Volume 35, No. 1 (2025)

ON THE INVARIANTS OF PARTIAL PARALLELIZABLE MANIFOLDS AND INTEGRABLE PARTIAL TRIVIAL STRUCTURES IN TANGENT BUNDLE

In this paper, the concepts of “partial parallelizable manifold” and “partial trivial tangent bundle” are fundamental notions. The aim of this paper is to obtain some properties about partial parallelizable manifolds (M, ρp) and their relations to integrable partial trivial structures in tangent bundle. In the study of the couple (M, ρp) an important operation on ρp is the construction of other geometric and topological objects: G(M)−invariants, homotopy invariants and their relations with Pontryagin classes of the tangent bundle TM. We give a description of Pontryagin algebra of the partial trivial tangent vector bundle of M: if F is a foliation of M, we show that TM is null in dimension d > 2 (m − p), where m = dim M and p = dim F . The case of integrable partial trivial structures is considered. If the subbundle E (ρp) generated from ρp is integrable, then there is a G(M)-invariant [M(ρp)] associated with (M, ρp). Finally, we give some examples.

APPLICATIONS OF GENERALIZED CLOSED SETS DEFINED USING THE FUZZY CLOSURE OF THE α-INTERIOR

 In this paper, a new type of fuzzy generalized closed set is defined via fuzzy α-open sets, called fgαθ-closed set. Complements of fgαθ -closed sets, called fgαθ -open sets form a family that is not a topology. Using this new type of generalized closed sets, the corresponding open functions and closed function are introduced and studied. A new separation property generalizing the T2-property of fuzzy topological space is shown to be invariant to bijective fgαθ -open functions. Many examples are given, to illustrate the new concepts and to distinguish these from known related concepts.

CONFORMAL TRANSFORMATIONS IN ELECTRO-COMPLEX RANDERS SPACES

In this paper we investigate a class of complex Finsler metrics of Randers type defined by \(F(z,{\mu}) = {\alpha}(z,{\mu}) + |{\beta}(z,{\mu})|\) where \({\alpha}(z,{\mu}) = \sqrt{a_{\overline{ij}}{\mu}^i{\overline{\mu}}^j}\) is a Hermitian norm induced by a positive-definite Hermitian metric and \({\beta}(z,{\mu}) = {\frac{e}{m}}A_i(z){\mu}^i\) is a complex 1-form. Motivated by the analogies with electromagnetic interactions we refer to these as electro-complex  Randers metric and we explore conformal deformations of such spaces.

NEW TOOLS FOR CODE CONVERSION AND OPTIMIZATION

Optimization aims to make code either faster or smaller in size. The applications presented in this paper accomplish mostly storage-related optimizations. Most CSS frameworks provide users with classes, that require a longer syntax to incorporate than CSS attributes. As such, a tool was proposed to enable attribute usage over class usage in CSS. Within the same context of web development, it can be stated that repetitive code inspections hinder development, as they needlessly increase development time. As such, a tool was made to show all the properties of an HTML element when the developer hovers over it. The tool is only required in development, and can be removed seamlessly when the application is released. An HTML generator is also introduced in this paper, which aims to provide a starting point for a web application, using a custom lightweight (attribute-based) CSS framework. This HTML generator takes paragraphed text as input and outputs an HTML page, leaving the possibility for conversion to other formats as well (RTF/PDF/...). Given that AI tries to enable natural language programming, such a tool can be adjacent to its endeavors. A JSON-HTML converter is also introduced. Since JSON is more lightweight than XML, a JSON-HTML converter can help compress static HTML pages, to use less network. The converter enables conversion to and from HTML. Besides the JSON-HTML converter, a separate tool introduces the possibility of shortening function/ variable names to optimize large code files. Another aspect of this tool is making code contents less accessible to parties that might handle with it.

STRONG IDEALS IN QI-ALGEBRAS

 The notion of QI-algebras was introduced in 2017 as a generalization of the concept of BI-algebras. In this article, the concept of strong ideal in QI-algebra is and its properties are observed. Also, we study some properties of weak ideals and α-ideals in a QI-algebra. Moreover, we prove that the direct product of a family of QI-algebras is a QI-algebra.

PERFECT TOTIENT GROUPS

Let G be a finite group and let φ(G)=|{a ∈ G | o(a)=exp(G)}|, where o(a) denotes the order of a in G and exp(G) denotes the exponent of G. We say that G is a perfect totient group (or a PT-group, in short) if \(|G|=\sum_{i=1}^{c_G}\varphi^i(G)\), where \(c_G=\min\{m\in\mathbb{N}^*\mid \varphi^m(G)=1\}\). In this note, several results concerning PT-groups are presented.