Trees of integral triangles with given rectangular defect

AbstractThe rectangular defect of a triangle with side lengths a, b and c is a2+b2−c2 where a,b≤c. For a given integer d we examine the set PIT(d) consisting of all primitive integral triangles with rectangular defect equal to d. There are simple transformations τ1, τ2 and τ3 which produce new elements of PIT(d) from any triangle with defect d. They determine a partial ordering on PIT(d) in which applying any τi moves upward. We will show that the poset PIT(d) has finitely many components and that each of these components is isomorphic to one of two rooted trees T or T˜ (where T is the regular rooted tree of valence three and T˜ is a subtree of it). It follows that the minimal elements of PIT(d) form a finite set from which any triangle in PIT(d) can be uniquely obtained by applying a finite sequence of the τi’s.In order to prove these statements we will analyze a larger poset Σ(d) which contains copies of both PIT(d) and its inverse −PIT(d) as subposets. The elements of Σ(d) are equivalence classes of solutions to the equation x12+x22+x32−2x2x1−2x2x3=d. The key result will assert that the complement of ±PIT(d) in Σ(d) is a finite poset, denoted by Core(d). The proof of this key result is very different according to whether d is nonpositive (the obtuse case) or d is positive (the acute case), and the two cases must be analyzed separately. In the obtuse case we will see that the components of Core(d) are singletons while in the acute case they are poset segments or poset circuits (these are the finite connected posets in which each element has at most two neighbors). For all values of d the analysis of Σ(d) will produce algorithms for constructing both Core(d) and the minimal elements of PIT(d)

Determination of quantum symmetries for higher ADE systems from the modular T matrix

We show that the Ocneanu algebra of quantum symmetries, for an ADE diagram (or for higher Coxeter-Dynkin systems, like the Di Francesco - Zuber system) is, in most cases, deduced from the structure of the modular T matrix in the A series. We recover in this way the (known) quantum symmetries of su(2) diagrams and illustrate our method by studying those associated with the three genuine exceptional diagrams of type su(3), namely E5, E9 and E21. This also provides the shortest way to the determination of twisted partition functions in boundary conformal field theory with defect lines.Comment: 30 pages, 16 figures. Several misprints have been corrected. We added several references and the appendix has been enlarged (one section on essential paths and one section devoted to open problems). This article will appear in the Journal of Mathematical Physic

Ricci curvature on polyhedral surfaces via optimal transportation

The problem of defining correctly geometric objects such as the curvature is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor, with the classical Ricci curvature, when the space is a smooth manifold. Lin, Lu & Yau, Jost & Liu have used and extended this notion for graphs giving estimates for the curvature and hence the diameter, in terms of the combinatorics. In this paper, we describe a method for computing the coarse Ricci curvature and give sharper results, in the specific but crucial case of polyhedral surfaces

Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics

Kinetically constrained lattice models of glasses introduced by Kob and Andersen (KA) are analyzed. It is proved that only two behaviors are possible on hypercubic lattices: either ergodicity at all densities or trivial non-ergodicity, depending on the constraint parameter and the dimensionality. But in the ergodic cases, the dynamics is shown to be intrinsically cooperative at high densities giving rise to glassy dynamics as observed in simulations. The cooperativity is characterized by two length scales whose behavior controls finite-size effects: these are essential for interpreting simulations. In contrast to hypercubic lattices, on Bethe lattices KA models undergo a dynamical (jamming) phase transition at a critical density: this is characterized by diverging time and length scales and a discontinuous jump in the long-time limit of the density autocorrelation function. By analyzing generalized Bethe lattices (with loops) that interpolate between hypercubic lattices and standard Bethe lattices, the crossover between the dynamical transition that exists on these lattices and its absence in the hypercubic lattice limit is explored. Contact with earlier results are made via analysis of the related Fredrickson-Andersen models, followed by brief discussions of universality, of other approaches to glass transitions, and of some issues relevant for experiments.Comment: 59 page

Efficient analytical and numerical modeling for nondestructive evaluation

NDT (nondestructive testing) and NDE (nondestructive evaluation) are the low-cost methods with great reliability, sensitivity and high operational speed which involve the identification and characterization of damages without cutting apart or altering the material. Efficient modeling or simulation can reduce the time cost for experiment with the accurate predictions for practical NDE/T problems. This dissertation presents the efficient analytical and numerical modeling for the NDE/T problems. In the first part, an efficient model is developed to simulate the multilayered biaxial anisotropic material with different orientations, which is a popular structure in composites that are widely used in the aerospace industry, by using the effective medium theory. We analyze the multilayered anisotropic medium with different rotations based on the transmission line theory to derive the reflection and transmission coefficients in the matrix form. An equivalent model is used to extract the effective permittivity, permeability, and orientation angle, for a multilayered biaxial anisotropic medium. Analytical expressions for the effective parameters and orientation angle are derived for the low frequency (LF) limit. The model also gives a non-magnetic effective anisotropic layer if each layer is non-magnetic anisotropic dielectric. A good agreement is achieved by comparing the effective parameters extracted with and without the low frequency approximation. We show that the frequency independent equivalent model is valid for the frequency up to 10 GHz. In the second part, the adaptive cross approximation (ACA) and multilevel adaptive cross approximation (MLACA) algorithms are presented to accelerate the boundary element method (BEM) for the 3D eddy current NDE problems involving arbitrary shapes. The Stratton-Chu formula, which does not have the low frequency breakdown issue, has been selected for modeling. The equivalent electric and magnetic surface currents are expanded with the Rao-Wilton-Glisson (RWG) vector basis functions while the normal component of the magnetic field is expanded with pulse basis functions. The ACA algorithm has the advantage of purely algebraic and kernel independent. The MLACA algorithm compresses the rank deficient matrices with the ACA and the butterfly algorithm. We improve the efficiency of the MLACA by truncating the integral kernels after a certain distance and applying the multi-stage (level) algorithm adaptively based on the criteria for the different operators to further decrease the memory and CPU time requirements while keeping almost the same accuracy comparing with the traditional MLACA. The proposed method is especially helpful to deal with the large solution domain issue of the BEM for the eddy current problems. Numerical predictions are compared with the analytical, the semi-analytical predictions, and the experimental results for the 3D eddy current NDE problems of practical interests to demonstrate the robustness and efficiency of the proposed method