This paper introduces and investigates a novel generalization of operator similarity, termed (α,β)--almost similarity, which extends the concept of almost similar operators by incorporating two real parameters. We establish fundamental properties of this new equivalence relation, demonstrating that it forms an equivalence class on the space of bounded linear operators on a Hilbert space. Key results include the invariance of spectrum, point spectrum, and approximate point spectrum under this relation. The study also defines the class of (α,β)-𝔗 operators, an expansion of the classical 𝔗-operator concept, and explores its relationship with (α,β)--almost similarity. Furthermore, we analyze the connections between similarity, unitary equivalence, and (α,β)--almost similarity, providing conditions under which these relations coincide, particularly for self-adjoint and projection operators. The results contribute to the broader understanding of operator equivalence relations and their spectral implications.
Cantilever balconies are widely used in modern buildings due to their architectural flexibility and efficient use of space. However, their structural performance is highly sensitive to environmental loading because their bending resistance is concentrated at the fixed support. In real service conditions, these structures are simultaneously subjected to static loads, wind-induced aerodynamic forces, seasonal thermal effects, and seismic ground acceleration. Most conventional analyses treat these effects independently, which may underestimate cumulative deflection and lead to inaccurate serviceability predictions. This study develops a unified mathematical model to quantify the total tip deflection of a cantilever balcony subjected to combined static, wind, thermal, and seismic loading. The formulation is based on Euler–Bernoulli beam theory and linear elasticity assumptions. Closed-form analytical expressions are derived for each loading component and integrated using the principle of superposition to obtain a compact total deflection equation. Numerical simulations are performed for aluminum, steel, reinforced concrete, and carbon fiber composites under representative environmental conditions. Results show that thermal effects become dominant in high-temperature environments for materials with large coefficients of thermal expansion, while seismic effects become significant in regions with high peak ground acceleration. Among the materials considered, carbon fiber composites consistently exhibit the smallest total deflection due to their high stiffness and low thermal sensitivity, while reinforced concrete shows the largest deformation due to its lower elastic modulus. The proposed model provides a mathematically consistent framework for evaluating cantilever balcony performance under multi-hazard environmental loading and offers a useful decision-support tool for preliminary structural design and material selection.
This paper shows that every integer and odd integer are a sum of a prime and two squares. We solved this by transforming the problem of the number of such representations into finding the number of integer solutions to the corresponding Diophantine equation. To find integer solutions to nonlinear or higher-order Diophantine equations is very challenging, whereas counting the number of integer solutions for the linear Diophantine equations is comparatively easier. The methods involve the combinatorial approximation method to obtain the number of integer solutions of the Diophantine equation with the countable solution sets. First, the number of integer solutions to the linear Diophantine equations with integer sets or sequences as solution sets can be determined by combinatorial methods. Second, the approximate number of integer solutions of the linear Diophantine equations with integer sets or sequences as solution sets is obtained using the approximate method. Finally, for cases where the solution sets and the number of solution sets of the variables in the linear Diophantine equations are either the same or different, we propose the approximate combinatorial method to compute the approximate number of integer solutions of the linear Diophantine equations whose solution sets are integer subsets or sequences of integer subsets. The results are consistent with the existing conclusions. The purpose of this paper is to verify the adaptability and correctness of the approximate combinatorial methods for solving the number of integer solutions of the general Diophantine equations with countable solution sets.
We prove that rank-zero elliptic curves over generate positive-rank elliptic curves through the unit circle, via quadratic twisted models . This construction demonstrates rank evolution from zero to infinite rational points, complementing high-rank families. All rank-zero status and positive-rank emergence rigorously verified computationally. SMC(2020): 11G05, 14H52, 11Y50.
The study of operator theory, specifically the determination of the norm of general derivations in C* algebras, has attracted significant attention in recent years. This problem has been approached through various methods across different spaces, yielding several results. The norm of a general derivation is crucial for understanding the structure and behavior of derivations in the context of C* algebras, with implications for functional analysis and operator theory. Despite previous efforts, a complete understanding remains elusive. The purpose of this paper is to investigate this problem using the finite rank operator in the tensor product of C*-algebras. By leveraging the tensor product structure, we explore how finite rank operators can provide insights into the norm of general derivations. The research utilizes a mathematical framework that examines the behavior of bounded linear operators within the tensor product of Hilbert spaces and C*-algebras. Through this approach, we aim to advance existing knowledge and offer new results that could potentially contribute to the field. The final conclusion of the study confirms that the use of finite rank operators in the tensor product of C*-algebras offers a valuable method for approximating and understanding the norm of general derivations, providing a more refined approach than previous techniques.
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.
This paper mainly explores the precise asymptotic behavior near zero of positive weak solutions to the quasilinear elliptic equation involving Hardy potential and Sobolev critical exponent, which is expressed as under the conditions that , , , , and . The research shows that if is a positive radial weak solution of this equation, then there exists such that , where is the smallest root of the equation . This result accurately depicts the asymptotic characteristics of positive weak solutions of the equation near zero. Compared with previous relevant studies which only indicate that the solutions are bounded near zero, this study further clarifies the limiting situation of the solutions.
