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We show that for every graph without isolated edge, the edges can be assigned weights from {1,2,3} so that no two neighbors receive the same sum of incident edge weights. This solves a conjecture of Karo\'{n}ski, Luczak, and Thomason from…
We obtain Marcinkiewicz-Zygmund strong laws of large numbers for weighted sums of pairwise positively quadrant dependent random variables stochastically dominated by a random variable $X \in \mathscr{L}_{p}$, $1 \leqslant p < 2$. We use our…
The main goal of this article is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result of Vereshchagin saying that $\limsup_n C(x|n)$ equals $C^{0'}(x)$. Then we use the…
The article is concerned with the problem of the additivity of the tensor rank. That is for two independent tensors we study when the rank of their direct sum is equal to the sum of their individual ranks. The statement saying that…
``Behind every limit theorem, there is an inequality'' said Kolmogorov. We say ``for every inequality, there is an approximate inequality under approximate regularity conditions.'' Suppose $X, X'$ are independent and identically distributed…
An important tool for statistical research are moment inequalities for sums of independent random vectors. Nemirovski and coworkers (1983, 2000) derived one particular type of such inequalities: For certain Banach spaces $(\B,\|\cdot\|)$…
Let $\lambda$ be the Liouville function, defined as $\lambda(n) := (-1)^{\Omega(n)}$ where $\Omega(n)$ is the number of prime factors of $n$ with multiplicity. In 2021, Helfgott and Radziwi{\l}{\l} proved that $$\sum_{n\leq x} \frac{1}{n}…
Kagan and Shalaevski 1967 have shown that if the random variables $X_1,\dots,X_n$ are independent and identically distributed and the distribution of $\sum_{i=1}^n(X_i+a_i)^2$ $a_i\in \mathbb{R}$ depends only on $\sum_{i=1}^na_i^2$ , then…
Let $\mathcal{D}=(d_n)_{n=1}^\infty$ be a bounded sequence of integers with $d_n\ge 2$ and let $(i, j)$ be a pair of strictly positive numbers with $i+j=1$. We prove that the set of $x \in \RR$ for which there exists some constant $c(x) >…
In this paper we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset $A$ of $\mathbb{Z}_n\setminus \{0\}$ of size $k$ such that $\sum_{z\in A} z\not= 0$, it is…
In this paper we prove that $\mathbb{P}(|X| \geq \sqrt{\text{Var}(X)}) \geq 7/32$ for every finite Rademacher sum $X$, confirming a conjecture by Hitczenko and Kwapie{\'n} from 1994, and improving upon results from Burkholder, Oleszkiewicz,…
There has not been an established mathematical measure of evidence. Some Bayesians have argued that probability can be an objectively correct measure of ``rational degrees of belief,'' which we do not distinguish from evidence. However,…
Recently, Li obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture. As a consequence, he showed the positivity of this sum. His result was based on a sieving…
The classical Berry-Esseen error bound, for the normal approximation to the law of a sum of independent and identically distributed random variables, is here improved by replacing the standardised third absolute moment by a weak norm…
Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2^[n/2].…
We present a short proof of a conjecture proposed by I. Ra\c{s}a (2017), which is an inequality involving basic Bernstein polynomials and convex functions. This proof was given in the letter to I. Ra\c{s}a (2017). The methods of our proof…
In 2015 Zhi-Wei Sun proposed the conjecture that any integer $n > 1$ admits a partition $n = x + y$ with integers $x, y >0$ such that $x + ny$ and $x^2 + ny^2$ are simultaneously prime. To approach this conjecture we use the method of…
The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer $n$ greater than $5$ is the sum of three primes. The present paper proves this conjecture. Both the ternary Goldbach conjecture and the binary, or…
We give a new, elementary proof of a key inequality used by Rudelson in the derivation of his well-known bound for random sums of rank-one operators. Our approach is based on Ahlswede and Winter's technique for proving operator Chernoff…
W. T. Gower generalized Hindman's Finite sum theorem over $X_{k}=\left\{ \left(n_{1},n_{2},\ldots,n_{k}\right):n_{1}\neq0\right\} $ by showing that for any finite coloring of $X_{k}$ there exists a sequence such that the Gower subspace…