Reduced density matrices are central to describing observablesin many-body quantum systems. In electronic structure theory, thetwo-particle reduced density matrix (2-RDM) suffices to determinethe energy and other key properties. Recent work has used matrix completion,leveraging the low-rank structure of RDMs and approximate theoreticalmodels, to reconstruct the 2-RDM from partial data and thus reducethe computational cost. However, matrix completion is, in general,an under-determined problem. Revisiting Rosina’s theorem (RosinaM.,Queen’s Papers on Pure and Applied Mathematics, 1968, No. 11, 369), we here show that the matrix completion is unique under certainconditions, identifying the subset of 2-RDM elements that enablesits exact reconstruction from incomplete information. Building onthis, we introduce a hybrid quantum–stochastic algorithm thatachieves exact matrix completion, demonstrated through applicationsto the Fermi–Hubbard model.

Why metaphysics is not like mathematics

This review article examines the ideas and analyses put forth by Josip Slisko in his 2026 book on matchstick puzzles, which provide a basis for projecting them onto domains of study such as math cognition with implications for math education. The book is a truly significant one bridging these two domains, showing how an apparently simple puzzle form enfolds deep mathematical ideas and principles that, when fleshed out, put on display what fundamental mathematics is all about. Above all else, Slisko’s book has specific important implications for math education, which will be highlighted in this review article.

This paper presents a unified study of integer partition theory, a foundational area of number theory and combinatorics. The subject is situated within its historical development, from Euler’s generating function framework to Ramanujan’s profound congruences, and then advances beyond classical exposition by synthesizing these ideas with modern combinatorial perspectives. The work systematically develops essential tools—including Ferrers and Young diagrams, Durfee squares, and generating functions—within a single coherent framework. Classical theorems are rigorously proved using both Algebraic and Combinatorial Techniques, Highlighting the Complementary Nature of these approaches. A novel aspect of this paper lies in its integrative treatment of partitions across different number systems and its qualitative exploration of applications extending beyond pure mathematics, demonstrating how partition theory interfaces with broader mathematical structures. By combining historical insight, illustrative constructions, and formal proofs, this study not only consolidates foundational knowledge but also clarifies pathways toward contemporary research questions in partition theory. The results underscore the continuing relevance of integer partitions as a unifying language in modern mathematics and provide a pedagogically strong and research-oriented framework for future investigations in combinatorics and number theory.

The Collatz Conjecture is more than a theoretical problem in mathematics; it also holds significant value in practical computational contexts.Beyond its abstract mathematical nature, the conjecture serves as a versatile tool within Computer Engineering and Computer Science.The algorithm derived from the Collatz process has been applied in several emerging fields such as steganography, cryptology, data hiding, and digital watermarking, demonstrating its adaptability to real-world problems.This study investigates these application areas in detail and introduces a number of mathematical expressions formulated through the Collatz Procedure.These expressions have the potential to support both theoretical efforts aimed at proving the conjecture and practical implementtations across various domains.By bridging pure mathematics with applied computer science, the Collatz Conjecture continues to inspire interdisciplinary research and innovation.

This contribution is dedicated to the interdependence of higher order linear difference equations and generalized continued fractions in Banach algebras. It turns out that the computation of certain subdominant solutions of a higher order linear difference equation can be done more efficiently by considering its adjoint equation. The final result is a Pincherle‐type convergence criterion for generalized continued fractions in terms of the adjoint equation. To demonstrate the applicability of the method, it is applied to the Poincaré–Perron difference equation and selected problems from pure mathematics and applied stochastic processes.

Galois theory provides a fundamental algebraic framework for understanding the solvability and symmetry properties of polynomial equations and has played a significant role in the development of modern algebra. In recent years, its relevance has extended beyond pure mathematics into computational techniques and engineering-oriented applications. This paper presents a comparative study of the theoretical foundations of Galois theory and its integration with computational methods, emphasizing its applicability in contemporary engineering and scientific problems. The study highlights how algebraic structures and symmetry principles derived from Galois theory can support symbolic computation, algorithmic problem solving, and simplified modeling approaches in applied systems. Particular attention is given to its relevance in mathematical modeling and computational analysis, where structural insights contribute to efficient solution strategies. The paper aims to provide a concise and accessible overview that bridges classical theory with computational and engineering perspectives, making it useful for researchers and practitioners working in applied mathematics, computational engineering, and interdisciplinary scientific domains.

This bibliographical exploration begins with a particularly colourful and experimental edition of a notable mathematics textbook published in the mid-19th century, and compares it with several later editions. Connecting historical, technical, design and pedagogical perspectives, and drawing on aspects of interface theory, this paper considers how a once-maligned reconceptualization of a fundamental mathematics text has come to be understood as an early example of translating abstract knowledge from one form into another (in this case, from linguistic to visual), so as to better serve alternative learning styles. It also examines how that forward-thinking spirit has been carried through into a digitally-remediated form.

