Nonnegative Numbers Research Articles

Neimark–Sacker bifurcation of two second-order rational difference equations

We investigate the Neimark–Sacker bifurcation of the equilibrium of two special cases of the difference Equation x n + 1 = β x n x n − 1 + γ x n − 1 2 + δ x n B x n x n − 1 + C x n − 1 2 + D x n where the parameters β, γ, δ, B, C, D are non-negative numbers which satisfy B + C + D>0 and the initial conditions x − 1 and x 0 are arbitrary nonnegative numbers such that B x n x n − 1 + C x n − 1 2 + D x n > 0 for all n ≥ 0 . More precisely, we consider special cases where either γ = D = 0 or β = D = 0 . As we will show both equations exhibit Neimark–Sacker bifurcation, where one of equations ( γ = D = 0 ) probably exhibits Chenciner bifurcation, with two invariant curves, while another Equation ( β = D = 0 ) exhibits simple Neimark–Sacker bifurcation with one invariant curve. We will also obtain some global asymptotic stability result for each equation in the subset of the parametric region of local stability.

Normalized weighted cross correlation for multi-channel image registration

The normalized cross-correlation (NCC) is widely used for image registration due to its simple geometrical interpretation and being feature-agnostic. Here, after reviewing NCC definitions for images with an arbitrary number of dimensions and channels, we propose a generalization in which each pixel value of each channel can be individually weighted using real non-negative numbers. This generalized normalized weighted cross-correlation (NWCC) and its zero-mean equivalent (ZNWCC) can be used, for example, to prioritize pixels based on signal-to-noise ratio. Like a previously defined NWCC with binary weights, the proposed generalizations enable the registration of uniformly, but not necessarily isotropically, sampled images with irregular boundaries and/or sparse sampling. All NCC definitions discussed here are provided with discrete Fourier transform (DFT) formulations for fast computation. Practical aspects of NCC computational implementation are briefly discussed, and a convenient function to calculate the overlap of uniformly, but not necessarily isotropically, sampled images with irregular boundaries and/or sparse sampling is introduced, together with its DFT formulation. Finally, examples illustrate the benefit of the proposed normalized cross-correlation functions.

GeoCoDA: Recognizing and validating structural processes in geochemical data. A workflow on compositional data analysis in lithogeochemistry

Geochemical data are compositional in nature and are subject to the problems typically associated with data that are restricted to the real non-negative number space with constant-sum constraint, that is, the simplex. Geochemistry can be considered a proxy for mineralogy, comprised of atomically ordered structures that define the placement and abundance of elements in the mineral lattice structure. Based on the innovative contributions of John Aitchison, who introduced the logratio transformation into compositional data analysis, this contribution provides a systematic workflow for assessing geochemical data in a simple and efficient way, such that significant geochemical (mineralogical) processes can be recognized and validated. This workflow, called GeoCoDA and presented here in the form of a tutorial, enables the recognition of processes from which models can be constructed based on the associations of elements that reflect mineralogy. Both the original compositional values and their transformation to logratios are considered. These models can reflect rock-forming processes, metamorphism, alteration and ore mineralization. Moreover, machine learning methods, both unsupervised and supervised, applied to an optimized set of subcompositions of the data, provide a systematic, accurate, efficient and defensible approach to geochemical data analysis. The workflow is illustrated on lithogeochemical data from exploration of the Star kimberlite, consisting of a series of eruptions with five recognized phases.

Some Tauberian theorems for the weighted mean method of summability of double sequence

UDC 517.5 Let p = ( p j ) and q = ( q k ) be real sequences of nonnegative numbers with the property that P m = ∑ j = 0 m p j ≠ 0 and Q n = ∑ k = 0 n q k ≠ 0 for all m and n . Let ( P m ) and ( Q n ) be regulary varying positive indices. Assume that ( u m n ) is a double sequence of complex (real) numbers, which is ( N ¯ , p , q ; α , β ) summable with a finite limit, where ( α , β ) = ( 1,1 ) , ( 1,0 ) , or ( 0,1 ) . We present some conditions imposed on the weights under which ( u m n ) converges in Pringsheim's sense. These results generalize and extend the results obtained by authors in [Comput. Math. Appl., <strong>62</strong>, No. 6, 2609–2615 (2011)].

Sufficient Matrices: Properties, Generating and Testing

This paper investigates various aspects of sufficient matrices, one of the most relevant matrix classes introduced in connection with linear complementarity problems. We summarize the most important theoretical results and properties related to sufficient matrices. Based on these, we propose different construction rules that can be used to generate new matrices that belong to this class. A nonnegative number can be assigned to each sufficient matrix, which is called its handicap and works as a measure of sufficiency. The handicap plays a crucial role in proving convergence and complexity results for interior point algorithms for linear complementarity problems. For a particular sufficient matrix, called Csizmadia’s matrix, we give the exact value of the handicap, which is exponential in the size of the matrix. Another important topic that we address is deciding whether a matrix is sufficient. Tseng proved in 2000 that this decision problem is co-NP hard. We investigate three different algorithms for determining the sufficiency of a given matrix: Väliaho’s algorithm, a linear programming-based algorithm, and an algorithm that facilitates nonlinear programming reformulations of the definition of sufficiency. We tested the efficiency of these methods on our recently launched benchmark data set that consists of four different sets of matrices. In this paper, we give the description and most important properties of the benchmark set, which can be used in the future to compare the performance of different interior point algorithms for linear complementarity problems.

