Sequence Of Real Numbers Research Articles

Convex combinations of some convergent sequences

We consider the convex combinations , of a pair of sequences of real numbers and such that , converging to , and study the location of the limit inside the intervals , for every or for sufficiently large . We also investigate the same problem for the case of two corresponding sequences converging to . Among other results, we prove some, a bit, unexpected ones. Namely, for each , we determine the exact index at which the sequence changes the monotonicity, and we also determine the type of the monotonicity. A number of interesting remarks are also presented.

A New Notion of Convergence Defined by The Fibonacci Sequence: A Novel Framework and Its Tauberian Conditions

The Fibonacci sequence has broad applications in mathematics, where its inherent patterns and properties are utilized to solve various problems. The sequence often emerges in areas involving growth patterns, series, and recursive relationships. It is known for its connection to the golden ratio, which appears in numerous natural phenomena and mathematical constructs. In this research paper, we introduce new concepts of convergence and summability for sequences of real and complex numbers by using Fibonacci sequences, called Δ-Fibonacci statistical convergence, strong Δ-Fibonacci summability, and Δ-Fibonacci statistical summability. And, these new concepts are supported by several significant theorems, properties, and relations in the study. Furthermore, for this type of convergence, we introduce one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers.

Generalized discrete Grüss and related results with applications

Abstract Grüss inequality is subject of interest for many authors due to its effectiveness in predicting bounds in several quadrature problems. In the present article, we give weighted treatment of the discrete Čebyšev and Grüss type inequalities pertaining two n-tuples of real numbers in which the bounding constants are mobilised with bounding sequences of real numbers. As an application estimations of discrete Ostrowski type inequalities are provided. Finally, by practicing obtained results along with Jensen’s difference, a wide range of estimations are formalised by considering Jensen-Grüss differences.

Hereditary completeness of exponential systems [formula omitted] in their closed span in L2(a,b) and spectral synthesis

Suppose that {λn}n=1∞ is a sequence of distinct positive real numbers satisfying the conditions inf{λn+1−λn}>0, and ∑n=1∞λn−1<∞. We prove that the exponential system {eλnt}n=1∞ is hereditarily complete in the closure of the subspace spanned by {eλnt}n=1∞ in the space L2(a,b). We also give an example of a class of compact non-normal operators defined on this closure which admit spectral synthesis.

Learn2Extend: Extending Sequences by Retaining their Statistical Properties with Mixture Models

This paper addresses the challenge of extending general finite sequences of real numbers within a subinterval of the real line, maintaining their inherent statistical properties by employing machine learning. Our focus lies on preserving the gap distribution and pair correlation function of these point sets. Leveraging advancements in deep learning applied to point processes, this paper explores the use of an auto-regressive Sequence Extension Mixture Model (SEMM) for extending finite sequences, by estimating directly the conditional density, instead of the intensity function. We perform comparative experiments on multiple types of point processes, including Poisson, locally attractive, and locally repelling sequences, and we perform a case study on the prediction of Riemann ζ function zeroes. The results indicate that the proposed mixture model outperforms traditional neural network architectures in sequence extension with the retention of statistical properties. Given this motivation, we showcase the capabilities of a mixture model to extend sequences, maintaining specific statistical properties, i.e. the gap distribution, and pair correlation indicators.

Generalized Cesàro operators on weighted Dirichlet spaces

Given a complex Borel measure η∈M(D), we study the boundedness of the Cesàro-type operator Cη given byCη(f)(z)=∑n=0∞(∫Dwndη(w))(∑k=0nak)zn where f(z)=∑n=0∞anzn, acting on the weighted Dirichlet spaces Dρ consisting in functions f∈H(D) with ∑n=0∞|an|2(n+1)ρn<∞ where (ρn) is a sequence of positive real numbers. We get new estimates on Taylor coefficients of functions in the Dirichlet space D0 and extend the results known for Dα and positive measures ν supported in [0,1) to more general weights and general measures.

Typical Sequence of Real Numbers From the Unit Interval Has All Distribution Functions

This note is devoted to the study of typical properties (in Baire category sense) of sequences of real numbers in [0, 1]. We prove that the subset of sequences that have all distribution functions forms a residual set.

The intrinsic connection between the fractal dimension of a real number sequence and convergence or divergence of the series formed by it

In this paper, we mainly make research on the relationship between convergence or divergence of a class of series and the fractal dimension of the corresponding real number sequence. A new discriminant method named the fractal dimension test to judge convergence or divergence of certain positive series has been proposed. Some examples have also been provided to illustrate this test.

