Related papers: Infinitesimal and infinite numbers in applied math…
This article begins with a brief introduction to numerical relativity aimed at readers who have a background in applied mathematics but not necessarily in general relativity. I then introduce and summarise my work on the problem of treating…
Noether's calculus of invariant variations yields exact identities from functional symmetries. The standard application to an action integral allows to identify conservation laws. Here we rather consider generating functionals, such as the…
It is well known that general variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of unrelated problems, which arise in pure and applied sciences. In this paper, we present a number…
In this paper we provide a rigorous mathematical foundation for continuous approximations of a class of systems with piece-wise continuous functions. By using techniques from the theory of differential inclusions, the underlying piece-wise…
Asymptotic statistical theory for estimating functions is reviewed in a generality suitable for stochastic processes. Conditions concerning existence of a consistent estimator, uniqueness, rate of convergence, and the asymptotic…
We give a survey on classical and recent applications of dynamical systems to number theoretic problems. In particular, we focus on normal numbers, also including computational aspects. The main result is a sufficient condition for…
The discovery of the infinite integer leads to a partition between finite and infinite numbers. Construction of an infinitesimal and infinitary number system, the Gossamer numbers. Du Bois-Reymond's much-greater-than relations and…
Quasi-Noether differential systems are more general than variational systems and are quite common in mathematical physics. They include practically all differential systems of interest, at least those that have conservation laws. In this…
This short paper proposes to learn models of satisfiability modulo theories (SMT) formulas during solving. Specifically, we focus on infinite models for problems in the logic of linear arithmetic with uninterpreted functions (UFLIA). The…
The concept of random dynamical system is a comparatively recent development combining ideas and methods from the well developed areas of probability theory and dynamical systems. Due to our inaccurate knowledge of the particular physical…
We present an overview of some recent developments in the theory of generalized formal series, grounded in diffeological geometric framework. These constructions aim to offer new tools for understanding infinite-dimensional phenomena in…
A geometric setup for control theory is presented. The argument is developed through the study of the extremals of action functionals defined on piecewise differentiable curves, in the presence of differentiable non-holonomic constraints.…
Many mathematical models utilize limit processes. Continuous functions and the calculus, differential equations and topology, all are based on limits and continuity. However, when we perform measurements and computations, we can achieve…
This is a tutorial introduction to the functional analysis mathematics needed in many physical problems, such as in waves in continuous media. Functional analysis takes us beyond finite matrices, allowing us to work with infinite sets of…
While nonlinear optical spectroscopy is becoming more commonly used to study the excited states of nonlinear-optical systems, a general theory of inhomogeneous broadening is rarely applied in lieu of either a simple Lorentzian or Gaussian…
A mathematical smooth function means that the function has continuous derivatives to a certain degree C(k). We call it a k-smooth function or a smooth function if k can grow infinitively. Based on quantum physics, there is no such smooth…
Classical mathematics (involving such notions as infinitely small/large and continuity) is usually treated as fundamental while finite mathematics is treated as inferior which is used only in special applications. We first argue that the…
For continuous-time dynamical systems with reversible trajectories, the nowhere-vanishing eigenfunctions of the Koopman operator of the system form a multiplicative group. Here, we exploit this property to accelerate the systematic…
Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to…
We consider adaptive control problem in presence of nonlinear parametrization of uncertainties in the model. It is shown that despite traditional approaches require for domination in the control loop during adaptation, it is not often…