Related papers: Smarandache Function Applied to Perfect Numbers
In this note, we derive a finite summation formula and an infinite summation formula involving Harmonic numbers of order up to some order by means of several definite integrals
In this paper, we introduce the notion of quasi-$F$-splitting for rings in mixed characteristic. By comparing quasi-$F$-splitting with perfectoid purity, we obtain a new inversion of adjunction-type result. Furthermore, we study the…
We calculate the partition function for "composite particles". For any finite number of states d, and in the following two cases: 1)all states have the same energy, 2)the energy is linearly distributed over the states, we transform the…
On any metric space, I provide an intrinsic characterization of those complex-valued functions which are uniform limits of Lipschitz functions. There are applications to function theory on complete Riemannian manifolds and, in particular,…
We explore some integrals associated with the Riesz function and establish relations to other functions from number theory that have appeared in the literature. We also comment on properties of these functions.
The problem as to when two noncommuting observables are considered to have the same value arises commonly, but shows a nontrivial difficulty. Here, an answer is given by establishing the notion of perfect correlations between noncommuting…
We systematically find conditions which yield locally uniform convergence in the Fourier inversion formula in one and higher dimensions. We apply the gained knowledge to the complex inversion formula of the Laplace transform to extend known…
In order to prove irrationality of \sqrt{2} by using only decimal expansions (and not fractions), we develop in detail a model of real numbers based on infinite decimals and arithmetic operations with them.
A new problem is studied, the concept of exactness of a second order nonlinear ordinary differential equations is established. A method is constructed to reduce this class into a first order equations. If the second order equation is not…
We study arithmetic inequalities for multiplicative, sub(super)-multiplicative, sub(super)-homogeneous functions. Applications for the classical arithmetic functions are pointed out.
We extend the asymptotic Samuel function of an ideal to a filtration and show that many of the good properties of this function for an ideal are true for filtrations. There are, however, interesting differences, which we explore. We study…
In this paper, we prove Smale's mean value conjecture by making use of quasiconformal deformations and holomorphic motions.
This work provides a complete characterization of congruent numbers in terms of Pythagorean triples. Specifically, we show that every congruent number can be written as $$\frac{nm\left(m-n\right)\left(m+n\right)}{\sigma^2}$$ were as…
A technique is presented for finding the classical Lagrange function corresponding to a given dispersion relation. This allows us to study the classical analogue of the Standard-Model Extension. Developments are discussed.
New estimates on the maximal function associated to the linear Schrodinger equation are established
This paper introduced a way of fractal to solve the problem of taking count of the integer partitions, furthermore, using the method in this paper some recurrence equations concerning the integer partitions can be deduced, including the…
In this paper, we study the asymptotic behavior of sums of functions of the increments of a given semimartingale, taken along a regular grid whose mesh goes to 0. The function of the $i$th increment may depend on the current time, and also…
We survey old and recent results on the problem of finding a complete set of rules describing the behavior of the power function, i.e. the function which takes a cardinal $\kappa$ to the cardinality of its power $2^\kappa$.
We study integral estimates of maximal functions for Schr\"odinger means.
We show two results of mean value problem, Smale's mean value problem is comprehensively solved in this paper.