By Furui S., Sandhi M.M. (eds.)
Read or Download Advances in Speech Signal Processing PDF
Similar mathematics_1 books
- Dickson Polynomials, 0th Edition
- Non-commuting Variations in Mathematics and Physics: A Survey (Interaction of Mechanics and Mathematics)
- Error Estimates for Well-Balanced Schemes on Simple Balance Laws: One-Dimensional Position-Dependent Models (SpringerBriefs in Mathematics)
- A Course in Mathematics for Students of Physics, vol. 1
- International Mathematics Tournament of Towns, Book 4: 1993-1997 (Enrichment Series, Volume 15)
Additional info for Advances in Speech Signal Processing
Each of the systems of two equations in two unknowns Xo and Yo admits of only trivial integral solutions, Xo = 1, Yo = 0 and Xo = - 1, Yo = 0 respectively. Hence, when A is equal to the square of an integer, equation (51) has only trivial solutions in integers Xo = ± 1, Yo = ltnen A is a negative integer, equation (51) has the same trivial integral solutions. ) Let us now consider a more general equation o. x2 - A y2 = C (73) V4 . , = is irrational. We have already seen that when C = 1 this equation always possesses an infinite number of integral solutions (see Theorem III) .
In which the positive integers Zo, z 1, Z2, ... zo> ZI > Z2 > ... > Zn > ... hold for them. But positive integers cannot form an infinite and monotonically decreasing sequence as there cannot be more than Zo terms in it. We have thus come to a contradiction by assuming 53 that equation (94) has at least one solution in integers x, y, Z, XYZ·=I= o. This serves as a proof that equation (94) does not have a solution.. Accordingly equation (93) has no solutions in positive integers [x, y, z] either, since, if otherwise, if [x, y, z] were a solution of equation (93), then [x, y, Z2] would be a solution to (94).
Anyn = c (81) where n is an integer greater than two and all the numbers ao, a2, ... , an and c are integers. At the beginning of this century, A. Thue proved that this equation possesses only a finite number of solutions in integers x and y, with the possible exception of cases when the homogeneous left-hand side is a power (1) of a homoqeneous linear binomial ab (ax + by)" = Co or (2) of a homogeneous quadratic trinomial (ax 2 + bxy + cy2)" = Co In both these instances integral solutions can exist only if Co is the nth power of some integer and, consequently, if equation (81) reduces to an equation of- the first or of the second degree respectively.