scholarly journals On the specification property and synchronization of unique q-expansions

2020 ◽  
pp. 1-47
Author(s):  
RAFAEL ALCARAZ BARRERA
Keyword(s):  
Positive Integer ◽  
Real Numbers ◽  
Specification Property ◽  
Greedy Expansion ◽  
Dynamical Properties

Abstract Given a positive integer M and $q \in (1, M+1]$ we consider expansions in base q for real numbers $x \in [0, {M}/{q-1}]$ over the alphabet $\{0, \ldots , M\}$ . In particular, we study some dynamical properties of the natural occurring subshift $(\boldsymbol{{V}}_q, \sigma )$ related to unique expansions in such base q. We characterize the set of $q \in \mathcal {V} \subset (1,M+1]$ such that $(\boldsymbol{{V}}_q, \sigma )$ has the specification property and the set of $q \in \mathcal {V}$ such that $(\boldsymbol{{V}}_q, \sigma )$ is a synchronized subshift. Such properties are studied by analysing the combinatorial and dynamical properties of the quasi-greedy expansion of q. We also calculate the size of such classes as subsets of $\mathcal {V}$ giving similar results to those shown by Blanchard [ 10 ] and Schmeling in [ 36 ] in the context of $\beta $ -transformations.

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 Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
G. M. Moremedi ◽  
I. P. Stavroulakis
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Difference Equations ◽  
Delay Difference Equation ◽  
Real Numbers ◽  
First Order ◽  
All Solutions ◽  
Delay Difference ◽  
Constant Argument

Consider the first-order delay difference equation with a constant argument Δxn+pnxn-k=0,  n=0,1,2,…, and the delay difference equation with a variable argument Δxn+pnxτn=0,  n=0,1,2,…, where p(n) is a sequence of nonnegative real numbers, k is a positive integer, Δx(n)=x(n+1)-x(n), and τ(n) is a sequence of integers such that τ(n)≤n-1 for all n≥0 and limn→∞τ(n)=∞. A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.

scholarly journals ON EXISTENCE OF POSITIVE SOLUTIONS OF NEUTRAL DIFFERENCE EQUATIONS

1994 ◽  
Vol 25 (3) ◽  
pp. 257-265
Author(s):  
J. H. SHEN ◽  
Z. C. WANG ◽  
X. Z. QIAN
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Difference Equations ◽  
Real Numbers ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Existence Of Positive Solutions

Consider the neutral difference equation \[\Delta(x_n- cx_{n-m})+p_nx_{n-k}=0, n\ge N\qquad (*) \] where $c$ and $p_n$ are real numbers, $k$ and $N$ are nonnegative integers, and $m$ is positive integer. We show that if \[\sum_{n=N}^\infty |p_n|<\infty \qquad (**) \] then Eq.(*) has a positive solution when $c \neq 1$. However, an interesting example is also given which shows that (**) does not imply that (*) has a positive solution when $c =1$.

scholarly journals Dynamics of a Higher-Order System of Difference Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Qi Wang ◽  
Qinqin Zhang ◽  
Qirui Li
Keyword(s):  
Positive Integer ◽  
Positive Solutions ◽  
Difference Equations ◽  
Initial Conditions ◽  
Higher Order ◽  
Order System ◽  
Real Numbers ◽  
Precise Description ◽  
System Of Difference Equations ◽  
Positive Real

Consider the following system of difference equations:xn+1(i)=xn-m+1(i)/Ai∏j=0m-1xn-j(i+j+1)+αi,xn+1(i+m)=xn+1(i),x1-l(i+l)=ai,l,Ai+m=Ai,αi+m=αi,i,l=1,2,…,m;n=0,1,2,…,wheremis a positive integer,Ai,αi,i=1,2,…,m, and the initial conditionsai,l,i,l=1,2,…,m,are positive real numbers. We obtain the expressions of the positive solutions of the system and then give a precise description of the convergence of the positive solutions. Finally, we give some numerical results.

scholarly journals Generators for locally compact groups

1993 ◽  
Vol 36 (3) ◽  
pp. 463-467 ◽  
Author(s):  
Joan Cleary ◽  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Positive Integer ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Real Numbers ◽  
Locally Compact ◽  
Hausdorff Group

It is proved that if G is any compact connected Hausdorff group with weight w(G)≦c, ℝ is the topological group of all real numbers and n is a positive integer, then the topological group G × ℝn can be topologically generated by n + 1 elements, and no fewer elements will suffice.

