Related papers: Calculating power integral bases by solving relati…
We write a computer program that uses the recursion relation to calculate wave function in the harmonic-oscillator potential for specified values of E/hv (with its deviation 0.001) containing only even numbers of v (0,2,4,...). In this…
In the design flow of integrated circuits, chip-level verification is an important step that sanity checks the performance is as expected. Power grid verification is one of the most expensive and time-consuming steps of chip-level…
A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for…
Robust solving of critical large power flow cases (with 50k or greater buses) forms the backbone of planning and operation of any large connected power grid. At present, reliable convergence with applications of existing power flow tools to…
This paper surveys and illustrates geometric methods for constructing normal bases allowing efficient finite field arithmetic. These bases are constructed using the additive group, the multiplicative group and the Lucas torus. We describe…
We introduce the notion of differential largeness for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of "tame" differential fields. We state several…
We develop a method for evaluation of A. Einstein's strength of systems of partial differential and difference equations based on the computation of Hilbert-type dimension polynomials of the associated differential and difference field…
We present new quasi-exactly solvable models with inverse quartic, sextic, octic and decatic power potentials, respectively. We solve these models exactly via the functional Bethe ansatz method. For each case, we give closed-form solutions…
The method of separation of variables can be used to solve many separable linear partial differential equations (LPDEs). Moreover, variable separation solutions usually are some trigonometric series. In the paper, base on some ideas of this…
The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical…
The skew-harmonic numbers are the partial sums of the alternating harmonic series, i.e. the expansion of log(2). We evaluate in closed form various power series and numerical series with skew-harmonic numbers. This provides a simultaneous…
We present a numerical algorithm for finding real non-negative solutions to polynomial equations. Our methods are based on the expectation maximization and iterative proportional fitting algorithms, which are used in statistics to find…
Loop calculations involve the evaluation of divergent integrals. Usually [1] one computes them in a number of dimensions different than four where the integral is convergent and then one performs the analytical continuation and considers…
Linear algebraic primitives are at the core of many modern algorithms in engineering, science, and machine learning. Hence, accelerating these primitives with novel computing hardware would have tremendous economic impact. Quantum computing…
We establish some upper bounds for the number of integer solutions to the Thue inequality $|F(x , y)| \leq m$, where $F$ is a binary form of degree $n \geq 3$ and with non-zero discriminant $D$, and $m$ is an integer. Our upper bounds are…
Let $n$ be a positive integer and let $0 < \alpha < n.$ In this paper, we continue our study of the integral equation $$ u(x) = \int_{R^{n}} \frac{u(y)^{(n+\alpha)(n-\alpha)}{|x - y|^{n-\alpha}}dy.$$ We mainly consider singular solutions in…
We obtain highly accurate solutions to the Thomas-Fermi equations for atoms and atoms in very strong magnetic fields. We apply the Pad\'e-Hankel method, numerical integration, power series with Pad\'e and Hermite-Pad\'e approximants and…
It is a classical fact that the exponential function is solution of the integral equation $ \int_0^X f(x)dx + f(0) =f(X)$. If we slightly modify this equation to $ \int_0^X f(x)dx+f(0)=f(\alpha X)$ with $\alpha\in ]0,1[$, it seems that no…
A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the…
In this note we consider differential equations driven by a signal $x$ which is $\gamma$-H\"older with $\gamma>1/3$, and is assumed to possess a lift as a rough path. Our main point is to obtain existence of solutions when the coefficients…