By Weidong H., Yulong M.

Best mathematics books

Infinite Ascent: A Short History of Mathematics

In limitless Ascent, David Berlinski, the acclaimed writer of the arrival of the set of rules, A travel of the Calculus, and Newton’s present, tells the tale of arithmetic, bringing to existence with wit, beauty, and deep perception a 2,500-year-long highbrow adventure.

Berlinski makes a speciality of the 10 most crucial breakthroughs in mathematical history–and the boys in the back of them. listed here are Pythagoras, intoxicated by way of the magical importance of numbers; Euclid, who gave the area the very notion of an evidence; Leibniz and Newton, co-discoverers of the calculus; Cantor, grasp of the endless; and Gödel, who in a single outstanding facts put every little thing in doubt.

The elaboration of mathematical wisdom has intended not anything under the unfolding of human cognizance itself. together with his unequalled skill to make summary principles concrete and approachable, Berlinski either tells an engrossing story and introduces us to the total energy of what surel

Additional info for 3-D wavelet transform for very low bit-rate video coding

Sample text

Further, with x = 111, y = signt (t^ 0), equation (7) gives h(t) = h(\t\ sign t) = H\t\) + Msignt) = h(\t|), or h(t) = h(\t\) for all t # 0 . Note that, by definition, sign t = 1 if t > 0, sign t= — 1 if f < 0 and sign 0 = 0, but here the sign function is applied only to t # 0. Compared to (8), we get the following. An application to information theory 27 Theorem 3. The general solution h:M*-> R of (7) is given by h(x) = /(log | x |) /or a// rea/ x # 0, (9) w/zere / is aw arbitrary solution of (3).

Therefore the following series is convergent on L(N): /(x) = / + xC + ^ C 2 + " + ^ C ' l + . . 2 (x2*0), (48) nl where / is the identity operator on N. The series (48) defines /(x) = e*c. (49) Then (49) is a solution of (47) for all x, y in R+ as is proved by the usual manipulation of convergent series. However, the general solution of (47) is not of the form (49). An extensive literature has been devoted to the problem of finding regularity conditions for a solution / of (47) so that it reduces to (48).

Conversely, let D be a linear operator mapping S9 a subspace of N9 into N. An interesting question is to decide when there exists a solution / of (50), with /(0) = /, and for which (51) holds with Dn = D(n). If Q = N and if D is bounded, f(x) = exD provides a solution for which / is continuous. There exist deeper results when 3 is dense in N and when D is a closed linear operator. For the existence of a solution / of (50), with /(0) = / and with lim x - 0 /(x)n — n = 0 for all n in N such that (51) holds, a necessary and sufficient condition is the existence of real numbers M and w such that, for every real X > w, (A/ — £>)"J exists and satisfies the inequalities Exercises and further results 1.