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"This book brings together many important recent developments in the analysis of singular perturbation and hysteresis phenomena in an accessible and reasonably comprehensive fashion. To bridge a gap between practitioners of these phenomena, the editors conducted a workshop in April 2002 at University College Cork to provide a forum for experts in both fields to share their interests and knowledge. For this book, the editors have compiled research from those practitioners in areas such as reacting systems, semiconductor lasers, shock phenomena in economic modeling, and fluid mechanics, all with an emphasis on hysteresis and singular perturbations."

"Pada tesis ini dibahas mengenai sistem predator-prey dengan tingkat kelahiran prey jauh lebih kecil dibandingkan tingkat kematian predator dan mengasumsikan bahwa populasi prey terbatas, sehingga ditambahkan faktor kapasitas lingkungan ke populasi prey. Karena perbedaan rentang waktu regenerasi antara predator dan prey relatif tinggi, sistem memiliki struktur fast-slow. Struktur fast-slow pada model dianalisis dengan menggunakan geometric singular perturbation theory (GSPT) yang membagi sistem menjadi fast subsystem dan slow subsystem. Dalam tesis ini, dipelajari keberadaan solusi ekuilibrium dan stabilitasnya, dan juga perilaku solusi di sekitar critical manifold. Selanjutnya, dengan menggunakan fungsi entry-exit diperoleh hubungan antara solusi fast subsystem dan slow subsystem secara analitis.

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In this thesis we discuss the predator-prey system with the birth rate of the prey is much smaller compared to the predator mortality rate where it is assumed that the population of prey is limited. Therefore, an environmental carrying capacity factor is added to the prey population. Due to the difference in regeneration’s timescales between predator and prey, some solutions of the system may have a fast-slow structure. The fast-slow structure of the model is analyzed using the geometric singular perturbations theory (GSPT) which divides the system into fast and slow subsystem. In this thesis, we study the existence of the equilibrium solutions and their stability, and also the behaviour of the solutions near critical manifold. Furthermore, using an entry-exit function we obtain the connection between the solutions of the slow subsystem and the fast subsystem analytically."

"This is the first book published in English devoted solely to the boundary function method, which is one of the asymptotic methods. This method provides an effective and simple way to obtain asymptotic approximations for the solutions of certain ordinary and partial differential equations containing small parameters in front of the highest derivatives. These equations, called singularly perturbed equations, are often used in modeling. In addition to numerous examples, the book includes discussions on singularly perturbed problems from chemical kinetics and heat conduction, semiconductor device modeling, and mathematical biology. The book also contains a variety of original ideas and explicit calculations previously available only in journal literature, as well as many concrete applied problems illustrating the boundary function method algorithms. Quite general asymptotic results described in the book are rigorous in the sense that, along with the asymptotic algorithms, in most cases the theorems on estimation of the remainder terms are presented. A survey of results of Russian mathematicians on the subject is provided; many of these results are not well known in the West."

"The authors give a comprehensive treatment of the Harnack inequality for non-negative solutions to p-laplace and porous medium type equations, both in the degenerate (p>2 or m>1) and in the singular range (1"

"ABSTRAKPendeteksian topik adalah proses untuk menemukan topik atau pokok pembahasan utama dalam suatu kumpulan dokumen. Untuk data yang besar, pendeteksian topik dengan manual sulit atau bahkan tidak mungkin dilakukan. Sehingga, dibutuhkan metode otomatis yang dikenal dengan istilah Topic Detection and Tracking (TDT). Pada penelitian ini metode TDT yang digunakan untuk masalah pendeteksian topik adalah fuzzy C-means (FCM). FCM bekerja cukup baik pada dimensi data yang rendah, tetapi gagal pada dimensi data yang tinggi. Pada metode fuzzy c-means umumnya dilakukan inisialisasi random yang menyebabkan data konvergen ke satu pusat (centre of gravity) sehingga topik-topik yang dihasilkan antara satu dengan yang lainnya sama. Untuk mengatasi masalah tersebut dibutuhkan inisialisasi yang tidak random, yaitu dengan menggunakan inisialisasi berbasis singular value decomposition (SVD). Hasil akurasi dari metode ini menunjukkan adanya peningkatan lebih baik dibandingkan dengan metode FCM dengan inisialisasi random. Dengan nilai akurasi terbaik untuk FA Cup adalah 0,923, untuk US Elections adalah 0,661 dan untuk Super Tuesday adalah 0,727.

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ABSTRACTTopic detection is the process of finding the main topic or topic in a document. For large data, manual topic detection is difficult or even impossible. Thus, it takes an automatic method known as Topic Detection and Tracking (TDT). In this research the TDT method used for topic detection problem is fuzzy C-means (FCM). FCM works reasonably well on low data dimensions, but fails on high data dimensions. In the method of fuzzy c-means is generally done random initialization that causes data convergent to one center (center of gravity) so that the topics generated from one another are equal. To solve this problem requires non-random initialization, ie by using a singular value decomposition (SVD) based initialization. The accuracy of this method shows a better improvement compared to the FCM method with random initialization. With the best accuracy value for the FA Cup is 0.923, for US Elections is 0.661 and for Super Tuesday is 0.727."

"Perturbations: Theory and Methods gives a thorough introduction to both regular and singular perturbation methods for algebraic and differential equations. Unlike most introductory books on the subject, this one distinguishes between formal and rigorous asymptotic validity, which are commonly confused in books that treat perturbation theory as a bag of heuristic tricks with no foundation. The meaning of "uniformity" is carefully explained in a variety of contexts. All standard methods, such as rescaling, multiple scales, averaging, matching, and the WKB method are covered, and the asymptotic validity (in the rigorous sense) of each method is carefully proved. First published in 1991, this book is still useful today because it is an introduction. It combines perturbation results with those known through other methods. Sometimes a geometrical result (such as the existence of a periodic solution) is rigorously deduced from a perturbation result, and at other times a knowledge of the geometry of the solutions is used to aid in the selection of an effective perturbation method. Dr. Murdock's approach differs from other introductory texts because he attempts to present perturbation theory as a natural part of a larger whole, the mathematical theory of differential equations. He explores the meaning of the results and their connections to other ways of studying the same problems."

"Most of the results are closely connected with PDEs, and the authors' approach presents a powerful method for studying the asymptotic behavior of the solutions of initial-boundary value problems for corresponding PDEs."