Related papers: Riemann surfaces for integer counting processes
For any analytically finite Riemann surface, the Teichm\"uller modular group is countable, but it is not easy to find an analytically infinite Riemann surface for which the Teichm\"uller modular group is countable. In this paper, we show…
We conjecture the full asymptotic expansion of a product of Riemann zeta functions, evaluated at the non-trivial zeros of the zeta function, with shifts added in each argument. By taking derivatives with respect to these shifts, we form a…
We describe a method to perform functional operations on probability distributions of random variables. The method uses reproducing kernel Hilbert space representations of probability distributions, and it is applicable to all operations…
We derive recursions for the probability distribution of random sums by computer algebra. Unlike the well-known Panjer-type recursions, they are of finite order and thus allow for computation in linear time. This efficiency is bought by the…
The problem of estimating the probability of a random process reaching a certain level is well known. In this article, two-sided estimates are established for the probability that a regenerative process reaches a high level. Two auxiliary…
The crosscap number of a knot is an invariant describing the non-orientable surface of smallest genus that the knot bounds. Unlike knot genus (its orientable counterpart), crosscap numbers are difficult to compute and no general algorithm…
Two discrete dynamical systems are discussed and analyzed whose trajectories encode significant explicit information about a number of problems in combinatorial probability, including graphical enumeration on Riemann surfaces and random…
Could elementary complex analysis, which covers the topics such as algebra of complex numbers, elementary complex functions, complex differentiation and integration, series expansions of complex functions, residues and singularities, and…
In this paper, we introduce and develop the circle embedding method. This method hinges essentially on a combinatorial-geometric structure which we choose to call circles of partition. We provide applications in the context of problems that…
We find a system of two polynomial equations in two unknowns, whose solution allows to give an explicit expression of the conformal representation of a simply connected three sheeted compact Riemann surface onto the extended complex plane.…
We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. We propose a wide class of recursive estimation procedures for the general…
We extend work of Heath-Brown and Salberger, based on the determinant method, to provide a uniform upper bound for the number of integral points of bounded height on an affine surface, which are subject to a polynomial congruence condition.…
We study countable graphs that -- up to isomorphism and with probability one -- arise from a random process, in a similar fashion as the Rado graph. Unlike in the classical case, we do not require that probabilities assigned to pairs of…
For nice functions, invariant means over integral currents (certain generalized surfaces), can be uniquely defined.
The Riemann surface for polylogarithms of half-integer index, which has the topology of an infinite dimensional hypercube, is studied in relation to one-dimensional KPZ universality in finite volume. Known exact results for fluctuations of…
We realize the Apollonian group associated to an integral Apollonian circle packings, and some of its generalizations, as a group of automorphisms of an algebraic surface. Borrowing some results in the theory of orbit counting, we study the…
We prove that every orientable infinite type surface without boundary and finite genus has a Riemann surface structure such that its modular group of quasiconformal homeomorphisms is countable.
Given a covering of the projective line with ramifications defined over a number field, we define a plain model of the algebraic curve realizing the Riemann existence theorem for this covering, and bound explicitly the defining equation of…
We present an overview of the role of generating functions in quantum mechanical contexts, mainly in the modern theory of polarization and in the study of quantum phase transitions. Generating functions enable the derivation of moments and…
We give a method of counting the number of curves with a given type of singularity in a suitably ample linear series on a smooth surface using punctual Hilbert schemes. The types of singulaties for which our methods suffice include the…