Sequence Of Real Numbers Research Articles

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.

Quantitative tame properties of differentiable functions with controlled derivatives

We show that differentiable functions, defined on a convex body K⊆Rd, whose derivatives are controlled by a suitable given sequence of positive real numbers share many properties with polynomials. The role of the degree of a polynomial is hereby played by an integer associated with the given sequence of reals, the diameter of K, and a real parameter linked to the C0-norm of the function. We give quantitative information on the size of the zero set, show that it admits a local parameterization by Sobolev functions, and prove an inequality of Remez-type. From the latter, we deduce several consequences, for instance, a bound on the volume of sublevel sets and a comparison of Lp-norms reversing Hölder’s inequality. The validity of many of the results only depends on the derivatives up to some finite order; the order can be specified in terms of the given data.

Form of the solutions of difference equations via Lie symmetry analysis and Fibonacci numbers

In this paper, we give the form of the solutions of the following rational difference equation where an and bn are sequences of real numbers, by using Lie symmetry analysis and associated with Fibonacci numbers, respectively. We find an interesting relation between the exact solution of Equation (1) and the classical Fibonacci sequence.

Discretization of inverse scattering on a half line

AbstractWe solve inverse scattering problem for Schrödinger operators with compactly supported potentials on the half line. We discretize S‐matrix: We take the value of the S‐matrix on some infinite sequence of positive real numbers. Using this sequence obtained from S‐matrix, we recover uniquely the potential by a new explicit formula, without the Gelfand–Levitan–Marchenko equation.

INTRODUCING THE CARD EDITING APPLICATION OF THE LIMSTROM DIDACTIC GAME

Game-based learning plays an important role in all education and learning processes, including higher education nowadays. In the course, Mathematical Analysis, offered during the first semester, all students studying informatics learn about sequences of real numbers, convergence, and limits. These concepts are of key importance to understand advanced mathematical concepts and successfully complete advanced courses. Based on the benefits of game-based learning we came up with the idea of LimStorm that helps study groups of 4-10 students to learn and practice the limits of sequences of real numbers. In our paper we introduce a self-developed application which is suitable for generating decks of mathematical didactic card games, for example the deck of LimStorm. Keywords: game-based learning, didactic game, active learning methods, limit of the sequence, card games

On Soft Lebesgue Measure

In this paper, we introduce the concept of soft intervals, soft ordering and sequences of soft real numbers, and some of their structural properties are studied. The notion of soft Lebesgue measure on the soft real numbers has been introduced. Also, a correspondence relationship has been established between the soft Lebesgue measure and the classical Lebesgue measure. Furthermore, we have studied some exciting results and relations between the soft Lebesgue measure and the Lebesgue measure of soft real sets.

A New Notion of Paranorm Intuitionistic Fuzzy Zweier I<sub>3</sub>-Convergent Triple Sequence Spaces

In this paper, we define and introduce the notion of paranorm intuitionistic fuzzy Zweier I3-convergent triple sequence spaces ZI3(μ,υ) (p) and ZIO3(μ,υ) (p) for p = (pnmj ) a triple sequence of positive real numbers. Besides, the fuzzy topology on these spaces is studied.

A localized Erdős-Kac theorem for $\omega_y(p+a)$

Let $\omega_y(n)$ denote the number of distinct prime divisors of $n$ less than $y$. Suppose $y_n$ is an increasing sequence of positive real numbers satisfying $\log y_n = o(\log\log n)$. In this paper, we prove an Erd\"{o}s-Kac theorem for the distribution of $\omega_{y_n}(p+a)$, where $p$ runs over all prime numbers and $a$ is a fixed integer. We also highlight the connection between the distribution of $\omega_y(p-1)$ and Ihara's conjectures on Euler-Kronecker constants.

Minimal gaps and additive energy in real-valued sequences

Abstract We study the minimal gap statistic for sequences of the form $\left( \alpha x_n \right)_{n = 1}^{\infty}$ where $\left( x_n \right)_{n = 1}^{\infty}$ is a sequence of real numbers, and its connection to the additive energy of $\left( x_n \right)_{n = 1}^{\infty}$. Inspired by a recent paper of Aistleitner, El-Baz and Munsch we show conditionally on the Lindelöf Hypothesis that if the additive energy is of lowest possible order then for almost all α, the minimal gap $\delta_{\min}^{\alpha} (N) = \min \left\{\alpha x_m - \alpha x_n \bmod \ 1 : 1 \leq m \neq n \leq N \right\}$ is close to that of a random sequence, a result Rudnick showed for integer-valued sequences. We also show unconditional results in this direction, as well as some converse theorems about sequences with large additive energy.

