined as ∑vw∈EGdv+γdw+γ−1/2, where γ is a nonnegative real number, EG is the edge set of G, and dt denotes the degree of an arbitrary vertex t in G. Soon after the innovation of the variable connectivity index, its various chemical applications have been reported in different papers. However, to the best of the authors’ knowledge, mathematical properties of the variable connectivity index, for γ&gt;0, have not yet been discussed explicitly in any paper. The main purpose of the present paper is to fill this gap by studying this topological index in a mathematical point of view. More precisely, in this paper, we prove that the star graph has the minimum variable connectivity index among all trees of a fixed order n, where n≥4.

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

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.

A survey on pairwise compatibility graphs

Let T be an edge weighted tree and dmin, dmax be two non-negative real numbers where The pairwise compatibility graph (PCG) of T for dmin, dmax is a graph G such that each vertex of G corresponds to a distinct leaf of T and two vertices are adjacent in G if and only if the weighted distance between their corresponding leaves lies within the interval A graph G is a PCG if there exist an edge weighted tree T and suitable dmin, dmax such that G is a PCG of T. The class of pairwise compatibility graphs was introduced to model evolutionary relationships among a set of species. Since not all graphs are PCGs, researchers become interested in recognizing and characterizing PCGs. In this paper, we review the results regarding PCGs and some of its variants.

An improved oscillation result for advanced differential equations

In this paper we study the advanced differential equations of the form x′(t)−q(t)xσ(t)=0,t≥t0,where q is a function of nonnegative real numbers and σ is a function of positive real numbers such that σ(t)>t, for all t≥t0. Based on a recursive sequence we have constructed, we improve the well known oscillation condition lim supt→∞∫tσ(t)q(s)ds>1 by lim supt→∞∫tσ(t)q(s)ds≥1.An example illustrating our results is also given.

Degree of approximation by means of hexagonal Fourier series

Let $f$ be a continuous function which is periodic with respect to the hexagon lattice, and let $A$ be a lower triangular infinite matrix of nonnegative real numbers with nonincreasing rows. The degree of approximation of the function $f$ by matrix means $T_{n}^{\left( A\right) }\left( f\right) $ of its hexagonal Fourier series is estimated in terms of the modulus of continuity of $f.$

Notes on a General Sequence

Abstract Let {rn}n ∈𝕅 be a strictly increasing sequence of nonnegative real numbers satisfying the asymptotic formula rn ~ αβn , where α, β are real numbers with α &gt; 0 and β &gt; 1. In this note we prove some limits that connect this sequence to the number e. We also establish some asymptotic formulae and limits for the counting function of this sequence. All of the results are applied to some well-known sequences in mathematics.

Existence of Oscillatory Solutions of Linear First order Neutral Delay Difference Equations

This paper is concerned with the oscillatory behaviour of first order neutral delay difference equation of the form, and p(e), q(e) are sequence of non-negative real numbers. Sufficient conditions involving oscillation of all solutions are established. These conditions improve all previous well known results in the literature. Examples illustrating the significance of the results.

Positivity of mixed multiplicities of filtrations

The theory of mixed multiplicities of filtrations by m-primary ideals in a ring is introduced in [Cutkosky, P. Sarkar and H. Srinivasan, Trans. Amer. Math. Soc. 372 (2019) 6183–6211]. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed multiplicities of filtrations must be nonnegative real numbers and give examples to show that they could be zero or even irrational. When R is analytically irreducible, and I ( 1 ) , … , I ( r ) are filtrations of R by m R -primary ideals, we show that all of the mixed multiplicities e R ( I ( 1 ) [ d 1 ] , … , I ( r ) [ d r ] ; R ) are positive if and only if the ordinary multiplicities e R ( I ( i ) ; R ) for 1 ⩽ i ⩽ r are positive. We extend this to modules and prove a simple characterization of when the mixed multiplicities are positive or zero on a finitely generated module.

Joint Uplink/Downlink Discrete Sum Rate Maximization for Full-Duplex Multicell Heterogeneous Networks

Consider multicell heterogeneous networks (HetNets), where each cell consists of a number of small (micro/pico) base stations (SBSs) and the SBSs perform coordinated multipoint (CoMP) transmission/reception to serve a number of downlink/uplink (DL/UL) users over the same frequency band. The SBSs work in full-duplex (FD) mode and the DL/UL users are half-duplex (HD). Due to co-channel transmissions, the DL and the UL are mutually interfered. This motivates us to consider a joint DL/UL beamformer design for maximizing the system sum rate. Different from the conventional sum rate maximization (SRM), where the user's rate is allowed to be any nonnegative real number, herein we focus on the discrete-rate case - each user's rate is constrained to a prescribed discrete-rate set. This discrete-rate constraint is motivated by the fact that practical communication systems are designed for a finite number of transmission rates, owing to the finite number of modulation and coding schemes (MCS). In view of this, our SRM problem is coined as discrete SRM (DSRM). The DSRM problem is essentially a mixed-integer nonlinear program, and NP-hard. To tackle it, two different approaches are proposed. The first approach is built upon the simplex reformulation of the discrete rate and the mirror descent (MD) algorithm. The second approach exploits the correspondence between the rate and minimum mean-square error (MMSE), and employs the block-coordinate descent (BCD) method to find an approximate solution. Simulation results demonstrate the superior rate performance of the proposed designs as compared with the weighted MMSE (WMMSE) design with naive rate quantization.

Global Dynamics of Certain Mix Monotone Difference Equation via Center Manifold Theory and Theory of Monotone Maps

We investigate the global dynamics of the following rational difference equation of second order\begin{equation*}x_{n+1}=\frac{Ax_{n}^{2}+Ex_{n-1}}{x_{n}^{2}+f},\quad n=0,1,\ldots ,\end{equation*}where the parameters $A$ and $E$ are positive real numbers and the initial conditions $x_{-1}$ and $x_{0}$ are arbitrary non-negative real numbers such that $x_{-1}+x_{0}&gt;0$. The transition function associated with the right-hand side of this equation is always increasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric values. The unique feature of this equation is that the second iterate of the map associated with this transition function changes from strongly competitive to strongly cooperative. Our main tool for studying the global dynamics of this equation is the theory of monotone maps while the local stability is determined by using center manifold theory in the case of the nonhyperbolic equilibrium point.

The average Laplacian polynomial of a graph

Let G=(V(G),E(G)) be a graph of order n=|V(G)| and size m=|E(G)|, and let Σ(G)={σ:E(G)→{−1,+1}} be the set of sign functions defined on the edges of G. Then, it is well known that μ(G,x)=12m∑σ∈Σ(G)det(xIn−A(Gσ)), where μ(G,x) is the matching polynomial of G, and A(Gσ) is the adjacency matrix of the signed graph Gσ=(G,σ). Motivated by this result, in this paper, we introduce the average Laplacian polynomial of G, which is defined as ψ¯(G,x)=12m∑σ∈Σ(G)det(xIn−L(Gσ)), where L(Gσ) is the Laplacian matrix of the signed graph Gσ. We find that this polynomial is closely related to the structure of the graph. For example, its coefficients can be expressed naturally in terms of the TU-subgraphs of G, and the multiplicity of 0 as a root is equal to the number of tree components in G. The relations between the average Laplacian polynomial of G and other polynomials, especially, the matching polynomial, are also investigated in this paper. Based on the relations, we prove that the roots of the average Laplacian polynomial of any graph are non-negative real numbers.