Arbitrary Real Numbers Research Articles

Well-posedness and exponential stability for a plate equation with time-varying delay and past history

This paper is concerned with a class of plate equation with past history and time-varying delay in the internal feedback $$\begin{aligned} u_{tt}+\alpha \Delta ^2 u-\int \limits ^t_{-\infty }g(t-s)\Delta ^2 u(s)\mathrm{d}s+\mu _1u_t+\mu _2u_t(t-\tau (t))+f(u)=h(x), \end{aligned}$$ defined in a bounded domain of $${\mathbb {R}}^n$$ $$(n\ge 1)$$ with some suitable initial data and boundary conditions. For arbitrary real numbers $$\mu _1$$ and $$\mu _2$$ , we proved the global well-posedness of the problem. Results on stability of energy are also proved under some restrictions on $$\mu _1$$ , $$\mu _2$$ and $$h(x)=0$$ .

Sampling Streams of Pulses With Unknown Shapes

This paper develops a theory for sampling and perfectly reconstructing streams of short pulses of unknown shapes in the continuous-time domain. The single pulse is modelled as the delayed version of a wavelet sparse signal, which is normally not band-limited. As the delay can be an arbitrary real number, it is difficult to develop an exact sampling result for this type of signals. We manage to achieve the exact reconstruction of the pulses by using only the knowledge of the Fourier transform of the signal at specific frequencies. We further introduce a multichannel acquisition system that uses a new family of compact-support sampling kernels for extracting the Fourier information from the samples. The shape of the kernel is independent of the wavelet basis in which the pulse is sparse, and hence the same acquisition system can be used with pulses that are sparse on different wavelet bases. By exploiting the fact that pulses have short duration and that the sampling kernels have compact support, we finally propose a local and sequential algorithm to reconstruct streaming pulses from the samples.

Generalized spatial Talbot effect based on all-dielectric metasurfaces

The Talbot effect is among the most fundamental wave propagation effect universal across the frequency spectrum and has been used extensively in both spatial and temporal domain for wide variety of applications. In this work, a generalized spatial Talbot effect is presented based on all-dielectric Huygens' metasurface, which when combined with the conventional Fractional Talbot effect, enables the scaling of aperture periodicities by any arbitrary real number. This is achieved by allowing and engineering the phase discontinuity distributions in space on the aperture using phase engineered all-dielectric reflectionless metasurfaces, in conjunction with free-space propagation. Specific implementations using Huygens' metasurfaces are proposed and their operation to achieve such non-integer scalings of the input aperture period is demonstrated using numerical results based on Fourier propagation method.

Diophantine inequalities involving a prime and an almost-prime

We prove that there are infinitely many solutions of|λ0+λ1p+λ2P3|<p−1131, where λ0 is an arbitrary real number and λ1,λ2∈R with λ2≠0 and 0>λ1λ2 not in Q. This improves a result by Harman.

Expression and behavior of the solutions of some rational recursive sequences

We obtain in this paper the expression of the solutions of the following recursive sequences: urn:x-wiley:mma:media:mma3953:mma3953-math-0001 where the initial conditions are arbitrary real numbers. Also, we study the behavior of the solution of these equations. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Qualitative Properties of Difference Equation of Order Six

In this paper we study the qualitative properties and the periodic nature of the solutions of the difference equation x n + 1 = α x n - 2 + β x n - 2 2 γ x n - 2 + δ x n - 5 , n = 0 , 1 , . . . , where the initial conditions x - 5 , x - 4 , x - 3 , x - 2 , x - 1 , x 0 are arbitrary positive real numbers and α , β , γ , δ are positive constants. In addition, we derive the form of the solutions of some special cases of this equation.

Integral expression for a topological charge in the Faddeev–Niemi nonlinear sigma model

We have introduced Faddeev–Niemi type variables for static SU(3) Yang–Mills theory. The variables suggest that a nonlinear sigma model whose sigma fields take values in SU(3)/(U(1) × U(1)) and SU(3)/(SU(2) × U(1)) may be relevant to infrared limit of the theory. Shabanov showed that the energy functional of the nonlinear sigma model is bounded from below by certain functional. However, Shabanov’s functional is not homotopy invariant, and its value can be an arbitrary real number—therefore it is not a topological charge. Since the third homotopy group of SU(3)/(U(1) × U(1)) is isomorphic to the group of integer numbers, there is a non-trivial topological charge (given by the isomorphism). We apply Novikov’s procedure to obtain integral expression for this charge. The resulting formula is analogous to the Whitehead’s realization of the Hopf invariant.

On the Nonlinear Difference Equation

In this paper, we investigate some qualitative behavior of the solutions of the difference equation where the coefficients a, b and ci are positive real numbers, and where the initial conditions are arbitrary positive real numbers.

