This paper establishes a comprehensive theoretical framework for the approximation and decomposition of functions within the square-integrable space L^2([0,1]) by utilizing fractal structures. We focus on investigating the structural and approximation properties of functional spaces, denoted as V_n, which are generated through the translations and rescalings of a family of generating functions B over the elements of a fractal partition Gamma. Within this general setting, classical techniques such as Fourier series decomposition emerge naturally as special cases of our proposed decomposition scheme.

A central result of this study concerns the necessary conditions for universal approximation when the family consists of a single generator B. We prove a significant rigidity condition: the system constitutes a dense fractal system in L^2 if and only if B is constant almost everywhere. Furthermore, we derive explicit quantitative bounds for the speed of convergence when the approximation spaces are spanned by piecewise polynomials and trigonometric polynomials adapted to the fractal structure. By extending the classical theorems of Jackson and Bernstein–Walsh to the fractal domain, we demonstrate that the error decay depends on the regularity of the target function and the geometry of the structure. Specifically, for analytic functions, the error decays exponentially based on the scaling factor and structure level, while for differentiable functions, the error decays polynomially with respect to the degree and geometrically with the level. Finally, we discuss the representation of functions as series expansions and extend the fundamental density results to general metric spaces endowed with a finite Radon measure.

In recent years, fuzzy metric spaces and fuzzy partial metric spaces have attracted considerable attention due to their ability to generalize classical metric structures and model uncertainty in a more flexible and nuanced manner. In particular, fuzzy partial metrics extend fuzzy metrics by allowing different self-distances for points. Considering that each point's self-distance may vary in fuzzy partial metric spaces, the concept of an open ball is defined accordingly, differing from its definition in classical fuzzy metric spaces. One of the central problems in this area is the construction and comparison of topologies induced by different notions of open balls. In the existing literature, there are two commonly used notions of open balls, and it has been shown that a topology can be induced using only one of these concepts. In this study, we introduce a new notion of open balls in fuzzy partial metric spaces and demonstrate that a topology can be constructed from a fuzzy partial metric via the proposed concept. We further compare this notion with other variants of open balls and examine the corresponding induced topologies. Additionally, we investigate fundamental properties of this topology, including completeness, metrizability, and compactness, providing insight into how classical topological properties extend to the fuzzy partial metric setting.

Many of the properties and aspects of nature should be understood in an inherently dynamical context. Some of them as a product or producers of dynamism, while others are to be understood as the lack of some changes that are expected to happen, as patterns of resistance to change. For this, it is pivotal to have a formal understanding of the basic unity of change, i.e., transformations. We define a transformation as a duple t = (K,s) of a set K of characteristics and a finite sequence s of entities with no constant steps, such that there is a characteristic c∈K that remains constant in each step. The concept of composition of two transformations is defined as the duple formed by the union of the sets of characteristics and the concatenation of the sequences. If we label with V the space of transformations, a subspace of V is any set S⊂V that is closed under composition of elements. Let <e> be the set of all the transformations that have e as the first element of the sequence; then, <e> is a subspace of V. It is proved that there is a category W whose set of objects is the space of entities. The set of morphisms Mor(e₁,e₂) is the set of transformations of e₁ into e₂. For every triple of entities e₁, e₂, and e₃, a suitable restriction of the composition is the composition of morphisms between Mor(e₁,e₂) and Mor(e₂,e₃). Each identity morphism is a trivial transformation with only one step. Finally, we approach the subject of patterns of resistance to change in terms of this theory of transformations, thus refining our understanding of the wholeness and unity of nature.

In my programme of covariant compactification, I seek to explain all the fundamental fields of physics as geometric quantities. This is based on the work of Kaluza and Klein to extend general relativity.

In this talk, I explore how the geometry of product spacetimes determines their diffeomorphism and covariance groups, and the general linear and (pseudo-)orthogonal actions these induce on tangent spaces. I show how tensors decompose and gauge fields arise when transformation groups are non-linearly realised.

I explain how the topology of factor spaces determines their harmonics, which are manifested as matter fields. With an appropriate choice of geometry, these can be identified with the known fermions, as each harmonic can be labelled by its quantum numbers.

Unlike most post-1960 Kaluza–Klein theories, unitary gauge transformations do not act directly on the additional dimensions. Instead, orthogonal diffeomorphisms on compact factor spaces have a ‘dragging action’ on fields defined on these spaces. The induced actions on multiplets of harmonics are unitary representations of the orthogonal groups. This includes spinor representations. This allows us to identify multiplets of harmonics with spinors, on which the unitary gauge transformations act.

