We prove a lower bound in a law of the iterated logarithm for sums of the form ${\sum }_{k=1}^N{a}_{k}f(n_{k}x+c_k)$ where f satisfies certain conditions and the $n_k$ satisfy the Hadamard gap condition $n_{k+1}/n_k\ge q>1$.

Let df be a Hilbert-space-valued martingale difference sequence. The paper is devoted to a new, elementary proof of the estimate $\parallel {\sum }_{k=0}^{\infty }d{f}_k{\parallel }_p\le C_p\parallel ({\sum }_{k=0}^{\infty }\left(\right|d{f}_k\left|²\right|{ℱ}_{k-1}{\left)\right)}^{1/2}{\parallel }_p+\parallel ({\sum }_{k=0}^{\infty }|d{f}_k{{|}^p)}^{1/p}{\parallel }_p,$ with $C_p=O(p/lnp)$ as p → ∞.

For a t-norm T on a bounded lattice $(L,\le )$, a partial order ${\le }_T$ was recently defined and studied. In [11], it was pointed out that the binary relation ${\le }_T$ is a partial order on $L$, but $(L,{\le }_T)$ may not be a lattice in general. In this paper, several sufficient conditions under which $(L,{\le }_T)$ is a lattice are given, as an answer to an open problem posed by the authors of [11]. Furthermore, some examples of t-norms on $L$ such that $(L,{\le }_T)$ is a lattice are presented.

In this paper, we propose the general methods, yielding uninorms on the bounded lattice $(L,\le ,0,1)$, with some additional constraints on $e\in L\setminus \{0,1\}$ for a fixed neutral element $e\in L\setminus \{0,1\}$ based on underlying an arbitrary triangular norm $T_e$ on $[0,e]$ and an arbitrary triangular conorm $S_e$ on $[e,1]$. And, some illustrative examples are added for clarity.

Let ℳ be a hyperfinite finite von Nemann algebra and ${\left({ℳ}_k\right)}_{k\ge 1}$ be an increasing filtration of finite-dimensional von Neumann subalgebras of ℳ. We investigate abstract fractional integrals associated to the filtration ${\left({ℳ}_k\right)}_{k\ge 1}$. For a finite noncommutative martingale $x={\left(x_k\right)}_{1\le k\le n}\subseteq L₁\left(ℳ\right)$ adapted to ${\left({ℳ}_k\right)}_{k\ge 1}$ and 0 < α < 1, the fractional integral of x of order α is defined by setting $I^{\alpha }x={\sum }_{k=1}^n{\zeta }_k^{\alpha }d{x}_k$ for an appropriate sequence ${\left({\zeta }_k\right)}_{k\ge 1}$ of scalars. For the case of a noncommutative dyadic martingale in L₁() where is the type II₁ hyperfinite factor...

For any positive integer $k\ge 3$, it is easy to prove that the $k$-polygonal numbers are $a_n\left(k\right)=(2n+n(n-1)(k-2))/2$. The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet $L$-functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums $S(a_n\left(k\right){\overline{a}}_m\left(k\right),p)$ for $k$-polygonal numbers with $1\le m,n\le p-1$, and give an interesting computational formula for it.

We extend Champernowne’s construction of normal numbers to base b to the ${\mathbb{Z}}^d$ case and obtain an explicit construction of a generic point of the ${\mathbb{Z}}^d$ shift transformation of the set ${0,1,...,b-1}^{{\mathbb{Z}}^d}$.

Let $S_n^{\left(2\right)}$ denote the iterated partial sums. That is, $S_n^{\left(2\right)}=S_1+S_2+\cdots +S_n$, where $S_i=X_1+X_2+\cdots +X_i$. Assuming $X_1,X_2,...,X_n$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities $$p_n^{\left(2\right)}:=\mathbb{P}\left(\underset{1\le i\le n}{max}{S}_i^{\left(2\right)}lt;0\right)\le c\sqrt{\frac{𝔼|S_{n+1}|}{(n+1)𝔼|X_1|}},$$ with $c\le 6\sqrt{30}$ (and $c=2$ whenever $X_1$ is symmetric). The converse inequality holds whenever the non-zero $min(-X_1,0)$ is bounded or when it has only finite third moment and in addition $X_1$ is squared integrable. Furthermore, $p_n^{\left(2\right)}\asymp n^{-1/4}$ for any non-degenerate squared integrable, i.i.d., zero-mean $X_i$. In contrast, we show that for any $0lt;\gamma lt;1/4$ there exist integrable,...

We consider Weil sums of binomials of the form $W_{F,d}\left(a\right)={\sum }_{x\in F}\psi (x^d-ax)$, where F is a finite field, ψ: F → ℂ is the canonical additive character, $gcd(d,|F^{\times }\left|\right)=1$, and $a\in F^{\times }$. If we fix F and d, and examine the values of $W_{F,d}\left(a\right)$ as a runs through $F^{\times }$, we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $|F^{\times }|$). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n...

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.

We investigate the lattice $𝐋\left(𝐕\right)$ of subspaces of an $m$-dimensional vector space $𝐕$ over a finite field $\mathrm{GF}\left(q\right)$ with a prime power $q=p^n$ together with the unary operation of orthogonality. It is well-known that this lattice is modular and that the orthogonality is an antitone involution. The lattice $𝐋\left(𝐕\right)$ satisfies the chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when $𝐋\left(𝐕\right)$ is orthomodular....