The general connectivity indices of fluoranthene-type benzenoid systems

The general connectivity index Rα(G) of a graph G=(V,E) is defined as Rα(G)=∑(u,v)∈E(G)(dudv)α, where (u, v) is an edge of G, du and dv denote the degrees of the vertices u and v, respectively, and α ≠ 0 is an arbitrary real number. In this paper, we give the expressions for computing the general connectivity indices of fluoranthene-type benzenoid systems, and we determine the extremal values of Rα(G) in f-benzenoid systems with h hexagons for some real number α. Especially, the extremal values of Rα(G) in cata-catacondensed fluoranthene-type benzenoid systems with h hexagons were studied.

The form of solutions and periodicity for some systems of third‐order rational difference equations

In this paper, we deal with the system that has solutions and the periodicity character of the following systems of rational difference equations with order three urn:x-wiley:mma:media:mma3547:mma3547-math-0001 with initial conditions x−2,x−1,x0,y−2,y−1, and y0 that are arbitrary nonzero real numbers. Some numerical examples will be given to illustrate our results. Copyright © 2015 John Wiley &amp; Sons, Ltd.

On the dynamics of the nonlinear rational difference equation xn+1=Axn+Bxn−k+Cxn−l+bxn−k/dxn−k−exn−l

Abstract In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation x n + 1 = Ax n + Bx n - k + Cx n - l + bx n - k dx n - k - ex n - l , n = 0 , 1 , 2 , … where the coefficients A , B , C , b , d , e ∈ ( 0 , ∞ ) , while k and l are positive integers. The initial conditions x - l , … , x - k , … , x - 1 , x 0 are arbitrary positive real numbers such that k l . Some numerical examples will be given to illustrate our results.

L2+ε -Esimates on Exponential Decay of Correlations in Equilibrium States of Classical Continuous Systems of Point Particles

We present and prove L 2+e -estimates on exponential decay of correlations in equilibrium states of classical continuous systems of point particles interacting via an exponentially decaying pair potential of interaction, where e is arbitrary small and positive real number. The obtained estimates exhibit not only the explicit dependence on the distance between the areas of the equilibrium classical systems between which the correlations are estimated but also on the volume of these areas, which can be used in the future for the investigation of the corresponding non-equilibrium and dynamic systems.

On the Dynamics of the Solutions of the Rational Recursive Sequences

In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation xn+1 = Axn +Bxn−k + Cxn−l +Dxn−σ + bxnxn−kxn−l dxn−k − exn−l , n = 0, 1, 2, ..... where the coefficients A,B,C,D, b, d, e ∈ (0,∞), while k, l and σ are positive integers. The initial conditions x−σ,..., x−l,..., x−k, ..., x−1, x0 are arbitrary positive real numbers such that k < l < σ. Some numerical examples will be given to illustrate our results.

Clusters and various versions of wiener-type invariants

The Wiener type invariant W(λ)(G) of a simple connected graph G is defined as the sum of the terms d(u, ν ׀G)λ over all unordered pairs {u,ν} of vertices of G, where d(u, ν ׀G)λ denotes the distance between the vertices u and v in G and λ is an arbitrary real number. The cluster G1{G2} of a graph G1 and a rooted graph G2 is the graph obtained by taking one copy of G1 and |V (G1)| copies of G2, and by identifying the root vertex of the i-th copy of G2 with the i-th vertex of G1, for i = 1; 2; … ; |V (G1)|. In this paper, we study the behavior of three versions of Wiener type invariant under the cluster product. Results are applied to compute several distance-based topological invariants of bristled and bridge graphs by specializing components in clusters.

Functional stochastic classical mechanics

The time irreversibility problem is the dichotomy of the reversible microscopic dynamics and the irreversible macroscopic physics. This problem was considered by Boltzmann, Poincare, Bogolyubov and many other authors and though some researchers claim that the problem is solved, it deserves a further study. In this paper an attempt is performed of the following solution of the irreversibility problem: a formulation of microscopic dynamics is suggested which is irreversible in time. In this way the contradiction between the reversibility of microscopic dynamics and irreversibility of macroscopic dynamics is avoided since both dynamics in the proposed approach are irreversible. A widely used notion of microscopic state of the system at a given moment of time as a point in the phase space and also a notion of trajectory and microscopic equation of motion does not have an immediate physical meaning since arbitrary real numbers are non observable. In the approach presented in this paper the physical meaning is attributed not to an individual trajectory but only to a bunch of trajectories or to the distribution function on the phase space. The fundamental equation of the microscopic dynamics in the proposed “functional” stochastic classical mechanics is not the Newton equation but the Liouville or Fokker-Planck-Kolmogorov equation for the distribution function of the single particle. Solutions of the Liouville equation have the property of delocalization which accounts for irreversibility. It is shown that the Newton equation in this approach appears as an approximate equation describing the dynamics of the average values of the position and momenta for not too long time intervals. Corrections to the Newton equation are computed.

