Related papers: A discrete Schwarzian derivative via circle packin…
High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of…
We extend the classical notion of the spherical depth in \mathbb{R}^k, to the important setup of data on a Riemannian manifold. We show that this notion of depth satisfies a set of desirable properties. For the empirical version of this…
Suppose that a transcendental meromorphic function in the plane has finitely many critical values, while its multiple points have bounded multiplicities, and its inverse function has finitely many transcendental singularities. Using the…
We give an interpretation of the holographic correspondence between two-dimensional $BF$ theory on the punctured disk with gauge group ${\rm PSL}(2,\mathbb R)$ and Schwarzian quantum mechanics in terms of a Drinfeld-Sokolov reduction. The…
These are notes on discrete mathematics for computer scientists. The presentation is somewhat unconventional. Indeed I begin with a discussion of the basic rules of mathematical reasoning and of the notion of proof formalized in a natural…
Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. In Euclidean space it is possible for every circle in such a packing to have integer radius of curvature,…
We consider the Schwarzian derivative $S_f$ of a complex polynomial $f$ and its iterates. We show that the sequence $S_{f^n}/d^{2n}$ converges to $-2(\partial G_f)^2$, for $G_f$ the escape-rate function of $f$. As a quadratic differential,…
This paper reviews a class of univariate piecewise polynomial functions known as discrete splines, which share properties analogous to the better-known class of spline functions, but where continuity in derivatives is replaced by (a…
The Schwarzian derivative parametrizes the fibres of the space of complex projective structures on a surface as vector bundle over its Teichm\"uller space. We study its behaviour on long complex projective tubes, and get estimates for the…
Just as for non-abelian gauge theories at strong coupling, discrete lattice methods are a natural tool in the study of non-perturbative quantum gravity. They have to reflect the fact that the geometric degrees of freedom are dynamical, and…
The discrete cosine transform is a valuable tool in analysis of data on undirected rectangular grids, like images. In this paper it is shown how one can define an analogue of the discrete cosine transform on triangles. This is done by…
We derive the general rules of functional integration in the theories of the Schwarzian type, and evaluate explicitly the functional integrals assigning correlation functions in the SYK model.
The Bayesian approach to inverse problems provides a practical way to solve ill-posed problems by augmenting the observation model with prior information. Due to the measure-theoretic underpinnings, the approach has raised theoretical…
A discrete formulation of the real-time path integral as the expectation value of a functional of paths with respect to a complex probability on a sample space of discrete valued paths is explored. The formulation in terms of complex…
Signal scaling is a fundamental operation of practical importance in which a signal is enlarged or shrunk in the coordinate direction(s). Scaling or magnification is not trivial for signals of a discrete variable since the signal values may…
A new set of discrete ordinates is proposed for one-dimensional radiative transfer in spheres with central symmetry. The set is structured with un-normalized circular functions. This resulted in a conservative and closed set of discrete…
We consider a Schwarzschild type solution in the discrete Regge calculus formulation of general relativity quantized within the path integral approach. Earlier, we found a mechanism of a loose fixation of the background scale of Regge…
Motivated by extending the functional stochastic calculus, to important functionals to which it does not apply, a notion of functional derivative along a curve is introduced. This new setting is developed by incorporating path-dependent…
In the context of the complex-analytic structure within the unit disk centered at the origin of the complex plane, that was presented in a previous paper, we show that the complete Fourier theory of integrable real functions is contained…
An abstract theory of ultradifferentiable sheafs is developed. Moreover, various applications to the theory of linear partial differential equations, differential geometry and, in particular, CR geometry are discussed.