Topological Semi-Conjugacies Between Adding Machines

Given an increasing sequence of positive integers { m n } \left \{ {{m_n}} \right \} , a non-decreasing sequence of positive integers { b n } \left \{ {{b_n}} \right \} , and a measurable, measure-preserving ergodic transformation τ \tau on a probability space ( Ω , F , μ ) \left ( {\Omega ,\mathcal {F},\mu } \right ) , the a.s. convergence of the moving averages T n ( f ) = b n − 1 ∑ k = m n + 1 m n + b n f ( τ k ) {T_n}\left ( f \right ) = b_n^{ - 1}\sum \nolimits _{k = {m_n} + 1}^{{m_n} + {b_n}} {f\left ( {{\tau ^k}} \right )} is considered, for f ∈ L p ( Ω ) f \in {L_p}\left ( \Omega \right ) . A counterexample is constructed in the case of polynomial-like { m n } \left \{ {{m_n}} \right \} .

Exact formulas for moments of sums of classical parking functions

We construct sequences of positive integers which are solutions of the equation x2+y2=z2. We introduce Mouanda’s choice functions which allow us to construct galaxies of sequences of positive integers. We give many examples of galaxies of numbers. We show that the equation x2n+y2n=z2n (n ≥2) has no integer solutions. We prove that the equation xn+yn=zn (n ≥3) has no solutions in N. We introduce the notion of the planetary representation of a galaxy of numbers which allow us to predict the structure, laws of the universe and life in every planet system of every galaxy of the universe. We show that every multiverse contains a finite number of universes.

Continued Fraction Expansion of Algebraic Numbers

On repeated difference sequences of a permutation of N

Variations on a game of Gale (II): Markov strategies

In this paper, we are connected with oscillation of third-order linear neutral delay difference equation of the form where . Here c 1(σ) and c 2(σ) are the sequence of positive integers, p(σ) and q(σ) are positive and real sequences such that q(σ) ≥ q 0 > 1 and p(σ) ≠ 0, ς, η are positive integers, such that ς > η.

Spectral properties of a class of Moran measures

For a non-negative integer r, the r-orbit of a vertex v in a connected graph G of order n is the set of vertices at distance r from v. A sequence of positive integers with is called an irregular orbital dominating sequence of G if for every pair i, j of integers with and G contains distinct vertices such that We investigate graphs that possess and graphs do not possess an irregular orbital dominating sequence. It is shown that a non-trivial tree has an irregular orbital dominating sequence if and only if it is neither a star, a path of order 2, nor a path of order 6.

Let be an array of rowwise pairwise negatively quadrant dependent random variables with mean zero and as , where is a sequence of positive integers. The complete qth moment convergence results for and are established which are much stronger than the complete convergence results obtained in the literature. As applications of our main results, we obtain some complete moment theorems for the dependent bootstrap samples arising from a sequence of independent and identically distributed random variables.

<p>One proves that if the trailing digit of a(n) is not zero for any n, then sum of a(n)/Rev Ca(n)) is divergent.</p>

Let be the collection of homogeneous product Moran sets determined by two sequences of positive integers , and two sequences of positive numbers , . We obtain the maximal and minimal values of the Hausdorff and packing dimensions of elements in .

Given an arithmetical function f : ℕ → {−1, 1}, consider the function hf (n) = ∑ν = 1n − 1f(ν)f(n − ν). Let λ be the Liouville function defined by λ(n) = (−1)Ω(n), where Ω(n) stands for the number of prime divisors of n counting their multiplicity. We prove that if qν is a sequence of positive integers with a corresponding sequence of primitive real characters χν (mod qν ) such that L(s, χν) has a Siegel zero βν = 1 − 1/(ην log qν), ην > exp e30, then there exist a positive constant c and a function e(n) tending to 0 as n → ∞ and such that |hλ(n)|/n ≤ c/ log log ην + e(n) uniformly for all n ∈ [qν10 , qν(log log ην)/3] such that (n, qν) = 1.

Let C be a smooth projective curve of genus $$g\ge 2$$ . Fix an integer $$r\ge 0$$ , and let $$\underline{k}=(k_1,\ldots ,k_n)$$ be a sequence of positive integers with $$\sum _{i=1}^n k_i =g-1$$ . In this paper, we study n-pointed curves $$(C,p_1,\ldots ,p_n)$$ such that the line bundle $$L:=O_C\left( \sum _{i=1}^n k_i p_i\right) $$ is a theta-characteristic with $$h^0\left( C,L\right) \ge r+1$$ and $$h^0\left( C,L\right) \equiv r+1 \,\mathrm {(mod\,2)}$$ . We prove that they describe a sublocus $${\mathcal {G}}^r_g(\underline{k})$$ of $${\mathcal {M}}_{g,n}$$ having codimension at most $$g-1+\frac{r(r-1)}{2}$$ . Moreover, for any $$r\ge 0$$ , $$\underline{k}$$ as above, and g greater than an explicit integer g(r) depending on r, we present irreducible components of $${\mathcal {G}}^r_g(\underline{k})$$ attaining the maximal codimension in $${\mathcal {M}}_{g,n}$$ , so that the bound turns out to be sharp.

Combinatorics A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.