By Robert Gilman, Alexei G. Myasnikov, Vladimir Shpilrain, Sean Cleary

This quantity grew out of 2 AMS meetings held at Columbia college (New York, long island) and the Stevens Institute of expertise (Hoboken, NJ) and offers articles on a wide selection of subject matters in workforce thought. Readers will discover a number of contributions, together with a set of over a hundred and seventy open difficulties in combinatorial team idea, 3 first-class survey papers (on limitations of hyperbolic teams, on mounted issues of loose crew automorphisms, and on teams of automorphisms of compact Riemann surfaces), and several other unique study papers that characterize the range of present tendencies in combinatorial and geometric workforce thought. The booklet is a wonderful reference resource for graduate scholars and learn mathematicians attracted to numerous elements of team idea

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**Example text**

Now that we have pentagonal numbers, we can define m-gonal numbers for any m ~ 3 in a similar manner. The two most common types of 40 12. Classical Partition Function and Euler's Product Formula polygonal numbers are the triangular numbers, ~ _ n(n+ 1) 2 n - ' and , of course, the square numbers , On = n 2 • In general , we may deduce geometrically the formula for the nth m-gonal number: mn _ (2)~ - m - + n -_ n(mn-2n-m+4) 2 . n -l Identities for triangular and square numbers similar to Euler 's product formula can also be derived from Jacobi's triple product identity.

1. ~~ . 6) Definition. 9) 9. Two Euler's Identities and Two q-Exponential Functions 31 Analogously, we can define another q-exponential function using El. Definition. Another q-analogue of the classical exponential function is EX = q ~ qj(j-1 )/2 x j [j]! J J=O = (1 + (1 _ q)x)oo . 10) q Let us study some properties of the two q-exponential functions. The classical exponential function is unchanged under differentiation. Its two q-analogues have similar behavior. J -[j]! J -[j]! J [ . -I]! ' J=O J=l J=l J J=O J and , t qj(j-1)/2 D~~j = [J] .

D Like the Pascal rule, many identities involving binomial coefficients have their q-analogues. Imagine that we have m + n balls, and they are placed into two groups, one with m and one with n of them. Each way of choosing k balls from all m + n of them corresponds in a one-to-one manner to a way of choosing j balls from the group with m balls and choosing k - j balls from the group of n balls , with j running from 0 to k. ). 3) Example. 1. Let V = IF;:+n and let Vm C V be a fixed subspace with dim Vm = m .