Search results for: incompleteness theorems

Authors: Abhilash Sahu, Amit Priyadarshi

Abstract:

In this paper, we will discuss the existence of weak solutions of the Kirchhoff type boundary value problem on the Sierpinski gasket. Where S denotes the Sierpinski gasket in R² and S₀ is the intrinsic boundary of the Sierpinski gasket. M: R → R is a positive function and h: S × R → R is a suitable function which is a part of our main equation. ∆p denotes the p-Laplacian, where p > 1. First of all, we will define a weak solution for our problem and then we will show the existence of at least two solutions for the above problem under suitable conditions. There is no well-known concept of a generalized derivative of a function on a fractal domain. Recently, the notion of differential operators such as the Laplacian and the p-Laplacian on fractal domains has been defined. We recall the result first then we will address the above problem. In view of literature, Laplacian and p-Laplacian equations are studied extensively on regular domains (open connected domains) in contrast to fractal domains. In fractal domains, people have studied Laplacian equations more than p-Laplacian probably because in that case, the corresponding function space is reflexive and many minimax theorems which work for regular domains is applicable there which is not the case for the p-Laplacian. This motivates us to study equations involving p-Laplacian on the Sierpinski gasket. Problems on fractal domains lead to nonlinear models such as reaction-diffusion equations on fractals, problems on elastic fractal media and fluid flow through fractal regions etc. We have studied the above p-Laplacian equations on the Sierpinski gasket using fibering map technique on the Nehari manifold. Many authors have studied the Laplacian and p-Laplacian equations on regular domains using this Nehari manifold technique. In general Euler functional associated with such a problem is Frechet or Gateaux differentiable. So, a critical point becomes a solution to the problem. Also, the function space they consider is reflexive and hence we can extract a weakly convergent subsequence from a bounded sequence. But in our case neither the Euler functional is differentiable nor the function space is known to be reflexive. Overcoming these issues we are still able to prove the existence of at least two solutions of the given equation.

Keywords: Euler functional, p-Laplacian, p-energy, Sierpinski gasket, weak solution

Procedia PDF Downloads 211

Authors: David C. Ni

Abstract:

The current efforts for Grand Unification Theory (GUT) can be classified into General Relativity, Quantum Mechanics, String Theory and the related formalisms. In the geometric approaches for extending General Relativity, the efforts are establishing global and local invariance embedded into metric formalisms, thereby additional dimensions are constructed for unifying canonical formulations, such as Hamiltonian and Lagrangian formulations. The approaches of extending Quantum Mechanics adopt symmetry principle to formulate algebra-group theories, which evolved from Maxwell formulation to Yang-Mills non-abelian gauge formulation, and thereafter manifested the Standard model. This thread of efforts has been constructing super-symmetry for mapping fermion and boson as well as gluon and graviton. The efforts of String theory currently have been evolving to so-called gauge/gravity correspondence, particularly the equivalence between type IIB string theory compactified on AdS5 × S5 and N = 4 supersymmetric Yang-Mills theory. Other efforts are also adopting cross-breeding approaches of above three formalisms as well as competing formalisms, nevertheless, the related symmetries, dualities, and correspondences are outlined as principles and techniques even these terminologies are defined diversely and often generally coined as duality. In this paper, we firstly classify these dualities from the perspective of physics. Then examine the hierarchical structure of classes from mathematical perspective referring to Coleman-Mandula theorem, Hidden Local Symmetry, Groupoid-Categorization and others. Based on Fundamental Theorems of Algebra, we argue that rather imposing effective constraints on different algebras and the related extensions, which are mainly constructed by self-breeding or self-mapping methodologies for sustaining invariance, we propose a new addition, momentum-angular momentum duality at the level of electromagnetic duality, for rationalizing the duality algebras, and then characterize this duality numerically with attempt for addressing some unsolved problems in physics and astrophysics.

Keywords: general relativity, quantum mechanics, string theory, duality, symmetry, correspondence, algebra, momentum-angular-momentum

Procedia PDF Downloads 360

Authors: J. Shamash

Abstract:

The high-school curriculum in algebra deals mainly with the solution of different types of equations. However, modern algebra has a completely different viewpoint and is concerned with algebraic structures and operations. A question then arises: What might be the relevance and contribution of an abstract algebra course for developing expertise and mathematical perspective in secondary school mathematics instruction? This is the focus of this paper. The course Algebra: From Equations to Structures is a carefully designed abstract algebra course for Israeli secondary school mathematics teachers. The course provides an introduction to algebraic structures and modern abstract algebra, and links abstract algebra to the high-school curriculum in algebra. It follows the historical attempts of mathematicians to solve polynomial equations of higher degrees, attempts which resulted in the development of group theory and field theory by Galois and Abel. In other words, algebraic structures grew out of a need to solve certain problems, and proved to be a much more fruitful way of viewing them. This theorems in both group theory and field theory. Along the historical ‘journey’, many other major results in algebra in the past 150 years are introduced, and recent directions that current research in algebra is taking are highlighted. This course is part of a unique master’s program – the Rothschild-Weizmann Program – offered by the Weizmann Institute of Science, especially designed for practicing Israeli secondary school teachers. A major component of the program comprises mathematical studies tailored for the students at the program. The rationale and structure of the course Algebra: From Equations to Structures are described, and its relevance to teaching school algebra is examined by analyzing three kinds of data sources. The first are position papers written by the participating teachers regarding the relevance of advanced mathematics studies to expertise in classroom instruction. The second data source are didactic materials designed by the participating teachers in which they connected the mathematics learned in the mathematics courses to the school curriculum and teaching. The third date source are final projects carried out by the teachers based on material learned in the course.

