Theorem Research Articles

Certain exact many-body results for Hubbard model ground states testable in small quantum dot arrays

We present several interesting phenomena related to flatband ferromagnetism in the Hubbard model. The first is a mathematical theorem stating certain conditions under which a flatband ferromagnetic must necessarily be degenerate with a nonferromagnetic state. This theorem is generally applicable and geometry-independent, but holds only for a small number of holes in an otherwise filled band. The second phenomenon is a peculiar example where the intuition fails that particles prefer to doubly occupy low-energy states before filling higher-energy states. Lastly, we show a pattern of ferromagnetism which appears in small pentagonal and hexagonal plaquettes at filling factors of roughly 3/10 and 1/4. These examples require only a small number of lattice sites, and may be observable in quantum dot arrays currently available as laboratory spin qubit arrays.

IMPLEMENTATION OF VECTOR AUTO-REGRESSION MODELS IN TOURISM: STATE OF THE ART ANALYSIS AND FURTHER DEVELOPMENT

Purpose The dissertation focuses on time series analysis and is based on several research strategies and methods. The methodology used in the research process was published in four papers as part of international scientific journals indexed in the Web of Science database. Since tourism is one of the most lagged industries in science there is need for new and innovative approaches in key tourist sector determinants modelling and forecasting. This doctoral thesis introduces an extension of time series methodology that focuses on investigating and testing the normal distribution of residuals, as a key adequacy prerequisite of econometric models. This issue has not systematically been considered in quantitative approaches in tourism. The motivation for research of the doctoral thesis are multidimensional: to filter previous research on time series in tourism and to theoretically and empirically improve and redesign time series methodology and methods for tourism. Both issues were successfully presented in one of the published papers. Finally, tourism forecasts should be based on reliable models as evident, from the most recent shocks, ex-ante tourism forecasting has to be considered crucial in evaluating model efficiency. The dissertation aimed to research and develop appropriate econometric models able to capture the specifics of multiple interactions in the tourism market. The research seeks to develop econometric models for the Republics of Slovenia and Croatia, two countries whose economic development is predicated on tourism. Four goals and four specific objectives have been specified during the research process: 1) To introduce an improved time series approach in cointegrated panels. The first specific objective (SO1) is to test at least ten econometric modelling structures that reduce cycle breaks. 2) To examine previous theoretical thinking regarding the cointegration of time series, cross-sectional data, and panels. The second specific objective (SO2) is to outline at least 250 previous empirical studies for the tourism industry. 3) To examine cointegration in tourism data for Slovenia and Croatia. The third objective (SO3) is to model at least three econometric time series equations and mathematical theorems/ lemmas for the tourism industry. 4) To improve and better understand unit root tests in tourism. The specific objective (SO4) is to approach the design of at least three stable and innovative models.

A Review on Methods for Determining the Vibratory Damping Ratio

This article aims to popularize the methods for determining the vibratory damping ratio, to explain the various mathematical and physical theorems related to the establishment of literal expressions. Vibration damping is an essential parameter to reduce the dynamic responses of structures. The study aimed at its determination is necessary and essential for the safeguard of buildings and human lives during the earthquake. Among the main methods studied in this article, the free vibration attenuation method seems to be easy to implement but requires a state-of-the-art device to capture the responses. In addition to this device, the other methods require other equipment for the vibration of the system and the transformation of the responses in the frequency domain.

Hausdorff Distance and Similarity Measures for Single-Valued Neutrosophic Sets with Application in Multi-Criteria Decision Making

Hausdorff distance is one of the important distance measures to study the degree of dissimilarity between two sets that had been used in various fields under fuzzy environments. Among those, the framework of single-valued neutrosophic sets (SVNSs) is the one that has more potential to explain uncertain, inconsistent and indeterminate information in a comprehensive way. And so, Hausdorff distance for SVNSs is important. Thus, we propose two novel schemes to calculate the Hausdorff distance and its corresponding similarity measures (SMs) for SVNSs. In doing so, we firstly develop the two forms of Hausdorff distance between SVNSs based on the definition of Hausdorff metric between two sets. We then use these new distance measures to construct several SMs for SVNSs. Some mathematical theorems regarding the proposed Hausdorff distances for SVNSs are also proven to strengthen its theoretical properties. In order to show the exact calculation behavior and distance measurement mechanism of our proposed methods in accordance with the decorum of Hausdorff metric, we utilize an intuitive numerical example that demonstrate the novelty and practicality of our proposed measures. Furthermore, we develop a multi-criteria decision making (MCDM) method under single-valued neutrosophic environment using the proposed SMs based on our defined Hausdorff distance measures, called as a single-valued neutrosophic MCDM (SVN-MCDM) method. In this connection, we employ our proposed SMs to compute the degree of similarity of each option with the ideal choice to identify the best alternative as well as to perform an overall ranking of the alternatives under study. We then apply our proposed SVN-MCDM scheme to solve two real world problems of MCDM under single-valued neutrosophic environment to show its effectiveness and application.

