Closure properties of solutions to heat inequalities

We prove that if $u_1,u_2 : (0,\infty) \times \R^d \to (0,\infty)$ are sufficiently well-behaved solutions to certain heat inequalities on $\R^d$ then the function $u: (0,\infty) \times \R^d \to (0,\infty)$ given by $u^{1/p}=u_1^{1/p_1} * u_2^{1/p_2}$ also satisfies a heat inequality of a similar type provided $\tfrac{1}{p_1} + \tfrac{1}{p_2} = 1 + \tfrac{1}{p}$. On iterating, this result leads to an analogous statement concerning $n$-fold convolutions. As a corollary, we give a direct heat-flow proof of the sharp $n$-fold Young convolution inequality and its reverse form.Comment: 12 page

Synthetic Aperture Radar (SAR) data processing

The available and optimal methods for generating SAR imagery for NASA applications were identified. The SAR image quality and data processing requirements associated with these applications were studied. Mathematical operations and algorithms required to process sensor data into SAR imagery were defined. The architecture of SAR image formation processors was discussed, and technology necessary to implement the SAR data processors used in both general purpose and dedicated imaging systems was addressed

Relativistic MHD with Adaptive Mesh Refinement

This paper presents a new computer code to solve the general relativistic magnetohydrodynamics (GRMHD) equations using distributed parallel adaptive mesh refinement (AMR). The fluid equations are solved using a finite difference Convex ENO method (CENO) in 3+1 dimensions, and the AMR is Berger-Oliger. Hyperbolic divergence cleaning is used to control the $\nabla\cdot {\bf B}=0$ constraint. We present results from three flat space tests, and examine the accretion of a fluid onto a Schwarzschild black hole, reproducing the Michel solution. The AMR simulations substantially improve performance while reproducing the resolution equivalent unigrid simulation results. Finally, we discuss strong scaling results for parallel unigrid and AMR runs.Comment: 24 pages, 14 figures, 3 table

On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics 225

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

Intertwining relations for one-dimensional diffusions and application to functional inequalities

International audienceFollowing the recent work [13] fulfilled in the discrete case, we pro- vide in this paper new intertwining relations for semigroups of one-dimensional diffusions. Various applications of these results are investigated, among them the famous variational formula of the spectral gap derived by Chen and Wang [15] together with a new criterion ensuring that the logarithmic Sobolev inequality holds. We complete this work by revisiting some classical examples, for which new estimates on the optimal constants are derived

Detonation of Meta-stable Clusters

We consider the energy accumulation in meta-stable clusters. This energy can be much larger than the typical chemical bond energy (~;;1 ev/atom). For example, polymeric nitrogen can accumulate 4 ev/atom in the N8 (fcc) structure, while helium can accumulate 9 ev/atom in the excited triplet state He2* . They release their energy by cluster fission: N8 -> 4N2 and He2* -> 2He. We study the locus of states in thermodynamic state space for the detonation of such meta-stable clusters. In particular, the equilibrium isentrope, starting at the Chapman-Jouguet state, and expanding down to 1 atmosphere was calculated with the Cheetah code. Large detonation pressures (3 and 16 Mbar), temperatures (12 and 34 kilo-K) and velocities (20 and 43 km/s) are a consequence of the large heats of detonation (6.6 and 50 kilo-cal/g) for nitrogen and helium clusters respectively. If such meta-stable clusters could be synthesized, they offer the potential for large increases in the energy density of materials

Predicting language diversity with complex network

Evolution and propagation of the world's languages is a complex phenomenon, driven, to a large extent, by social interactions. Multilingual society can be seen as a system of interacting agents, where the interaction leads to a modification of the language spoken by the individuals. Two people can reach the state of full linguistic compatibility due to the positive interactions, like transfer of loanwords. But, on the other hand, if they speak entirely different languages, they will separate from each other. These simple observations make the network science the most suitable framework to describe and analyze dynamics of language change. Although many mechanisms have been explained, we lack a qualitative description of the scaling behavior for different sizes of a population. Here we address the issue of the language diversity in societies of different sizes, and we show that local interactions are crucial to capture characteristics of the empirical data. We propose a model of social interactions, extending the idea from, that explains the growth of the language diversity with the size of a population of country or society. We argue that high clustering and network disintegration are the most important characteristics of models properly describing empirical data. Furthermore, we cancel the contradiction between previous models and the Solomon Islands case. Our results demonstrate the importance of the topology of the network, and the rewiring mechanism in the process of language change

Entropic Uncertainty Relations in Quantum Physics

Uncertainty relations have become the trademark of quantum theory since they were formulated by Bohr and Heisenberg. This review covers various generalizations and extensions of the uncertainty relations in quantum theory that involve the R\'enyi and the Shannon entropies. The advantages of these entropic uncertainty relations are pointed out and their more direct connection to the observed phenomena is emphasized. Several remaining open problems are mentionedComment: 35 pages, review pape

Ca2+-induced changes in energy metabolism and viability of melanoma cells

Cancer cells are characterized by a high rate of glycolysis, which is their primary energy source. We show here that a rise in intracellular-free calcium ion (Ca2+), induced by Ca2+-ionophore A23187, exerted a deleterious effect on glycolysis and viability of B16 melanoma cells. Ca2+-ionophore caused a dose-dependent detachment of phosphofructokinase (EC 2.7.1.11), one of the key enzymes of glycolysis, from cytoskeleton. It also induced a decrease in the levels of glucose 1,6-bisphosphate and fructose 1,6-bisphosphate, the two stimulatory signal molecules of glycolysis. All these changes occurred at lower concentrations of the drug than those required to induce a reduction in viability of melanoma cells. We also found that low concentrations of Ca2+-ionophore induced an increase in adenosine 5′-triphosphate (ATP), which most probably resulted from the increase in mitochondrial-bound hexokinase, which reflects a defence mechanism. This mechanism can no longer operate at high concentrations of the Ca2+-ionophore, which causes a decrease in mitochondrial and cytosolic hexokinase, leading to a drastic fall in ATP and melanoma cell death. The present results suggest that drugs which are capable of inducing accumulation of intracellular-free Ca2+ in melanoma cells would cause a reduction in energy-producing systems, leading to melanoma cell death. © 1999 Cancer Research Campaig