Applied mathematics represents a blend of methods of pure mathematics and knowledge of problems from a field to which those methods can be applied toward the advancement of the field. Notably, the fields of engineering and life sciences have been significantly advanced by formulating their problems in the language of mathematics and using rigorous mathematical methods to enable intuitive thinking to be replaced by exact models to which formally proved mathematical propositions can be applied. This paper suggests that among the fields to which knowledge and methods of mathematics can be applied, especially in the age of technology which supports experimental problem solving, is mathematics teacher education. The paper builds on the ideas about mathematics as an experimental science that span from ancient to modern times. It provides three major illustrations of different levels of contextual and conceptual intricacy reflecting on the author’s work as a mathematician-teacher educator. This conceptual reflection alludes to a case in point that seeing mathematics pedagogy through technology-enhanced lens can sometimes give way for the emergence of new mathematical knowledge through learners’ unwitting entry into the substance of the discipline. Examples of such unexpected entries provided in the paper include the discovery of generalized Golden Ratios as strings of numbers of different lengths, Fibonacci-like polynomials stemming from a rearranged Pascal’s triangle, and symmetrical vs. asymmetrical location of the roots of the polynomials depending on the sums of their coefficients.

This study presents a comprehensive exploration of inner product spaces and their completion into Hilbert spaces, examining their foundational roles in both pure mathematics and modern machine learning. Inner product spaces introduce geometric notions such as orthogonality, angle, and norm, while Hilbert spaces, being complete IPS, extend these ideas to infinite dimensional settings. This paper develops key theoretical concepts including the Cauchy–Schwarz inequality, Bessel's inequality, Parseval's identity, the polarization identity, and orthogonal projections. The discussion further explores the functional enrichment that Hilbert spaces provide over normed and Banach spaces, particularly in contexts requiring convergence and projection-based optimization. The practical relevance of Hilbert spaces is demonstrated through their role in machine learning algorithms such as Support Vector Machines (SVM), Principal Component Analysis (PCA), and kernel methods using Reproducing Kernel Hilbert Spaces (RKHS). Numerical simulations and MATLAB visualizations are employed to aid understanding and demonstrate the application of inner product theory in data driven tasks such as classification and dimensionality reduction. The results show how the geometry of Hilbert spaces naturally supports core operations in machine learning, making them indispensable in theoretical development and algorithmic design.

The Journal invites original, high-quality research papers in all areas of pure mathematics (including theoretical probability, mathematical physics and mathematical statistics).Papers should be of moderate length (15-35 pages) and wide interest; those with good introductions explaining the context, meaning and value of the results are preferred.Only papers highly rated by assessors can be accepted.

This comprehensive study investigates the mathematical competency of teachers at Mindanao State University, assessing their proficiency, pedagogical strategies, and preparedness for effective mathematics instruction. Employing a mixed-methods approach, the research combines quantitative assessments, such as standardized tests and performance evaluations, with qualitative insights. Findings reveal that while many teachers have a solid grasp of fundamental mathematics, there are significant gaps in advanced topics and innovative teaching strategies. Competency levels vary across departments and years of experience. The study emphasizes the need for continuous development to improve teachers' mathematical skills, which in turn enhances student engagement and performance. It recommends specialized training programs and curriculum enhancements, advocating for institutional support to foster lifelong learning among educators, ultimately aiming to elevate the quality of mathematics instruction and student achievement at the university.

Introduction: Classical orthogonal polynomials such as the Legendre, Laguerre, Hermite, and Chebyshev polynomials have a wide range of applications across various domains of science and engineering. Aim: The aim of this research paper is to present selected recent developments in the theory of orthogonal polynomials, with particular emphasis on their analytical properties and their role in approximation theory. Methods: The study employs a combined quantitative-qualitative research methodology. It is based on the analysis of relevant scientific literature, from which data significant to the subject matter were collected, interpreted, and systematically examined. Results: The results indicate that contemporary research in probability theory, graph theory, coding theory, and related areas increasingly relies on the theory of orthogonal polynomials. This paper provides a theoretical analysis of the connection between orthogonal polynomials and their applications in specific computational problems. Fundamental properties of these polynomials are presented and illustrated through selected examples that contribute to a clearer understanding of their structure and practical relevance. Conclusion: The approximation of the transfer function of a low-pass filter can be achieved through a straightforward adaptation of orthogonal Jacobi polynomials. Furthermore, Gaussian quadrature represents a powerful numerical technique for the approximation of definite integrals, utilizing optimally chosen nodes and weight functions to achieve high accuracy with minimal computational effort. Orthogonal polynomials thus serve as a significant link between pure mathematics and engineering disciplines, highlighting the importance of further study on their properties and applications.

