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We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…

Coin flipping is a cryptographic primitive in which two distrustful parties wish to generate a random bit in order to choose between two alternatives. This task is impossible to realize when it relies solely on the asynchronous exchange of…

These notes describe some results on dice comparisons when changing the numbers on the faces while the sum of all the face stay the same.

In this paper we deal with a classical problem in elementary number theory, namely repeating decimals. We show how the digits of the period of the decimal representation of any fraction $\frac{k}{m}$, where $k$ and $m$ are positive integers…

This paper proposes new derivations of three well-known sorting algorithms, in their functional formulation. The approach we use is based on three main ingredients: first, the algorithms are derived from a simpler algorithm, i.e. the…

We show that any permutation of ${1,2,...,N}$ can be written as the product of two involutions. As a consequence, any permutation of the elements of an array can be performed in-place in parallel in time O(1). In the case where the…

Let $A(n,m)$ denote the Eulerian numbers, which count the number of permutations on $[n]$ with exactly $m$ descents. It is well known that $A(n,m)$ also counts the number of permutations on $[n]$ with exactly $m$ excedances. In this report,…

This article is a gentle introduction to the mathematical area known as circle packing, the study of the kinds of patterns that can be formed by configurations of non-overlapping circles. The first half of the article is an exposition of…

Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…

I propose two simple ways of generating the partitions of (n+1) from the partitions of n. A recurrence relation for P(n+1), the number of partitions of (n+1), in terms of P(n) and Q(n), where Q(n) denotes the number of partitions of n…

A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a…

Inflation is painful, for firms, customers, employees, and society. But careful study of periods of hyperinflation point to ways that firms can adapt. In particular, companies need to think about how to change prices regularly and cheaply,…

Using modular forms we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients $1$, $2$, $3$ or $6$.

We generalize the problem of coin flipping to more than two outcomes and parties. We term this problem dice rolling, and study both its weak and strong variants. We prove by construction that in quantum settings (i) weak N-sided dice…

A quantum money scheme enables a trusted bank to provide untrusted users with verifiable quantum banknotes that cannot be forged. In this work, we report an experimental demonstration of the preparation and verification of unforgeable…

Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion…

The set of common numerical and analytical problems is introduced in the form of the generalized multidimensional discrete Poisson equation. It is shown that its solutions with square-summable discrete derivatives are unique up to a…

Decision-making methods very often use the technique of comparing alternatives in pairs. In this approach, experts are asked to compare different options, and then a quantitative ranking is created from the results obtained. It is commonly…

We look at two possible routes to classical behavior for the discrete quantum random walk on the line: decoherence in the quantum ``coin'' which drives the walk, or the use of higher-dimensional coins to dilute the effects of interference.…

We count the number of subsets of $\{1,2,\cdots,n\}$ under different conditions and study the sequence obtained as we let $n$ increase.