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.

The anomaly of convex bodies

1953 ◽  
Vol 49 (1) ◽  
pp. 54-58 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Euclidean Space ◽  
Real Number ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Real Numbers

I write X for the point (x1, x2, …, xn) of n-dimensional Euclidean space Rn. The coordinates x1, x2, …, xn are real numbers. The origin (0, 0,…, 0) is denoted by O. If t is a real number, tX denotes the point (tx1, tx2, …, txn); in particular, − X is the point (−x1, −x2,…, −xn). Also X + Y denotes the point {x1 + y1, x2 + y2, …, xn + yn).

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.

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 On Fourier transform multipliers in Lp

1972 ◽  
Vol 13 (2) ◽  
pp. 219-223
Author(s):  
G. O. Okikiolu
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Characteristic Function ◽  
Lebesgue Measure ◽  
Real Numbers ◽  
Measurable Functions ◽  
Image Position

We denote by R the set of real numbers, and by Rn, n ≧ 2, the Euclidean space of dimension n. Given any subset E of Rn, n ≧ 1, we denote the characteristic function of E by xE, so that XE(x) = 0 if x ∈ E; and XE(X) = 0 if x ∈ Rn/E.The space L(Rn) Lp consists of those measurable functions f on Rn such that is finite. Also, L∞ represents the space of essentially bounded measurable functions with ║f║>0; m({x: |f(x)| > x}) = O}, where m represents the Lebesgue measure on Rn The numbers p and p′ will be connected by l/p+ l/p′= 1.

On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

2020 ◽  
Vol 70 (3) ◽  
pp. 641-656
Author(s):  
Amira Khelifa ◽  
Yacine Halim ◽  
Abderrahmane Bouchair ◽  
Massaoud Berkal
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Difference Equations ◽  
Fibonacci Numbers ◽  
Higher Order ◽  
Real Numbers ◽  
Lucas Numbers ◽  
Rational Difference Equations ◽  
Initial Values ◽  
Fibonacci And Lucas Numbers

AbstractIn this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$where n, k∈ ℕ0, the initial values x−k, x−k+1, …, x0, y−k, y−k+1, …, y0, z−k, z−k+1, …, z1 and z0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

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 ON THE SPECTRA OF PISOT NUMBERS

2011 ◽  
Vol 54 (1) ◽  
pp. 127-132 ◽  
Author(s):  
TOUFIK ZAIMI
Keyword(s):  
Real Number ◽  
Fractional Part ◽  
Pisot Number ◽  
Real Numbers ◽  
Pisot Numbers

AbstractLet θ be a real number greater than 1, and let (()) be the fractional part function. Then, θ is said to be a Z-number if there is a non-zero real number λ such that ((λθn)) < for all n ∈ ℕ. Dubickas (A. Dubickas, Even and odd integral parts of powers of a real number, Glasg. Math. J., 48 (2006), 331–336) showed that strong Pisot numbers are Z-numbers. Here it is proved that θ is a strong Pisot number if and only if there exists λ ≠ 0 such that ((λα)) < for all$\alpha \in \{ \theta ^{n}\mid n\in \mathbb{N}\} \cup \{ \sum\nolimits_{n=0}^{N}\theta ^{n}\mid \mathit{\}N\in \mathbb{N}\}$. Also, the following characterisation of Pisot numbers among real numbers greater than 1 is shown: θ is a Pisot number ⇔ ∃ λ ≠ 0 such that$\Vert \lambda \alpha \Vert <\frac{1}{% 3}$for all$\alpha \in \{ \sum\nolimits_{n=0}^{N}a_{n}\theta ^{n}\mid$an ∈ {0,1}, N ∈ ℕ}, where ‖λα‖ = min{((λα)), 1 − ((λα))}.

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