Related papers: Discretisation for odd quadratic twists
We consider linear inverse problems under white noise. These types of problems can be tackled with, e.g., iterative regularisation methods and the main challenge is to determine a suitable stopping index for the iteration. Convergence…
An interesting family of geometric integrators for Lagrangian systems can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators. In this…
Symmetric tridiagonal matrices appear ubiquitously in mathematical physics, serving as the matrix representation of discrete random Schr\"odinger operators. In this work we investigate the top eigenvalue of these matrices in the large…
Large-deviations theory deals with tails of probability distributions and the rare events of random processes, for example spreading packets of particles. Mathematically, it concerns the exponential fall-of of the density of thin-tailed…
A discrete version of the two-dimensional inverse scattering problem is considered. On this basis, algebraic transformations for the two-dimensional finite-difference Schredinger equation are elaborated.
In this work, we study the optimal discretization error of stochastic integrals, in the context of the hedging error in a multidimensional It\^{o} model when the discrete rebalancing dates are stopping times. We investigate the convergence,…
Near full-null degenerate singular points of analytic vector fields, asymptotic behaviors of orbits are not given by eigenvectors but totally decided by nonlinearities. Especially, in the case of high full-null degeneracy, i.e., the lowest…
Recent progress in holographic correspondence uncovered remarkable relations between key characteristics of the theories on both sides of duality and certain integrable models. In this note we revisit the problem of the role of certain…
We apply duality theory to discretized convex minimization problems to obtain computable guaranteed upper bounds for the distance of given discrete functions and the exact discrete minimizer. Furthermore, we show that the discrete duality…
Based on a new idea of factorization, we prove an improved discrete Rellich inequality and discuss its optimality. We also give a conjecture on improved higher order discrete Hardy-like inequalities and formulate an open problem for the…
There exists an extensive and fairly comprehensive discrete analytic function theory which is based on circle packing. This paper introduces a faithful discrete analogue of the classical Schwarzian derivative to this theory and develops its…
Most of the literature on the solution of linear ill-posed operator equations, or their discretization, focuses only on the infinite-dimensional setting or only on the solution of the algebraic linear system of equations obtained by…
Ordinary differential equations (ODEs) and ordinary difference systems (O$\Delta$Ss) invariant under the actions of the Lie groups $\mathrm{SL}_x(2)$, $\mathrm{SL}_y(2)$ and $\mathrm{SL}_x(2)\times\mathrm{SL}_y(2)$ of projective…
Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. This paper considers discrete Tracy-Widom…
We present an analytical technique to compute the probability of rare events in which the largest eigenvalue of a random matrix is atypically large (i.e.\ the right tail of its large deviations). The results also transfer to the left tail…
Discrete tomography is concerned with the reconstruction of images that are defined on a discrete set of lattice points from their projections in several directions. The range of values that can be assigned to each lattice point is…
We develop the approach to the problem of integrable discretization based on the notion of $r$--matrix hierarchies. One of its basic features is the coincidence of Lax matrices of discretized systems with the Lax matrices of the underlying…
N.Katz's middle convolution algorithm provides a description of rigid connections on the projective line with regular singularities. We extend the algorithm by adding the Fourier transform to it. The extended algorithm provides a…
In this paper, the weak convergence about the discretization error of stochastic iterated integrals in the Skorohod sense are studied, while the integrands and integrators of iterated integrals are supposed to be semimartingales with jumps.…
Matrix perturbation inequalities, such as Weyl's theorem (concerning the singular values) and the Davis-Kahan theorem (concerning the singular vectors), play essential roles in quantitative science; in particular, these bounds have found…