Principles Of Algebra Research Articles

The Intersection of Algebra and Cryptography: Enhancing Information Security through Mathematical Foundations

The rapid advancements in digital technologies have necessitated the development of robust information security measures. This paper explores the intersection of algebra and cryptography, focusing on how algebraic principles can enhance cryptographic techniques to provide stronger security foundations. By leveraging mathematical structures such as groups, rings, and fields, we can address critical challenges in encryption, secure communications, and data privacy. This study reviews key algebraic methods used in contemporary cryptographic protocols, including elliptic curve cryptography, homomorphic encryption, and lattice-based cryptography, and demonstrates their practical applications through detailed case studies. Our comparative analysis highlights the superior performance and security of algebra-based cryptographic solutions compared to traditional methods. Finally, we discuss the emerging trends and future directions in algebraic cryptography, emphasizing the potential of these mathematical foundations to address the evolving threats in information security.

Enhancing the acquisition of basic algebraic principles using algebra tiles

Introducing initial algebraic principles poses a significant challenge, often compounded by the inherent abstract nature of algebra. This article introduces an innovative pedagogical approach that promotes the use of algebra tiles, a didactic manipulative material formed by a collection of geometric pieces symbolizing distinct algebraic monomials. Additionally, this article includes the findings of a quasi-experimental study that applied this inventive teaching method. This research was carried out across two separate classes of 15 students (10 boys and five girls) of first year of compulsory secondary education. One group of students adhered to the conventional teaching approach (the control group), while the other class embraced the proposed methodology using algebra tiles (the experimental group). The disparities in algebraic proficiency observed between these two student groups, as assessed through various examinations conducted during the intervention were statistically significant, with the experimental group consistently achieving superior results. Moreover, mathematical and algebraic errors of students were assessed using eight distinct indicators. In all cases, the experimental group demonstrated lower error percentages, and these errors showed a marked decrease as the intervention progressed. In summary, this innovative methodology markedly enhanced students’ comprehension of algebra, their knowledge, and their motivation while significantly reducing mathematical errors.

An averaging model for analysis and interpretation of high-order genetic interactions.

While combinatorial genetic data collection from biological systems in which quantitative phenotypes are controlled by active and inactive alleles of multiple genes (multi-gene systems) is becoming common, a standard analysis method for such data has not been established. The currently common approaches have three major drawbacks. First, although it is a long tradition in genetics, modeling the effect of an inactive allele (a null mutant allele) contrasted against that of the active allele (the wild-type allele) is not suitable for mechanistic understanding of multi-gene systems. Second, a commonly-used additive model (ANOVA with interaction) mathematically fails in estimation of interactions among more than two genes when the phenotypic response is not linear. Third, interpretation of higher-order interactions defined by an additive model is not intuitive. I derived an averaging model based on algebraic principles to solve all these problems within the framework of a general linear model. In the averaging model: the effect of the active allele is contrasted against the effect of the inactive allele for easier mechanistic interpretations; there is mathematical stability in estimation of higher-order interactions even when the phenotypic response is not linear; and interpretations of higher-order interactions are intuitive and consistent-interactions are defined as the mean effects of the last active genes added to the system. Thus, the key outcomes of this study are development of the averaging model, which is suitable for analysis of multi-gene systems, and a new, intuitive, and mathematically and interpretationally consistent definition of a genetic interaction, which is central to the averaging model.

Refinement of String Theory Results based on Ideas of Scientific Philosophy

As is known, A. Einstein's key method was unification through symmetry. He was able to formulate a symmetry unifying space and time. He was also able to formulate a generalized symmetry describing acceleration and gravity. However, Einstein's third approach to creating a great unifying theory was unsuccessful because he lacked a symmetry that would unify gravity and light. He even wrote: "I am sure that in order to make real progress, it is necessary to elicit from nature one more some general principle". This paper attempts to prove that this general principle is the principles of algebra and arithmetic.