And in contrast to perceived wisdom about modern Kaluza–Klein theories, symmetry restoration does not take place at high energies. Instead, it occurs at the zero-curvature ‘decompactification limit’, in which all dimensions appear on the same footing.

Introduction: Zero-divisors are essential for understanding the structural intricacies of commutative rings and their algebraic properties. A zero-divisor in a commutative ring R is a non-zero element a such that there exists b ≠ 0 with ab = 0. Beyond detecting non-reduced behavior, zero-divisors encode how ideals and annihilators interact, and they often reflect decomposition phenomena in residue rings such as Z_n.

Methods: We investigate zero-divisors using a graph-theoretic approach by constructing the zero-divisor graph Γ(R). Vertices are the non-zero zero-divisors of R, and two distinct vertices are adjacent exactly when their product is zero. We examine how quotients and finite direct products affect adjacency, and we compare rings using standard graph invariants (connectivity, diameter, clique number, chromatic number, and girth). Equitable partitions and spectral information (adjacency and Laplacian eigenvalues) are used as complementary descriptors to compare graphs efficiently and to highlight regularities in zero-product relations for finite examples.

Results: The combined viewpoint reveals patterns that distinguish rings with similar zero-divisor sets but different multiplication. We discuss zero-divisors in Noetherian and Artinian rings, links to algebraic geometry via annihilators, and relevance to applications in cryptography and network optimization where zero-product constraints naturally arise. Representative examples from Z_n and polynomial quotients over Z_n illustrate how modular arithmetic governs adjacency and clustering, and how product decompositions lead to predictable changes in distances and colorings.

Conclusions: By bridging foundational theory with recent advances, this work highlights open problems and proposes practical methodologies for studying zero-divisors, supporting further developments in commutative algebra and its applications.

Introduction: The set of positive integers admits a latent hierarchical structure beyond the familiar linear ordering. We introduce the Canonical Triple-Graph (CTG), a fixed directed graph on the positive integers defined by an admissible associator on odd integers: n = (2^k m - 1)/3, where 2^k m ≡ 1 (mod 3). This relation is interpreted algebraically as defining edges that exist a priori within a predetermined structure, not as steps of an iterative or dynamical process.

Method: We analyze the structural properties of the admissible associator and its induced adjacencies. Every odd integer not divisible by 3 admits infinitely many admissible exponents forming blocks of associates. These blocks decompose uniquely into canonical triples of the affine form (n, 4n+1, 16n+5), which expose uniform self-similarity throughout the structure. Even integers integrate canonically via 2-adic factorization, forming deterministic vertical pillars above their odd parts.

Results: The CTG forms a directed graph (forest) on the positive integers. The distinguished root 1 generates Block(1), an infinite set of odd integers each serving as the root of its own infinite self-similar tree with identical local structure governed by canonical triples. We prove structural completeness via a block-closure principle: every block-closed component must contain an element of Block(1), ensuring all odd integers belong to trees rooted in Block(1). The structure is acyclic with unique parenthood within the CTG.

Conclusions: The CTG provides a purely structural framework comprising infinitely many self-similar trees, all rooted in Block(1), organizing positive integers independent of numerical magnitude or dynamical interpretation. This reframes classical iteration questions as structural position and component membership within an a priori fixed combinatorial forest.

This paper develops a rigorous mathematical framework for understanding modern neural networks as infinite-dimensional dynamical systems acting on function spaces rather than on finite vectors. Neural operators—networks designed to learn mappings between spaces of functions, such as those arising in fluid mechanics or material science—are typically analysed numerically but lack a coherent theoretical foundation that connects their stability, generalization, and controllability. This study proposes a synthesis of nonlinear functional analysis, differential topology, and control theory to address this gap.

By modeling neural operators as nonlinear semigroups on Banach and Hilbert manifolds, the paper defines learning as a topological flow evolving in an infinite-dimensional state space. Training, in this formulation, is not merely optimization but a process of steering trajectories through a geometrically constrained function landscape. Stability is shown to depend not only on loss minimization but on the topological structure of attractors induced by the network architecture and regularization scheme.

The paper introduces a notion of topological generalization, where the ability of a neural operator to extrapolate across unseen physical regimes is governed by the homotopy class of its learned solution manifold. Using tools from degree theory and nonlinear spectral analysis, the study demonstrates how certain architectural choices enforce global geometric constraints that prevent chaotic overfitting even in highly underdetermined learning problems.