The Banach spaces is the most popular and well-known spaces in the subject of pure mathematics. Many mathematicians have worked on different concepts from different angles in the field of functional analysis due to wide range of utilizations in quantum mechanics, optimization, and numerical analysis. In this paper, we explore the new iterative approach, namely HK-iteration method in Banach spaces. In addition, we investigate the some new convergence results in Banach Spaces via newly introduced concept with an application to a first-order delay differential equation. For the paper quality and reader’sinterest, numerical example and performance comparison via newly introduced concept have been added.

A Special Issue of the Indian Journal of Pure and Applied Mathematics

In this work we employ machine learning to understand structured mathematical data involving finite groups and derive a theorem about necessary properties of generators of finite simple groups. We create a database of all two-generated subgroups of the symmetric group on n -objects and conduct a classification of finite simple groups among them using shallow feed-forward neural networks. We show that this neural network classifier can decipher the property of simplicity with varying accuracies depending on the features. Our neural network model leads to a natural conjecture concerning the generators of a finite simple group. We subsequently prove this conjecture. This new toy theorem comments on the necessary properties of generators of finite simple groups. We show this explicitly for a class of sporadic groups for which the result holds. Our work further makes the case for a machine motivated study of algebraic structures in pure mathematics and highlights the possibility of generating new conjectures and theorems in mathematics with the aid of machine learning.

In this study, we present a new family of number sequences called the generalized hyperbolic Pierre numbers, defined within the bidimensional Clifford algebra of hyperbolic numbers. This algebraic framework enables the extension of classical sequence theory into the hypercomplex domain, offering both structural and analytical enrichment. As notable special cases, we examine the hyperbolic Pierre numbers and hyperbolic Pierre Lucas numbers, analyzing their algebraic properties, characteristic behaviors, and mutual relationships in detail. We derive closed-form expressions through Binet-type formulas, construct generating functions that reflect the recursive nature of the sequences, and establish summation identities that reveal deeper arithmetic patterns. Furthermore, we develop matrix representations for each sequence, providing a compact and elegant algebraic tool for modeling and manipulating their evolution. This research contributes to the broader theory of hypercomplex number sequences and proposes a novel approach to generalizing classical sequences within Clifford algebraic systems. The results have potential applications not only in pure mathematics but also in fields such as cryptography, numerical analysis, modeling of symmetric structures, solving differential equations, and algebraic representation of physical systems. In this context, the study lays a solid foundation for advanced investigations into the reinterpretation of number sequences in hyperbolic spaces.

Several studies have observed that the presentation tasks in mobile devices in a conversational rather than a formal style may produce a personalization effect and benefit performance. Put another way, students’ cognitive structures tend to function better when they are involved in personalized contexts. However, little attention has been paid to the personalization principle from mathematics contexts, particularly in the southern hemisphere. Framed by the personalization principle, this mixed methods study investigated the effects of presentation mode in a task involving explicitly laying out the theories and obtaining results through logical proofs, ensuring no ambiguity, which entangles the practice of mathematicians. To this end, this mixed methods study selected a convenience sample of 162 pre-service teachers (PSTs) enrolled in an advanced mathematics module in a large public university located in south-eastern South Africa. Following random assignment, eighty-five (n<sub>1 </sub>= 85) PSTs were presented with material in conversational tone and seventy-seven (n<sub>2 </sub>= 77) PSTs were presented with material in formal tone. Quantitative analysis revealed, among other things, a significant difference in the responses by the two groups of PSTs, <i>t</i>(160) = 4.83, <i>p</i> < .001, and <i>d</i> = .16. In addition, qualitative analysis of what PSTs say about the functions of pure mathematicians in response to the presentation of the prompt in different contexts showed that personalized contexts foster performance. Considering the limitations of this study, a discussion of the consequences the results of this paper might have on the direction of future mathematics education research is provided.

The role of pure mathematics in resolving complex biological problems: Applications to F1-ATPase, achievements, and future directions.

This paper explores the influence of number theory concepts on the behavior and security of cryptographic hash functions. Hash functions play a critical role in modern cryptography, ensuring data integrity, authentication, and digital signatures. While they are primarily designed using principles from algebra and complexity theory, number theory significantly contributes to their construction and security analysis. Key number-theoretic conceptssuch as modular arithmetic, prime number distributions, and discrete logarithmsunderpin many hash function designs, especially in schemes that rely on structured algebraic inputs or are constructed from hard mathematical problems. We analyze how these mathematical foundations affect essential properties like collision resistance, pre-image resistance, and avalanche behavior. Additionally, we examine how number-theoretic attacks (e.g., those exploiting modular congruences or integer factorization) pose potential threats to certain classes of hash functions. The paper concludes by highlighting current research trends leveraging advanced number-theoretic techniques to enhance hash function robustness, emphasizing the ongoing interplay between pure mathematics and practical cryptographic design.