Research Article
AStudy of Quasi-filtrations on a Semiring and Extension of the Generalized Samuel Number w̅f(g)to Quasi-filtrations on a Semiring
Issue:
Volume 14, Issue 6, December 2025
Pages:
161-165
Received:
24 August 2025
Accepted:
15 September 2025
Published:
10 November 2025
Abstract: In this work, we develop a theory of asymptotic growth for quasi-filtrations in the framework of commutative semirings. A quasi-filtration g = (Gn)n ∈ ℕ ∪ {+∞} is a family of submonoids of a semiring B that satisfies the conditions of a quasi-graduation and is also decreasing for indices n ≥ 1. This structure generalizes the notion of a filtration by using submonoids instead of the more restrictive semi-ideals. In this context, we extend the Samuel number wf(J) of a pair of semi-ideals (I, J) to a quasi-filtration f and a submonoid J, denoted wf(J), and also to a pair of quasi-filtrations (f, g), which we will denote wf(g). The central question is the existence of generalized Samuel number, denoted w̅f(g), which is defined as the limit of the ratio wf(Gn) ⁄ n as n tends to infinity. Our main result provides a positive answer to this question under certain conditions. We demonstrate that if the quasi-filtration f is regular, then wf(J) exists for any submonoid J, and the generalized Samuel number w̅f(g) is well-defined for any quasi-filtration g that is Approximable by Powers of submonoids (AP). Finally, we study the algebraic properties of this number. We prove that this number is non-negative and positively homogeneous under certain conditions. These results constitute an essential step towards the development of analytical tools for non-symmetrizable algebraic structures.
Abstract: In this work, we develop a theory of asymptotic growth for quasi-filtrations in the framework of commutative semirings. A quasi-filtration g = (Gn)n ∈ ℕ ∪ {+∞} is a family of submonoids of a semiring B that satisfies the conditions of a quasi-graduation and is also decreasing for indices n ≥ 1. This structure generalizes the notion of a filtration ...
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Research/Technical Note
Modular Resolution by Polyseries
Issue:
Volume 14, Issue 6, December 2025
Pages:
166-175
Received:
4 August 2025
Accepted:
16 August 2025
Published:
10 November 2025
Abstract: We study the modular resolution method using a new tool called a Polyserie introduced by Wildberger N.J. & Rubine D. to prove an equivalence theorem of the existence and the unicity of the solution of the equation of the form Ht2(x) – Ht(x)+t ≡ 0 (mod tn). These equations admits solutions for a fixed t but when t changes in a range of several values, then the equation becomes parametric and hence the solution should be parametric too. Prof. Wildberger N.J has studied such equations using Catalan numbers sequence using the recurrence formula between their terms Cn+1 = ∑k=0n Ck Cn-k, which allow us to easily prove the main result of the present article. We study Wildberger’s Polynumber Sequences, Binomial Chu-Vandermonde Identity, Truncated Polyseries and finally Modular Resolution as Application.
Abstract: We study the modular resolution method using a new tool called a Polyserie introduced by Wildberger N.J. & Rubine D. to prove an equivalence theorem of the existence and the unicity of the solution of the equation of the form Ht2(x) – Ht(x)+t ≡ 0 (mod tn). These equations admits solutions for a fixed t but when t changes in a range of several value...
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Research Article
AXE-Filtrations of Semi-modules, Generalized Samuel Number v̅φ(θ) andValuative Reduction
Issue:
Volume 14, Issue 6, December 2025
Pages:
176-181
Received:
19 September 2025
Accepted:
7 October 2025
Published:
10 November 2025
Abstract: In this work, we develop an asymptotic theory for axe-filtrations within the broader framework of semi-modules, aiming to generalize the classical theory of filtrations from rings to these algebraic structures where addition is not necessarily invertible. This extension is crucial as semi-modules and semirings appear naturally in fields like geometry and computer science. Our first contribution is the introduction of the concept of an axe-filtration on a semi-module, which adapts the notion of a sequence of powers of an ideal in a ring. We then define a generalized Samuel number, denoted v̅φ(θ), designed to measure the relative asymptotic growth between two distinct axe-filtrations, φ and θ. The main result of this paper establishes a fundamental theorem: the existence of this Samuel number is guaranteed under the condition that one axe-filtration, φ, is a valuative reduction of the other, θ. This key finding extends the classical results of D. Rees by connecting the asymptotic behavior of these generalized filtrations to the well-established theory of discrete valuations. By doing so, we lay the groundwork for a robust asymptotic theory applicable to semi-modules. This work provides new analytical tools for studying non-symmetrizable algebraic structures and opens avenues for further research into the geometric and algebraic properties of systems described by semirings.
Abstract: In this work, we develop an asymptotic theory for axe-filtrations within the broader framework of semi-modules, aiming to generalize the classical theory of filtrations from rings to these algebraic structures where addition is not necessarily invertible. This extension is crucial as semi-modules and semirings appear naturally in fields like geomet...
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Research Article
The Mirror Harmony Mod-9 Law: Residue Classification of Brahmam Mirror Numbers
Brahmam Odugu*
,
Vraj Magtarpara,
Nishant Lakhani,
Jainam Narola,
Tanmay Amrutiya
Issue:
Volume 14, Issue 6, December 2025
Pages:
182-186
Received:
5 November 2025
Accepted:
17 November 2025
Published:
19 December 2025
DOI:
10.11648/j.pamj.20251406.14
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Views:
Abstract: This paper develops the concept of Brahmam Mirror Numbers (BMNs), a class of integers defined through a mirror-product operation in which a number is multiplied by its digit-reversed counterpart. When this product forms a decimal palindrome, the number is classified as a BMN. This operation connects digit reversal, reflective symmetry, and multiplicative structure, creating an interesting foundation for studying digit-based transformations. In this work, we identify a central modular identity governing these behaviours, referred to as the Mirror Harmony Mod-9 Law. Through digit-sum properties and congruence arguments, we show that for any integer, the mirror product is always congruent to the square of the number modulo nine. As a result, every mirror product must lie within the set of quadratic residues modulo nine: {0, 1, 4, 7}. This restriction holds universally and remains valid whether or not the number itself is a BMN. To examine the frequency and structure of BMNs, we conducted a complete computational scan of integers up to 200 million and identified 1246 BMNs. These findings confirm their rarity and demonstrate the residue pattern predicted by the modular identity. Additional observations highlight how mirror products behave under mod-11 alternating-digit rules, suggesting deeper interactions between digit symmetry and modular behaviour. We also explore a polynomial interpretation in which digit strings are treated as self-reciprocal forms, offering an algebraic viewpoint on palindromic mirror products. Overall, the results presented here provide a unified framework for understanding Brahmam Mirror Numbers, combining modular theory, computational evidence, and structural analysis. This study opens pathways for further research in digit-based number theory, density heuristics, modular classification, and potential applications in reflective arithmetic and digital computation.
Abstract: This paper develops the concept of Brahmam Mirror Numbers (BMNs), a class of integers defined through a mirror-product operation in which a number is multiplied by its digit-reversed counterpart. When this product forms a decimal palindrome, the number is classified as a BMN. This operation connects digit reversal, reflective symmetry, and multipli...
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