Sequence Of Real Numbers Research Articles

On an approach to the analysis of oscillatory solutions of nonlinear delay difference equations

The objective of this article is to establish an oscillation criterion for all solutions of nonlinear difference equation where is a sequence of nonnegative real numbers and is not necessarily monotone sequence. Also, an example is presented to illustrate the significance of the main result.

ECH capacities and the Ruelle invariant

The ECH capacities are a sequence of real numbers associated to any symplectic four-manifold, which are monotone with respect to symplectic embeddings. It is known that for a compact star-shaped domain in \({{\mathbb {R}}}^4\), the ECH capacities asymptotically recover the volume of the domain. We conjecture, with a heuristic argument, that generically the error term in this asymptotic formula converges to a constant determined by a “Ruelle invariant” which measures the average rotation of the Reeb flow on the boundary. Our main result is a proof of this conjecture for a large class of toric domains. As a corollary, we obtain a general obstruction to symplectic embeddings of open toric domains with the same volume. For more general domains in \({{\mathbb {R}}}^4\), we bound the error term with an improvement on the previously known exponent from 2/5 to 1/4.

Fock and Hardy spaces: Clifford Appell case

AbstractIn this paper, we study a specific system of Clifford–Appell polynomials and, in particular, their product. Moreover, we introduce a new family of quaternionic reproducing kernel Hilbert spaces in the framework of Fueter regular functions. The construction is based on a general idea which allows us to obtain various function spaces by specifying a suitable sequence of real numbers. We focus on the Fock and Hardy cases in this setting, and we study the action of the Fueter mapping and its range.

The distribution of spacings of real‐valued lacunary sequences modulo one

Let ( a n ) n = 1 ∞ $(a_{n})_{n=1}^{\infty }$ be a lacunary sequence of positive real numbers. Rudnick and Technau showed that for almost all α ∈ R $\alpha \in \mathbb {R}$ , the pair correlation of ( α a n ) n = 1 ∞ $(\alpha a_{n})_{n=1}^{\infty }$ mod 1 is Poissonian. We show that all higher correlations and hence the nearest-neighbour spacing distribution are Poissonian as well, thereby extending a result of Rudnick and Zaharescu to real-valued sequences.

On Triple Difference Sequences of Real Numbers in Neutrosophic Normed Spaces

The aim of this article is to investigate triple $\Delta $-statistical convergent sequences in a neutrosophic normed space (NNS). Also, we examine the notions of $\Delta $-statistical limit points and $\Delta $-statistical cluster points and prove their important features.

Fast and Longest Rollercoasters

For kge 3, a k-rollercoaster is a sequence of numbers whose every maximal contiguous subsequence, that is increasing or decreasing, has length at least k; 3-rollercoasters are called simply rollercoasters. Given a sequence of distinct real numbers, we are interested in computing its maximum-length (not necessarily contiguous) subsequence that is a k-rollercoaster. Biedl et al. (in: ICALP, volume 107 of LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, pp 18:1–18:15, 2018) have shown that each sequence of n distinct real numbers contains a rollercoaster of length at least lceil n/2rceil for n>7, and that a longest rollercoaster contained in such a sequence can be computed in O(nlog n)-time (or faster, in O(n log log n) time, when the input sequence is a permutation of {1,ldots ,n}). They have also shown that every sequence of ngeqslant (k-1)^2+1 distinct real numbers contains a k-rollercoaster of length at least frac{n}{2(k-1)}-frac{3k}{2}, and gave an O(nklog n)-time (respectively, O(n klog log n)-time) algorithm computing a longest k-rollercoaster in a sequence of length n (respectively, a permutation of {1,ldots ,n}). In this paper, we give an O(nk^2)-time algorithm computing the length of a longest k-rollercoaster contained in a sequence of n distinct real numbers; hence, for constant k, our algorithm computes the length of a longest k-rollercoaster in optimal linear time. The algorithm can be easily adapted to output the respective k-rollercoaster. In particular, this improves the results of Biedl et al. (2018), by showing that a longest rollercoaster can be computed in optimal linear time. We also present an algorithm computing the length of a longest k-rollercoaster in O(n log ^2 n)-time, that is, subquadratic even for large values of kle n. Again, the rollercoaster can be easily retrieved. Finally, we show an Omega (n log k) lower bound for the number of comparisons in any comparison-based algorithm computing the length of a longest k-rollercoaster.

Curlicues generated by circle homeomorphisms

We investigate the curves in the complex plane which are generated by sequences of real numbers being the lifts of the points on the orbit of an orientation preserving circle homeomorphism. Geometrical properties of these curves such as boundedness, superficiality, local discrete radius of curvature are linked with dynamical properties of the circle homeomorphism which generates them: rotation number and its continued fraction expansion, existence of a continuous solution of the corresponding cohomological equation and displacement sequence along the orbit.

