Random algebraic construction of extremal graphs

In this expository paper, we present a motivated construction of large graphs not containing a given complete bipartite subgraph. The key insight is that the algebraic constructions yield very non-smooth probability distributions.Comment: 8 page

Quantum Hall Ground States, Binary Invariants, and Regular Graphs

Extracting meaningful physical information out of a many-body wavefunction is often impractical. The polynomial nature of fractional quantum Hall (FQH) wavefunctions, however, provides a rare opportunity for a study by virtue of ground states alone. In this article, we investigate the general properties of FQH ground state polynomials. It turns out that the data carried by an FQH ground state can be essentially that of a (small) directed graph/matrix. We establish a correspondence between FQH ground states, binary invariants and regular graphs and briefly introduce all the necessary concepts. Utilizing methods from invariant theory and graph theory, we will then take a fresh look on physical properties of interest, e.g. squeezing properties, clustering properties, etc. Our methodology allows us to `unify' almost all of the previously constructed FQH ground states in the literature as special cases of a graph-based class of model FQH ground states, which we call \emph{accordion} model FQH states

Scattering of Massless Particles: Scalars, Gluons and Gravitons

In a recent note we presented a compact formula for the complete tree-level S-matrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimension. In this paper we show that a natural formulation also exists for a massless colored cubic scalar theory. In Yang-Mills, the formula is an integral over the space of n marked points on a sphere and has as integrand two factors. The first factor is a combination of Parke-Taylor-like terms dressed with U(N) color structures while the second is a Pfaffian. The S-matrix of a U(N)xU(N') cubic scalar theory is obtained by simply replacing the Pfaffian with a U(N') version of the previous U(N) factor. Given that gravity amplitudes are obtained by replacing the U(N) factor in Yang-Mills by a second Pfaffian, we are led to a natural color-kinematics correspondence. An expansion of the integrand of the scalar theory leads to sums over trivalent graphs and are directly related to the KLT matrix. We find a connection to the BCJ color-kinematics duality as well as a new proof of the BCJ doubling property that gives rise to gravity amplitudes. We end by considering a special kinematic point where the partial amplitude simply counts the number of color-ordered planar trivalent trees, which equals a Catalan number. The scattering equations simplify dramatically and are equivalent to a special Y-system with solutions related to roots of Chebyshev polynomials.Comment: 31 page

Exploiting chordal structure in polynomial ideals: a Gr\"obner bases approach

Chordal structure and bounded treewidth allow for efficient computation in numerical linear algebra, graphical models, constraint satisfaction and many other areas. In this paper, we begin the study of how to exploit chordal structure in computational algebraic geometry, and in particular, for solving polynomial systems. The structure of a system of polynomial equations can be described in terms of a graph. By carefully exploiting the properties of this graph (in particular, its chordal completions), more efficient algorithms can be developed. To this end, we develop a new technique, which we refer to as chordal elimination, that relies on elimination theory and Gr\"obner bases. By maintaining graph structure throughout the process, chordal elimination can outperform standard Gr\"obner basis algorithms in many cases. The reason is that all computations are done on "smaller" rings, of size equal to the treewidth of the graph. In particular, for a restricted class of ideals, the computational complexity is linear in the number of variables. Chordal structure arises in many relevant applications. We demonstrate the suitability of our methods in examples from graph colorings, cryptography, sensor localization and differential equations.Comment: 40 pages, 5 figure

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

On the Minimum Roots of the Adjoint Polynomials of Unicyclic Graphs

引入伴随多项式是为了从补图的角度研究色多形式,图的伴随多项式的极小根可用于判定色等价图.β(g)表示图g的伴随多项式的极小根.n表示n个顶点的单圈图的集合.分别确定了具有MAX{β(g)|g∈Ωn}和MIn{β(g)|g∈Ωn}的所有单圈图.The adjoint polynomial was introduced for solving the chromaticity problem of the complements of graphs.The minimum roots of the adjoint polynomials of graphs can be applied to sort out graphs that are not chromatically equivalent.Let β(G) be the minimum root of the adjoint polynomial of the graph G.Denote by n the set of all unicyclic graphs on n vertices.All graphs with max{β(G)|G ∈Ωn}(resp.min{β(G)|G ∈Ωn}) are determined.supportedbyNSFC(11061027;11161037); theNaturalScienceFoundationofQinghaiProvince(2011-Z-907;2011-Z-911

Identifying the parametric occurrence of multiple steady states for some biological networks

We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem using tools from Symbolic Computation. We describe how progress was made to achieve semi-algebraic descriptions of the multistationarity regions of parameter space, and compare symbolic results to numerical methods. The biological networks studied are models of the mitogen-activated protein kinases (MAPK) network which has already consumed considerable effort using special insights into its structure of corresponding models. Our main example is a model with 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment. The model also imposes positivity conditions on all variables and parameters. We apply combinations of symbolic computation methods designed for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition, as well as a simplification technique adapted from Gaussian elimination and graph theory. We are able to determine multistationarity of our main example over a 2-dimensional parameter space. We also study a second MAPK model and a symbolic grid sampling technique which can locate such regions in 3-dimensional parameter space.Comment: 60 pages - author preprint. Accepted in the Journal of Symbolic Computatio