Related papers: To Infinity and Beyond: Some ODE and PDE Case Stud…
We consider a system of charged particles moving on the real line driven by electrostatic interactions. Since we consider charges of both signs, collisions might occur in finite time. Upon collision, some of the colliding particles are…
Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool…
Numerical simulation of ordinary differential equations (ODEs) can be challenging when the system exhibits high accelerations and rapidly changing dynamics. Under these conditions the ODE solver often needs to take very small time steps in…
Blow-up in second and fourth order semi-linear parabolic partial differential equations (PDEs) is considered in bounded regions of one, two and three spatial dimensions with uniform initial data. A phenomenon whereby singularities form at…
The time evolution of dynamical systems is frequently described by ordinary differential equations (ODEs), which must be solved for given initial conditions. Most standard approaches numerically integrate ODEs producing a single solution…
We examine numerical rounding errors of some deterministic solvers for systems of ordinary differential equations (ODEs). We show that the accumulation of rounding errors results in a solution that is inherently random and we obtain the…
Positive linear programs (LP), also known as packing and covering linear programs, are an important class of problems that bridges computer science, operations research, and optimization. Despite the consistent efforts on this problem, all…
Numerical solving differential equations with fractional derivatives requires elimination of the singularity which is inherent in the standard definition of fractional derivatives. The method of integration by parts to eliminate this…
Partiality is a natural phenomenon in computability that we cannot get around. So, the question is whether we can give the areas where partiality occurs, that is, where non-termination happens, more structure. In this paper we consider…
Through Borel summation, one can often reconstruct an analytic solution of a problem from its asymptotic expansion. We view the effectiveness of Borel summation as a regularity property of the solution, and we show that the solutions of…
In this paper, we investigate the problem of synthesizing computable functions of infinite words over an infinite alphabet (data $\omega$-words). The notion of computability is defined through Turing machines with infinite inputs which can…
In previous literature, backward error analysis was used to find ordinary differential equations (ODEs) approximating the gradient descent trajectory. It was found that finite step sizes implicitly regularize solutions because terms…
Checking infinite-state systems is frequently done by encoding infinite sets of states as regular languages. Computing such a regular representation of, say, the set of reachable states of a system requires acceleration techniques that can…
We first describe a general class of optimization problems that describe many natural, economic, and statistical phenomena. After noting the existence of a conserved quantity in a transformed coordinate system, we outline several instances…
We consider linear programming (LP) problems in infinite dimensional spaces that are in general computationally intractable. Under suitable assumptions, we develop an approximation bridge from the infinite-dimensional LP to tractable finite…
I describe my path to unconventionality in my exploration of theoretical and applied aspects of computation towards revealing the algorithmic and reprogrammable properties and capabilities of the world, in particular related to applications…
In the last decade, the scientific community has devolved its attention to the deployment of data-driven approaches in scientific research to provide accurate and reliable analysis of a plethora of phenomena. Most notably, Physics-informed…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing…
We obtain bounded for all $t$ solutions of ordinary differential equations as limits of the solutions of the corresponding Dirichlet problems on $(-L,L)$, with $L \rightarrow \infty$. We derive a priori estimates for the Dirichlet problems,…
Since many real-world problems arising in the fields of compiler optimisation, automated software engineering, formal proof systems, and so forth are equivalent to the Halting Problem--the most notorious undecidable problem--there is a…