Extremal interpolation with the least value of the norm of the second derivative in

In this paper we formulate a general problem of extreme functional interpolation of real-valued functions of one variable (for finite differences, this is the Yanenko–Stechkin–Subbotin problem) in terms of divided differences. The least value of the -th derivative in , , needs to be calculated over the class of functions interpolating any given infinite sequence of real numbers on an arbitrary grid of nodes, infinite in both directions, on the number axis for the class of interpolated sequences for which the sequence of -th order divided differences belongs to . In the present paper this problem is solved in the case when . The indicated value is estimated from above and below using the greatest and the least step of the grid of nodes.

Conditions under which convergence of a sequence or its certain subsequences follows from deferred Cesàro summability

Let (un:n=1,2,...) be a sequence of real or complex numbers. We aim in this paper to determine necessary and/or sufficient conditions under which convergence of a sequence (un) or its certain subsequences follows from summability by deferred Ces?ro means. We also investigate the limiting behavior of deferred moving averages of (un). The conditions in our theorems are one-sided if (un) is a sequence of real numbers, and two-sided if (un) is a sequence of complex numbers. The theory developed in this paper should be useful for developing more interesting and useful results in connection with other sophisticated summability means as well as to extend to other spaces like ordered linear spaces.

Parabolicity of Zero-Twist Tight Flute Surfaces and Uniformization of the Loch Ness Monster

In this paper we study the zero-twist flute surface and we associate to each one of them a sequence of positive real numbers. Conversely, we associate to each sequence of positive real numbers $\mathbf{x}=(x_{n})_{n\in\mathbb{N}_{0}}$ a torsion-free Fuchsian group $\Gamma_{\mathbf{x}}$ such that the convex core of $\mathbb{H}^2/\Gamma_{\mathbf{x}}$ is isometric to a zero-twist tight flute surface $S_{\mathbf{x}}$. Moreover, we prove that the Fuchsian group $\Gamma_{\mathbf{x}}$ is first kind if and only if the series $\sum x_{n}$ diverges. As consequence of the recent work of Basmajian, Hakobian and {S}ari{c}, we obtain that the zero-twist flute surface $S_{\mathbf{x}}$ is of parabolic type if and only $\sum x_{n}$ diverges. In addition, we present an uncountable family of hyperbolic surfaces homeomorphic to the Loch Ness Monster. More precisely, we associate to each sequence $\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}$, where $y_{n}=(a_{n},b_{n},c_{n},d_{n},e_{n})\in \mathbb{R}^5$ and $a_n\leq b_n \leq c_n \leq d_n \leq e_n \leq a_{n+1}$, a Fuchsian group $G_{\mathbf{y}}$ such that $\mathbb{H}^2/G_{\mathbf{y}}$ is homeomorphic to the Loch Ness Monster.

On a certain characterisation of the semigroup of positive natural numbers with multiplication

Abstract In this paper we continue our investigation concerning the concept of a liken. This notion has been defined as a sequence of non-negative real numbers, tending to infinity and closed with respect to addition in ℝ. The most important examples of likens are clearly the set of natural numbers ℕ with addition and the set of positive natural numbers ℕ* with multiplication, represented by the sequence ( ln ( n + 1 ) ) n = 0 ∞ \left( {\ln \left( {n + 1} \right)} \right)_{n = 0}^\infty . The set of all likens can be parameterized by the points of some infinite dimensional, complete metric space. In this space of likens we consider elements up to isomorphism and define properties of likens as such that are isomorphism invariant. The main result of this paper is a theorem characterizing the liken ℕ* of natural numbers with multiplication in the space of all likens.

Characterisations of bounded linear and compact operators on the generalised Hahn space

We establish the characterisations of the classes of bounded linear operators from the generalised Hahn sequence space hd, where d is an unbounded monotone increasing sequence of positive real numbers, into the spaces w0, w and w? of sequences that are strongly summable to zero, strongly summable and strongly bounded by the Ces?ro method of order one. Furthermore, we prove estimates for the Hausdorff measure of noncompactness of bounded linear operators from hd into w, and identities for the Hausdorff measure of noncompactness of bounded linear operators from hd to w0, and use these results to characterise the classes of compact operators from hd to w and w0. Finally, we provide an example for an application of our results.

A phase transition for repeated averages

Let x1,…,xn be a fixed sequence of real numbers. At each stage, pick two indices I and J uniformly at random, and replace xI, xJ by (xI+xJ)/2, (xI+xJ)/2. Clearly, all the coordinates converge to (x1+⋯+xn)/n. We determine the rate of convergence, establishing a sharp “cutoff” transition answering a question of Jean Bourgain.

On (f, ρ)−Statistical convergence and strong (f, ρ)−summability of order α

Abstract The main object of this article is to introduce the concepts of (f, ρ)− statistical convergence of order α and strong (f, ρ)− summability of order α of sequences of real numbers and give some inclusion relations between these spaces.