An interval feature discrete grey-Markov model based on data distributions and applications

Data sequences represent complex changes in external environments or internal systems, and patterns of some index sequences have become more difficult to characterize using real number sequences. Considering the imbalances between regional development levels in terms of socioeconomic indicators, a discrete grey-Markov model based on interval distribution characteristics is proposed in this paper. Based on the classic discrete grey model, a difference coefficient sequence and median sequences of the upper and lower information domains, respectively, are introduced to establish differential equations for the representation and prediction of different development level trends. Furthermore, a combination of discrete grey models with the Markov model is used to modify the nonstationary data sequences and further improve the new model prediction accuracy. The parameter estimation and solution of the proposed model are organized using analytical methods, and the accuracy of the model is investigated. By fitting and predicting examples of private car ownership and milk production in representative unbalanced development regions, the results of the proposed model are compared with those of four other prediction models to verify its effectiveness. The results demonstrate that the prediction accuracy of the proposed model is better than those of other grey models in nonstationary data sequences. By capturing the interval features, this model achieves the goal of characterization of interval distribution information and further modification of the nonstationary series.

A completely hyperexpansive completion problem for weighted shifts on directed trees with one branching vertex

Let ? = {?k}nk =0 be given a finite sequence of positive real numbers. The completely hyperexpansive completion problem seeks equivalence conditions for the existence of a completely hyperexpansive weighted shift W?? such that ? ? ??. Let T?,? be a directed tree consisting of one branching vertex, ? branches and a trunk of length ?, and let T?,?,p be a subtree of T?,? whose members consist of the p-generation family from branching vertex. Suppose S? is the weighted shift acting on the tree T?,?. This object S? on the tree T?,? has been applied to the several topics. Recently, Exner-Jung-Stochel-Yun studied the subnormal completion problem for weighted shifts on T?,? in 2018. In this paper we discuss the completely hyperexpansive completion problem for weighted shifts on T?,? as a counterpart of the subnormal completion problem for S?.

How Are Students' Prior Knowledge Differentiate Analytical Thinking Process in Identifying the Convergence of Real Number Sequences?

How Are Students' Prior Knowledge Differentiate Analytical Thinking Process in Identifying the Convergence of Real Number Sequences?

A remark on the spectrum of fractional difference operator

This work computes the spectrum of the backward fractional difference operator over various Banach spaces and . Note that recently, the spectrum of the fractional difference operator , ( being either a constant or strictly decreasing sequence of positive real numbers satisfying some conditions) on the sequence space has been computed by Baliarsingh P. [On the spectrum of fractional difference operator. Linear Multilinear Algebra. 2021;1–13] and it has been noticed that the results obtained are unified and more general, but lack of accuracy for the case . In this work, we improve the results by providing the sharper estimations and some illustrative examples. Also, individual sets describing the point, the residual and the continuous spectrum of the operator on the Banach space c and are computed.

On Statistical Convergence of Uncertain Sequence of Fuzzy Numbers

In this paper, we extend the various ideas of convergence such as convergence in distribution, in mean, and in measure along with the convergence uniformly almost surely of the uncertain fuzzy sequence of real numbers in statistical context. We also establish some interesting results among these convergence.

Double-Execution Strategies Using Path Signatures

We employ the expected signature of equity and foreign exchange markets to derive an optimal double-execution trading strategy. The signature of a path of a stochastic process is a sequence of real numbers that provides a full description of the evolution of the process. The double-execution strategy maximizes the wealth (in units of the domestic currency) of an investor who liquidates a block of shares in a foreign stock market. Our approach is model agnostic because we do not specify the dynamics of the market. We prove that the optimal strategy is a linear combination of the terms in the expected signature of the market and employ high-frequency data from Nasdaq and for various currencies to compute the signature of the market. Data for 10 stocks and 4 currency pairs are employed to implement the strategy. Our results show that the performance of the signature-based double-execution strategy is superior than the performance of the benchmarks. In most cases, the outperformance increases further when the signature of the market is enhanced with the price dynamics of the SPY, a tracker of the Standard & Poor's 500 index.