Proving The Fermat Last Theorem for Case q≤n

Fermat's Last Theorem is a well-known classical theorem in mathematics. Andrew Willes has proven this theorem using the modular elliptic curve. However, the proposed proof is difficult for mathematicians and researchers to understand. For this reason, in this study, we provide evidence of several properties of Fermat's Last Theorem with a simple concept. We use Newton's Binomial Theorem, well-known in Fermat's time. In this study, we prove Fermat's Last Theorem for case . We also use the Newton’s Binomial theorem to verify several cases .

INTELLIGENT TECHNOLOGIES IN AGRO-INDUSTRY COMPLEX IN THE WORLD

Most of the results achieved using AI technologies are superior to humans: in 1997, a computer beat the then world chess champion, and more recently, in 2016, other computers beat the world's best go and poker players. Computers prove or help prove mathematical theorems; knowledge is generated automatically, based on machine learning methods, and with the help of large data sets, the size of which is calculated in terabytes (from the 10th to the 12th power) and even petabytes (to the 10th to the 15th power).

Hilbert solution, iterative algorithms, convergence theoretical results, and error bound for the fractional Langevin model arising in fluids with Caputo’s independent derivative

Studying and analyzing the random motion of a particle immersed in a liquid represented in the Langevin fractional model by Caputo’s independent derivative is one of the aims of applied physics. In this article, we will attend to a new, accurate, and comprehensive numerical solution to the aforementioned model using the reproducing kernel Hilbert approach. Basically, numerical and exact solutions of the fractional Langevin model are represented using an infinite/finite sum, simultaneously, in the Σ2Ξ space. The proof has been sketched for many mathematical theorems such as independence, convergence, error behavior, and completeness of the solution. A sufficient set of tabular results and two-dimensional graphs are shown, and absolute/relative error graphs that express the dynamic behavior of the fractional parameters α,β are utilized as well. From an analytical and practical point of view, we noticed that the simulation process and the iterative approach are appropriate, easy, and highly efficient tools for solving the studied model. In conclusion, what we have carried out is presented with a set of recommendations and an outlook on the most important literature used.

Can Mathematical Modelling Be Taught and Learned in Primary Mathematics Classrooms: A Systematic Review of Empirical Studies

STEM education has been promoted in schools worldwide to cultivate students’ 21st-century skills. Mathematical modelling is a valuable method for developing STEM education. However, in this respect, more attention is given to secondary level or above compared with kindergarten or primary level. Teaching mathematics at the primary level is closely related to authentic problems, which is a crucial characteristic of mathematical modelling activities. After screening 239 publications from various databases, we reviewed 10 empirical studies on mathematical modelling at the primary level. In this systematic review, we analysed the following three aspects: (1) the use of professional development intervention methods/strategies to enhance the intervention effects and the competencies of primary teachers to utilize mathematical modelling; (2) the effects of mathematical modelling on primary students and methods of improving their mathematical modelling skills; and (3) methods used to assess the modelling skills of primary school teachers and students. The results indicate that professional development interventions can enhance the teaching quality of mathematical modelling. The components of the interventions should include an introduction to the pedagogy of mathematical modelling, clarifying the role of the teacher and the student in mathematical modelling activities. Through mathematical modelling, students can generate mathematical ideas, explore mathematical theorems independently, develop critical thinking, and improve their metacognitive and communicative skills. The competency of mathematical modelling is often determined using formative assessments of teachers and students. Because limitations still exist in conducting primary-level modelling activities, schools should utilise more standardised assessment methods, provide universal teacher training, and grant more opportunities for primary school students to participate in mathematical modelling activities. The lack of research on cross-cultural contexts should draw the attention of future research.

Identifying the Argumentations and the Mathematical Theorems of a Geometric Task

A task is presented to demonstrate the process of understanding the argumentations and the proof for a task, by looking on a figure and its mathematical expressions, without any verbal text.