Several recently published studies regarding flow problems propose schemes of high order of accuracy designed as evolution of traditional methods. A drawback common to these new schemes is the necessity to adopt uniform mesh refinement for solving sharp problems, by increasing the computational cost. Even the so called essentially non-oscillatory and weight essentially non-oscillatory methods suffer of the same drawback and are not suitable to cope with h-adaptive methods due to their definition on finite volumes necessarily of equal diameter. Therefore, in order to overcome the above drawback, the formulation of dynamically locally self h-adaptive processes is designed to achieve the dual purpose to increase the accuracy and to keep as small as possible the number of finite volumes. To define a locally h-adaptive finite volume (FV) scheme need two simple but important tools, namely a particular FV named Bridge FV positioned between two adjacent subdomains and the definition of suitable profiles approximating the fluxes on the FV faces. In this article a new FV method for the numerical solution of convective-diffusive 1D problems is developed. It is conservative, second order in time and space for equal FV, and allows the partitioning of the domain by equal or unequal finite volumes, thus dynamically locally self h-adaptive. The definition of the monotonic profiles is accomplished by means of cubic weighted ν-splines and Taylor expansions. The profile analysis respect to the numerical properties is conducted in the normalized plane with the velocity varying in time and space and gives the flux value on the FV faces. Moreover the flux is assigned by Upwind or by second order back-ward Characteristics if the estimated flux is outside of the unit square or the transformation into the normalized plane is not possible, respectively. The initial-boundary stability and convergence properties of the new method are examined in detail, also in presence of h-adaptivity. In addition, a generalization of the new scheme to 2D and 3D problems is presented. Finally, some numerical test are carried out to verify the properties of the new method, including two CFD problems.
Wilson’s Theorem states that if p is a prime number, then the product of the first (p - 1) positive integers, increased by one is divisible by p. This classical result in number theory was attributed to John Wilson by Edward Waring in 1770. The first known proof of Wilson’s Theorem was published in 1771 by the French mathematician Joseph-Louis Lagrange. Wilson’s Theorem has applications in primality testing, cryptography, and various other areas of mathematics. In this article, we shall prove that if p is a prime number, then for each natural number, j, that lies between 1 and (p - 2), the binomial coefficient (p - 1) choose j, decreased by the jth power of negative one, is divisible by p. As such, these new results can be viewed as extensions of Wilson’s Theorem. We shall also prove that if p is a prime number, then the square of the factorial of the ratio of (p - 1) and two, is congruent to either 1 modulo p or - 1 modulo p. Additionally, we shall prove that for all positive integers m bigger than or equal to 5, m is prime if and only if m divides the sum of either one or negative one and the square of the factorial of the ratio of (m - 1) and two. Next, we shall use some of the schemes developed earlier to investigate the behavior of the divisors of the sum of prime powers of relatively prime positive integers a, b. Lastly, we shall show that if p is a prime number and k is a positive integer, then the ratio of the sum of prime powers of a, b and the sum of a, b is not divisible by the kth power of p.
We study the two-parameter class of (M, k)-Quasi-∗-Parahyponormal operators on separable Hilbert spaces, which strictly enlarges the traditional parahyponormal and paranormal hierarchies. Analytically we prove three fundamental results: Every operator in the class has finite ascent and enjoys the single-valued extension property (SVEP); The Browder–Weyl partition holds, so Weyl’s theorem is valid; A non-trivial closed invariant subspace exists whenever the commutant contains a non-zero compact element. Complementing these theorems, we introduce a proposed computational framework that realises the abstract operators as large weighted-shift matrices, verifies the defining quadratic inequality, and computes eigenvalues as well aaccelerated pseudospectra. Together, the analytic results and the computational framework deepen the spectral theory of (M, k)-Quasi-∗-Parahyponormal Operators and supply the first large-scale numerical evidence for their structural properties.
The study of almost periodic functions occupies an important place in functional analysis and the theory of differential equations, beginning with the classical works of H. Bohr, A. S. Besikovich and B. M. Levitan. Almost periodic functions, being a generalization of periodic functions, are characterized by the fact that they retain their structure under shifts, without being strictly periodic. On the other hand, entire functions are functions of a complex variable that are analytic in the entire complex plane. Their behavior, especially their growth and the location of their zeros, is studied in detail in the theory of functions of a complex variable. Of particular interest is the study of entire functions whose values on the real axis are almost periodic in the sense of Bohr. The question of under what conditions an entire function takes on values on the real axis that form a uniformly almost periodic function is a non-trivial problem at the intersection of function theory and spectral analysis. Such conditions can be formulated through the properties of the spectrum of the function, through the conditions on the coefficients of the Fourier series, and also through the growth properties of the function itself. These functions find application in spectral theory, quantum mechanics, oscillation theory, and other areas of mathematics and physics. In this section, we study the problems of approximation of functions f(x)∈B by entire functions of finite degree with arbitrary Fourier exponents. We establish necessary and sufficient conditions for functions f(x)∈B to belong to the class of entire functions of bounded degree.