Novel weight-adaptive fusion grey prediction model based on interval sequences and its applications

With the increasing complexity of systems, it is difficult to describe a huge amount of information accurately with real numbers. To make predictions using uncertain information, this study uses interval numbers as descriptors and proposes a weight-adaptive fusion grey prediction model based on interval sequences (WAFGM(1,N) model). Unlike most interval prediction models; the proposed model uses the adaptive weight and compensation coefficient to directly predict interval grey numbers without converting interval sequences into real number sequences. Moreover, it improves the four errors of the traditional GM(1,N) model, thereby further improving model accuracy. First, by fully considering the integrity and interaction of the upper and lower boundaries of the interval, this study introduced a weight parameter to comprehensively characterize the development trend of the interval, and by optimizing the background value and adjusting the compensation coefficient, the univariate and multivariate weight-adaptive grey prediction models for interval prediction were constructed, respectively. Second, the interval development and driving coefficients, compensation coefficient, and weight parameter were estimated and optimized using the least squares method, accumulation method, and particle swarm algorithm, respectively. Finally, the WAFGM(1,N) model was constructed by combining the two models. The validity and applicability of the WAFGM(1,N) model were proved using two examples: the gross agricultural output value of Northeast China and the production of dairy products in the golden milk source areas.

SOME ROUGH HAUSDORFF LIMIT LAWS FOR SEQUENCES OF SETS

In this study, we observe the change of roughness degree for the rough Hausdorff convergence of a sequence consisting of the product of a sequence of sets and a sequence of real numbers. Then we prove that the rough Hausdorff convergence is preserved under the operators of addition, union, Cartesian product and convex hull.

Finding structure in sequences of real numbers via graph theory: a problem list

We investigate a method of generating a graph $G=(V,E)$ out of an ordered list of $n$ distinct real numbers $a_1, \dots, a_n$. These graphs can be used to test for the presence of interesting structure in the sequence. We describe sequences exhibiting intricate hidden structure that was discovered this way. Our list includes sequences of Deutsch, Erd\H{o}s, Freud & Hegyvari, Recaman, Quet, Zabolotskiy and Zizka. Since our observations are mostly empirical, each sequence in the list is an open problem.

Dynamic Time Warping Under Product Quantization, With Applications to Time-Series Data Similarity Search

The similarity search on sensor data generated by a myriad of sensing devices is a frequently encountered problem in the era of the Internet of Things (IoT). This sensor data generally appear in the form of time series, a temporally ordered sequence of real numbers obtained regularly in time. It has been widely accepted that the dynamic time warping (DTW) currently is the most prevalent similarity measure in the time-series mining community, mainly due to its flexibility and broad applicability. However, calculating DTW between two time series has quadratic time complexity, leading to unsatisfactory efficiency when performing the similarity search over the large time-series data set. The main contribution of this article is to propose a method called product quantization (PQ)-based DTW (PQDTW) for fast time-series approximate similarity search under DTW. The PQ, a well-known approximate nearest neighbor search approach, is used in PQDTW. Nevertheless, the conventional PQ is developed with the Euclidean distance and is not designed for DTW. To solve this problem, the DTW barycenter averaging (DBA) technique is utilized to adapt the PQ for DTW before using it. We employ PQDTW along with the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">filter-and-refine</i> framework to efficiently and accurately perform the time-series similarity search. Our method can reasonably reduce many DTW computations in the filtering phase; thus, the query process is accelerated. We compare PQDTW with related popular algorithms using public time-series data sets. Experimental results verify that the proposal achieves the best tradeoff between query efficiency and retrieval accuracy compared to the competitors.

LINEAR RECURRENCE RELATONS AND ORDINARY GENERATING FUNCTIONS APPLIED ON MODELING PROCESSES IN CONTROL THEORY

In this paper we apply multistep recurrence relations, as one of very simple and useful mathematical models. It is an efficient tool for solving many problems in mathematics, science, and technics. We also use generating functions, as a connection between real number sequences and real functions, and as a very smooth and efficient connection between the discrete mathematics and (continual) mathematical analysis. We present an application of multistep homogenous linear recurrence relations for modelling some processes in the control theory. Further on, we use the ordinary generating function aiming to find appropriate formulae for calculating members of an appropriate recurrence sequence. Finally, we show the application of this novel mathematical approach on one real example in the control theory.