Keywords: abstract algebra , linking abstract algebra and school mathematics, school algebra, secondary school mathematics, teacher professional development

Procedia PDF Downloads 122

Authors: Jin Liang, James H. Liu, Ti-Jun Xiao

Abstract:

It is known that there exist many physical phenomena where abrupt or impulsive changes occur either in the system dynamics, for example, ad-hoc network, or in the input forces containing impacts, for example, the bombardment of space antenna by micrometeorites. There are many other examples such as ultra high-speed optical signals over communication networks, the collision of particles, inventory control, government decisions, interest changes, changes in stock price, etc. These are impulsive phenomena. Hence, as a combination of the traditional initial value problems and the short-term perturbations whose duration can be negligible in comparison with the duration of the process, the systems with impulsive conditions (i.e., impulsive systems) are more realistic models for describing the impulsive phenomenon. Such a situation is also suitable for the delay systems, which include some of the past states of the system. So far, there have been a lot of research results in the study of impulsive systems with delay both in finite and infinite dimensional spaces. In this paper, we investigate the periodicity of solutions to the nonautonomous impulsive evolution equations with infinite delay in Banach spaces, where the coefficient operators (possibly unbounded) in the linear part depend on the time, which are impulsive systems in infinite dimensional spaces and come from the optimal control theory. It was indicated that the study of periodic solutions for these impulsive evolution equations with infinite delay was challenging because the fixed point theorems requiring some compactness conditions are not applicable to them due to the impulsive condition and the infinite delay. We are happy to report that after detailed analysis, we are able to combine the techniques developed in our previous papers, and some new ideas in this paper, to attack these impulsive evolution equations and derive periodic solutions. More specifically, by virtue of the related transition operator family (evolution family), we present a Poincaré operator given by the nonautonomous impulsive evolution system with infinite delay, and then show that the operator is a condensing operator with respect to Kuratowski's measure of non-compactness in a phase space by using an Amann's lemma. Finally, we derive periodic solutions from bounded solutions in view of the Sadovskii fixed point theorem. We also present a relationship between the boundedness and the periodicity of the solutions of the nonautonomous impulsive evolution system. The new results obtained here extend some earlier results in this area for evolution equations without impulsive conditions or without infinite delay.

Keywords: impulsive, nonautonomous evolution equation, optimal control, periodic solution

Procedia PDF Downloads 221

Authors: Razaq A. Olaitan, Ayansina Ayanlade

Abstract:

Understanding aerosol properties is the main goal of global research in order to lower the uncertainty associated with climate change in the trends and magnitude of aerosol particles. In order to identify aerosol particle types, optical properties, and the relationship between aerosol properties and particle concentration between 2019 and 2021, a study conducted in Ilorin, Nigeria, examined the aerosol robotic network's ground-based sun/sky scanning radiometer. The AERONET algorithm version 2 was utilized to retrieve monthly data on aerosol optical depth and angstrom exponent. The version 3 algorithm, which is an almucantar level 2 inversion, was employed to retrieve daily data on single scattering albedo and aerosol size distribution. Excel 2016 was used to analyze the data's monthly, seasonal, and annual mean averages. The distribution of different types of aerosols was analyzed using scatterplots, and the optical properties of the aerosol were investigated using pertinent mathematical theorems. To comprehend the relationships between particle concentration and properties, correlation statistics were employed. Based on the premise that aerosol characteristics must remain constant in both magnitude and trend across time and space, the study's findings indicate that the types of aerosols identified between 2019 and 2021 are as follows: 29.22% urban industrial (UI) aerosol type, 37.08% desert (D) aerosol type, 10.67% biomass burning (BB), and 23.03% urban mix (Um) aerosol type. Convective wind systems, which frequently carry particles as they blow over long distances in the atmosphere, have been responsible for the peak-of-the-columnar aerosol loadings, which were observed during August of the study period. The study has shown that while coarse mode particles dominate, fine particles are increasing in seasonal and annual trends. Burning biomass and human activities in the city are linked to these trends. The study found that the majority of particles are highly absorbing black carbon, with the fine mode having a volume median radius of 0.08 to 0.12 meters. The investigation also revealed that there is a positive coefficient of correlation (r = 0.57) between changes in aerosol particle concentration and changes in aerosol properties. Human activity is rapidly increasing in Ilorin, causing changes in aerosol properties, indicating potential health risks from climate change and human influence on geological and environmental systems.