LACUNARY STATISTICAL HARMONIC SUMMABILITY

In this paper, the concepts of lacunary (H,1) summability, lacunary strongly harmonically summability, lacunary statistical (H,1) summability, lacunary statistical logarithmic convergence of sequences of real numbers are introduced and relations between these concepts are investigated.

AI-statistical limit points and AI-statistical cluster points

In this paper using a non-negative regular summability matrix A and a non trivial admissible ideal I of subsets of N we have introduced the notion of AI-statistical limit point as a generalization of A-statistical limit point of sequences of real numbers. We have also studied some basic properties of the sets of all AI-statistical limit points and AI-statistical cluster points of real sequences including their interrelationship. Also introducing additive property of AI-density zero sets we have established AI-statistical analogue of some completeness theorems of R.

Digital Image Watermarking Technique Using Discrete Wavelet Transform and Discrete Cosine Transform

Digital watermarking has been proposed as a viable solution to the need of copyright protection and authentication of multimedia data in a networked environment, since it makes possible to identify the author, owner, distributor or authorized consumer of a document. In this paper a new watermarking technique to add a code to digital images is presented: the method operates in the frequency domain embedding a pseudo-random sequence of real numbers in a selected set of DCT coefficient. And a new method for digital image watermarking which does not require the original image for watermark detection. The watermark is added in select coefficients with significant image energy in the transform domain in order to ensure non- erasability of the watermark. Advantages of the proposed method include: improved resistance to attacks on the watermark, implicit visual masking utilizing the time-frequency localization property of wavelet transform and a robust definition for the threshold which validates the watermark.. Experimental results demonstrate that this proposed technique is robust to most of the signal processing techniques and geometric distortions.

Analysis on the related factors of China's technological innovation ability using greyness relational degree

PurposeIn order to make grey relational analysis applicable to the interval grey number, this paper discusses the model of grey relational degree of the interval grey number and uses it to analyze the related factors of China's technological innovation ability.Design/methodology/approachFirst, this paper gives the definitions of the lower bound domain, the value domain, the upper bound domain of interval grey number and the generalized measure and the generalized greyness of interval grey number. Then, based on the grey relational theory, this paper proposes the model of greyness relational degree of the interval grey number and analyzes its relationship with the classical grey relational degree. Finally, the model of greyness relational degree is applied to analyze the related factors of China's technological innovation ability.FindingsThe results show that the model of greyness relational degree has strict theoretical basis, convenient calculation and easy programming and can be applied to the grey number sequence, real number sequence and grey number and real number coexisting sequence. The relational order of the four related factors of China's technological innovation ability is research and development (R&D) expenditure, R&D personnel, university student number and public library number, and it is in line with the reality.Practical implicationsThe results show that the sequence values of greyness relational degree have large discreteness, and it is feasible and effective to analyze the related factors of China's technological innovation ability.Originality/valueThe paper succeeds in realizing both the model of greyness relational degree of interval grey number with unvalued information distribution and the order of related factors of China's technological innovation ability.

The Modulus of p-Variation and Its Applications

Let \(\nu \) be a nondecreasing concave sequence of positive real numbers and \(1\le p<\infty \). In this note, we introduce the notion of modulus of p-variation for a function of a real variable, and show that it serves in at least two important problems, namely, the uniform convergence of Fourier series and computation of certain K-functionals. Using this new tool, we first define a Banach space, denoted \(V_p[\nu ]\), that is a natural unification of the Wiener class \(BV_p\) and the Chanturiya class \(V[\nu ]\). Then we prove that \(V_p[\nu ]\) satisfies a Helly-type selection principle which enables us to characterize continuous functions in \(V_p[\nu ]\) in terms of their Fejér means. We also prove that a certain K-functional for the couple \((C,BV_p)\) can be expressed in terms of the modulus of p-variation, where C denotes the space of continuous functions. Next, we obtain equivalent optimal conditions for the uniform convergence of the Fourier series of all functions in each of the classes \(C\cap V_p[\nu ]\) and \(H^\omega \cap V_p[\nu ]\), where \(\omega \) is a modulus of continuity and \(H^\omega \) denotes its associated Lipschitz class. Finally, we establish sharp embeddings into \(V_p[\nu ]\) of various spaces of functions of generalized bounded variation. As a by-product of these latter results, we infer embedding results for certain symmetric sequence spaces.

Large deviation principle for linear processes generated by real stationary sequences under the sub-linear expectation

Let be the linear processes, where is an absolutely summable sequence of real numbers and is a sequence of real stationary independent random variables under the sub-linear expectation space In the framework of sub-linear expectation space, the Beveridge–Nelson decomposition of linear processes and the sub-additivity of capacity are used to establish the large deviation principle for linear processes in the sense of upper probability.