Cyclic reduction densities for elliptic curves

For an elliptic curve E defined over a number field K, the heuristic density of the set of primes of K for which E has cyclic reduction is given by an inclusion–exclusion sum delta _{E/K} involving the degrees of the m-division fields K_m of E over K. This density can be proved to be correct under assumption of GRH. For E without complex multiplication (CM), we show that delta _{E/K} is the product of an explicit non-negative rational number reflecting the finite entanglement of the division fields of E and a universal infinite Artin-type product. For E admitting CM over K by a quadratic order mathcal {O}, we show that delta _{E/K} admits a similar ‘factorization’ in which the Artin type product also depends on mathcal {O}. For E admitting CM over overline{K} by an order mathcal {O}not subset K, which occurs for K=textbf{Q}, the entanglement of division fields over K is non-finite. In this case we write delta _{E/K} as the sum of two contributions coming from the primes of K that are split and inert in mathcal {O}. The split contribution can be dealt with by the previous methods, the inert contribution is of a different nature. We determine the ways in which the density can vanish, and provide numerical examples of the different kinds of densities.

Spectrum-Efficient Analog Pulse Index Modulation for High-Volume Wireless Data Telemetry

Spectrum-efficient, high-density, and high-speed data transmission are essential to store and exchange sensitive information through Internet-of-Things (IoT) networks. However, the resource-constrained IoT devices and the limited existing frequency spectrum pose a significant challenge to employing the more efficient technique for the next-generation technology. This paper presents an innovative pulse index modulation (PIM) architecture with a randomization technique for analog pulse-based packet-less high-volume data transmission. The pulse-based compression simplifies the encoding and decoding schemes for short-range wireless telemetry. The pulse randomization technique creates the pulse blinding by reordering the pulse positions and adds an extra level of physical layer security. This paper describes the design, simulation, and prototype development of PIM to validate the feasibility and compatibility of pulse-based data telemetry using the standard architecture. Simulation results show that the proposed PIM increases the data compression ratio by 2.61 times and improves the processing time by 5.56 time compared to the Huffman coding for a 3-bit system. Test results show that the BER is 7.5×10-4 at 10.0 dB SNR, which satisfies the lower bound for data communication and increases the transmission speed k-times by supporting 2k*k bits data using 2k number of pulses, where k is a non-negative integer number. The proposed PIM has the potential to improve data rate with added security to support the next-generation IoT networks.

On a sum involving small arithmetic function and the integral part function

Let f be a “small” arithmetic function in the sense that f=g⁎1 and g(n)≪n−j, where j is a fixed non-negative number. In this paper, we study the sum∑n⩽xf([x/n])[x/n]k as x→∞, where [⋅] denotes the integral part function and k is a fixed non-negative number. Our results generalize the very recent work of Stucky, also combine and generalize the original two types of sums studied by Bordellès-Dai-Heyman-Pan-Shparlinski.

Gevrey regularity of the solutions of inhomogeneous nonlinear partial differential equations

In this article, we are interested in the Gevrey properties of the formal power series solutions in time of some inhomogeneous nonlinear partial differential equations with analytic coefficients at the origin of Cn+1. We systematically examine the cases where the inhomogeneity is s-Gevrey for any s≥0, in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy with respect to a nonnegative rational number sc fully determined by the Newton polygon of a convenient associated linear partial differential equation: for any s≥sc, the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s<sc, the formal solutions are generically sc-Gevrey. In the latter case, we give an explicit example in which the solution is s'-Gevrey for no s'<sc. As a practical illustration, we apply our results to the generalized Burgers-Korteweg-de Vries equation.

A strong Borel–Cantelli lemma for recurrence

Consider a dynamical systems $([0,1], T, \mu )$ which is exponentially mixing for $L^1$ against bounded variation. Given a non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k (x)$ such that $\mu (B(x, r_k(x)) = m_k$. It is prove

On Hankel matrices and the symmetric nonnegative inverse eigenvalue problem

An n-by-n real symmetric matrix is said to be a Hankel matrix if , for each and . The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a list of real numbers is the spectrum of an n-by-n symmetric nonnegative matrix H. In this paper, we search for conditions on the list for the matrix H to be Hankel. For n = 3, sufficient conditions are established. In particular, a necessary and sufficient condition is obtained if Λ is a list of three nonnegative numbers. Also, if , we give conditions for realizability by a Hankel matrix. Finally, we present a special type of list that can serve as the spectrum of a Hankel nonnegative matrix with positive trace. Several of our results are constructive and provide a Hankel realizing matrix.

