Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of k-colorings or the number of independent sets of a graph and also the partition functions of certain “spin glass” models of statistical physics such as the Ising model. Building on earlier work by Dyer and Greenhill [Random Structures Algo...

Since the celebrated work of Jerrum, Sinclair, and Vigoda, we have known that the permanent of a {0, 1} matrix can be approximated in randomized polynomial time by using a rapidly mixing Markov chain to sample perfect matchings of a bipartite graph. A separate strand of the literature has pursued the possibility of an alternate, algebraic polynomial-time approximation scheme. These schemes work...

We consider two well-known algebraic proof systems: Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR), a system introduced in [2] which combines together PC and Resolution. Moreover we introduce an algebraic proof system PCRk , which combines together Polynomial Calculus (PC) and k-DNF Resolution (RESk). This is a natural generalization to RESk of PCR. In the paper we study...

We consider the numerical solution of time-dependent partial differential equations with random coefficients. A spectral approach, called stochastic finite element method, is used to compute the statistical characteristics of the solution. This method transforms a stochastic partial differential equation into a coupled system of deterministic equations by means of a Galerkin projection onto a g...

Algebraic geometry emerged from the study of subsets of C defined by polynomial equations, namely algebraic sets. Parallel to this, real algebraic geometry emerged from studying subsets of R defined by polynomial equations and inequalities, namely semi-algebraic sets. Like algebraic geometry, classical methods in real algebraic geometry provided useful but limited intuition on the relationship ...

This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra, leads to the reduced polynomial semiring – a structure that provides a basis for developing a tropical analogue to the classical theory of commutative algebra....

Let $G$ be a molecular graph with vertex set $V(G)$, $d_G(u, v)$ the topological distance between vertices $u$ and $v$ in $G$. The Hosoya polynomial $H(G, x)$ of $G$ is a polynomial $sumlimits_{{u, v}subseteq V(G)}x^{d_G(u, v)}$ in variable $x$. In this paper, we obtain an explicit analytical expression for the expected value of the Hosoya polynomial of a random benzenoid chain with $n$ hexagon...

So far in the course, we have seen three types of codes, summarized as follows: 1. Algebraic Codes • Work for worst-case errors. • Have polynomial time encoding and decoding algorithms 2. LDPCs • Work for worst-case errors as well, but a less many errors (smaller distance) compared to the Algebraic Codes. • Have linear time encoding and decoding algorithms and are simpler. 3. Polar Codes • Work...

We use unified algebraic methods to investigate the properties of polynomial Bezoutians with respect to a general basis. Not only can three known results be easily verified, but also some new properties of polynomial Bezoutians are obtained. Nonsymmetric Lyapunov-type equations of polynomial Bezoutians are also discussed. It turns out that most properties of classical Bezoutians can be analogou...

We use unified algebraic methods to investigate the properties of polynomial Bezoutians with respect to a general basis. Not only can three known results be easily verified, but also some new properties of polynomial Bezoutians are obtained. Nonsymmetric Lyapunov-type equations of polynomial Bezoutians are also discussed. It turns out that most properties of classical Bezoutians can be analogou...