scholarly journals A Bibliometric Analysis for Lebesgue Measure Integration in Optimization

2021 ◽  
Vol 2 (2) ◽  
pp. 54-63
Author(s):  
Endang Rusyaman ◽  
Devi Munandar ◽  
Diah Chaerani ◽  
Dwindi Agryanti Johar ◽  
Rizky Ashgi
Keyword(s):  
Lebesgue Measure ◽  
Integral Theory ◽  
Lebesgue Integral ◽  
Set Domain ◽  
Real Numbers ◽  
Search Results ◽  
Pure Mathematics ◽  
Bibliographic Data ◽  
Mathematical Problems ◽  
Integration Concept

In solving mathematical problems so far, Riemann's integral theory is quite adequate for solving pure mathematics and applications problems. But not all problems can be solved using this integration, such as a discontinuous function that is not Riemann's integration. Lebesgue integral is an integration concept based on measure and can solve finite and unlimited function problems and be solved in a more general set domain. One of the bases of this integration is the Lebesgues measure includes the set of real numbers, where the length of the interval is the endpoints. The alternative use of this integral is widely used in various studies such as partial differential equations, quantum mechanics, and probabilistic analysis, requiring the integration of arbitrary set functions. This paper will show a comprehensive bibliometric survey of peer-reviewed articles referring to Lebesgue measure in integration. Search results are obtained 832 papers in the google scholar database and 997 papers using Lebesgue measure integration in optimization. It can also be seen that the research have 4 clusters and 3 clusters respectively with scattered keywords for each cluster. Finally, using bibliographic data can be obtained Lebesgues measure in integration and optimization supports many of the research and provides productive citations to citing the study.

scholarly journals Approximation properties of -expansions II

2017 ◽  
Vol 38 (5) ◽  
pp. 1627-1641
Author(s):  
SIMON BAKER
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Measure Zero ◽  
Hausdorff Measures ◽  
Approximation Properties ◽  
Real Numbers ◽  
Positive Real ◽  
Transference Principle ◽  
Significant Step ◽  
Full Lebesgue Measure

Let $\unicode[STIX]{x1D6FD}\in (1,2)$ be a real number. For a function $\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{\geq 0}$, define $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ to be the set of $x\in \mathbb{R}$ such that for infinitely many $n\in \mathbb{N},$ there exists a sequence $(\unicode[STIX]{x1D716}_{i})_{i=1}^{n}\in \{0,1\}^{n}$ satisfying $0\leq x-\sum _{i=1}^{n}(\unicode[STIX]{x1D716}_{i}/\unicode[STIX]{x1D6FD}^{i})\leq \unicode[STIX]{x1D6F9}(n)$. In Baker [Approximation properties of $\unicode[STIX]{x1D6FD}$-expansions. Acta Arith. 168 (2015), 269–287], the author conjectured that for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1,2)$, the condition $\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n)=\infty$ implies that $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ is of full Lebesgue measure within $[0,1/(\unicode[STIX]{x1D6FD}-1)]$. In this paper we make a significant step towards proving this conjecture. We prove that given a sequence of positive real numbers $(\unicode[STIX]{x1D714}_{n})_{n=1}^{\infty }$ satisfying $\lim _{n\rightarrow \infty }\unicode[STIX]{x1D714}_{n}=\infty$, for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1.497,\ldots ,2)$, the set $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D714}_{n}\cdot 2^{-n})$ is of full Lebesgue measure within $[0,1/(\unicode[STIX]{x1D6FD}-1)]$. We also study the case where $\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n)<\infty$ in which the set $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ has Lebesgue measure zero. Applying the mass transference principle developed by Beresnevich and Velani in [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], we obtain some results on the Hausdorff dimension and the Hausdorff measure of $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$.

scholarly journals The Gauge Integral Theory in HOL4

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Zhiping Shi ◽  
Weiqing Gu ◽  
Xiaojuan Li ◽  
Yong Guan ◽  
Shiwei Ye ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Higher Order ◽  
Integral Theory ◽  
Order Logic ◽  
Cauchy Type ◽  
Lebesgue Integral ◽  
Higher Order Logic ◽  
Riemann Integral ◽  
Integrability Criterion ◽  
Class Of Functions

The integral is one of the most important foundations for modeling dynamical systems. The gauge integral is a generalization of the Riemann integral and the Lebesgue integral and applies to a much wider class of functions. In this paper, we formalize the operational properties which contain the linearity, monotonicity, integration by parts, the Cauchy-type integrability criterion, and other important theorems of the gauge integral in higher-order logic 4 (HOL4) and then use them to verify an inverting integrator. The formalized theorem library has been accepted by the HOL4 authority and will appear in HOL4 Kananaskis-9.