On the dimensional connection between a class of real number sequences and local fractal functions with a single unbounded variation point

In this paper, we investigate the connection between a class of real number sequences and local fractal functions in terms of fractal dimensions. Under certain conditions, we show that the Box dimension of the graph of a local fractal function with a single unbounded variation point is equal to that of its zero points set plus one. Several concrete examples of such functions whose Box dimension can take any numbers belonging to [1,2] have also been given. This work may provide new approaches to the construction of various local fractal functions with the required Box dimension in the future.

ROUGH LACUNARY STATISTICAL ANALYSIS OF ORDER OF η OF TRIPLE DIFFERENCE SEQUENCE SPACES

We generalized the concepts in the probability of rough lacunary statistical analysis of order η by introducing the difference operator Δ_γ^α of fractional order, where α is a proper fraction and γ=(γ_mnk ) is any fixed sequence of nonzero real or complex numbers. We study some properties of this operator involving lacunary sequence θ and arbitrary sequence p=(p_rst ) of strictly positive real numbers and investigate the topological structures of related with triple difference sequence spaces. The main focus of the present paper is to generalize rough lacunary statistical analysis of order η of triple difference sequence spaces and give the relations between statistical analysis and lacunary statistical analysis of order η.

Erdős-Ko-Rado theorem for bounded multisets

Let k,m,n be positive integers with k⩾2. A k-multiset of [n]m is a collection of k integers from the set {1,2,…,n} in which the integers can appear more than once but at most m times. A family of such k-multisets is called an intersecting family if every pair of k-multisets from the family have non-empty intersection. A finite sequence of real numbers (a1,a2,…,an) is said to be unimodal if there is some k∈{1,2,…,n}, such that a1⩽a2⩽…⩽ak−1⩽ak⩾ak+1⩾…⩾an. Given m,n,k, denote Ck,ℓ as the coefficient of xk in the generating function (∑i=1mxi)ℓ, where 1⩽ℓ⩽n. In this paper, we first show that the sequence of (Ck,1,Ck,2,…,Ck,n) is unimodal. Then we use this as a tool to prove that the intersecting family in which every k-multiset contains a fixed element attains the maximum cardinality for n⩾k+⌈k/m⌉. In the special case when m=1 and m=∞, our result gives rise to the famous Erdős-Ko-Rado Theorem and an unbounded multiset version for this problem given by Meagher and Purdy [11], respectively. The main result in this paper can be viewed as a bounded multiset version of the Erdős-Ko-Rado Theorem.

On the Lindelöf hypothesis for general sequences

AbstractIn a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindelöf hypothesis (LH) for general sequences that coincides with the usual LH for the Riemann zeta function in the case of the sequence of positive integers. They made two conjectures: that LH should hold for every admissible sequence of positive integers, and that LH should hold for the “generic” admissible sequence of positive real numbers. In this paper, we give counterexamples to the first conjecture, and show that the second conjecture can be either true or false, depending on the meaning of “generic”: we construct probabilistic processes producing sequences satisfying LH with probability 1, and we construct Baire topological spaces of sequences for which the subspace of sequences satisfying LH is meagre. We also extend the main result of Gonek, Graham, and Lee, stating that the Riemann hypothesis is equivalent to LH for the sequence of prime numbers, to the context of Beurling generalized number systems.

What is the Simplest Linear Ramp?

We discuss conditions under which a deterministic sequence of real numbers, interpreted as the set of eigenvalues of a Hamiltonian, can exhibit features usually associated to random matrix spectra. A key diagnostic is the spectral form factor (SFF) — a linear ramp in the SFF is often viewed as a signature of random matrix behavior. Based on various explicit examples, we observe conditions for linear and power law ramps to arise in deterministic spectra. We note that a very simple spectrum with a linear ramp is En ~ log n. Despite the presence of ramps, these sequences do not exhibit conventional level repulsion, demonstrating that the lore about their concurrence needs refinement. However, when a small noise correction is added to the spectrum, they lead to clear level repulsion as well as the (linear) ramp. We note some remarkable features of logarithmic spectra, apart from their linear ramps: they are closely related to normal modes of black hole stretched horizons, and their partition function with argument s = β + it is the Riemann zeta function ζ(s). An immediate consequence is that the spectral form factor is simply −ζ|(it)|2. Our observation that log spectra have a linear ramp, is closely related to the Lindelöf hypothesis on the growth of the zeta function. With elementary numerics, we check that the slope of a best fit line through |ζ(it)|2 on a log-log plot is indeed 1, to the fourth decimal. We also note that truncating the Riemann zeta function sum at a finite integer N causes the would-be-eternal ramp to end on a plateau.