On the Unification of the Foundations of Physics Based on the Ideas of Algebra and Arithmetic

Historically, scientific theory of knowledge relied on the deductive method of thinking, established in Euclid's geometry. Since the era of Descartes, there has been an evolution in the method of thinking based on the principles of algebra and arithmetic. This study analyzes the contribution of algebra and arithmetic to modern scientific philosophy, demonstrating their significance in forming the universal foundations of theoretical physics.

Sanskrit Scriptures (Ancient Indian History) and Their Scientific Relevance in Modern Science

Abstract: Ancient Indian Science has a plethora of knowledge which include Vedas, Upanishads, Bhagavad Gita, and Puranas but limited online resources available to everyone. to Indian Ancient History preserves the truths of modern science and many justifications of present questions raised in the field of sciences. Most of these scriptures were written in 500-1500 BCE, an era which is believed to have lacked scientific knowledge and reasonings. Many of the Indian transcripts testify data and proofs evident in the Ancient Indian Literature. The Vedas contain intricate details about celestial bodies, their movements, and the concept of time, reflecting an advanced understanding of astronomy. Mathematical concepts, including geometry and algebraic principles, are discernible in texts like the Shulba Sutras. Moreover, Ayurveda, an ancient Indian medical system documented in Sanskrit, provides holistic insights into healthcare, emphasizing a balance of physical, mental, and spiritual well-being. The scriptures also exhibit ecological awareness, prescribing sustainable practices for harmonious coexistence with nature. Sushruta Samhita and other scriptures provide in depth details of utilization of every plant part for cure of diseases. This study encompasses reasonings and specific citing of exact verses of scientific findings found years before modern scientists. Further research will contribute in expansion of knowledge and help delve deeper into Ancient Indian Text.

Roadmap for Simulating Quantum Circuits Utilising IBM’s Qiskit Library: Programming Approach

This paper explains the roadmap for running quantum circuit programs based on the Qiskit library in quantum simulators as well as real cloud-based IBM quantum computers. Qiskit is a free and open-source software development platform based on the Python programming language that is used in quantum programming. Qiskit acts as a link between quantum computing’s theoretical foundations and the practical aspects of programming and experimentation. It also allows users to experiment with and develop quantum algorithms, as well as simulate and execute them on simulators and real-world cloud-based quantum devices. It also simplifies the quantum programming process and allows a diverse range of people to participate in the exciting world of quantum computing. The paper, on the other hand, provides the mathematical foundation for analyzing quantum circuits and algorithms using linear algebra principles, as they provide the tools needed to describe and manipulate quantum states and operations. Furthermore, the paper shows quantum circuit design and implementation using real Qiskit codes.

SIMULASI GERBANG DASAR LOGIKA DALAM APLIKASI

It is no longer a secret that all science is currently developing and has technological advances that cannot be denied, especially in digital technology which is currently popular, one of which is a digital electronic system that is composed of logic gates so that it becomes a digital system formed from logic elements. the smallest is the logic gate (Logic Gate): OR, AND and NOT where the circuit work process on this logic gate uses Boolean algebra principles. In its implementation it is rather difficult to provide an understanding of logic gates manually or in theory to the public and the general public in understanding gate circuits electronics to understand the functions and uses of logic gates in electronic circuits with EWB (Electronic WorkBanch) as an electronic computer software that can be used to simulate the workings of an electronic circuit, both analog and digital. The EWB application is very helpful in providing an indirect understanding of understanding basic logic gate circuits as simulations in applications to reduce possible dangerous risks, with this application it makes it easy to know the conditions you want to design and create before implementing them in real form..

Prestress behaviour and ductility requirements in structures: solutions from a unified algebraic approach.

