Polynomially moving ergodic averages

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

Let K = (k 1, k 2, . . .) be a sequence of positive integers, and \({\Sigma_{K} = \prod_{n=1}^\infty {\bf Z}_{k_n}}\) be a topological group with a metric and an additive operation given. The adding machine f K is the addition-by-one map on Σ K . Buescu and Stewart (Ergodic Theory Dynam Syst 15:271–290, 1995), Block and Keesling (Topology Appl 140:151–161, 2004) and Banks (Ergodic Theory Dynam Syst 17:505–529, 1997) obtained several equivalent conditions for which two adding machines are topologically conjugate, respectively. In this paper, we have a further discussion about semi-conjugate relationship between adding machines, and give some necessary and sufficient conditions for an adding machine to be semi-conjugate to another one. Moveover, we also prove that a transitive translation on a compact subgroup of Σ K is isometrically conjugate to an adding machine.

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>

On repeated difference sequences of a permutation of N

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.

Spectral properties of a class of Moran measures

Given a sequence of positive integers \(p=(p_1,\dots ,p_n)\), let \(S_p\) denote the family of all sequences of positive integers \(x=(x_1,\ldots ,x_n)\) such that \(x_i\le p_i\) for all \(i\). Two families of sequences (or vectors), \(A,B\subseteq S_p\), are said to be \(r\) -cross-intersecting if no matter how we select \(x\in A\) and \(y\in B\), there are at least \(r\) distinct indices \(i\) such that \(x_i=y_i\). We determine the maximum value of \(|A|\cdot |B|\) over all pairs of \(r\)-cross-intersecting families and characterize the extremal pairs for \(r\ge 1\), provided that \(\min p_i>r+1\). The case \(\min p_i\le r+1\) is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Frankl, Furedi, Livingston, Moon, and Tokushige, and answers a question of Zhang. The special case \(r=1\) has also been settled recently by Borg.

On a conjecture concerning the Bruhat order

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