Related papers: New Indefinite Summation Formulas and Some Applica…
We show that for any sequence $f: {\bf N} \to \{-1,+1\}$ taking values in $\{-1,+1\}$, the discrepancy $$ \sup_{n,d \in {\bf N}} \left|\sum_{j=1}^n f(jd)\right| $$ of $f$ is infinite. This answers a question of Erd\H{o}s. In fact the…
Computing the permanent of a non-negative matrix is a computationally challenging, \#P-complete problem with wide-ranging applications. We introduce a novel permanental analogue of Schur's determinant formula, leveraging a newly defined…
Solutions of nonlinear functional equations are generally not expressed as a finite number of combinations and compositions of elementary and known special functions. One of the approaches to study them is, firstly, to find formal solutions…
We consider analytic functions of the Riemann zeta type, for which, if $s$ is a zero, so is $1-s$. We use infinite product representations of these functions, assuming their zeros to be of first order. We use exponential factors to…
Let $\tau$ denote the divisor function, and $f$ be any multiplicative function that satisfies some mild hypotheses. We establish the asymptotic formula or non-trivial upper bound for the shifted convolution sum $\sum_{n \leq…
Solving nonlinear SMT problems over real numbers has wide applications in robotics and AI. While significant progress is made in solving quantifier-free SMT formulas in the domain, quantified formulas have been much less investigated. We…
In this paper, we give explicit exponential estimates $\displaystyle |x(t)|\leq M e^{ -\gamma (t-t_0) }$, where $t\geq t_0$, $M>0$, for solutions of a linear scalar delay differential equation $$ \dot{x}(t)+\sum_{k=1}^m…
In this paper, by using the spectral theory of functions and properties of evolution semigroups, we establish conditions on the existence, and uniqueness of asymptotic 1-periodic solutions to a class of abstract differential equations with…
We establish quantitative compactness estimates for finite difference schemes used to solve nonlinear conservation laws. These equations involve a flux function $f(k(x,t),u)$, where the coefficient $k(x,t$ is $BV$-regular and may exhibit…
The theory of summability of divergent series is a major branch of mathematical analysis that has found important applications in engineering and science. It addresses methods of assigning natural values to divergent sums, whose…
The special case of the hypergeometric function $_{2}F_{1}$ represents the binomial series $(1+x)^{\alpha}=\sum_{n=0}^{\infty}(\:\alpha n\:)x^{n}$ that always converges when $|x|<1$. Convergence of the series at the endpoints, $x=\pm 1$,…
This paper is concerned with the study of the fractional finite sums theory. We present the classes of functions for which it is possible to characterize the constant related to the derivative of fractional sums (denominated by essence of a…
An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the…
This paper gives the existence and uniqueness results for solution of fractional differential equations with Hilfer derivative. Using some new techniques and generalizing the restrictive conditions imposed on considered function, the…
The Dirichlet lambda function $\lambda(s)$ is defined for $\mathrm{Re}(s) > 1$ by \[ \lambda(s) = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^s}. \] This function was initially studied by Euler on the real line, where he denoted it by $N(s)$. In…
Although being powerful, the differential transform method yet suffers from a drawback which is how to compute the differential transform of nonlinear non-autonomous functions that can limit its applicability. In order to overcome this…
We study sharp conditions for the existence and nonexistence of infinitely many nonnegative solutions to the problem $-\Delta_p u = \lambda f(u)$ in a bounded domain with Dirichlet boundary conditions, where $f$ is a continuous function…
This paper deals with the numerical methods for the reconstruction of source term in linear parabolic equation from final overdetermination. We assume that the source term has the form f(x)h(t) and h(t) is given, which guarantees the…
We derive new reduction formulas for the incomplete beta function and the Lerch transcendent in terms of elementary functions. As an application, we calculate some new integrals. Also, we use these reduction formulas to test the performance…
Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been…