This paper addresses the challenge of extending general finite sequences of real numbers within a subinterval of the real line, maintaining their inherent statistical properties by employing machine learning. Our focus lies on preserving the gap distribution and pair correlation function of these point sets. Leveraging advancements in deep learning applied to point processes, this paper explores the use of an auto-regressive \textit{Sequence Extension Mixture Model} (SEMM) for extending finite sequences, by estimating directly the conditional density, instead of the intensity function. We perform comparative experiments on multiple types of point processes, including Poisson, locally attractive, and locally repelling sequences, and we perform a case study on the prediction of Riemann $\zeta$ function zeroes. The results indicate that the proposed mixture model outperforms traditional neural network architectures in sequence extension with the retention of statistical properties. Given this motivation, we showcase the capabilities of a mixture model to extend sequences, maintaining specific statistical properties, i.e. the gap distribution, and pair correlation indicators.

Triangles are everywhere in the virtual world. The surface of nearly every graphical object is saved as a triangular mesh on a computer. Light effects and movements of virtual objects are computed on the basis of triangulations. Besides computer graphics, triangulated surfaces are used for the simulations of physical processes, like heating or cooling of objects or deformations. The numerical method for these simulations is often the finite element method, whose accuracy depends on the quality of the triangulation. The quality of a triangle is generally determined by computing its proximity to an equilateral triangle. Namely, the triangle's inner angles should neither be too small nor too big in order to obtain reliable numerical results. Therefore, one often improves the mesh quality before any simulation. The fact that we require triangulations for accurate simulations is the main motivation for our occupation with triangle transformations. We need a triangulation method that transforms each triangle into a more regular one. However, the transformation should not regularize a particular triangle too fast as this may inhibit that the regularity a neighboring triangles can achieve. At the same time, we would like to prove the efficacy of the transformation. a property often missed by the heuristic procedures used in practice. Besides the practical motivation, the transformation itself exhibits interesting properties which can nicely be proved by basic mathematics.

In this note we are interested in the rich geometry of the graph of a curve $\gamma_{a,b}: [0,1] \rightarrow \mathbb{C}$ defined as \begin{equation*} \gamma_{a,b}(t) = \exp(2\pi i a t) + \exp(2\pi i b t), \end{equation*} in which $a,b$ are two different positive integers. It turns out that the sum of only two exponentials gives already rise to intriguing graphs. We determine the symmetry group and the points of self intersection of any such graph using only elementary arguments and describe various interesting phenomena that arise in the study of graphs of sums of more than two exponentials.

Some methods based on simple regularizing geometric element transfor10 mations have heuristically been shown to give runtime efficient and quality effective 11 smoothing algorithms for meshes. We describe the mathematical framework and 12 a systematic approach to global optimization-based versions of such methods for 13 mixed volume meshes. In particular, we identify efficient smoothing algorithms for 14 certain algebraic mesh quality measures. We also provide explicit constructions of 15 potentially useful smoothing algorithms. 16

We want to introduce another smoothing approach by treating each geometric element as a player in a game: a quest for the best element quality. In other words, each player has the goal of becoming as regular as possible. The set of strategies for each element is given by all translations of its vertices. Ideally, he would like to quantify this regularity using a quality measure which corresponds to the utility function in game theory. Each player is aware of the other players' utility functions as well as their set of strategies, which is analogous to his own utility function and strategies. In the simplest case, the utility functions only depend on the regularity. In more complicated cases this utility function depends on the element size, the curvature, or even the solution to a differential equation. This article is a sketch of a possible game-theoretical approach to mesh smoothing and still on-going research.

We study the convergence of the parameter family of series: V α , β ( t ) = ∑ p p − α exp ( 2 π i p β t ) , α , β ∈ R > 0 , t ∈ [ 0 , 1 ) defined over prime numbers p and, subsequently, their differentiability properties. The visible fractal nature of the graphs as a function of α , β is analyzed in terms of Hölder continuity, self-similarity and fractal dimension, backed with numerical results. Although this series is not a lacunary series, it has properties in common, such that we also discuss the link of this series with random walks and, consequently, explore its random properties numerically.

