Cristina Ballantine

Colour visualization of Blaschke product mappings

A visualization of Blaschke product mappings can be obtained by treating them as canonical projections of covering Riemann surfaces and finding fundamental domains and covering transformations corresponding to these surfaces. A working tool is the technique of simultaneous continuation we introduced in previous papers. Here, we are refining this technique for some particular types of Blaschke products for which colouring pre-images of annuli centred at the origin allow us to describe the mappings with a high degree of fidelity. Additional graphics and animations are provided on the website of the project (http://math.holycross.edu/~cballant/complex/complex-functions.html).

Explicit Construction of Ramanujan Bigraphs

We construct explicitly an infinite family of Ramanujan graphs which are bipartite and biregular. Our construction starts with the Bruhat–Tits building of an inner form of \(\mathrm{SU}_{3}(\mathbb{Q}_{p})\). To make the graphs finite, we take successive quotients by infinitely many discrete co-compact subgroups of decreasing size.

Ramanujan bigraphs associated with

We use the representation theory of the quasisplit form G of SU(3) over a p-adic field to investigate whether certain quotients of the Bruhat--Tits tree associated to this form are Ramanujan bigraphs. We show that a quotient of the tree associated with G (which is a biregular bigraph) is Ramanujan if and only if G satisfies a Ramanujan type conjecture. This result is analogous to the seminal case of PGL(2) considered by Lubotzky-Phillips-Sarnak. As a consequence, the classification by Rogawski of the automorphic spectrum of U(3) implies the existence of certain infinite families of Ramanujan bigraphs.

SU(3)

Abstract Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of partitions into odd parts and the generating function for the number of odd divisors.

over a

Abstract Recently, G.E. Andrews and M. Merca considered specializations of the Rogers–Fine identity and obtained partition-theoretic interpretations of two truncated identities of Gauss solving a problem by V.J.W. Guo and J. Zeng. In this paper, we provide purely combinatorial proofs of these results.

p

We find accurate approximations for certain finite differences of the Euler zeta function, .x/. 2010 Mathematics Subject Classification: 41A60; 11M06; 26D15

-adic field

We use recent work of Jonah Blasiak (2012) to prove a stability result for the coefficients in the Kronecker product of two Schur functions: one indexed by a hook partition and one indexed by a rectangle partition. We also give bounds for the size of the partition starting with which the Kronecker coefficients are stable. Moreover, we show that once the bound is reached, no new Schur functions appear in the decomposition of Kronecker product, thus allowing one to recover the decomposition from the smallest case in which the stability holds.

Inequalities involving the generating function for the number of partitions into odd parts

The decomposition of an even power of the Vandermonde determinant in terms of the basis of Schur functions matches the decomposition of the Laughlin wavefunction as a linear combination of Slater wavefunctions and thus contributes to the understanding of the quantum Hall effect. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, we give recursive formulas for the coefficient of the Schur function sμ in the decomposition of an even power of the Vandermonde determinant in n + 1 variables in terms of the coefficient of the Schur function sλ in the decomposition of the same even power of the Vandermonde determinant in n variables if the Young diagram of μ is obtained from the Young diagram of λ by adding a tetris type shape to the top or to the left.