By Arnold V.I., et al.

This outstanding e-book is a party of the country of arithmetic on the finish of the millennium. Produced below the auspices of the foreign Mathematical Union (IMU), the quantity was once born as a part of the actions looking at the realm Mathematical 12 months 2000. the amount contains 30 articles written by way of one of the most influential mathematicians of our time. Authors of 15 contributions have been well-known in a variety of years through the IMU as recipients of the Fields Medal, from ok. F. Roth (Fields Medalist, 1958) to W. T. Gowers (Fields Medalist, 1998). The articles supply helpful reflections concerning the extraordinary mathematical growth we now have witnessed during this century and insightful speculations concerning the attainable improvement of arithmetic over the following century. a few articles formulate very important difficulties, demanding destiny mathematicians. Others pay specific homage to the well-known set of Hilbert difficulties posed 100 years in the past, giving enlightening observation. but different papers provide a deeply own standpoint, permitting singular perception into the minds and hearts of individuals doing arithmetic this present day. arithmetic: Frontiers and views is a different quantity that relates to a huge mathematical viewers of assorted backgrounds and degrees of curiosity. It deals readers real and unequaled perception into the fantastic global of arithmetic at this crucial juncture: the flip of the millennium. The paintings is a type of infrequent volumes that may be browsed, and when you do easily flick through it, you get an excellent experience of arithmetic this present day. but it can also be intensely studied on an in depth technical point for gaining perception into many of the nice difficulties on which mathematicians are presently operating. person contributors of mathematical societies of the IMU member international locations can buy this quantity on the AMS member fee whilst paying for without delay from the AMS.

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**Sample text**

J ^ ' "" 10 if ; & Ni and s ^ A . , otherwise We ihave t k a t ^S^H ^ 1, for all t, an*
*

1. Let \qn\£=1 integers be an increasing T^\ sequence and, for each n , let yn e [^]/=g^+i be so that \\yn\\(p) for some fixed p ^ 1. Then, for any choice of scalars integers {kn \£=i with q^ + l^k^ 3 " 1 ' * II t n=l of = 1, [ a n J * = 1 and ^ ^ n + i . 1 is as follows. 1'. 2. Let p > 1 and Ex < E2 < ' ' ' be finite subsets of the integers. For each n, let L^ be a linear operator from EnT^ into itself Then the operator L : 7 ^ -» 7 ^ , defined by oo x x € to - ZI Ln,En i ^ ^ ^ bounded if and only if sup ||Z

I is equivalent to Je^J^!. If, on the other hand, tp restricted to pairs of integers selected from Nx is equal to zero then \m;m ^ * ( p ), w m (fc(l)) > ir m (*(2)) > for any choice of p. Thus, as above, (2C)- 1 !! t