Related papers: The error in an alternating series
An extension of sinc interpolation on $\mathbb{R}$ to the class of algebraically decaying functions is developed in the paper. Similarly to the classical sinc interpolation we establish two types of error estimates. First covers a wider…
New cases of the multiplicity conjecture are considered.
The aim of this note is to provide a simple proof of some well-known identities and recurrences relating classical Bernoulli and Euler numbers by using the Abel sum of the divergent series $\sum_{n=0}^\infty (-1)^{n} (n+1)^k$, $k$ a…
We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge's theorem valid for higher-dimensional varieties, generalizing a uniform version…
We generalize Llarull's scalar curvature comparison to Riemannian manifolds admitting metric connections with parallel and alternating torsion and having a nonnegative curvature operator on 2-vectors. As a byproduct, we show that Euler…
This paper does three things: It proves a central limit theorem for novel permutation statistics (for example, the number of descents plus the number of descents in the inverse). It provides a clear illustration of a new approach to proving…
We put forward a new method of constructing the complete ordered field of real numbers from the ordered field of rational numbers. Our method is a generalization of that of A. Knopfmacher and J. Knopfmacher. Our result implies that there…
A new series representation of the Madelung constant is given. We represent Madelung constant as a sum of an exact term plus an exponentially fast converging series. The remarkable result is that even if the series part is discarded, one…
The unfolding problem formulation for correcting experimental data distortions due to finite resolution and limited detector acceptance is discussed. A novel validation of the problem solution is proposed. Attention is drawn to fact that…
This work belongs to the framework of inverse problems with linear model. The resolution of this type of problem consists in minimizing (possibly under constraints) a function of discrepancy between the measurements and a physical model of…
The aim of this paper is to derive a summation formula for the alternating infinite series and an expression for zeta function by using hyperbolic secant random variables. These identities involve Euler numbers and are obtained by computing…
Predictability is an emerging metric that quantifies the highest possible prediction accuracy for a given time series, being widely utilized in assessing known prediction algorithms and characterizing intrinsic regularities in human…
In this note we give a new proof of a closed formula for the multivariable generating series of diagonally colored Young diagrams. This series also describes the Euler characteristics of certain Nakajima quiver varieties. Our proof is a…
This paper extends the earlier work on an oscillating error correction technique. Specifically, it extends the design to include further corrections, by adding new layers to the classifier through a branching method. This technique is still…
Assuming the Riemann Hypothesis, we derive explicit bounds for the error terms in short interval analogues of the prime number theorem and Mertens' theorems using a smoothing argument. Our results improve upon previous bounds in both…
A new formula for the probability that a standard Brownian motion stays between two linear boundaries is proved. A simple algorithm is deduced. Uniform precision estimates are computed. Different implementations have been made available…
We prove the occurrence of a phase transition accompanied by the emergence of cycles of diverging lengths in the random interchange process on the hypercube.
A classical result of Euler states that the tangent numbers are an alternating sum of Eulerian numbers. A dual result of Roselle states that the secant numbers can be obtained by a signed enumeration of derangements. We show that both…
This article gives some properties of intervals in $\mathbb{R}$ and discusses some problems involving intervals for which the concept of outer measure on $\mathbb{R}$ provides a more efficient solution than an elementary approach. The outer…
A novel approach to an old symmetry problem is developed. A new proof is given for the following symmetry problem, studied earlier.