Transmission of Tuberculosis and Reproduction Number Using Lyapunov Function of SIR Model

This paper studies the mechanism of transmission of Tuberculosis (TB) caused by the mycobacterium tuberculosis bacteria. TB is an airborne disease that endangering human population globally. Mathematical modelling of tuberculosis was developed using the Susceptible-Infected-Recovered (SIR) model in this analysis. The model generated by the four-dimensional, nonlinear dynamic system is reduced to three dimensions, with some assumptions. Then, the model will be analyzed by creating a mathematical theorem that will prove the existence of a TB event, the disease-free equilibrium phase, and the TB endemic disease stage. By using the Lyapunov function method, the three theorems can be proved. The basic reproduction number R0 also can be obtained from the model. If the basic reproduction number R0 ≤ 1, then the disease-free equilibrium is global asymptotically stable and if R0 > 1, then the endemic equilibrium is asymptotically stable globally. Carrying out a simulation using data in a selected region, the result of the model successfully describes and forecast the number of TB cases and it may also be used for assessing the TB disease status.

Stability of the Just Society


 
 
 Political theorists have invested significant effort in studying the attributes of desirable social-moral states of affairs. Schaefer (forthcoming) aims to show that “static political theory” of this kind rests on shaky foundations. His argument revolves around an application of an abstruse mathematical theorem — Kakutani’s fixed point theorem — to the social-moral domain. We show that Schaefer has misunderstood the implications of this theorem for political theory. Theorists who wish to study the attributes of social-moral states of affairs should carry on, secure in the knowledge that Kakutani’s theorem poses no threat to this enterprise.
 
 

Mutual Information and Multi-Agent Systems.

We consider the use of Shannon information theory, and its various entropic terms to aid in reaching optimal decisions that should be made in a multi-agent/Team scenario. The methods that we use are to model how various agents interact, including power allocation. Our metric for agents passing information are classical Shannon channel capacity. Our results are the mathematical theorems showing how combining agents influences the channel capacity.

A Transfer Learning Framework for Power System Event Identification

The increasing uncertain components of power systems foster the wide applications of Machine Learning (ML) techniques. While traditional ML models demand a large set of data, data-scarce dilemmas exist for new meters, devices, and new grids. Further, for rich historical measurements, valuable data may still be limited, especially for targets like identifying system events that rarely occur in the power system. To enhance the event type differentiation and localization for a data-limited grid, we propose a Transfer Learning (TL) framework to transfer knowledge from a data-rich grid (source grid) to the target grid, using measurements from Phasor Measurement Units (PMUs). The transferring process is challenging because of (1) high-volume data with redundant information, (2) different measurement dimensionalities, (3) dissimilar data distributions, and (4) disjoint event-location-label spaces for two grids. To handle the challenges of (1) to (3), we propose a joint optimization to reduce dimensionality and maximize common knowledge in a shared low-dimensional feature space, where the commonality lies in the same dimensions and close data distributions. Such an optimization-based procedure is verified via rigid mathematical theorems given the same label space, i.e., event-type-label space. However, for event localization, challenge (4) obstructs the optimization. Therefore, we design a label space alignment method to relabel the event location by the event zone location and build an event zone estimation problem. Then, the framework is generalized to both tasks. Finally, comprehensive experiments demonstrate the advantages of the proposed methods over state-of-the-art transfer learning models.

Late-time-accelerated expansion esteemed from minisuperspace deformation

The effects of minisuperspace deformation on Einstein–Hilbert action along with ordinary and phantom scalar fields as the matter contents are investigated. It is demonstrated that late-time-accelerated expansion and phase transition (from decelerated to accelerated) are obtained as a consequence of minisuperspace deformation. Finally, a mathematical theorem for distinguishing valid descriptions of the noncommutative frames is suggested.

Combinatorial Interpretation of Numbers in the Generalized Padovan Sequence and Some of Its Extensions

There is ongoing research into combinatorial methods and approaches for linear and recurrent sequences. Using the notion of a board defined for the Fibonacci sequence, this work introduces the Padovan sequence combinatorial approach. Thus, mathematical theorems are introduced that refer to the study of the Padovan combinatorial model and some of its extensions, namely Tridovan, Tetradovan and its generalization (Z-dovan). Finally, we obtained a generalization of the Padovan combinatorial model, which was the main result of this research.

