Mathematical Omnibus: Thirty Lectures on Classic Mathematics doc

... chaotic. Continued fractions for rational numbers are always finite. Infinite periodic continued fractions correspond to “quadratic irrationalities”, that is, to roots of quadratic equations with rational ... 0. Proposition 1.3. Any rational number has a unique presentation as a finite continued fraction. Proof of existence. For an irreducible fraction p q , we shall prove the existence of a con...

Ngày tải lên: 08/03/2014, 12:20

... (1.1.8). It is an operation defined on the set M N  M . Two matrices of the same order are said to be conformable for addition and subtraction. Addition and subtraction are not defined for ... switching two rows, b) multiplying one of the rows by a nonzero constant c) multiply one row by a nonzero constant and adding it to another row, or d) combinations of a),b) and c). Equiva...

Ngày tải lên: 17/03/2014, 14:20

... even, respectively. Then prove that the functions f and g ◦ f are one-one but g is not one-one. Solution: By definiton, it is clear that f is indeed one-one and g is not one-one. But g ◦f(x) = g(f (x)) = g(2x) ... odd. Then prove that f is one- one. Is f onto? Solution: Let us use the contrapostive argument to prove that f is one-one. Let if possible f(x) = f(y), for some x, y ∈ N. Using the d...

Ngày tải lên: 31/03/2014, 12:20

... standard formulation of the corresponding optimization problem, one restricts attention to absolutely continuous measures admitting a rate of consumption, and the utility functional is a time–additive ... of utility functionals where utilities at different times depend on an index of satisfaction based on past consumption. The corresponding singular control problem raises new mathematica...

Ngày tải lên: 22/02/2014, 06:20

... theory 3P.T. Johnstone Stone spaces 4 W.H. Schikhof Ultrametric calculus 5J-P. Kahane Some random series of functions, second edition 6 H. Cohn Introduction to the construction of class fields 7J. ... The finitely many inequalities (i)-(iii) of the proposition are implications of the infinitely many reduction conditions in Definition 1. On the other hand, we will see later that we do not lo...

Ngày tải lên: 22/03/2014, 15:20

... contractions. We shall contend for only two cases of nouns, one kind of pronouns, one kind of verbs, that all are active; three modes, and as many tenses; that articles, adverbs, prepositions, conjunctions, ... definition is produced by contrast and comparison. Children soon learn the difference between a sweet apple and a sour one, a white rose and a red one, a hard seat and a soft one,...

Ngày tải lên: 22/03/2014, 16:22

Ngày tải lên: 12/05/2014, 08:11

... lao động trong những năm qua đã có sự chuyển dịch theo hướng tăng tỷ trọng lao động làm việc trong các ngành công nghiệp – xây dựng và dịch vụ, đồng thời giảm tỷ trọng lao động trong ngành nông ... NSNN, mà trong bất kì cơ chế nào và trong tời đại nào NSNN cũng phải thực hiện, vai trò của NSNN được xác định trên cơ sở bản chất kinh tế của NSNN. Sự hoạt động của Nhà nước trong các lĩnh ......

Ngày tải lên: 21/09/2012, 17:10

... difficult. In addition, among various types of validity, content and face validity are considered to be most important for teachers to consider when constructing his own tests (Harrison, 1987). 2.1.2.2. ... the festival season, people tranditionally give each other boxes of moon cakes, which are very rich in taste. 19 Part 4: Conclusion References Part 1 Introduction 1.1. Reasons for choo...

Ngày tải lên: 27/10/2012, 16:40