The control-theoretic implications are substantial. Training algorithms can be reinterpreted as feedback control laws acting on infinite-dimensional learning dynamics, opening the possibility of stability-certified and robustly controllable AI systems. This framework provides a mathematically principled route toward trustworthy machine learning for complex physical systems governed by partial differential equations.

By embedding AI within the deep structure of geometry, topology, and functional analysis, the paper reframes learning not as a black-box procedure but as a controllable mathematical process.

This paper introduces two novel concepts in module theory based on the structural behavior of annihilator ideals and their radicals: totally annihilator-dependent modules and a radical dependency relation between modules. These notions aim to generalize classical submodule dependence—such as linear or essential dependence—by incorporating the radical behavior of annihilator ideals into the algebraic analysis of modules over commutative Noetherian rings. Within the framework of radical dependency, denoted by $N_{1} \triangleleft_{rad} N_{2}$, we establish that two submodules are radically dependent if and only if their radical annihilators satisfy a mutual containment relation, specifically that either $\sqrt{Ann(N_{1})} \subseteq \sqrt{Ann(N_{2})}$ or $\sqrt{Ann(N_{2})} \subseteq \sqrt{Ann(N_{1})}$. We further provide a comprehensive characterization theorem for totally annihilator-dependent modules, proving that a finitely generated module $M$ satisfies this dependency condition if and only if its set of associated primes, $Ass(M)$, consists of a single prime ideal. Additionally, for finitely generated multiplication modules, we demonstrate that the radical of the annihilator of the sum of two nonzero submodules is equal to the sum of the radicals of their respective individual annihilators under the condition of a single associated prime. The study also explores the properties of radically supplemented modules and introduces the set $Z_{g}(M)$ to translate these dependency relations from submodules to module elements via annihilator radical equivalency classes. Several illustrative examples are provided to demonstrate the scope and limitations of these results, offering a new perspective for classifying modules through the lens of annihilator radicals and Krull dimension.

This study investigates a specific class of modules in terms of supplemented module theory, defined as $t$-$g$-radical supplemented modules. An $R$-module $M$ is called $t$-$g$-radical supplemented if every submodule of $M$ possesses a $g$-radical supplement that is also a $t$-summand of $M$. This research aims to provide a comprehensive structural analysis of these modules and establish several characterization theorems regarding their algebraic properties.

First, we establish the relationship between $t$-$g$-radical supplemented modules and classical supplemented modules. It is proved that if $M$ is a $t$-$g$-radical supplemented module such that $Rad_g(M)$ is a small submodule of $M$, then $M$ is necessarily a supplemented module. For finitely generated $R$-modules, we show that being $t$-$g$-radical supplemented implies being $t$-$g$-supplemented. One of the central results of this paper is the behavior of these modules under sums. We prove that a finite sum of $t$-$g$-radical supplemented modules remains $t$-$g$-radical supplemented. Furthermore, the preservation of this property under factor modules and homomorphic images is examined. Specifically, it is shown that for a distributive $t$-$g$-radical supplemented module $M$, every quotient module and every homomorphic image of $M$ inherits the $t$-$g$-radical supplemented property. Finally, we discuss the conditions under which every $t$-summand of $M$ is $t$-$g$-radical supplemented, particularly focusing on modules satisfying the $(D3)$ property or the Summand Sum Property (SSP).

Introduction
Classical information geometry provides a differential-geometric framework for statistical models based on structures such as statistical manifolds and the Fisher–Rao metric. However, time is typically treated as an external parameter, which limits the representation of evolving statistical systems and dynamic probabilistic structures.

Methods
We propose a conceptual extension called Time-Embedded Information Geometry (TEIG), in which temporal evolution is incorporated directly into the structure of the statistical model. Instead of treating time as an external variable, it is embedded as an additional coordinate within the model. This formulation enables a unified geometric representation that captures both parameter variation and temporal change, while preserving the underlying statistical structure.

Results
We show that this formulation remains compatible with the classical Fisher–Rao geometry and provides a consistent way to describe time-dependent statistical configurations. The proposed framework maintains structural coherence with existing theory while extending its expressive capability. In particular, when temporal effects are neglected, the framework reduces naturally to standard information geometry.

Conclusions
TEIG offers a geometrically consistent extension of Fisher–Rao-based models and provides a new perspective for analyzing time-dependent statistical systems. This approach may contribute to broader applications in mathematical modeling, dynamical systems, and computational science. Furthermore, it provides a flexible framework for integrating temporal dynamics into geometric modeling.