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We study the correlators of irregular vertex operators in two-dimensional conformal field theory (CFT) in order to propose an exact analytic formula for calculating numbers of partitions, that is: 1) for given $N,k$, finding the total…
Evaluating an expectation value of an arbitrary observable $A\in{\mathbb C}^{2^n\times 2^n}$ through na\"ive Pauli measurements requires a large number of terms to be evaluated. We approach this issue using a method based on Bell…
The powers of matrices with Stirling number-coefficients are investigated. It is revealed that the elements of these matrices have a number of properties of the ordinary Stirling numbers. Moreover, "higher order" Bell, Fubini and Eulerian…
For any $n$ and $k$, we provide an explicit (that is, computable in polynomial time) example of integer $B_k$-sequence of size $n$ consisting of elements bounded by $n^{k+o(k)}$.
In 1957 Leo Moser published a problem in American Mathematical Monthly asking whether knowing the set of all pairwise sums of five numbers one could determine the original numbers. Problem was quickly generalized as "Is it always possible…
The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…
The theorem of Bell states that certain results of quantum mechanics violate inequalities that are valid for objective local random variables. We show that the inequalities of Bell are special cases of theorems found ten years earlier by…
In a recent letter, new representations were proposed for the pair of sequences ($\gamma,\delta$), as defined formally by Bailey in his famous lemma. Here we extend and prove this result, providing pairs ($\gamma,\delta$) labelled by the…
We look at extensions of formulas given by Jovovic and recently proved by Dhar on integer partitions where the smallest part occurs at least $m$ times and on integer partitions with fixed differences between the largest and smallest parts…
The classical Bernoulli numbers $B_m$ can be expressed using Stirling numbers of the second kind, and M. Kaneko extended this framework by defining poly-Bernoulli numbers ${\mathbb B}_m^{(k)}$, for which explicit formulas using the Stirling…
For enumerative problems, i.e. computable functions f from N to Z, we define the notion of an effective (or closed) formula. It is an algorithm computing f(n) in the number of steps that is polynomial in the combined size of the input n and…
In 1985, Edward Nelson, who formulated the theory of stochastic mechanics, made an interesting remark on Bell's theorem. Nelson analysed the latter in the light of classical fields that behave randomly. He found that if a stochastic hidden…
In this paper, we prove that the numerical-semigroup-gap counting problem is #NP-complete as a main theorem. A numerical semigroup is an additive semigroup over the set of all nonnegative integers. A gap of a numerical semigroup is defined…
In the paper, the authors review some explicit formulas and establish a new explicit formula for Bernoulli and Genocchi numbers in terms of Stirling numbers of the second kind.
Scale invariant scattering suggests that all Bernoulli numbers B_{2n} can be naturally partitioned, i.e., written as particular finite sums of same-signed, monotonic, rational numbers. Some properties of these rational numbers are discussed…
We analyse the proof of Bell's inequality and demonstrate that this inequality is related to one particular model of probability theory, namely Kolmogorov measure-theoretical axiomatics, 1933. We found a (numerical) statistical correction…
The representation of any integer as the sum of two cubes to a fixed modulus is always possible if and only if the modulus is not divisible by seven or nine. For a positive non-prime integer N there is given an inductive way to find its…
Answering a question of P. Erdos from 1965, we show that for every eps>0 there is a set A of n integers with the following property: every subset A' of A with at least (1/3 + eps)n elements contains three distinct elements x,y,z with x + y…
In a 1977 paper, Ringel conjectured a formula for the genus of the $n$-prism $K_n\times K_2$ and verified its correctness for about five-sixths of all values $n$. We complete this calculation by showing that, with the exception of $n = 9$,…
This article, written in fond memory of Herbert Saul Wilf (June 13, 1931- Jan. 7, 2012), explores integer partitions where each part shows up a different number of times than the other parts (if it shows up at least once), thereby making a…