Two Inequalities

1953 ◽  
Vol 37 (322) ◽  
pp. 244-246 ◽  
Author(s):  
G.N. Watson
Keyword(s):  
Positive Integer ◽  
Real Numbers ◽  
Intrinsic Interest ◽  
Elementary Analysis ◽  
Order Of Magnitude

(I). The following inequality is a straight generdisation of one of the most important inequalities occurring in elementary analysis. It is consequently of some intrinsic interest, even though it has to do with a determinant. Let n be a positive integer (≥ 2) and let a, b, . . . , h be n real numbers (unrestricted as to sign) arranged in descending order of magnitude, and no two being equal. Let x be a positive number, which will be regarded as variable. Let the determinant

Oscillation of higher order neutral functional difference equations with positive and negative coefficients

2010 ◽  
Vol 60 (3) ◽  
Author(s):  
R. Rath ◽  
B. Barik ◽  
S. Rath
Keyword(s):  
Positive Integer ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Functional Difference ◽  
Real Numbers ◽  
Positive And Negative Coefficients ◽  
Forward Difference ◽  
Functional Difference Equation ◽  
Functional Difference Equations ◽  
Infinite Sequences

AbstractSufficient conditions are obtained so that every solution of the neutral functional difference equation $$ \Delta ^m (y_n - p_n y_{\tau (n)} ) + q_n G(y_{\sigma (n)} ) - u_n H(y_{\alpha (n)} ) = f_n , $$ oscillates or tends to zero or ±∞ as n → ∞, where Δ is the forward difference operator given by Δx n = x n+1 − x n, p n, q n, u n, f n are infinite sequences of real numbers with q n > 0, u n ≥ 0, G,H ∈ C(ℝ,ℝ) and m ≥ 2 is any positive integer. Various ranges of {p n} are considered. The results hold for G(u) ≡ u, and f n ≡ 0. This paper corrects, improves and generalizes some recent results.

scholarly journals A note on a functional inequality

1992 ◽  
Vol 15 (2) ◽  
pp. 413-415
Author(s):  
Horst Alzer
Keyword(s):  
Positive Integer ◽  
Complex Number ◽  
Functional Inequality ◽  
Entire Solutions ◽  
Real Numbers ◽  
Positive Real

We prove: Ifr1,…,rkare (fixed) positive real numbers with∏j=1krj>1, then the only entire solutionsφ:ℂ→ℂof the functional inequality∏j=1k|φ(rjz)|≥(∏j=1krj)|φ(z)|kareφ(z)=czn, wherecis a complex number andnis a positive integer.

scholarly journals Sharp conditions for the oscillation of delay difference equations

1989 ◽  
Vol 2 (2) ◽  
pp. 101-111 ◽  
Author(s):  
G. Ladas ◽  
Ch. G. Philos ◽  
Y. G. Sficas
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Sufficient Condition ◽  
Delay Difference Equation ◽  
Real Numbers ◽  
All Solutions ◽  
Existence Of Positive Solutions ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonnegative Sequence

Suppose that {pn} is a nonnegative sequence of real numbers and let k be a positive integer. We prove that limn→∞inf [1k∑i=n−kn−1pi]>kk(k+1)k+1 is a sufficient condition for the oscillation of all solutions of the delay difference equation An+1−An+pnAn−k=0,   n=0,1,2,…. This result is sharp in that the lower bound kk/(k+1)k+1 in the condition cannot be improved. Some results on difference inequalities and the existence of positive solutions are also presented.

scholarly journals On simultaneous rational approximation to a p-adic number and its integral powers, II

2021 ◽  
pp. 1-21
Author(s):  
Dzmitry Badziahin ◽  
Yann Bugeaud ◽  
Johannes Schleischitz
Keyword(s):  
Hausdorff Dimension ◽  
Positive Integer ◽  
Real Number ◽  
Prime Number ◽  
Rational Approximation ◽  
Real Numbers ◽  
The Real ◽  
Simultaneous Rational Approximation

Abstract Let $p$ be a prime number. For a positive integer $n$ and a real number $\xi$ , let $\lambda _n (\xi )$ denote the supremum of the real numbers $\lambda$ for which there are infinitely many integer tuples $(x_0, x_1, \ldots , x_n)$ such that $| x_0 \xi - x_1|_p, \ldots , | x_0 \xi ^{n} - x_n|_p$ are all less than $X^{-\lambda - 1}$ , where $X$ is the maximum of $|x_0|, |x_1|, \ldots , |x_n|$ . We establish new results on the Hausdorff dimension of the set of real numbers $\xi$ for which $\lambda _n (\xi )$ is equal to (or greater than or equal to) a given value.

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