This paper presents an algebraic approach that unifies both the elastic and limit theories of static structural analysis. This approach reveals several previously unpublished improvements, which are based on several dualities in the mathematical description. Firstly, we show a novel duality between the solutions to two different problems: the elastic solution to the internal forces of an externally loaded structure, and the nodal displacements induced by prestressing in one or several elements of the same structure. This duality is proven and discussed. The application of this solution to the limit state analysis is very productive, and includes the determination of the ductility requirements necessary to achieve full plastic behaviour, and the assessment of the prestress needed to limit or eliminate such requirements. The unified framework also allows to obtain the elasto-plastic deformed state at the beginning of the plastic structural collapse. We have also detailed the theoretical duality between two classes of structures—hyperstatic and hypostatic—which was derived from the linear algebra principles that define these solutions. Finally, we studied and exposed the dimensionality reduction of structural problems given by the singular value decomposition and the eigenvalues problem. An illustrative example that clearly illustrates all these points is provided.

Transform the Multivariate to Unitevariate System and Using Elimination and Duplication Matrices with Some Applications

In this work, we have transformed the multivariate to unitevariate system, and studied the elimination and duplication matrices, using proposed approach depends on some algebra principle. And, a connection has been made between these two matrices with each of the ( υec ) and ( υ ) operators for matrices. Several examples are given to illustrate the applications which is used to define and specialize the elimination and duplication matrices.

Some Notes on a Formal Algebraic Structure of Cryptology

Cryptology, since its advent as an art, art of secret writing, has slowly evolved and changed, above all since the middle of the last century. It has gone on to obtain a more solid rank as an applied mathematical science. We want to propose some annotations in this regard in this paper. To do this, and after reviewing the broad spectrum of methods and systems throughout history, and from the traditional classification, we offer a reordering in a more compact and complete way by placing the cryptographic diversity from the algebraic binary relations. This foundation of cryptological operations from the principles of algebra is enriched by adding what we call pre-cryptological operations which we show as a necessary complement to the entire structure of cryptology. From this framework, we believe that it is improved the diversity of questions related to the meaning, the fundamentals, the statute itself, and the possibilities of cryptological science.

On the Syllogistic of G. Boole

Introduction. This article focuses on the investigation of Boole’s theory of categorical syllogism, exposed in his book “The Mathematical analysis of Logic”. That part of Boolean legacy has been neglected in the prevailed investigations on the history of logic; the latter provides the novelty of the work presented.Methodology and sources. The formal reconstruction of the methods of algebraic presentation of categorical syllogism, as it is exposed in the original work of Boole, is conducted. The character of Boolean methods is investigated in the interconnections with the principles of symbolic algebra on the one hand, and with the principles of signification, taken from R. Whately, on the other hand. The approaches to signification, grounding the syllogistic theories of Boole and Brentano, are analyzed in comparison, wherefrom we explain the reasons why the results of those theories are different so much.Results and discussion. It is demonstrated here that Boole has borrowed the principles of signification from the Whately’s book “The Elements of Logic”. The interpreting the content of the terms as classes, being combined with methods of symbolic algebra, has determined the core features of Boolean syllogism theory and its unexpected results. In contrast to Whately, Boole conduct the approach to ultimate ends, overcoming the restrictions imposed by Aristotelean doctrine. In particular, he neglects the distinction of subject and predicate among the terms of proposition, the order of premises, and provide the possibility to draw conclusions with negative terms. At the same time Boole missed that the forms of inference, parallel to Bramantip and Fresison, are legitimate forms in his system. In spite of the apparent affinities between the Boolean and Brentanian theories of judgment, the syllogistics of Boole appeared to be more flexible. The drawing of particular conclusion from universal premises is allowable in Boolean theory, but not in Brentanian one; besides, in his theory is allowable the drawing of conclusion from two negative premises, which is prohibited in Aristotelian syllogistic.Conclusion. Boole consistently interpreted signification of terms as classes; being combine with methods symbolic algebra it led to very flexible syllogism theory with rich results.