In this work, we are interested in studying a particular class of Side Channel Attacks on elliptic curves defined over binary fields. Side Channel Attacks exploit physical leakages such as power consumption or electromagnetic emanations during cryptographic computations in order to recover secret information. Among these attacks, the one we focus on is known as Same Values Analysis (SVA). This method does not rely directly on distinguishing the sequence of operations, but rather on detecting situations where different inputs lead to identical intermediate values inside the formulas used for point addition and point doubling. Since these formulas are usually well known and publicly available, an adversary can exploit such collisions in order to reveal sensitive information, in particular the secret scalar used during scalar multiplication. The objective of our study is therefore to identify the points on elliptic curves that produce identical intermediate variables during addition or doubling steps, and to determine the algebraic conditions under which such coincidences occur. By analyzing these conditions, one can highlight vulnerabilities in scalar multiplication algorithms and evaluate their potential exploitation. We will focus our investigation on three important families of elliptic curves over binary fields: Weierstrass curves, Edwards curves, and Hessian curves. For completeness, we will also verify the effectiveness of the SVA attack on standardized curves recommended by NIST and SECG. The results of our analysis clearly show that all these curves are vulnerable to Same Values Analysis, which implies that their use in cryptographic protocols requires a careful reassessment of security guarantees.
In this paper, we introduce and systematically study the concept of pointwise biflatness in Banach algebras, which generalizes classical biflatness by localizing the homological structure to individual elements. Unlike global biflatness, this localized approach captures finer algebraic and module-theoretic behaviors that remain invisible under classical definitions. We prove that a pointwise biflatBanach algebra is super-amenable if and only if it possesses an identity element, providing a precise criterion linking local biflatness with classical amenability. Additionally, we explore the interrelations between pointwise biflatness, pointwise flatness, and pointwise amenability, clarifying their mutual influence and delineating boundaries between local and global homological properties. Applications to group algebras and classical Banach algebras are presented, illustrating scenarios where global biflatness fails but pointwise biflatness holds. These results provide concrete examples, highlight the practical relevance of the localized approach, and establish a foundation for further study in operator algebras, Segal algebras, and harmonic analysis. This study opens new directions for theoretical research and offers refined tools for understanding module structures, cohomologicalbehaviors, and approximation properties in Banach algebras.
We investigate the interplay between the Chinese Remainder Theorem and the theory of congruent numbers through a modular approach to expressing integers as sums of three cubes. By analyzing congruence systems arising from specific residue classes modulo 8 and modulo 9, we classify the possible integers likely to be representable as sums of cubes based on their modular residues. We also explore computational methods applying this modular framework to identify explicit cube decompositions within these classes.
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.
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.
This study explores the application of machine learning (ML) and artificial intelligence (AI) techniques to analyze unstructured textual data, focusing on topic modeling, sentiment detection, and behavioral prediction. We employ multinomial document models and unsupervised learning strategies to extract latent topics and evaluate the emotional and conversational drivers behind social media posts. A major contribution is the implementation of Behavior Dirichlet Probability Model (BDPM) which analyzes user moods and behaviors through unstructured textual data. The results validate the hypothesis of the model's ability to identify and guess behavior patterns with high accuracy, providing actionable insights for digital marketing strategies, techniques to enhance user interaction and mental wellness evaluation.
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.
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.
The Leech lattice occupies a central place in exceptional mathematics and coding theory, but because it is 24-dimensional, its geometric structure is difficult to visualize. We elucidate a coordinate system employed in Conway’s and Sloane’s Sphere Packings, Lattices and Groups to describe and to illustrate the environs of the Leech lattice’s deep holes but never explained in that book. By using this coordinate system for the deep holes, we are able to provide colorful diagrams for the environs of several deep holes. The most interesting of those deep holes is E83, and we describe a new E83 -based coordinate system for the Leech lattice. We enumerate the Delaunay cells of the Leech lattice immediately neighboring a Delaunay cell of type E83, neighboring in the sense of intersecting the latter cell in a 23-dimensional face. There are 729 such faces, but fewer isometry classes of neighboring cells, including both deep holes and shallow holes. We also enumerate the Delaunay cells of the Leech lattice in close proximity to a Delaunay cell of type E83, close in the sense of having all vertices less than √2.2 distant from the center of E83; the latter’s circumradius is √2. Neither of the two classes of Delaunay cells enumerated below is a subset of the other. We next consider conjectures about the Leech lattice and the several lattices with similar properties, most especially the lattices ℤ , A2, ℤ 2, D4, and E8. Other lattices sharing fewer optimality properties include the Coxeter-Todd lattice K12 and the Barnes-Wall lattice Λ16. The properties of greatest interest are those involving various measures of high degrees of symmetry, as well as various measures of high efficiency in packing and covering. Our culminating theorem is a partial classification of lattices that have very strong properties similar to those of the Leech lattice.