Efficient and Accurate Parallel Inversion of the Gamma Distribution

A method for parallel inversion of the gamma distribution is described. This is very desirable for random number generation in Monte Carlo simulations where gamma variates are required. Let $\alpha$ be a fixed but arbitrary positive real number. Explicitly, given a list of uniformly distributed random numbers our algorithm applies the quantile function (inverse CDF) of the gamma distribution with shape parameter $\alpha$ to each element. The result is, therefore, a list of random numbers distributed according to the said distribution. The output of our algorithm has accuracy close to a choice of single- or double-precision machine epsilon. Inversion of the gamma distribution is traditionally accomplished using some form of root finding. This is known to be computationally expensive. Our algorithm departs from this paradigm by using an initialization phase to construct, on the fly, a piecewise Chebyshev polynomial approximation to a transformation function, which can be evaluated very quickly during variate generation. The Chebyshev polynomials are high order, for good accuracy, and generated via recurrence relations derived from nonlinear second order ODEs. A novelty of our approach is that the same change of variable is applied to each uniform random number before evaluating the transformation function. This is particularly amenable to implementation on SIMD architectures, whose performance is sensitive to frequently diverging execution flows due to conditional statements (branch divergence). We show the performance of a CUDA GPU implementation of our algorithm (called Quantus) is within an order of magnitude of the time to compute the normal quantile function.

Commonly Shared Foundation of Mathematics, Information Science, Natural Science, Social Science, and Theology

Through a simple thought experiment, this paper shows that there must be a shared foundation of mathematics, information science, natural science, social science, and theology. The thought experiment is to ask a volunteer to write down an arbitrary real number between 0 and 1 with many digits. For example, 0.19823765010367129462…. would be one of such numbers. Then we analyze this experiment result by asking five simple questions: Is the real number a random real? Can the observed real number be produced by a computer? What laws of physics govern the real number creation process? How to predict which number the volunteer choose to write? What is the meaning of the real number creation actions? Without answering these questions, this paper proves that these five questions are fundamental to mathematics, information science, physics, social science, and theology respectively. These five lines of questioning are universally applicable for all human choices. Because these five lines of questions are closely related with each other, we conclude that there must be a commonly-shared logic foundation for mathematics, information science, physics, social science, and theology.

Two-parameter non-commutative Central Limit Theorem

En 1992, Speicher a montré que les mesures de probabilités jouant le rôle des lois gaussiennes dans les différentes théories des probabilités non-commutatives (probabilités fermioniques, probabilités libres à la Voiculescu, probabilités $q$-déformées à la Bożejko et Speicher) apparaissent toutes comme limites d’un Théorème de la limite centrale généralisé. Ceci concerne des suites de variables aléatoires non-commutatives (éléments d’une $\ast$-algèbre munie d’un état) choisies dans un ensemble d’éléments qui commutent ou anti-commutent deux-à-deux, avec les distributions limites respectives déterminées par la valeur moyenne des coefficients de commutation. Dans ce papier, nous dérivons une forme plus générale du Théorème de la limite centrale où les coefficients de commutation deux-à-deux sont des nombres réels arbitraires. Les statistiques gaussiennes classiques dépendent maintenant d’un second paramètre comme résultat du contrôle du premier et du second moment des coefficients de commutation. Une application donne le modèle de matrices aléatoires pour les statistiques $(q,t)$-gaussiennes, pour lesquelles il a été montré récemment qu’elles ont des profondes connexions avec les algèbres d’opérateurs, les fonctions spéciales, les polynômes orthogonaux, la physique mathématique et la théorie des matrices aléatoires.

A New Multidimensional Slow Continued Fraction Algorithm and Stepped Surface

We give a new algorithm of slow continued fraction expansion related to an arbitrary real cubic number field as a 2-dimensional version of the Farey map. Using our algorithm, we can find the generators of dual substitutions (so-called tiling substitutions) for any stepped surface for any cubic direction.

GLOBAL DYNAMICS OF A NON-AUTONOMOUS RATIONAL DIFFERENCE EQUATION

In this paper, we investigate the boundedness character, the periodic character and the global behavior of positive solutions of the difference equation xn+1 = pn + xn xn−1 , n = 0, 1, · · · , where {pn} is a two periodic sequence of nonnegative real numbers and the initial conditions x−1, x0 are arbitrary positive real numbers. AMS Mathematics Subject Classification: 39A10.