Prescription of finite Dirichlet eigenvalues and area on surface with boundary

In the present paper, we consider Dirichlet Laplacian on compact surface. We show that for a fixed surface with boundary X, a finite increasing sequence of real numbers 0<a1<a2<⋯<aN and a positive number A, there exists a metric g on X such that for any integer 1≤k≤N, we have λkD(X,g)=ak and Area(X,g)=A.

A note on the general moment problem

In this note we show that given an indeterminate Hamburger moment sequence, it is possible to perturb the first moment in such way that the obtained sequence remains an indeterminate Hamburger moment sequence. As a consequence we prove that every sequence of real numbers is a moment sequence for a signed discrete measure supported in \(\mathbb{R}_+\).

On the complete moment convergence of moving average processes generated by negatively dependent random variables under sub-linear expectations

&lt;abstract&gt;&lt;p&gt;The moving average processes $ X_k = \sum_{i = -\infty}^{\infty}a_{i+k}Y_{i} $ are studied, where $ \{Y_i, -\infty &amp;lt; i &amp;lt; \infty\} $ is a double infinite sequence of negatively dependent random variables under sub-linear expectations, and $ \{a_i, -\infty &amp;lt; i &amp;lt; \infty\} $ is an absolutely summable sequence of real numbers. We establish the complete moment convergence of a moving average process under proper conditions, extending the corresponding results in classic probability space to those in sub-linear expectation space.&lt;/p&gt;&lt;/abstract&gt;

On Gradual Borel Summability Method of Rough Convergence of Triple Sequences of Beta Stancu Operators

We define the concept of rough limit set of a triple sequence space of beta Stancu operators of Borel summability of gradual real numbers and obtain the relation between the set of rough limit and the extreme limit points of a triple sequence space of beta Stancu operators of Borel summability method of gradual real numbers. Finally, we investigate some properties of the rough limit set of beta Stancu operators under which Borel summable sequence of gradual real numbers are convergent. Also, we give the results for Borel summability method of series of gradual real numbers.

Can a single migrant per generation rescue a dying population?

We introduce a population model to test the hypothesis that even a single migrant per generation may rescue a dying population. Let (ck:k∈N) be a sequence of real numbers in (0,1). Let Xn be a size of the population at time n≥0. Then, Xn+1=Xn−Yn+1+1, where the conditional distribution of Yn+1 given Xn=k is a binomial random variable with parameters (k,c(k)). We assume that limk→∞⁡kc(k)=ρ exists. If ρ<1 the process is transient with speed 1−ρ. So for our model a single migrant per generation may rescue a dying population! If ρ>1 the process is positive recurrent. In the critical case ρ=1 the process is recurrent or transient according to how kc(k) converges to 1. When ρ=0 and under some regularity conditions, the support of the increments is eventually finite.

Complete convergence for moving average process generated by extended negatively dependent random variables under sub-linear expectations

. In this article, the complete convergence for the partial sum of the moving average process { X n = ∑ i = − ∞ ∞ a i Y i + n , n ≥ 1 } is established under some mild conditions, where { Y i , − ∞ < i < ∞ } is a sequence of extended negatively dependent and identically distributed random variables that is stochastically dominated by a random variable Y under a sub-linear expectation ( Ω , H , E ̂ ) , and { a i , − ∞ < i < ∞ } is an absolutely summable sequence of real numbers. These conclusions promote and improve the corresponding results of moving the average process under extended negatively dependent random variables from the traditional probability space to the sub-linear expectation space.

Plato’s Timaeus and optimal pentatonic scales

After a short review of Pythagorean theory of harmonic ratios and musical scales as it is described in Plato’s Timaeus treatise, the concept of ‘optimality of a sequence of (real) numbers with respect to Pythagorean ratios’ is defined and main theorem of this article proves that there are only three optimal sequences of length 6, which correspond to three well-known pentatonic scales which are used in many musical traditions (including Chinese, Japanese and others). It is also noted that a definition similar to our optimal scales has appeared in a treatise by Sadi-al-Din Urmavi, a thirteenth century Iranian musicologist.