Mathematics

The plane partition polynomial Qn(x) is the polynomial of degree n whose coefficients count the number of plane partitions of n indexed by their trace. Extending classical work of E.M. Wright, we develop the asymptotics of these polynomials inside the unit disk using the circle method.

A partition polynomial is a refinement of the partition number p(n) whose coefficients count some special partition statistic. Just as partition numbers have useful asymptotics so do partition polynomials. In fact, their asymptotics determine the limiting behavior of their zeros which form a network of curves inside the unit disk. An important new feature in their study requires a detailed analysis of. the “root dilogarithm” given as the real part of the square root of the usual dilogarithm.

We give a new proof of an old identity of Dixon (1865-1936) that uses tools from topological combinatorics. Dixon’s identity is re-established by constructing an infinite family of non-pure simplicial complexes ∆(n), indexed by the positive integers, such that the alternating sum of the numbers of faces of ∆(n) of each dimension is the left-hand side of the identity. We show that ∆(n) is shellable for all n. Then, using the fact that a shellable simplicial complex is homotopy equivalent to a wedge of spheres, we compute the Betti numbers of ∆(n) by counting (via a generating function) the number of facets of ∆(n) of each dimension that attach along their entire boundary in the shelling order. In other words, Dixon’s identity is re-established by using the Euler-Poincaré relation.

The plane partition polynomial Q_n(x) is the polynomial of degree n whose coefficients count the number of plane partitions of n indexed by their trace. Extending classical work of E.M. Wright, we develop the asymptotics of these polynomials inside the unit disk using the circle method.

Let A be the С-algebra given by the infinite tensor product of (2 x 2)-matrix algebras. A is the UHF algebra with invariant 2 and is also known as the CAR (canonical anticommutation relation) algebra. In principle, the methods developed for the study of inductive limit groups in [4, 5, 9, 16, 17] are applicable to the investigation of the representation theory of the unitary group of A. Let us see what this means in the framework given by Voiculescu [17] for the classification of the finite characters.

The starting point of this work was the announcement by SV Kerov and AM Vershik [11] that the finite characters of the inductive limit group U (oo) can all be obtained as limits of normalized characters of U (N), which we call the Asymptotic Character Formula or the" ergodic method". In Section 2, we give a detailed proof of this Theorem for representations in the non-negative signatures.

Rm+ r (z)= zr− 1Rm+ r− 1 (z)+ zr− 2Rm+ r− 2 (z)+···+ Rm (z) where the initial polynomials are polynomials over C with no common complex root. In this paper, we show that the zero attractor of the sequence of r-bonacci-related polynomials is a portion of a real algebraic curve in the complex plane together with a finite subset Σ⊂ C. In the special case of the r-bonacci polynomials Σ is empty.

Contemporary Mathematics Contemporary Mathematics Volume 517, 2010 Appell polynomials and their zero attractors Robert P. Boyer and William MY Goh Abstract. A polynomial family {pn (x)} is Appell if it is given by e xt g (t) = ∑ ∞ n= 0 pn (x) tn or, equivalently, pn (x)= pn− 1(x). If g (t) is an entire function, g (0)= 0, with at least one zero, the asymptotics of linearly scaled polynomials {pn (nx)} are described by means of finitely zeros of g, including those of minimal modulus.

Abstract Let Hm (z) be a sequence of polynomials whose generating function∑ m Hm (z) tm= N (t, z)/D (t, z) is rational with the denominator D (t, z)= A (z) tn+ B (z) t+ 1, where A (z) and B (z) are polynomials in z with complex coefficients and N (t, z) and D (t, z) do not have a trivial common factor. We show that the zero attractor of Hm (z) is a portion of an real algebraic curve together with a finite subset of the set of roots of the polynomial A (z) Rest (N (t, z), D (t, z)).

The representation theory of approximately finite-dimensional (AF) C*-alge~ bras was vigorously developed by Strätilä and Voiculescu. Their main objective was a study of the unitary representations о Г the unitary group (/(со), the direct limit of the classical unitary groups U (n); such a study had been previously suggested by Kirillov [7].

The representation theory of infinite wreath product groups is developed by means of the relationship between their group algebras and conjugacy classes with those of the infinite symmetric group. Further, since these groups are inductive limits of finite groups, their finite characters can be classified as limits of normalized irreducible characters of prelimit finite groups. This identification is called the “asymptotic character formula.” The K0-invariant of the group C∗-algebra is also determined.

Abstract It is well-known that the Fourier partial sums of a function exhibit the Gibbs phenomenon at a jump discontinuity. We study the same question for de la Vallée-Poussin sums. Here we find a new Gibbs function and a new Gibbs constant. When the function is continuous, a behavior similar to the Gibbs phenomenon also occurs at a kink. We call it the “generalized Gibbs phenomenon”. Let, where x 0 is a kink and where kn (g, x) represents Fourier partial sums and de la Vallée-Poussin sums.

Abstract. In the study of the asymptotic behavior of polynomials from partition theory, the determination of their leading term asymptotics inside the unit disk depends on a sequence of sets derived from comparing certain complexvalued functions constructed from polylogarithms, functions defined as

Abstract Let PL (n) be the number of all plane partitions of n while ppk (n) be the number of plane partitions of n whose trace is exactly k. We study the zeros of polynomial versions Qn (x) of plane partitions where Qn (x)=∑ ppk (n) xk. Based on the asymptotics we have developed for Qn (x) and computational evidence, we determine the limiting behavior of the zeros of Qn (x) as n→∞. The distribution of the zeros has a two-scale behavior which has order n2/3 inside the unit disk while has order n on the unit circle.

In the representation theory of inductive limit groups, wide classes of representations are studied via the dynamical system (X, G) attached to an AF C*-algebra, even for nonlocally compact groups [17]. Questions of factoriality and equivalence are translated into problems of ergodicity and equivalence of probability measures on X.

Introduction. Let G denote the universal-covering of the DeSitter group and C*(G) the group C*-algebra of G, ie, the enveloping C*-algebra of the involutive Banach algebra LX(G) (see [2]). The main goal of this paper is to give a complete description of the structure of C*(G). Briefly, the main result is that C*(G) is isomorphic to the restricted product of certain C*-algebras whose structures have concrete descriptions given by the extension theory of C. Delaroche [1].