AN INNOVATIVE STUDY ON SUM OF POWERS OF INTEGERS

The generalization of sum of integral powers of first n-natural numbers has been an interesting problem among the researchers in Analytical Number Theory for decades. This research article mainly focuses on the derivation of generalized result of this sum. More explicit formula has been derived in order to get the sum of any arbitrary integral powers of first n-natural numbers. Furthermore by using the fundamental principles of Combinatorics and Linear Algebra an attempt has been made to answer an interesting question namely: Is the sum of integral powers of natural numbers a unique polynomial? As a result it is depicted that this sum always equals a unique polynomial over natural numbers. Moreover some properties of the coefficients of this polynomial are derived.More importantly a recurrence relation which can give the formulas for sum of any positive integral powers of first n-natural numbers has been proposed and it is strongly believed that this recurrence relation is the most significant thing in this entire discussion

Design and engineering of a dual-mode absorption/emission molecular switch for all-optical encryption.

Photochemical reactions that produce a detectable change in the spectroscopic properties of organic chromophores can be exploited to harness the principles of Boolean algebra and design molecule-based logic circuits. Moreover, the logic processing capabilities of these photoactive molecules can be directed to protect, encode, and conceal information at the molecular level. We have designed a photochemical strategy to read, write and encrypt data in the form of optical signals. We have synthesized a supramolecular system based on the known dye resazurin, and investigated a series of photochemical transformations that can be used to regulate its absorption and emission properties upon illumination with ultraviolet or visible light. We have then examined the logic behaviour of the photochemistry involved, and illustrated its potential application in data encryption.

Cost-suitability land modeling for current and proposed transport infrastructure along Timiș-Cerna Corridor (Romania)

Cost-suitability land modeling is one of the mandatory conditions for the proposal of new transport development projects, in order to achieve optimal costs of design and construction. In the present study, we intend to present a model for assessing land suitability for current and proposed transport infrastructure along the Orient-East Med TEN-T Corridor, in the mountainous region of the Timiș-Cerna Corridor (Romania). The relevant factors for our analysis were outsourced from various thematic data sets and refer to lithology, morphological and morphometrical properties of landforms, soil characteristics and climatic setting of the study area. Conservation of natural environment and human habitat was also taken into consideration by means of analyzing land use patterns and spatial distribution of protected areas. The methodology is based on the principles of cartographic algebra and vector overlay analysis, currently implemented in various GIS applications. The results of our study aim for the identification of critical sectors along the current and proposed transport infrastructure, which intersect areas with high development cost and low suitability. Starting from our observations, specialists in transport networks and spatial planning could make use of dedicated computation algorithms to identify and describe optimal routes for future infrastructure projects that would connect the southern and western parts of the country.

Mathematics Embedded in Culture: The Weaves of Batak Basketry

A close look at the art of basket weaving of the Batak indigenous community in Palawan, Philippines, discloses the depth of mathematics present in the weaves of the baskets. Using white and black one-sided bamboo strips, Batak weavers are able to construct a basket of a given size that displays highly symmetric patterns. The presence of elaborate patterns in the baskets indicates masterful planning by the weaver and an inherent knowledge of algebraic and geometric principles even without formal training in mathematics. KEYWORDS: BATAK • BASKET • WEAVE • SYMMETRY • PATTERN

Investigation of Technological Parameters for Machining Toroidal Section of Solid Ceramic End Mills

Solid ceramic mills are a promising technological solution for cutting heat-resistant materials. Although nanostructured cutting ceramics has a number of valuable physical and mechanical properties providing high operational performance of mills, it has low strength. The design of a solid ceramic mills is formed by smooth surfaces without stress concentrators to ensure operability and to reduce the probability of failure. The most important structural element of solid ceramic end mill is a helical groove with a negative rake angle needed to increase surface strength. This groove can be machined with standard grinding wheels of type 1A1 or 1V1. However, the application of grinding wheels of a standard shape requires the use of reduced cutting parameters to prevent chipping on the forming sections of the cutter. Chipping can be caused by too short length of the contact line and the stress concentration at the end point of the profile of the grinding wheel, which is typically fully responsible for shaping the front surface and its transition to the tooth back. In this paper, in order to increase the contact line, we develop a new approach to designing grinding wheels for machining helical grooves of solid mills and determining rational technological and operational parameters for there use. A new approach is developed based on the fundamental principles of analytical geometry and linear algebra, numerical methods and basic axioms of the shaping theory. In this study, we have identified the key relationships between parameters determining the position of the grinding wheel relative to the workpiece and the geometrical parameters of a solid ceramic mills. The developed approach allowing to determine the specific design of the grinding wheel, depending on the initial parameters of the mill and the trajectory of the axis of the wheel relative to the mill, was implemented in the MathCAD environment.

