mathematical problems and proofs combinatorics, number theory, and geometry - kluwer academic

Ngày tải lên: 31/03/2014, 16:22

Ngày tải lên: 31/03/2014, 16:22

... 1 - . 11 1 by the 1-digit number 6 - 1 gives ( 6- l)(b- l)(b- 1 )-( 6- l)(b- l)(b- I)(, = bn - 1. Corollary. For any integer b and any positive integers m and n, we have bmn - ... m. Factor 315 - 1 and 324 - 1. Factor 512 - 1. Factor lo5 - 1, lo6 - 1 and lo8 - 1. Factor 233 - 1 and 221 - 1. Fact...

Ngày tải lên: 31/03/2014, 16:20

... , p r are prime, and if this expression is unique up to order and equiv- alence of primes. In other words, if a = q 1 ···q s is another expression of the same form, then r = s and we can find a ... factors, ie b is a unit; and similarly a is a unit, and so γ is a unit.  Lemma 1.5 A non-constant polynomial F (x) = a n x n + ··· + a 0 ∈ A[x] is prime in A[x] if and only if 374 1–25...

Ngày tải lên: 31/03/2014, 16:21

... one hand, presenting the abso- lute minimum required to understand and rigorously analyze the applications, and on the other hand, presenting a full-blown development of the relevant mathemat- ics. ... left-hand side of (1.2) and q j from the right-hand side of (1.2), and the statement now follows from the induction hypothesis. That proves the uniqueness part of Theorem 1.3. 1.2 Ideal...

Ngày tải lên: 03/04/2014, 12:23

... 1.2.4 (Hardy-Littlewood-Polya) Any vectors x and y in R n satisfy the inequality x T y ≤ [x] T [y]. We describe a proof of Fan’s Theorem in the exercises, using the above propo- sition and the following ... nonnegative entries, and each row and column summing to 1) and the set P n of permutation matrices (square matrices with all entries 0 or 1, and with exactly one entry 1 in eac...

Ngày tải lên: 31/03/2014, 15:13

... 1 Qn b qo +- 1 q1 + - q2 + -, and, specifically, Similarly from (e) above, we have 3x2 - lox + 7 - 9 - x3 - 5s' + 7~ - 3 (32 - 5) - 8 (32 - 7) * (g) ... = IU. (3112k - 1) = -1 ’(12k - 113)R = -1 . (-1 13)p = IN (3112k + 5) = (1% + 513)R = (-1 13)p = -IN. (3112k - 5) = -...

Ngày tải lên: 15/03/2014, 16:03

... Solution: A - c - D - E +A. D = Y Supplementary Problems 5.45 Write a minterm Boolean expression for the truth table in Fig. 5-5 1. Ans. X-B- C +x.B - c+A. B - C +A. B ... 5-5 7 NAND logic circuit 5.57 Draw an OR -AND logic circuit for the Boolean expression A. (B + C) * Ans. See Fig. 5-5 8. - E = Y. B C - A.(...

Ngày tải lên: 21/01/2014, 06:20