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"This book offers an original and informative view of the development of fundamental concepts of computability theory. The treatment is put into historical context, emphasizing the motivation for ideas as well as their logical and formal development. In Part I the author introduces computability theory, with chapters on the foundational crisis of mathematics in the early twentieth century, and formalism; in Part II he explains classical computability theory, with chapters on the quest for formalization, the Turing Machine, and early successes such as defining incomputable problems, c.e. (computably enumerable) sets, and developing methods for proving incomputability; in Part III he explains relative computability, with chapters on computation with external help, degrees of unsolvability, the Turing hierarchy of unsolvability, the class of degrees of unsolvability, c.e. degrees and the priority method, and the arithmetical hierarchy.This is a gentle introduction from the origins of computability theory up to current research, and it will be of value as a textbook and guide for advanced undergraduate and graduate students and researchers in the domains of computability theory and theoretical computer science."

"The 53 revised papers presented together with 6 invited lectures were carefully reviewed and selected with an acceptance rate of under 29,8%. The CiE 2012 Turing Centenary Conference will be remembered as a historic event in the continuing development of the powerful explanatory role of computability across a wide spectrum of research areas. The papers presented at CiE 2012 represent the best of current research in the area, and forms a fitting tribute to the short but brilliant trajectory of Alan Mathison Turing. Both the conference series and the association promote the development of computability-related science, ranging over mathematics, computer science and applications in various natural and engineering sciences such as physics and biology, and also including the promotion of related non-scientific fields such as philosophy and history of computing."

"The economic theory of general equilibrium underpins the most important models used in economic theory in general and in its more specialized areas such as macroeconomics, international trade, environmental economics, growth theory, and developmental economics. In Foundations of the Theory of General Equilibrium, leading academic scholar, Yves Balasko offers a good introduction to the economic theory of general equilibrium and makes use of various mathematical tools as intuitive and easy as possible. The second half of the book addresses properties of the general equilibrium model that are still at the frontier of current research. These properties deal with the characterization of economies with a unique equilibrium and, more generally, with the relationships between the number of equilibria and the fundamentals of an econom"

"Domain dari suatu fungsi peubah bernilai real adalah suatu persegi panjang tertutup dan terbatas di ruang berdimensi yang komponen-komponen dari titik ujungnya merupakan bilangan computable. Suatu fungsi disebut computable jika pertama fungsi tersebut memetakan barisan titik di ruang dimensi yang computable ke barisan bilangan real computable, kedua fungsi tersebut merupakan fungsi yang effectively uniformly continuous. Pada skripsi ini pertama–tama ditunjukkan cara memeriksa suatu barisan merupakan barisan computable. Kemudian diperlihatkan cara memeriksa suatu bilangan real merupakan bilangan computable. Pada akhirnya diperlihatkan cara memeriksa suatu fungsi peubah bernilai real merupakan fungsi yang computable.

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Domain of a computable real function of variable is a closed and bounded rectangle in dimension space. The compenents of the vertex of the rectangle are computable numbers. A real function of variable is called computable if : first it maps a computable sequence of point in dimension space to a computable sequence of real numbers, second it is an effectively uniformly continuous function. In this skripsi, first it is shown the method to check whether a sequence is computable or not. Next, some methods to determine whether a real number is computable are explored. Finally the way to prove that a real function of variable is computable is exposed."