The Rademacher sums are investigated in the Cesàro spaces $Ce{s}_p$ (1 ≤ p ≤ ∞) and in the weighted Korenblyum-Kreĭn-Levin spaces $K_{p,w}$ on [0,1]. They span l₂ space in $Ce{s}_p$ for any 1 ≤ p < ∞ and in $K_{p,w}$ if and only if the weight w is larger than $tlog{₂}^{p/2}(2/t)$ on (0,1). Moreover, the span of the Rademachers is not complemented in $Ce{s}_p$ for any 1 ≤ p < ∞ or in $K_{1,w}$ for any quasi-concave weight w. In the case when p > 1 and when w is such that the span of the Rademacher functions is isomorphic to l₂, this span is...

Let $m\ge 3$ be an integer. The so-calledis a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when $m\le 26$, the asymptotic behavior of the process is Gaussian, but for $m\ge 27$ it is no longer Gaussian and a limit $W^{DT}$ of a complex-valued martingale arises. In this paper, we consider the multitype branching process which is the continuous time version of...

Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies ${\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\partial }_j{f}_j{\left(\epsilon \right)\left|\right|}^{p}d\mu \left(\epsilon \right){\le }^p{\int }_{-1,1ⁿ}{\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\delta }_j\Delta f_j\left(\epsilon \right){\left|\right|}^{p}d\mu \left(\epsilon \right)d\mu \left(\delta \right)$, where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${{\partial }_j}_{j=1}^n$ and $\Delta ={\sum }_{j=1}^n{\partial }_j$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by ${ⁿ}_p\left(X\right)$, we show that ${ⁿ}_p\left(X\right)\le {\sum }_{k=1}^{n}1/k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

Let p be a prime, ℤₚ be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero kth powers in ℤₚ. The goal of this paper is to determine, for a given positive integer s, a value tₛ such that if |A| ≫ tₛ then every element of ℤₚ is a sum of s kth powers. We obtain $t₄=p^{22/39+ϵ}$, $t₅=p^{15/29+ϵ}$ and for s ≥ 6, $tₛ=p^{(9s+45)/(29s+33)+ϵ}$. For s ≥ 24 further improvements are made, such as $t_{32}=p^{5/16+ϵ}$ and $t_{128}=p^{1/4}$.

It is well known that, in forward inference in fuzzy logic, the generalized modus ponens is guaranteed by a functional inequality called the law of $T$-conditionality. In this paper, the $T$-conditionality for $T$-power based implications is deeply studied and the concise necessary and sufficient conditions for a power based implication $I^T$ being $T$-conditional are obtained. Moreover, the sufficient conditions under which a power based implication $I^T$ is $T^{*}$-conditional are discussed, this discussions...

The sequence of balancing numbers $\left(B_n\right)$ is defined by the recurrence relation $B_n=6B_{n-1}-B_{n-2}$ for $n\ge 2$ with initial conditions $B_0=0$ and $B_1=1.$ $B_n$ is called the $n$th balancing number. In this paper, we find all repdigits in the base $b,$ which are sums of four balancing numbers. As a result of our theorem, we state that if $B_n$ is repdigit in the base $b$ and has at least two digits, then $(n,b)=(2,5),(3,6)$. Namely, $B_2=6={\left(11\right)}_5$ and $B_3=35={\left(55\right)}_6.$

For each n ≥ 1, let $v_{n,k},k\ge 1$ and $u_{n,k},k\ge 1$ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let $X{ₙ}^B\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(v_{n,j}-v̅ₙ)$, $X{ₙ}^A\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(u_{n,j}-u̅̅ₙ)$, t ≥ 0, and $Xₙ=X{ₙ}^B-X{ₙ}^A$. The main result gives conditions under which the weak convergence $Xₙ\stackrel{}{\to }X$, where X is a Lévy process, implies $X{ₙ}^B\stackrel{}{\to }{X}^B$ and $X{ₙ}^A\stackrel{}{\to }{X}^A$, where $X^B$ and $X^A$ are mutually independent Lévy processes and $X=X^B-X^A$.

We study sums and products in a field. Let $F$ be a field with $\mathrm{ch}\left(F\right)\ne 2$, where $\mathrm{ch}\left(F\right)$ is the characteristic of $F$. For any integer $k\ge 4$, we show that any $x\in F$ can be written as $a_1+\cdots +a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1\cdots a_k=1$, and that for any $\alpha \in F\setminus \left\{0\right\}$ we can write every $x\in F$ as $a_1\cdots a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1+\cdots +a_k=\alpha$. We also prove that for any $x\in F$ and $k\in \{2,3,\cdots \}$ there are $a_1,\cdots ,a_{2k}\in F$ such that $a_1+\cdots +a_{2k}=x=a_1\cdots a_{2k}$.

We show that every operator from ${\ell }_s$ to ${\ell }_p\otimes ̂{\ell }_q$ is compact when 1 ≤ p,q < s and that every operator from ${\ell }_s$ to ${\ell }_p\widehat{\otimes ̂}{\ell }_q$ is compact when 1/p + 1/q > 1 + 1/s.

Let $X={\sum }_{i=1}^k{a}_i{U}_i$, $Y={\sum }_{i=1}^k{b}_i{U}_i$, where the $U_i$ are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and $a_i,b_i$ are real constants. We prove that if $b{²}_i$ is majorized by $a{²}_i$ in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp $L²-L^p$ Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts...