Related papers: Discretization and superintegrability all rolled i…
We discuss integrable discretizations of 3-dimensional cyclic systems, that is, orthogonal coordinate systems with one family of circular coordinate lines. In particular, the underlying circle congruences are investigated in detail, and…
The Abelian Sandpile Model (ASM) is a game played on a graph realizing the dynamics implicit in the discrete Laplacian matrix of the graph. The purpose of this primer is to apply the theory of lattice ideals from algebraic geometry to the…
Semialgebraic splines are functions that are piecewise polynomial with respect to a cell decomposition into sets defined by polynomial inequalities. We study bivariate semialgebraic splines, formulating spaces of semialgebraic splines in…
In this note we discuss a class of hyperelliptic curves introduced by Abel in a 1826 paper. After some indications of the context in which he introduced them and a description of his main result we give some results on the moduli space of…
We consider a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation. We derive the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality…
Abel's continuity theorem for power series is a great tool to compute sums and prove properties of special functions. However, apart from two basic examples, this no longer occupies a place in Analysis courses. This note is an invitation to…
The result of performing integrations over connection type variables in the path integral for the discrete field theory may be poorly defined in the case of non-compact gauge group with the Haar measure exponentially growing in some…
Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the…
For a model convection-diffusion problem, we address the presence of oscillatory discrete solutions, and study difficulties in recovering standard approximation results for its solution. We justify the presence of non-physical oscillations…
We analyze the method for calculation of properties of non-relativistic quantum systems based on exact diagonalization of space-discretized short-time evolution operators. In this paper we present a detailed analysis of the errors…
We present a novel framework based on semi-bounded spatial operators for analyzing and discretizing initial boundary value problems on moving and deforming domains. This development extends an existing framework for well-posed problems and…
The systematic method for the conversion of first class constraints to the equivalent set of Abelian one based on the Dirac equivalence transformation is developed. The representation for the corresponding matrix performing this…
Mixed-dimensional partial differential equations arise in several physical applications, wherein parts of the domain have extreme aspect ratios. In this case, it is often appealing to model these features as lower-dimensional manifolds…
A general prescription for the treatment of constrained quantum motion is outlined. We consider in particular constraints defined by algebraic submanifolds of the quantum state space. The resulting formalism is applied to obtain solutions…
To solve the path integral for quantum gravity, one needs to regularise the space-times that are summed over. This regularisation usually is a discretisation, which makes it necessary to give up some paradigms or symmetries of continuum…
We give a detailed general description of a recent geometrical discretisation scheme and illustrate, by explicit numerical calculation, the scheme's ability to capture topological features. The scheme is applied to the Abelian Chern-Simons…
Let $C$ be a nodal curve and $L$ be an invertible sheaf on $C$. Let $\alpha_{L}:C\dashrightarrow J_{C}$ be the degree-$1$ rational Abel map, which takes a smooth point $Q\in C$ to $\left[ m_{Q}\otimes L\right] $ in the Jacobian of $C$. In…
Lavrent'ev regularization for the autoconvolution equation was considered by J. Janno in {\itshape Lavrent'ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution…
We develop a general framework for Abel maps associated with a family $X/S$ of integral curves using derived algebraic geometry. For compactified Picard schemes, our approach yields relative quasi-smooth derived enhancements of the Quot…
We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalisation of multiple polylogarithms, we construct our set of integration kernels ensuring…