Keywords: aerosol loading, aerosol types, health risks, optical properties

Procedia PDF Downloads 21

Authors: Wedad Albalawi

Abstract:

The mathematical inequalities have been the core of mathematical study and used in almost all branches of mathematics as well in various areas of science and engineering. The inequalities by Hardy, Littlewood and Polya were the first significant composition of several science. This work presents fundamental ideas, results and techniques, and it has had much influence on research in various branches of analysis. Since 1934, various inequalities have been produced and studied in the literature. Furthermore, some inequalities have been formulated by some operators; in 1989, weighted Hardy inequalities have been obtained for integration operators. Then, they obtained weighted estimates for Steklov operators that were used in the solution of the Cauchy problem for the wave equation. They were improved upon in 2011 to include the boundedness of integral operators from the weighted Sobolev space to the weighted Lebesgue space. Some inequalities have been demonstrated and improved using the Hardy–Steklov operator. Recently, a lot of integral inequalities have been improved by differential operators. Hardy inequality has been one of the tools that is used to consider integrity solutions of differential equations. Then, dynamic inequalities of Hardy and Coposon have been extended and improved by various integral operators. These inequalities would be interesting to apply in different fields of mathematics (functional spaces, partial differential equations, mathematical modeling). Some inequalities have been appeared involving Copson and Hardy inequalities on time scales to obtain new special version of them. A time scale is an arbitrary nonempty closed subset of the real numbers. Then, the dynamic inequalities on time scales have received a lot of attention in the literature and has become a major field in pure and applied mathematics. There are many applications of dynamic equations on time scales to quantum mechanics, electrical engineering, neural networks, heat transfer, combinatorics, and population dynamics. This study focuses on Hardy and Coposon inequalities, using Steklov operator on time scale in double integrals to obtain special cases of time-scale inequalities of Hardy and Copson on high dimensions. The advantage of this study is that it uses the one-dimensional classical Hardy inequality to obtain higher dimensional on time scale versions that will be applied in the solution of the Cauchy problem for the wave equation. In addition, the obtained inequalities have various applications involving discontinuous domains such as bug populations, phytoremediation of metals, wound healing, maximization problems. The proof can be done by introducing restriction on the operator in several cases. The concepts in time scale version such as time scales calculus will be used that allows to unify and extend many problems from the theories of differential and of difference equations. In addition, using chain rule, and some properties of multiple integrals on time scales, some theorems of Fubini and the inequality of H¨older.

Keywords: time scales, inequality of hardy, inequality of coposon, steklov operator

Procedia PDF Downloads 56

Authors: Henda Gnakate Biba, Ndassa Mouafon Issa

Abstract:

Dictionaries modeled on the Western model [tonic accent languages] are not suitable and do not account for tonal languages phonologically, which is why the [prosodic and phonological] ethnographic dictionary was designed. It is a glossary that expresses the tones and the rhythm of words. It recreates exactly the speaking or singing of a tonal language, and allows the non-speaker of this language to pronounce the words as if they were a native. It is a dictionary adapted to tonal languages. It was built from ethnomusicological theorems and phonological processes, according to Jean. J. Rousseau 1776 hypothesis /To say and to sing were once the same thing/. Each word in the French dictionary finds its corresponding language, ekaη. And each word ekaη is written on a musical staff. This ethnographic dictionary is also an inventive, original and innovative research thesis, but it is also an inventive, original and innovative research thesis. A contribution to the theoretical, musicological, ethno musicological and linguistic conceptualization of languages, giving rise to the practice of interlocution between the social and cognitive sciences, the activities of artistic creation and the question of modeling in the human sciences: mathematics, computer science, translation automation and artificial intelligence. When you apply this theory to any text of a folksong of a world-tone language, you do not only piece together the exact melody, rhythm, and harmonies of that song as if you knew it in advance but also the exact speaking of this language. The author believes that the issue of the disappearance of tonal languages and their preservation has been structurally resolved, as well as one of the greatest cultural equations related to the composition and creation of tonal, polytonal and random music. The experimentation confirming the theorization designed a semi-digital, semi-analog application which translates the tonal languages of Africa (about 2,100 languages) into blues, jazz, world music, polyphonic music, tonal and anatonal music and deterministic and random music). To test this application, I use a music reading and writing software that allows me to collect the data extracted from my mother tongue, which is already modeled in the musical staves saved in the ethnographic (semiotic) dictionary for automatic translation ( volume 2 of the book). Translation is done (from writing to writing, from writing to speech and from writing to music). Mode of operation: you type a text on your computer, a structured song (chorus-verse), and you command the machine a melody of blues, jazz and, world music or, variety etc. The software runs, giving you the option to choose harmonies, and then you select your melody.

Keywords: music, language, entenglement, science, research

Procedia PDF Downloads 35