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Dimitris Vartziotis

Publications and source records attributed to Dimitris Vartziotis.

At least 19 recordsLinked to original sources
arXiv2026

Semantic Field Theory: Historical Origin, Higher-Order Interaction, and Stabilized Semantic Inference

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.

cs.CL↗
arXiv2026

Rigorous Geometric Obstructions for Fourier Curves Generated by Prime Numbers

We study planar curves defined by finite Fourier series of the form $F_n(t)=\sum_{p\le n} v_p(n!)\, e^{i p t}$, 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 $n\to\infty$. 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 $n\log\log n$. 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.

math.GM↗
arXiv2026

Spectral Geometry of Fourier Curves with Prime Frequencies: A Comparative Experimental Study

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 $F_n(t)=\sum_{p\le n} v_p(n!)\, e^{i p t},~t\in[-\pi,\pi]$, 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 Cram\'er 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.

math.GM↗
arXiv2026

Fourier Series Generated by Additive Prime Factor Functions

We introduce a rigorous arithmetic--spectral construction associating planar geometric objects with additive prime factor statistics. Let $\mathrm{sopfr}(n)$ denote the sum of prime factors of $n$, counted with multiplicity, and define the summatory function $B(x) = \sum_{n \le x} \mathrm{sopfr}(n)$. It is known that $B(x) \sim \frac{\pi^2 x^2}{12 \log x}$ as $x \to \infty$. We show that $B(n)$ admits an exact prime-indexed decomposition $B(n) = \sum_{p \le n} p\, v_p(n!)$, where $v_p(n!)$ denotes the $p$-adic valuation of $n!$. This identity motivates the definition of a sparse prime-indexed Fourier series $F_n(t) = \sum_{p \le n} v_p(n!) e^{i p t}$, 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.

math.GM↗
arXiv2026

Language as Mathematical Structure: Examining Semantic Field Theory Against Language Games

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.

cs.CL↗
arXiv2023

Learn2Extend: Extending sequences by retaining their statistical properties with mixture models

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.

cs.LG↗
arXiv2023

An Angular Transformation of Triangles

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.

math.HO↗
arXiv2020

Smoothing Game

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.

cs.GT↗
arXiv2018

On the symmetry of finite sums of exponentials

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.

math.NT↗
arXiv2017

Existence of an attractor for a geometric tetrahedron transformation

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 $\mbox{Sl}(3,\mathbb{R}) / \mbox{So}(3,\mathbb{R})$. 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.

math.DG↗
arXiv2017

Fractal curves from prime trigonometric series

We study the convergence of the parameter family of series $$V_{α,β}(t)=\sum_{p}p^{-α}\exp(2πi p^βt),\quad α,β\in \mathbb{R}_{>0},\; t \in [0,1)$$ 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. We also discuss the link of this series to random walks and consequently, explore numerically its random properties.

math.DS↗
arXiv2017

On sums of prime factors

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 $$ \sum_{n \leq x} \rm{sopfr}(n) \sim \frac{π^2}{12} \frac{x^2}{\log x},$$ 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 \cite{rj12} which was brought to our attention after the completion of the first version of this manuscript.

math.NT↗
arXiv2016

The fractal nature of an approximate prime counting function

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 are prime number indexed basis entities taken from the discrete or continuous Fourier basis.

math.NT↗
arXiv2016

Curvature transformation

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

cs.GR↗
arXiv2014

A geometric mesh smoothing algorithm related to damped oscillations

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.

math.NA↗