Related papers: A product representation of pi
We give a completely elementary study for averages of short correlations of so-called sieve functions (a pretty general class of arithmetic functions).
Although the characterization of ring derivations has an extensive literature, up to now, all of the characterizations have had the following form: additivity and another property imply that the function in question is a derivation. The aim…
The note contains a short elementary proof of Cayley's formula for labeled trees.
We give three derivations of Polya's approximation for the expected range of a simple random walk in one dimension. This result allows for an estimation of the volatility of a financial instrument from the difference between the high and…
A result about projections of Gibbs measures from a particular class arising in economic modeling is proved.
Our main result shows that for every odd natural number n > 1 there is a partition of the rational numbers, x/2n in lowest terms, in the unit interval, the product of whose Gamma values takes the form of powers of 2 and pi.
In this short note we introduce a new metric on certain finite groups. It leads to a class of groups for which the element orders satisfy an interesting inequality. This extends the class CP_2 studied in our previous paper [16].
This note deals with a simultaneous approximation of several matrices by a finite family of diagonalizable matrices satisfying an additional condition for the spectrum of a matrix product. That is the simplicity of all eigenvalues.
Bisets can be considered as categories. This note uses this point of view to give a simple proof of a Mackey-like formula expressing the tensor product of two induced bimodules.
This note gives a simple approach to q-analogues of some results associated with Abel polynomials.
This note aims to investigate the tensor product of two given Hilbert quasi *-algebras and its properties. The construction proposed in this note turns out to be again a Hilbert quasi *-algebra, thus interesting representability properties…
We show that Pinney's equation [2] with a constant coefficient can be reduced to its linear part by a simple change of variables. Also, Pinney's original solution is simplified slightly.
We present a generalization of a formula of higher order derivatives and give a short proof.
This short note present a "proof" of $P\neq NP$. The "proof" with double quotation marks is to indicate that we do not know whether the proof is correct or not (We're confused because we do know in which we make the mistakes).
Let $\xi$ be an algebraic number and let $\alpha,\beta\in \mathbb Q[\xi]$. An explicit formula for the coordinates of the product $\alpha\beta$ is given in terms of the coordinates of $\alpha$ and $\beta$ and the companion matrix of the…
This brief note concerns the invertibility of certain alternant matrices. In particular those that consisting of polynomials and products of polynomials and logarithms are shown to be invertible under appropriate conditions on the degrees…
By changing variables in a suitable way and using dominated convergence methods, this note gives a short proof of Stirling's formula and its refinement.
In this paper, we give a short proof of a relation generalizing many identities for Bernoulli numbers.
The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.
Stirling's formula is a powerful asymptotic approximation of the factorial function. Many well-known proofs of this formula are grounded in integral calculus. In this paper, we present an alternative proof of Stirling's formula using only…