A new approach to fuzzy partial metric spaces

In this study, we aim to introduce the notion of fuzzy partial metric spaces which is a generalization of crisp partial metric spaces in the fuzzifying view with the distance between ordinary points. For this aim, we first present the concept of fuzzy partial metric spaces by considering the distance as non-negative, upper semi-continuous, normal and convex fuzzy numbers by giving examples. We obtain some useful inequalities under some restrictions in fuzzy partial metric spaces. Then we discuss the relationships with the other metric structures and we point out Banach's fixed point theorem as an application of the proposed properties and relations. Finally, we show that fuzzy partial metric spaces induce some $\alpha$-level topology, Lowen fuzzy topology, and fuzzifying topology.

Weighted inequalities for discrete iterated kernel operators

AbstractWe develop a new method that enables us to solve the open problem of characterizing discrete inequalities for kernel operators involving suprema. More precisely, we establish necessary and sufficient conditions under which there exists a positive constant C such that holds for every sequence of nonnegative numbers where U is a kernel satisfying certain regularity condition, and and are fixed weight sequences. We do the same for the inequality We characterize these inequalities by conditions of both discrete and continuous nature.

Left-Invertibility of Rank-One Perturbations

For each isometry V acting on some Hilbert space and a pair of vectors f and g in the same Hilbert space, we associate a nonnegative number c(V; f, g) defined by $$\begin{aligned} c(V; f,g) = (\Vert f\Vert ^2 - \Vert V^*f\Vert ^2) \Vert g\Vert ^2 + |1 + \langle V^*f , g\rangle |^2. \end{aligned}$$We prove that the rank-one perturbation \(V + f \otimes g\) is left-invertible if and only if $$\begin{aligned} c(V;f,g) \ne 0. \end{aligned}$$We also consider examples of rank-one perturbations of isometries that are shift on some Hilbert space of analytic functions. Here, shift refers to the operator of multiplication by the coordinate function z. Finally, we examine \(D + f \otimes g\), where D is a diagonal operator with nonzero diagonal entries and f and g are vectors with nonzero Fourier coefficients. We prove that \(D + f\otimes g\) is left-invertible if and only if \(D+f\otimes g\) is invertible.

Two types irregular labelling on dodecahedral modified generalization graph

Irregular labelling on graph is a function from component of graph to non-negative natural number such that the weight of all vertices, or edges are distinct. The component of graph is a set of vertices, a set of edges, or a set of both. In this paper we study two types of irregular labelling on dodecahedral modified generalization graph. We determined the total vertex irregularity strength and the modular irregularity strength of dodecahedral modified generalized graph. These results are important because there many classes of graph have the same structure with modified dodecahedral graphs. These results can be used to determine the total vertex irregularity strength and the modular irregularity strength of other graphs that have the similar structure with modified dodecahedral graph.

An Iterative Approach for the Solution of the Constrained OWA Aggregation Problem with Two Comonotone Constraints

In this paper, first, we extend the analytical expression of the optimal solution of the constrained OWA aggregation problem with two comonotone constraints by also including the case when the OWA weights are arbitrary non-negative numbers. Then, we indicate an iterative algorithm that precisely indicates whether a constraint in an auxiliary problem is either biding or strictly redundant. Actually, the biding constraint (or two biding constraints, as this case also may occur) are essential in expressing the solution of the initial constrained OWA aggregation problem.

Polar degree and vanishing cycles

We prove that the polar degree of an arbitrarily singular projective hypersurface can be decomposed as a sum of non-negative numbers which quantify local vanishing cycles of two different types. This yields lower bounds for the polar degree of any singular projective hypersurface.

Dynamics of a second order three species nonlinear difference system with exponents

In this paper, we study the persistence, boundedness, convergence, invariance and global asymptotic behavior of the positive solutions of the second order difference system xn+1 = α1 + ae−xn−1 + byne−yn−1 , (0.1) yn+1 = α2 + ce−yn−1 + dzne−zn−1, zn+1 = α3 + he−zn−1 + jxne−xn−1, n = 0, 1, 2,.... Here xn, yn, zn can be considered as population densities of three species such that the population density of xn, yn, zn depends on the growth of yn, zn, xn respectively with growth rate b, d, j respectively. The positive real numbers α1, α2, α3 are immigration rate of xn, yn, zn respectively, while a, c, h denotes the growth rate of xn, yn, zn respectively, and the initial values x−1, y−1, z−1, x0, y0, z0 are nonnegative numbers.

A Cesàro average for an additive problem with an arbitrary number of prime powers and squares

In this paper we extend and improve all the previous results known in literature about weighted average, with Cesàro weight, of representations of an integer as sum of a positive arbitrary number of prime powers and a non-negative arbitrary number of squares. Our result includes all cases dealt with so far and allows us to obtain the best possible outcome using the chosen technique.

The non-existence of a Hamel-basis and the general solution of Cauchy's functional equation for nonnegative numbers

The non-existence of a Hamel-basis and the general solution of Cauchy's functional equation for nonnegative numbers