Related papers: Rational Solutions of First-Order Algebraic Ordina…
These notes aim to provide a classical approach to solving some conformable differential equations based on prior knowledge of how to solve ordinary differential equations. That is, using the methods of separation of variables, homogeneous…
An algebraic linear ordering is a component of the initial solution of a first-order recursion scheme over the continuous categorical algebra of countable linear orderings equipped with the sum operation and the constant 1. Due to a general…
A special class of autonomous algebraic differential equations is studied. No equations in the class have any entire transcendental solutions. In a sense, for almost all equations in the class, transcendental meromorphic solutions can also…
Robust optimization (RO) is one of the key paradigms for solving optimization problems affected by uncertainty. Two principal approaches for RO, the robust counterpart method and the adversarial approach, potentially lead to excessively…
Exactly solving first-order constraints (i.e., first-order formulas over a certain predefined structure) can be a very hard, or even undecidable problem. In continuous structures like the real numbers it is promising to compute approximate…
This work provides a method(an algorithm) for solving the solvable unary algebraic equation $f(x)=0$ ($f(x)\in\mathbb{Q}[x]$) of arbitrary degree and obtaining the exact radical roots. This method requires that we know the Galois group as…
We present an explicit basis for orders of arbitrary level N>1 in definite rational quaternion algebras. These orders have applications to computations of spaces of elliptic and quaternionic modular forms.
Let $f(t,y,y')=\sum_{i=0}^n a_i(t,y)y'^i=0$ be an irreducible first order ordinary differential equation with polynomial coefficients. Eremenko in 1998 proved that there exists a constant $C$ such that every rational solution of…
We study an optimization problem with the feasible set being a real algebraic variety $X$ and whose parametric objective function $f_u$ is gradient-solvable with respect to the parametric data $u$. This class of problems includes Euclidean…
When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find…
The aim of this work is to develop general optimization methods for finite difference schemes used to approximate linear differential equations. The specific case of the transport equation is exposed. In particular, the minimization of the…
This thesis focuses on developing and analyzing accelerated and inexact first-order methods for solving or finding stationary points of various nonconvex composite optimization (NCO) problems. The main tools mainly come from variational and…
Complexity and decidability of logics is a major research area involving a huge range of different logical systems. This calls for a unified and systematic approach for the field. We introduce a research program based on an algebraic…
In the first paper (part I) of this series of two, we introduce four novel definitions of the ODT problems: three for size-constrained trees and one for depth-constrained trees. These definitions are stated unambiguously through executable…
This paper presents a novel approach to automatically solving arithmetic word problems. This is the first algorithmic approach that can handle arithmetic problems with multiple steps and operations, without depending on additional…
For each $n$, let RD$(n)$ denote the minimum $d$ for which there exists a formula for the general polynomial of degree $n$ in algebraic functions of at most $d$ variables. In this paper, we recover an algorithm of Sylvester for determining…
We study the rational solutions of the Abel equation $x'=A(t)x^3+B(t)x^2$ where $A,B\in C[t]$. We prove that if $deg(A)$ is even or $deg(B)>(deg(A)-1)/2$ then the equation has at most two rational solutions. For any other case, an upper…
We explore in detail a method to solve ordinary differential equations using feedforward neural networks. We prove a specific loss function, which does not require knowledge of the exact solution, to be a suitable standard metric to…
Using an optimization algorithm to solve a machine learning problem is one of mainstreams in the field of science. In this work, we demonstrate a comprehensive comparison of some state-of-the-art first-order optimization algorithms for…
When a system of first order linear ordinary differential equations has eigenvalues of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The cost of representing…