scholarly journals On Soft Lebesgue Measure

2020 ◽  
Author(s):  
Sujay Goldar ◽  
Subhasis Ray
Keyword(s):  
Lebesgue Measure ◽  
Real Numbers

In this article, we introduce the concept of soft intervals, soft ordering and sequences of soft real numbers and study their properties and some interesting results. Also the notion of soft Lebesgue measure on the soft real numbers has been introduced. A correspondence relationship between the soft Lebesgue measure and the Lebesgue measure has been established.

scholarly journals Socratic as Mathematics Learning Application for Differential Equations Concept

2021 ◽  
Vol 3 (1) ◽  
pp. 97-110
Author(s):  
Wanda Nugroho Yanuarto ◽  
Anton Jaelani ◽  
Joko Purwanto ◽  
Mohamad Ikram Zakaria
Keyword(s):  
Differential Equations ◽  
Industrial Revolution ◽  
Mathematics Learning ◽  
Relevant Literature ◽  
Technology Application ◽  
Search Results ◽  
Source Selection ◽  
Learning Mathematics ◽  
Teachers And Students ◽  
Mathematical Problems

Technological improvements in the era of the Industrial Revolution 4.0 can be applied to learning mathematics as a medium of improving teacher performance. In addition, a pandemic situation forces teachers and students to implement online learning. One of the learning media that can help teachers in learning mathematics online is Socratic. The Socratic system generally consists of four basic components: acting humanely, thinking humanely, thinking rationally and acting rationally. For example, the Socratic can provide alternative solutions to the differential equations problems. Mathematics problems can be recorded by the Socratic through three main features:1) home screen, which is used to take pictures of solving the problem needed; 2) search results, the image that has been recorded will be searched for a solution through Socratic's artificial intelligence in internet database, and 3) an explanation, the search results that have been obtained, have their own explanation. Teachers and students as users may find the best solution for each explanation by the Socratic. Meanwhile, this study uses four stages of the literature review process:1) search for relevant literature, 2) evaluation and source selection, 3) identification of themes, and 4) outline of the structure of the writing. This paper systematically investigates the use of the Socratic as a technology application that can assist mathematical problems.

scholarly journals The discrepancy of some real sequences

2003 ◽  
Vol 93 (2) ◽  
pp. 268
Author(s):  
H. Kamarul Haili ◽  
R. Nair
Keyword(s):  
Real Number ◽  
Lebesgue Measure ◽  
Fractional Part ◽  
Real Numbers ◽  
The Real ◽  
Almost Everywhere

Let $(\lambda_n)_{n\geq 0}$ be a sequence of real numbers such that there exists $\delta > 0$ such that $|\lambda_{n+1} - \lambda_n| \geq \delta , n = 0,1,...$. For a real number $y$ let $\{ y \}$ denote its fractional part. Also, for the real number $x$ let $D(N,x)$ denote the discrepancy of the numbers $\{ \lambda _0 x \}, \cdots , \{ \lambda _{N-1} x \}$. We show that given $\varepsilon > 0$, 9774 D(N,x) = o ( N^{-\frac{1}{2}}(\log N)^{\frac{3}{2} + \varepsilon})9774 almost everywhere with respect to Lebesgue measure.

scholarly journals Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Dug Hun Hong ◽  
Jae Duck Kim
Keyword(s):  
Lebesgue Measure ◽  
Upper Bound ◽  
Convex Functions ◽  
Hölder Inequality ◽  
Concave Functions ◽  
Lebesgue Integral ◽  
Reverse Hölder ◽  
Special Case

The classical Hölder inequality shows an interesting upper bound for Lebesgue integral of the product of two functions. This paper proposes Hölder type inequalities and reverse Hölder type inequalities for Sugeno integrals under usual multiplication operations for nonincreasing concave or convex functions. One of the interesting results is that the inequality, (S)∫01f(x)pdμ1/p(S)∫01g(x)qdμ1/q≤p-q/p-p-q+1∨q-p/q-q-p+1(S)∫01f(x)g(x)dμ, where 1<p<∞,1/p+1/q=1 and μ is the Lebesgue measure on R, holds if f and g are nonincreasing and concave functions. As a special case, we consider Cauchy-Schwarz type inequalities for Sugeno integrals involving nonincreasing concave or convex functions. Some examples are provided to illustrate the validity of the proposed inequalities.

scholarly journals On the metric theory of the optimal continued fraction expansion

1997 ◽  
Vol 56 (1) ◽  
pp. 69-79
Author(s):  
R. Nair
Keyword(s):  
Lebesgue Measure ◽  
Continued Fraction ◽  
Natural Numbers ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Special Case

Suppose kn denotes either φ(n) or φ(rn) (n = 1, 2, …) where the polynomial φ maps the natural numbers to themselves and rk denotes the kth rational prime. Let denote the sequence of convergents to a real numbers x for the optimal continued fraction expansion. Define the sequence of approximation constants byIn this paper we study the behaviour of the sequence for all most all x with respect to Lebesgue measure. In the special case where kn = n (n = 1, 2, …) these results are due to Bosma and Kraaikamp.

