An exact general remeshing scheme applied to physically conservative voxelization

We present an exact general remeshing scheme to compute analytic integrals of polynomial functions over the intersections between convex polyhedral cells of old and new meshes. In physics applications this allows one to ensure global mass, momentum, and energy conservation while applying higher-order polynomial interpolation. We elaborate on applications of our algorithm arising in the analysis of cosmological N-body data, computer graphics, and continuum mechanics problems. We focus on the particular case of remeshing tetrahedral cells onto a Cartesian grid such that the volume integral of the polynomial density function given on the input mesh is guaranteed to equal the corresponding integral over the output mesh. We refer to this as "physically conservative voxelization". At the core of our method is an algorithm for intersecting two convex polyhedra by successively clipping one against the faces of the other. This algorithm is an implementation of the ideas presented abstractly by Sugihara (1994), who suggests using the planar graph representations of convex polyhedra to ensure topological consistency of the output. This makes our implementation robust to geometric degeneracy in the input. We employ a simplicial decomposition to calculate moment integrals up to quadratic order over the resulting intersection domain. We also address practical issues arising in a software implementation, including numerical stability in geometric calculations, management of cancellation errors, and extension to two dimensions. In a comparison to recent work, we show substantial performance gains. We provide a C implementation intended to be a fast, accurate, and robust tool for geometric calculations on polyhedral mesh elements.Comment: Code implementation available at https://github.com/devonmpowell/r3

Second Quantization of the Wilson Loop

Treating the QCD Wilson loop as amplitude for the propagation of the first quantized particle we develop the second quantization of the same propagation. The operator of the particle position $\hat{\cal X}_{\mu}$ (the endpoint of the "open string") is introduced as a limit of the large $N$ Hermitean matrix. We then derive the set of equations for the expectation values of the vertex operators \VEV{ V(k_1)\dots V(k_n)} . The remarkable property of these equations is that they can be expanded at small momenta (less than the QCD mass scale), and solved for expansion coefficients. This provides the relations for multiple commutators of position operator, which can be used to construct this operator. We employ the noncommutative probability theory and find the expansion of the operator $\hat{\cal X}_\mu$ in terms of products of creation operators $a_\mu^{\dagger}$. In general, there are some free parameters left in this expansion. In two dimensions we fix parameters uniquely from the symplectic invariance. The Fock space of our theory is much smaller than that of perturbative QCD, where the creation and annihilation operators were labelled by continuous momenta. In our case this is a space generated by $d = 4$ creation operators. The corresponding states are given by all sentences made of the four letter words. We discuss the implication of this construction for the mass spectra of mesons and glueballs.Comment: 41 pages, latex, 3 figures and 3 Mathematica files uuencode

Parallelized Rigid Body Dynamics

Physics engines are collections of API-like software designed for video games, movies and scientific simulations. While physics engines often come in many shapes and designs, all engines can benefit from an increase in speed via parallelization. However, despite this need for increased speed, it is uncommon to encounter a parallelized physics engine today. Many engines are long-standing projects and changing them to support parallelization is too costly to consider as a practical matter. Parallelization needs to be considered from the design stages through completion to ensure adequate implementation. In this project we develop a realistic approach to simulate physics in a parallel environment. Utilizing many techniques we establish a practical approach to significantly reduce the run-time on a standard physics engine

Silhouette-based gait recognition using Procrustes shape analysis and elliptic Fourier descriptors

This paper presents a gait recognition method which combines spatio-temporal motion characteristics, statistical and physical parameters (referred to as STM-SPP) of a human subject for its classification by analysing shape of the subject's silhouette contours using Procrustes shape analysis (PSA) and elliptic Fourier descriptors (EFDs). STM-SPP uses spatio-temporal gait characteristics and physical parameters of human body to resolve similar dissimilarity scores between probe and gallery sequences obtained by PSA. A part-based shape analysis using EFDs is also introduced to achieve robustness against carrying conditions. The classification results by PSA and EFDs are combined, resolving tie in ranking using contour matching based on Hu moments. Experimental results show STM-SPP outperforms several silhouette-based gait recognition methods

Self-intersection local times of random walks: Exponential moments in subcritical dimensions

Fix $p>1$, not necessarily integer, with $p(d-2)<d$. We study the $p$-fold self-intersection local time of a simple random walk on the lattice $\Z^d$ up to time $t$. This is the $p$-norm of the vector of the walker's local times, $\ell_t$. We derive precise logarithmic asymptotics of the expectation of $\exp\{\theta_t \|\ell_t\|_p\}$ for scales $\theta_t>0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $\theta_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the {\it Gagliardo-Nirenberg inequality}. As a corollary, we obtain a large-deviation principle for $\|\ell_t\|_p/(t r_t)$ for deviation functions $r_t$ satisfying t r_t\gg\E[\|\ell_t\|_p]. Informally, it turns out that the random walk homogeneously squeezes in a $t$-dependent box with diameter of order $\ll t^{1/d}$ to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.Comment: 15 pages. To appear in Probability Theory and Related Fields. The final publication is available at springerlink.co

Basic principles of hp Virtual Elements on quasiuniform meshes

In the present paper we initiate the study of $hp$ Virtual Elements. We focus on the case with uniform polynomial degree across the mesh and derive theoretical convergence estimates that are explicit both in the mesh size $h$ and in the polynomial degree $p$ in the case of finite Sobolev regularity. Exponential convergence is proved in the case of analytic solutions. The theoretical convergence results are validated in numerical experiments. Finally, an initial study on the possible choice of local basis functions is included

Digital technologies for virtual recomposition : the case study of Serpotta stuccoes

The matter that lies beneath the smooth and shining surface of stuccoes of the Serpotta family, who used to work in Sicily from 1670 to 1730, has been thoroughly studied in previous papers, disclosing the deep, even if empirical, knowledge of materials science that guided the artists in creating their master- works. In this work the attention is focused on the solid perspective and on the scenographic sculpture by Giacomo Serpotta, who is acknowledged as the leading exponent of the School. The study deals with some particular works of the artist, the so-called "teatrini" (Toy Theater), made by him for the San Lorenzo Oratory in Palermo. On the basis of archive documents and previous analogical photogrammetric plotting, integrated with digital solutions and methodologies of computer- based technologies, the study investigates and interprets the geometric-formal genesis of the examined works of art, until the prototyping of the whole scenic apparatus.peer-reviewe