COMPETENCE APPROACH TO TEACHING MATHEMATICS AND HIGHER MATHEMATICS FOR THE STUDENTS MAJORING IN 181 FOOD TECHNOLOGIES SPECIALITY ON THE EXAMPLE OF APPLICATION OF KRAMER'S RULE

The relevance of the chosen research topic is that the formation of mathematical skills of students majoring in 181 Food Technology is a component of their success in professional activities. An important basis is the ability to solve problems using equations and their systems. One of the tools for solving quadratic systems of linear algebraic equations is Cramer's rule, and Cramer's theorem is one of the key theorems of higher mathematics. The aim of the article is to discuss a new methodological approach to the study of Cramer's theorem and its application for students majoring in 181 Food Technology educational and professional degree of bachelor. To achieve the goal, the following methods were used: analysis, synthesis, generalization, abstraction. The article proposes to introduce Cramer's rule for cases n = 2 and n = 3 within the discipline «Mathematics» with appropriate practical application in professional problems and to continue the study of the topic in the general case within the discipline «Higher Mathematics». This can be realized if we consider Cramer's rule for systems of two linear algebraic equations with two unknowns and for systems of three linear algebraic ones. equations with three unknowns with practical application for solving problems of professional orientation. In our opinion, such a competence approach has a number of advantages. First, in addition to the classical methods of addition, substitution, graphical method of solving systems of two linear equations with two unknowns, students will master another method of solving, and therefore additional benefits in preparing for the state final certification in mathematics. Secondly, when studying the discipline «Higher Mathematics» it is possible to consider the topic «Cramer's Theorem» as a generalization of the study on this topic in the discipline «Mathematics», which will increase motivation for learning and connections between disciplines. Third, it places additional emphasis on the formation of mathematical skills of students majoring in 181 Food Technology and the practical application of mathematics to solve professional problems in junior high school.

A MATHEMATICAL MODEL TO ADDRESS OUT-OF-SCHOOL CHILDREN MENACE FOR ACTUALIZATION OF SUSTAINABLE DEVELOPMENT IN NIGERIA

The rising number of out-of-school children (OOSC) constitutes a major obstacle to growth and development in Nigeria. Despite various institutional frameworks and policy initiatives, Nigeria accounts for the highest number of OOSC worldwide with one out of every five OOSC globally residing in Nigeria. In an attempt to characterize dynamics of OOSC and how it could be tackled to fount sustainable development in Nigeria, a new mathematical model was formulated. The validity of the model was examined using some mathematical theorems and the model equilibria were derived. The inclusive schooling ratio, an analytic parameter that quantified the extent to which the rising OOSC was being tackled to fount development, was computed. The stability properties of the model were studied via stability theory of differential equations based on the derived inclusive schooling ratio. Sensitivity analysis was conducted for some major parameters following the normalized forward sensitivity index approach to examine the relative importance of the model parameters to OOSC expansion and contraction. Numerical simulation was later conducted to justify the theoretical results and the results of the simulation showed that efforts to fount development through minimization of OOSC were fruitful if the inclusive schooling ratio was greater than one otherwise the menace of OOSC persisted. The policy implication of the result is that tackling the menace of OOSC to fount sustainable development in Nigeria is a long-term process and any policies designed to pursue the course must be sustained.

Effect of suspended particles in a porous medium layer heated from below saturating a Jeffrey fluid: A Mathematical Theorem

In this study, the influence of suspended particles on thermal convection in a porous layer saturating a Jeffrey fluid is examined. Linear stability theory based on normal modes is employed to derive a mathematical theorem on thermal convection in a porous layer saturating a Jeffrey fluid which states that the viscoelastic thermal convection at marginal state cannot manifest as stationary convection if the thermal Rayleigh number R, the medium permeability parameter Pl, the Jeffrey parameter λ_3 and suspended particles parameter B, satisfy the inequality

Vortex flow, a couple important theorems, and an introduction to distributions

We study vortex flow and perform a number of textbook calculations to highlight inconsistencies between results when using familiar, and widely important, mathematical theorems; these include Stokes’ theorem and a theorem that often enables interchanging field notions like irrotational and conservative, and their connection to scalar potentials and their associated exact differentials. The purpose is to use the familiar setting of a fluid vortex to go beyond conventional functional analysis and introduce students to the theory of distributions. After briefly doing this, we revisit vortex flow using a distributional approach to show how all prior inconsistencies disappear, emphasizing the significance of distributions in physics.

The Shape Parameter in the Shifted Surface Spline—An Easily Accessible Approach

In this paper, we present an easily accessible approach to finding a suitable shape parameter in the shifted surface spline for function interpolation. We aim at helping more readers, including mathematicians and non-mathematicians, to use our method to solve practical problems. Hence, some highly complicated mathematical theorems and definitions are avoided. The major requirement, as in our previous approach, that the data points should be evenly spaced in the domain is also relaxed. This means that the data points are purely scattered without restrictions. The drawback is that the shape parameter thus obtained is not exactly the same as the theoretically predicted optimal value, which can always be achieved by using our previous rigorous approach. However, experiments show that the gap is quite small and the final interpolation errors thus obtained are satisfactory.