On cavitation in Elastodynamics

Motivated by the works of Ball (1982) and Pericak-Spector and Spector (1988), we investigate singular solutions of the compressible nonlinear elastodynamics equations. These singular solutions contain discontinuities in the displacement field and can be seen as describing fracture or cavitation. We explore a definition of singular solution via approximating sequences of smooth functions. We use these approximating sequences to investigate the energy of such solutions, taking into account the energy needed to open a crack or hole. In particular, we find that the existence of singular solutions and the finiteness of their energy is strongly related to the behavior of the stress response function for infinite stretching, i.e. the material has to display a sufficient amount of softening. In this note we detail our findings in one space dimension

Singular limiting induced from continuum solutions and the problem of dynamic cavitation

In the works of K.A. Pericak-Spector and S. Spector [Pericak-Spector, Spector 1988, 1997] a class of self-similar solutions are constructed for the equations of radial isotropic elastodynamics that describe cavitating solutions. Cavitating solutions decrease the total mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions (for polyconvex energies) due to point-singularities at the cavity. To resolve this paradox, we introduce the concept of singular limiting induced from continuum solution (or slic-solution), according to which a discontinuous motion is a slic-solution if its averages form a family of smooth approximate solutions to the problem. It turns out that there is an energetic cost for creating the cavity, which is captured by the notion of slic-solution but neglected by the usual entropic weak solutions. Once this cost is accounted for, the total mechanical energy of the cavitating solution is in fact larger than that of the homogeneously deformed state. We also apply the notion of slic-solutions to a one-dimensional example describing the onset of fracture, and to gas dynamics in Langrangean coordinates with Riemann data inducing vacuum in the wave fan

Holomorphic subgraph reduction of higher-point modular graph forms

Modular graph forms are a class of modular covariant functions which appear in the genus-one contribution to the low-energy expansion of closed string scattering amplitudes. Modular graph forms with holomorphic subgraphs enjoy the simplifying property that they may be reduced to sums of products of modular graph forms of strictly lower loop order. In the particular case of dihedral modular graph forms, a closed form expression for this holomorphic subgraph reduction was obtained previously by D'Hoker and Green. In the current work, we extend these results to trihedral modular graph forms. Doing so involves the identification of a modular covariant regularization scheme for certain conditionally convergent sums over discrete momenta, with some elements of the sum being excluded. The appropriate regularization scheme is identified for any number of exclusions, which in principle allows one to perform holomorphic subgraph reduction of higher-point modular graph forms with arbitrary holomorphic subgraphs.Comment: 38 pages; v2: publication versio

Probing Quark Gluon Plasma by Heavy Flavors

The drag and diffusion coefficients of charm and bottom quarks propagating through quark gluon plasma (QGP) have been evaluated within the framework of perturbative Quantum Chromodynamics (pQCD). Both radiative and collisional processes of dissipation are included in evaluating these transport coefficients. The dead cone as well as the LPM effects on radiative energy loss of heavy quarks have also been considered. The Fokker Planck equation has been solved to study the dissipation of heavy quarks momentum in QGP. The nuclear suppression factor, $R_{\mathrm AA}$ and the elliptic flow $v_2^{HF}$ of the non-photonic electrons resulting from the semi-leptonic decays of hadrons containing charm and bottom quarks have been evaluated for RHIC and LHC nuclear collision conditions. We find that the observed $R_{\mathrm AA}$ and $v_2$ at RHIC can be reproduced simultaneously within the pQCD framework.Comment: To appear in the proceedings of WPCF, 2011, Tokyo, Japa

A branch and bound algorithm to optimize the representation of tabular decision processe.

Decision situations have various aspects: knowledge acquisition and structuring, knowledge representation, knowledge validation and decision making. It has been recognized in literature that decision tables can play an important role in each of these stages. It is however not necessary to use only one representation formalism during the whole life cycle of an intelligent system. Likewise it is possible that different formats of the same formalism serve different purposes in the development process.Important in this respect is the search for automated and, if possible, optimized transitions between different formats of a formalism and between various formalisms. In this paper a branch and bound algorithm is presented that transforms expanded decision tables, that, because of their explicit enumeration of all decision cases primarily serve an acquisition and verification function, into optimized contracted decision tables, primarily used as target representation of a decision process. An optimal contracted decision table is a contracted decision table with a condition order which results in the minimum number of contracted decision columns.

Measuring radial flow of partonic and hadronic phases in relativistic heavy ion collision

It has been shown that the thermal photon and the lepton pair spectra can be used to estimate the radial velocity of different phases of the matter formed in nuclear collisions at ultra-relativistic energies. We observe a non-monotonic variation of the flow velocity with invariant mass of the lepton pair which is indicative of two different sources of thermal dilepton sources at early and late stage of the dynamically evolving system. We also show that the study of radial velocity through electromagnetic probes may shed light on the nature of the phase transition from hadrons to QGP.Comment: 0ne figure added, minor modification in tex

Non-factorizable contribtion to Bd0ˉ→π0D0\bar{B_{d}^0} \to \pi^0 D^{0}

The decay modes of the type $B \to \pi \, D$ are dynamically different. For the case $\bar{B_{d}^0} \to \pi^- D^{+}$ there is a substantial factorized contribution which dominates. In contrast, the decay mode $\bar{B_{d}^0} \to \pi^0 D^{0}$ has a small factorized contribution, being proportional to a very small Wilson coefficient combination. In this paper we calculate the relevant Wilson coefficients at one loop level in the heavy quark limits, both for the $b$-quark and the $c$-quark. We also emphasize that for the decay mode $\bar{B_{d}^0} \to \pi^0 D^{0}$ there is a sizeable non-factorizable contribution due long distance interactions, which dominate the amplitude. We estimate the branching ratio for this decay mode within our framework, which uses the heavy quark limits, both for the $b$- and the $c$-quarks. In addition, we treat energetic light ($u,d,s$) quarks within a variant of Large Energy Effective Theory and combine this with a new extension of chiral quark models. For reasonable values of the model dependent parameters of our model can account for at least 3/4 of the amplitude needed to explain the experimental branching ratio $\simeq 2.6 \times 10^{-4}$.Comment: 23 pages, 4 figures, 39 reference