We present a novel approach to mesh smoothing, called GETOpt, by using a variation of the Geometric Element Transformation Method (GETMe) which falls within the framework of global optimization-based schemes and mesh quality improvement methods. We introduce a global mesh quality measure, which is inspired by rigorous convergence results concerning the size of the geometric elements (non-degeneracy and comparable sizing) and whose gradient flow yields a combination of GETMe and Laplace smoothing. Under this scheme, it can be shown rigorously that the tetrahedral mesh never degenerates. Furthermore, numerical tests show that the sizing of the mesh elements become more comparable and that the FEM gives better approximations in case the PDE problem being solved demonstrates isotropic physical behavior. More specifically, we present in this article an example where the mesh is used to solve Poisson's equation using the standard Galerkin finite element method with piecewise linear and order 2 functions on the low-order mesh.

This work contributes to the study of the non-trivial roots of the Riemann zeta function. In view of the Hilbert-Polya conjecture a series of self-adjoint operators on a Hilbert space is constructed whose eigenvalues approximate these roots.

This work contributes to the analysis of linear geometric polygon transformations with the aim to force a desired shape by an iterative procedure.

We introduce a smoothing algorithm for triangle, quadrilateral, tetrahedral and hexahedral meshes whose centerpiece is a simple geometric triangle transformation. The first part focuses on the mathematical properties of the element transformation. In particular, the transformation gives rise directly to a continuous model given by a system of coupled damped oscillations. Derived from this physical model, adaptive parameters are introduced and their benefits presented. The second part discusses the mesh smoothing algorithm based on the element transformation and its numerical performance on example meshes.

We suggest a way to produce higher-energy superposition states in a circular system of quantum wells. This is inspired by a link to convergence results for geometric transformations of polygons using circulant Hermitian matrices.

Mesh smoothing improves mesh quality by node relocation without altering mesh topology. Such methods play a vital role in finite element mesh improvement with a direct consequence on the quality of the discretized solution. In this work, an improved version of the recently proposed geometric element transformation method (GETMe) for mesh smoothing is presented. Key feature is the introduction of adaptive concepts, which improve the resulting mesh quality, reduce the number of parameters, and enhance the parallelization capabilities. Implementational aspects are discussed and results of a more efficient version are presented, which demonstrate that GETMe adaptive smoothing yields high quality meshes, is particularly fast, and has a comparably low memory profile. Furthermore, results are compared to those of other state-of-the-art smoothing methods.

Prime number related fractal polygons and curves are derived by combining two different aspects. One is an approximation of the prime counting function build on an additive function. The other is prime number indexed basis entities taken from the discrete or continuous Fourier basis.

We analyze the dynamical properties of a tetrahedron transformation on the space of non-degenerate tetrahedra which can be identified with the non-compact globally symmetric 8-dimensional space SL(3,R)/SO(3,R). We establish the existence of a local attractor which coincides with the set of regular tetrahedra and identify conditions which imply that the basin of attraction is the entire space. In numerical tests, these conditions are fulfilled for a large set of random tetrahedra.

We study the arithmetic function sopfr(n) (OEIS A001414) which gives the sum of prime factors (with repetition) of a number n. In particular we obtain the asymptotic formula ∑_n ≤ xsopfr(n) ∼π^2/12x^2/log x, which holds as well for the function sopf(n) (OEIS A008472) that just gives the sum of distinct prime factors of n. This asymptotic formula was already stated by R. Jakimcyuk which was brought to our attention after the completion of the first version of this manuscript.

A transformation based on mean curvature is introduced which morphs triangulated surfaces into round spheres.

We describe a simple geometric transformation of triangles which leads to an efficient and effective algorithm to smooth triangle and tetrahedral meshes. Our focus lies on the convergence properties of this algorithm: we prove the effectivity for some planar triangle meshes and further introduce dynamical methods to study the dynamics of the algorithm which may be used for any kind of algorithm based on a geometric transformation.

The signed volume function for polyhedra can be generalized to a mean volume function for volume elements by averaging over the triangulations of the underlying polyhedron. If we consider these up to translation and scaling, the resulting quotient space is diffeomorphic to a sphere. The mean volume function restricted to this sphere is a quality measure for volume elements. We show that the gradient ascent of this map regularizes the building blocks of hybrid meshes consisting of tetrahedra, hexahedra, prisms, pyramids, and octahedra, that is, the optimization process converges to regular polyhedra. We show that the ( normalized) gradient flow of the mean volume yields a fast and efficient optimization scheme for the finite element method known as the geometric element transformation method. Furthermore, we shed some light on the dynamics of this method and the resulting smoothing procedure both theoretically and experimentally.