Explore the vast range of titles available.

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 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.

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.

Semantic Field Theory (SFT) has developed from a philosophical critique of strong anti-formalist readings of language games into a proposed computational model class for lexical semantics, higher order composition, and stabilized interpretation. This paper reconstructs that evolution and gives SFT a sharper mathematical core suitable for independent evaluation in computational linguistics and representation learning. The central proposal is that a tractable level of linguistic organization can be modeled through lexical representations expressed as semantic fields, through contextual deformation of those fields, through interaction terms defined over subsets of tokens, and through stabilization governed by semantic energy dynamics. The paper contributes five formal elements. First, it defines a semantic field model as a tuple consisting of a semantic space, a lexical field lifting, a contextual deformation map, an interaction complex, and an interpretation functional. Second, it proves a Gaussian product closure result showing that multiplicative field interactions have explicit centers, precisions, and compatibility factors. Third, it generalizes the three-word problem by using Mobius inversion on the subset lattice to isolate irreducible semantic interactions of arbitrary order. Fourth, it introduces an order spectrum that measures how much field mass is explained at each interaction order. Fifth, it formulates stabilized interpretation as minimization of an energy functional associated with the sentence and gives existence, descent, and stability conditions. A small worked example shows how a three-word summer day triple can be represented by Gaussian semantic fields, implemented in Python, and summarized by a flow diagram. The result is not a completed theory of natural language meaning and does not replace social, pragmatic, or normative accounts of language.

We study planar curves defined by finite Fourier series of the form \\(F_n(t)=_p n v_p(n!)\\, e^i p t\\), where the frequencies are the prime numbers and \\(v_p(n!)\\) denotes the exponent of the prime \\(p\\) in the factorization of \\(n!\\). We establish several rigorous obstructions to uniform geometric regularity as \\(nınfty\\). In particular, we prove that the curve lengths grow without bound, that neither the first nor the second derivatives remain uniformly bounded, and that the diameters grow at least on the order of \\(n n\\). As a consequence, the covering numbers of the curves satisfy explicit quantitative lower bounds. These results provide a rigorous explanation for the complex geometric behavior observed in numerical investigations of this model.

We present a comparative experimental study of planar curves arising from a Fourier series whose frequencies are the prime numbers, together with several randomized control models. Starting from the series \\(F_n(t)=_p n v_p(n!)\\, e^i p t,~tın[- ]\\), introduced and motivated in a companion work, we investigate the geometric complexity of the associated planar curves obtained by sampling in the complex plane. To test whether the observed multiscale behavior reflects arithmetic structure or can be reproduced as a generic consequence of sparsity or density, we compare the prime frequency model with randomized alternatives, including random frequency sets, a Cramér type random model, and a shuffled coefficient model. Using consistent box counting protocols and Monte Carlo ensembles, we observe stable scale dependent behavior for the prime frequency curves that is not reproduced by the randomized models. All results are experimental and are presented as evidence motivating further theoretical investigation.

We introduce a rigorous arithmetic--spectral construction associating planar geometric objects with additive prime factor statistics. Let \\(sopfr(n)\\) denote the sum of prime factors of \\(n\\), counted with multiplicity, and define the summatory function \\(B(x) = _n x sopfr(n)\\). It is known that \\(B(x) ^2 x^212 x\\) as \\(x ınfty\\). We show that \\(B(n)\\) admits an exact prime-indexed decomposition \\(B(n) = _p n p\\, v_p(n!)\\), where \\(v_p(n!)\\) denotes the \\(p\\)-adic valuation of \\(n!\\). This identity motivates the definition of a sparse prime-indexed Fourier series \\(F_n(t) = _p n v_p(n!) e^i p t\\), which we investigate from analytic and geometric perspectives. We establish precise norm identities, relate the construction to circulant Hermitian polygon transformations whose eigenpolygons are discrete Fourier modes, and examine the planar geometry arising from sampled curves. All geometric observations are explicitly experimental. The results provide a rigorous arithmetic foundation for prime-related Fourier geometry and motivate further theoretical and experimental investigations.

Large language models (LLMs) offer a new empirical setting in which long-standing theories of linguistic meaning can be examined. This paper contrasts two broad approaches: social constructivist accounts associated with language games, and a mathematically oriented framework we call Semantic Field Theory. Building on earlier work by the author, we formalize the notions of lexical fields (Lexfelder) and linguistic fields (Lingofelder) as interacting structures in a continuous semantic space. We then analyze how core properties of transformer architectures-such as distributed representations, attention mechanisms, and geometric regularities in embedding spaces-relate to these concepts. We argue that the success of LLMs in capturing semantic regularities supports the view that language exhibits an underlying mathematical structure, while their persistent limitations in pragmatic reasoning and context sensitivity are consistent with the importance of social grounding emphasized in philosophical accounts of language use. On this basis, we suggest that mathematical structure and language games can be understood as complementary rather than competing perspectives. The resulting framework clarifies the scope and limits of purely statistical models of language and motivates new directions for theoretically informed AI architectures.

The geometric element transformation method (GETMe) is a geometry-based smoothing method for mixed and non-mixed meshes. It is based on a simple geometric transformation applicable to elements bounded by polygons with an arbitrary number of nodes. The transformation, if applied iteratively, leads to a regularization of the polygons. Global mesh smoothing is accomplished by averaging the new node positions obtained by local element transformations. Thereby, the choice of transformation parameters as well as averaging weights can be based on the element quality which leads to high quality results. In this paper, a concept of an enhanced transformation approach is presented and a proof for the regularizing effect of the transformation based on eigenpolygons is given. Numerical examples confirm that the GETMe approach leads to superior mesh quality if compared to other geometry-based methods. In terms of quality it can even compete with optimization-based techniques, despite being conceptually significantly simpler.

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.