Cartesian closed bicategories: type theory and coherence

In this thesis I lift the Curry--Howard--Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type theory to prove a coherence result for cartesian closed bicategories. Cartesian closed bicategories---2-categories `up to isomorphism' equipped with similarly weak products and exponentials---arise in logic, categorical algebra, and game semantics. I show that there is at most one 2-cell between any parallel pair of 1-cells in the free cartesian closed bicategory on a set and hence---in terms of the difficulty of calculating---bring the data of cartesian closed bicategories down to the familiar level of cartesian closed categories. In fact, I prove this result in two ways. The first argument is closely related to Power's coherence theorem for bicategories with flexible bilimits. For the second, which is the central preoccupation of this thesis, the proof strategy has two parts: the construction of a type theory, and the proof that it satisfies a form of normalisation I call local coherence. I synthesise the type theory from algebraic principles using a novel generalisation of the (multisorted) abstract clones of universal algebra, called biclones. Using a bicategorical treatment of M. Fiore's categorical analysis of normalisation-by-evaluation, I then prove a normalisation result which entails the coherence theorem for cartesian closed bicategories. In contrast to previous coherence results for bicategories, the argument does not rely on the theory of rewriting or strictify using the Yoneda embedding. Along the way I prove bicategorical generalisations of a series of well-established category-theoretic results, present a notion of glueing of bicategories, and bicategorify the folklore result providing sufficient conditions for a glueing category to be cartesian closed.

Accelerating Scientific Computing in the Post-Moore’s Era

Novel uses of graphical processing units for accelerated computation revolutionized the field of high-performance scientific computing by providing specialized workflows tailored to algorithmic requirements. As the era of Moore’s law draws to a close, many new non–von Neumann processors are emerging as potential computational accelerators, including those based on the principles of neuromorphic computing, tensor algebra, and quantum information. While development of these new processors is continuing to mature, the potential impact on accelerated computing is anticipated to be profound. We discuss how different processing models can advance computing in key scientific paradigms: machine learning and constraint satisfaction. Significantly, each of these new processor types utilizes a fundamentally different model of computation, and this raises questions about how to best use such processors in the design and implementation of applications. While many processors are being developed with a specific domain target, the ubiquity of spin-glass models and neural networks provides an avenue for multi-functional applications. This also hints at the infrastructure needed to integrate next-generation processing units into future high-performance computing systems.

The benefit of combining teacher-direction with contrasted presentation of algebra principles

Teacher-directed and self-directed learning have been compared across various contexts. Depending on the settings and the presentation of material, mixed benefits are found; the specific circumstances under which either condition is advantageous are unclear. We combined and reanalyzed data from two experimental studies investigating the effects of contrasted versus sequential presentation of materials on learning the principles of algebraic addition and multiplication in sixth-grade classrooms. In both studies, students were presented the same structured materials that differed only in whether the principles were explained by the teacher (n = 154) or inferred by the students (n = 91). We found short- and medium-term advantages of combining teacher-direction with contrasted presentation of algebra principles. An examination of aptitude-treatment interactions shows that particularly students with lower reasoning abilities benefited from teacher-direction in the demanding contrasted condition. Based on these findings, we discuss the particular circumstances under which teacher-directed instruction reveals its advantages.