Related papers: A note on unique continuation for discrete harmoni…
We prove that subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some asymptotically small sets on spheres, are bounded from above everywhere. It follows that subharmonic functions of…
In this article, a novel method to compute all discrete polyharmonic functions in the quarter plane for models with small steps, zero drift and a finite group is proposed. A similar method is then introduced for continuous polyharmonic…
A novel basis of discrete analytic polynomials on a rhombic lattice is introduced and the associated convolution product is studied. A class of discrete analytic functions that are rational with respect to this product is also described.
A discretized massless wave equation in two dimensions, on an appropriately chosen square lattice, exactly reproduces the solutions of the corresponding continuous equations. We show that the reason for this exact solution property is the…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
Transition from discrete to continuous Fourier series is studied for the functions becoming singular in the transition. Conditions are specified when summing replacement by integration is inadmissible.
A lattice action for QED is considered, where the derivatives in the Dirac operator are replaced by one-sided lattice differences. A systematic expansion in the lattice spacing of the one-loop contribution to the fermion self energy, vacuum…
We propose a necessary and sufficient condition for a real-valued function on the real line to be a characteristic function of a probability measures. The statement is given in terms of harmonic functions and completely monotonic functions.
This note contains the complete mathematical proof of the main Theorem of the paper "How continuous measurements in finite dimension are actually discrete" (quant-ph/0702068), thus showing that in finite dimension any measurement with…
The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions…
We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete…
We prove the existence of solitary waves in a lattice where all particles interact with each other by pair-wise repulsive forces that decay with distance. The variational existence proof is based on constrained optimization and provides a…
We propose a model for the quantum harmonic oscillator on a discrete lattice which can be written in supersymmetric form, in contrast with the more direct discretization of the harmonic oscillator. Its ground state is easily found to be…
It has been unknown whether Hirota's discrete Korteweg-de Vries equation and the lattice sine-Gordon equation have the consistency around a cube (CAC) property. In this paper, we show that they have the CAC property. Moreover, we also show…
We study the existence of separable infinite harmonic functions in any cone of R N vanishing on its boundary under the form u(r, $\sigma$) = r --$\beta$ $\omega$($\sigma$). We prove that such solutions exist, the spherical part $\omega$…
We prove that functions defined on a lattice in a finite dimensional torus with bounded finite differences can be smoothly extended to the whole torus, and relate the bounds on the extension's derivatives with bounds on the original…
A new realization of the conformal algebra is studied which mimics the behaviour of a statistical system on a discrete albeit infinite lattice. The two-point function is found from the requirement that it transforms covariantly under this…
A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was only partially resolved, but standard…
The analogy between monodromy in dynamical (Hamiltonian) systems and defects in crystal lattices is used in order to formulate some general conjectures about possible types of qualitative features of quantum systems which can be interpreted…
We prove an inequality for functions on the discrete cube extending the edge-isoperimetric inequality for sets. This inequality turns out to be equivalent to the following claim about random walks on the cube: Subcubes maximize 'mean first…