Modern Matematikte Analiz ve Uygulamalı Yöntemler: Ölçü Teorisi, Diferansiyel Denklemler, Yaklaşım Yöntemleri Ve Makine Öğrenmesi

This academic work, titled Modern Mathematical Analysis and Applied Methods: Measure Theory, Differential Equations, Approximation Methods, and Machine Learning, comprehensively covers theoretical and applied aspects of modern mathematics. The section on measure theory presents classical and fuzzy measure concepts, including definitions, properties, and examples. It further explores lattice theory and lattice-valued fuzzy measures. In approximation theory, newly defined Bernstein-Kantorovich polynomials and the (λ, μ)-Bernstein-Durrmeyer-Stancu operator sequences are introduced with detailed properties, central moments, and Voronovskaja-type convergence theorems, supported by graphical illustrations. The differential equations segment tackles eigenvalue problems, particularly Sturm-Liouville problems with transmission conditions, presenting Green's function expansions, Parseval's equality, and the Carleman formula. Additionally, numerical techniques such as finite difference schemes and double Laplace transform methods are discussed for elliptic partial differential equations, linking rigorous functional analysis frameworks with computational approaches. In an applied section, a comparative analysis of logistic regression and k-nearest neighbors classifiers is conducted on gout disease patient data, highlighting the significant influence of uric acid levels and age on disease prediction. The book bridges formal mathematical theories and practical machine learning tools, offering a multidimensional resource for researchers and graduate students in mathematics and related fields.