A NOTE ON INHOMOGENEOUS DIOPHANTINE APPROXIMATION IN BETA-DYNAMICAL SYSTEM

2014 ◽  
Vol 91 (1) ◽  
pp. 34-40 ◽  
Author(s):  
YUEHUA GE ◽  
FAN LÜ
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Real Number ◽  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Positive Function ◽  
Real Numbers ◽  
Image Position

AbstractWe study the distribution of the orbits of real numbers under the beta-transformation$T_{{\it\beta}}$for any${\it\beta}>1$. More precisely, for any real number${\it\beta}>1$and a positive function${\it\varphi}:\mathbb{N}\rightarrow \mathbb{R}^{+}$, we determine the Lebesgue measure and the Hausdorff dimension of the following set:$$\begin{eqnarray}E(T_{{\it\beta}},{\it\varphi})=\{(x,y)\in [0,1]\times [0,1]:|T_{{\it\beta}}^{n}x-y|<{\it\varphi}(n)\text{ for infinitely many }n\in \mathbb{N}\}.\end{eqnarray}$$

scholarly journals Dynamical behavior of alternate base expansions

2021 ◽  
pp. 1-34
Author(s):  
ÉMILIE CHARLIER ◽  
CÉLIA CISTERNINO ◽  
KARMA DAJANI
Keyword(s):  
Dynamical System ◽  
Invariant Measure ◽  
Lebesgue Measure ◽  
Dynamical Behavior ◽  
Base Case ◽  
Real Numbers ◽  
Absolutely Continuous ◽  
Dynamical Behaviors ◽  
Unique Measure ◽  
Dynamical Properties

Abstract We generalize the greedy and lazy $\beta $ -transformations for a real base $\beta $ to the setting of alternate bases ${\boldsymbol {\beta }}=(\beta _0,\ldots ,\beta _{p-1})$ , which were recently introduced by the first and second authors as a particular case of Cantor bases. As in the real base case, these new transformations, denoted $T_{{\boldsymbol {\beta }}}$ and $L_{{\boldsymbol {\beta }}}$ respectively, can be iterated in order to generate the digits of the greedy and lazy ${\boldsymbol {\beta }}$ -expansions of real numbers. The aim of this paper is to describe the measure-theoretical dynamical behaviors of $T_{{\boldsymbol {\beta }}}$ and $L_{{\boldsymbol {\beta }}}$ . We first prove the existence of a unique absolutely continuous (with respect to an extended Lebesgue measure, called the p-Lebesgue measure) $T_{{\boldsymbol {\beta }}}$ -invariant measure. We then show that this unique measure is in fact equivalent to the p-Lebesgue measure and that the corresponding dynamical system is ergodic and has entropy $({1}/{p})\log (\beta _{p-1}\cdots \beta _0)$ . We give an explicit expression of the density function of this invariant measure and compute the frequencies of letters in the greedy ${\boldsymbol {\beta }}$ -expansions. The dynamical properties of $L_{{\boldsymbol {\beta }}}$ are obtained by showing that the lazy dynamical system is isomorphic to the greedy one. We also provide an isomorphism with a suitable extension of the $\beta $ -shift. Finally, we show that the ${\boldsymbol {\beta }}$ -expansions can be seen as $(\beta _{p-1}\cdots \beta _0)$ -representations over general digit sets and we compare both frameworks.

scholarly journals Decision problems for linear recurrences involving arbitrary real numbers

2021 ◽  
Vol Volume 17, Issue 3 ◽  
Author(s):  
Eike Neumann
Keyword(s):  
Euclidean Space ◽  
Real Number ◽  
Lebesgue Measure ◽  
Decision Problems ◽  
Real Numbers ◽  
Linear Recurrences ◽  
Initial Values ◽  
Problem Instances ◽  
Low Order

We study the decidability of the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem for linear recurrences with real number initial values and real number coefficients in the bit-model of real computation. We show that for each problem there exists a correct partial algorithm which halts for all problem instances for which the answer is locally constant, thus establishing that all three problems are as close to decidable as one can expect them to be in this setting. We further show that the algorithms for the Positivity Problem and the Ultimate Positivity Problem halt on almost every instance with respect to the usual Lebesgue measure on Euclidean space. In comparison, the analogous problems for exact rational or real algebraic coefficients are known to be decidable only for linear recurrences of fairly low order.

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