Mathematics الرياضيات

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Dynamic Set Theory and Extensions on Lagrange Squares: New Theorems, Philosophical Insights and Applications
(Al-Quds University, 2025-05-14) Ahmad Zeyad Mahmoud Al-Awawdeh; أحمد زياد محمود العواودة
This thesis presents an innovative mathematical and philosophical development by establishing a connection between Lagrange Four Square Theorem and five well-known sequences: the sequence of natural numbers, square numbers, cubic numbers, triangular numbers, and the Fibonacci sequence. This integration resulted in the construction of five new sequences, each of which has been rigorously analyzed and accompanied by fourteen original theorems, all proven within a strict methodological framework.. This thesis reflects a multifaceted approach that bridges the rigorous structure of number theory with the philosophical perspective of mathematics. It introduces a novel principle in the philosophy of mathematics, termed "coexistence of opposites within a single entity", which opens new horizons for understanding contradiction and integration within the mathematical framework itself. Furthermore, the research establishes connections with fundamental issues in the philosophy of mathematics. The developed sequences were also utilized in various applied domains, including a reimagining of the classical game of chess based on the framework of Lagrange’s foursquare theorem, as well as the resolution of real-world problems in fields such as physics, cryptography, economics, and engineering—demonstrating the practical value of the proposed theorems. In the concluding part of the thesis, a new theoretical framework proposed under the title ―Dynamic Set Theory,‖ which reimagines mathematical sets as mutable entities rather than fixed ones—contrasting with the classical perspective established by Cantor. The logical, philosophical, and algebraic foundations of this approach presented, positioning it as a fertile ground for future research.
Optimal Homotopy Asymptotic Method for Solving Multidimensional First Order Systems of Partial Differential Equations
(Al-Quds University, 2024-07-15) Malak Tawfiq Abdelfattah Meqbil; ملاك توفيق عبدالفتاح مقبل
In this thesis, a semi-analytic approximating method, namely Optimal Homotopy Asymptotic Method (OHAM), which is developed from the Homotopy Analysis Method (HAM), is used to find continuous approximate solutions for linear and non-linear first-order systems of partial differential equations. Within this work, the geometrical topological homotopy concept is used to construct the algorithm for solving such systems. A homotopy equation that depends on an embedding parameter belonging to interval [0, 1] is assumed. As the parameter varies from 0 to 1 the solution of the homotopy equation (which is assumed to be a power series of the embedding parameter) varies continuously from a solution, which is easy to find, to the exact solution. The approximate continuous solution is obtained by truncating the series and using a finite number of its terms. Least Squares Method is used to determine the so-called control-convergence parameters that appear in the approximate solution. The derived algorithm is applied to solve some examples and the obtained solutions are compared with exact solutions. The results confirm the validity of OHAM and reveal that OHAM is effective, simple, and explicit. Moreover, it is independent of the small parameters required in perturbation methods. Furthermore, the convergence domain of OHAM is easily modifiable, depending on the convergence-control parameters that appear in the approximated solutions, enhancing its versatility.
A Generlized Defenition of the Fractional Derivative with Application in Newtonian Mechanics
(Al-Quds University, 2023-05-27) Samah Mousa Kalaf Bajali; سماح موسى خلف بجالي
This work studies the proposed new generalized fractional derivative (GD) definition, showing that the index law 𝐷𝛼𝐷𝛽𝑓(𝑡) = 𝐷𝛼+𝛽𝑓(𝑡); 0 < 𝛼,𝛽 ≤ 1 works for a differentiable function expanded by a Taylor series. (GD) is applied for some functions, the results are compared with Caputo fractional derivative. The solutions of some fractional differential equation are obtained via the (GD) operator. A comparison with the conformable derivative (CD) is also discussed. Newtonian Mechanics is discussed in the light of the fractional calculus. يدرس هذا العمل التعريف الجديد المعمم للاشتقاق الكسري (GD)، والذي يوضح أن قانون المؤشر D^α D^β f(t) = D^(α+β) f(t); 0 < α,β ≤ 1 تنطبق على جميع الاقترانات القابلة للتفاضل موسعة بواسطة سلسلة تايلور. يتم تطبيق (GD) على بعض الاقترانات ومقارنة النتائج مع مشتقة Caputo الكسرية. يتم الحصول على حلول بعض المعادلات التفاضلية الكسرية عبر عامل التشغيل (GD). كما تمت مناقشة النتائج مع تعريف (CD) للاشتقاق الكسري. كما تمت مناقشة ميكانيكا نيوتن في ضوء حساب التفاضل والتكامل الكسري.
Multivariate Time Series with Application On Recurrent Neural Networks
(Al-Quds University, 2020-06-12) Safa Nader Mustafa Shanaa; صفاء نادر مصطفى شناعة
Multivariate time series data in practical applications, such as health care, geosciences, engineering, and biology. This thesis introduces a survey study of time series analysis to recurrent neural networks research, an analytic domain that has been essential for understanding and predicting the behavior of variables across many diverse fields, in this research the following were investigated. First, the characteristics and preliminaries of time series data are investigated and discussed, including various time series models, specially, Autoregressive Models such as, AR, MA, ARMA, and ARIMA. Frequently one wishes to fit a parametric model to time-series data and determine accurate values of the parameters and reliable estimates for the uncertainties in those parameters. It is important to gain a thorough understanding of the noise and develop appropriate methods for parameter estimation, so that various approaches of parameter estimates will be considered in this thesis, such as, yules walker method, least square method, method of moments and maximum likelihood approach. Second, different time series modeling techniques are surveyed that can address various topics of interest to artificial neural networks researchers, including describing the pattern of change in a variable, modeling seasonal effects, assessing the immediate and long-term impact of a salient event, and forecasting future values. The structure of the artificial neural networks especially the recurrent neural networks were discussed in details in this research, concerning on GRUs and LSTMs, and their properties, also some difficulties that arises in recurrent neural networks such as vanished gradient and the overfitting were discussed. To illustrate these approaches, an illustrative application based on Monte Carlo and bootstrapping methods is used throughout the research, constructing a one layer hidden recurrent neural networks and applied back-propagation, for the purpose of comparison, the variance of error in each method was estimated.
Linear Regression Models Assuming a Stable Distribution with Applications
(Al-Quds University, 2020-08-31) Layla Khaled Mahmoud Lahaleeh; ليلى خالد محمود اللهاليه

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