Related papers: Some results for conjugate equations
We have shown previously that functionally fitted Runge-Kutta (FRK) methods can be studied using a convenient collocation framework. Here, we extend that framework to functionally fitted Runge-Kutta-Nystr\"om (FRKN) methods, shedding…
In this paper, we consider mixed sums of generalized polygonal numbers. Specifically, we obtain a finiteness condition for universality of such sums; this means that it suffices to check representability of a finite subset of the positive…
In this pedagogical note I will discuss one-loop integrals, where (i) different regions of the integration region lead to divergences and (ii) where these divergences cancel in the sum over all regions. These integrals cannot be calculated…
Motivated by applications in data science, we study partial differential equations on graphs. By a classical fixed-point argument, we show existence and uniqueness of solutions to a class of nonlocal continuity equations on graphs. We…
In this paper we study a class of anisotropic equations with a lower order term whose coefficients lay in Marcinkiewicz spaces. We prove some regularity results for local solutions requiring any control on the norm of the coefficients.
The global existence of classical solutions to strongly coupled parabolic systems is shown to be equivalent to the availability of an iterative scheme producing a sequence of solutions with uniform continuity in the BMO norms. Amann's…
We consider a beam equation in presence of a leading degenerate operator which is not in divergence form. We impose clamped conditions where the degeneracy occurs and dissipative conditions at the other endpoint. We provide some conditions…
We consider the limit set of generalised iterated function systems. Under the assumption of a natural potential, the so called cylinder function, we prove the existence of the invariant probability measure satisfying the equilibrium state.…
We introduce a hypergoemetirc series with two complex variables, which generalizes Appell's, Lauricella's and Kemp\'e de F\'eriet's hypergeometric series, and study the system of differential equations that it satisfies. We determine the…
Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of…
In a Hilbert space setting, we study the stability properties of the regularized continuous Newton method with two potentials, which aims at solving inclusions governed by structured monotone operators. The Levenberg-Marquardt…
This paper considers hyperbolic wave equations with non-local in time conditions involving integrals with respect to time. It is shown that regularity of the solution can be achieved for complexified problem with integral conditions…
We establish the global $C^{1, \alpha}$-regularity for functions in solution classes, whenever ellipticity constants are sufficiently close. As an application, we derive the global regularity result concerning the parabolic normalized…
The method of exhaustion is generalized to a simple formula that can be used to integrate functions under very general conditions, provided that the integral exists. Both a geometric proof (following the usual procedure for the method of…
We propose and analyze a regularization approach for structured prediction problems. We characterize a large class of loss functions that allows to naturally embed structured outputs in a linear space. We exploit this fact to design…
Multi-class systems having possibly both finite and infinite classes are investigated under a natural partial exchangeability assumption. It is proved that the conditional law of such a system, given the vector of the empirical measures of…
The Riccati equation method is used for study the behavior of solutions of the systems of two linear first order ordinary differential equations. All types of oscillation and regularity of these system are revealed. A generalization of…
We consider functional equations driven by linear fractional transformations, which are special cases of de Rham's functional equations. We consider Hausdorff dimension of the measure whose distribution function is the solution. We give a…
The recent significant enrichment of the Order Completion Method for nonlinear Systems of PDEs resulted in the global existence of generalized solutions to a large class of such equations. In this paper we investigate the existence and…
In the context of the correspondence between real functions on the unit circle and inner analytic functions within the open unit disk, that was presented in previous papers, we show that the constructions used to establish that…