A new polynomial analogue of the Rogers–Ramanujan identities is proven. Here the product-side of the Rogers–Ramanujan identities is replaced by a partial theta sum and the sum-side by a weighted sum over Schur polynomials.

A Monte Carlo method is presented for checking when a multivariate formal (noncommuting variables) polynoial vanishes. The method uses O(n) random bits to achieve an error upper bound of 1=2, where n is the total degree of the polynomial. It is applied to produce a certiicate of a digital image which for an error upper bound of 1=2 has size O(n + q) log(n + q)) bits where n is the image resolut...

Throughout this section, G = (V,E) is a multi-graph. A cut of G is a bipartition (S, S̄) of the vertex set of G. The capacity of a cut is the number of edges having one endpoint on both sides of the cut. A min-cut is a cut of minimum capacity. A minimum cut can be computed with the help of maxflow computations. For some vertex s and every other vertex t, one computes the minimum cut separating s...

We compute the sequence of colengths in the cocharacters of the ∗polynomial identities of the ∗-simple algebra M2(F )⊕M2(F ), char(F ) = 0.

To any cleft Hopf Galois object, i.e., any algebra H obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two “universal algebras” AH and U α H . The algebra A α H is obtained by twisting the multiplication of H with the most general two-cocycle σ formally cohomologous to α. The cocycle σ takes values in the field of rational functions on H. By construct...

We extract a paradigm for derandomizing tests for polynomial identities from the recent AKS primality testing algorithm. We then discuss its possible application to other tests.

The core idea of stochastic stability is that thermodynamic observables must be robust under small (random) perturbations of the quenched Gibbs measure. Combining this idea with the cavity field technique, which aims to measure the free energy increment under addition of a spin to the system, we sketch how to write a stochastic stability approach to diluted mean field spin glasses which explici...

We construct link invariants using the D2n subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between small modular categories involving the even parts of the D2n planar algebras. We discuss the origins of these ...

The algebra Mn(K) of the matrices n×n over a field K can be regarded as a Z-graded algebra. In this paper, it is proved that if K is an infinite field, all the Z-graded polynomial identities of Mn(K) follow from the identities: x = 0, |α(x)| ≥ n, xy = yx, α(x) = α(y) = 0, xyz = zyx, α(x) = −α(y) = α(z), where α is the degree of the corresponding variable. This is a generalization of a result of...

We find a number of new combinatorial identities for, and interpretations of evaluations of, the topological Tutte polynomials of Las Vergnas, L(G), and of and Bollobás and Riordan, R(G), as well as for the classical Tutte polynomial T (G). For example, we express R(G) and T (G) as a sum of chromatic polynomials, show that R(G) counts non-crossing graph states and k-valuations, and reformulate ...