Low rank matrix recovery from rank one measurements

We study the recovery of Hermitian low rank matrices $X\in {\mathbb{C}}^{n\times n}$ from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form $a_j{a}_j^{?}$ for some measurement vectors $a_1,\dots ,a_m$, i.e., the measurements are given by $b_j=\mathrm{tr}(X{a}_j{a}_j^{?})$. The case where the matrix $X=x{x}^{?}$ to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements $b_j=|〈x,a_j〉{|}^2$) via the PhaseLift approach, which has been introduced recently. We derive bounds for the number m of measurements that guarantee successful uniform recovery of Hermitian rank r matrices, either for the vectors $a_j$, $j=1,\dots ,m$, being chosen independently at random according to a standard Gaussian distribution, or $a_j$ being sampled independently from an (approximate) complex projective t-design with $t=4$. In the Gaussian case, we require $m\ge Crn$ measurements, while in the case of 4-designs we need $m\ge \mathit{Cr}n\log ?(n)$. Our results are uniform in the sense that one random choice of the measurement vectors $a_j$ guarantees recovery of all rank r-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 4-designs generalizes and improves a recent bound on phase retrieval due to Gross, Krahmer and Kueng. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.     

Complete Caps in with Both N and q Odd

In this work, an inductive method to construct complete caps in affine spaces is provided. Using this tool, for 20-5838ad6544cd/jcd21564-math-0004.png"> odd and q odd, complete caps smaller than all already known ones are obtained.     

Solitary wave solution of the generalized Hirota–Satsuma coupled KdV system

In this research, we find the exact traveling wave solutions involving parameters of the generalized Hirota–Satsuma couple KdV system according to the modified simple equation method with the aid of Maple 16. When these parameters are taken special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the modified simple equation method provides an effective and a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Comparison between our results and the well-known results will be presented.     

Two Ecdysteroids Isolated from Micropropagated Lychnis flos-cuculi and the Biological Activity of Plant Material

Genetically uniform plant material, derived from Lychnis flos-cuculi propagated in vitro, was used for the isolation of 20-hydroxyecdysone and polypodine B and subjected to an evaluation of the antifungal and antiamoebic activity. The activity of 80% aqueous methanolic extracts, their fractions, and isolated ecdysteroids were studied against pathogenic Acanthamoeba castellani. Additionally, a Microtox^® acute toxicity assay was performed. It was found that an 80% methanolic fraction of root extract exerts the most potent amoebicidal activity at IC₅₀ of 0.06 mg/mL at the 3rd day of treatment. Both ecdysteroids show comparable activity at IC₅₀ of 0.07 mg/mL. The acute toxicity of 80% fractions at similar concentrations is significantly higher than that of 40% fractions. Crude extracts exhibited moderate antifungal activity, with a minimum inhibitory concentration (MIC) within the range of 1.25�2013;2.5 mg/mL. To the best of our knowledge, the present report is the first to show the biological activity of L. flos-cuculi in terms of the antifungal and antiamoebic activities and acute toxicity. It is also the first isolation of the main ecdysteroids from L. flos-cuculi micropropagated, ecdysteroid-rich plant material.     

Multiplication formulas and isomorphism theorem of ıSchur superalgebras

We introduce the notion of ?Schur superalgebra, which can be regarded as a type B/C counterpart of the q-Schur superalgebra (of type A) formulated as centralizer algebras of certain signed q-permutation modules over Hecke algebras. Some multiplication formulas for ?Schur superalgebra are obtained to construct their canonical bases. Furthermore, we established an isomorphism theorem between the ?Scuhr superalgebras and the q-Schur superalgebras of type A, which helps us derive semisimplicity criteria of the ?Schur superalgebras.     

Inhomogeneous ‘longitudinal’ plane waves in a deformed elastic material

By definition, a homogeneous isotropic compressible Hadamard material has the property that an infinitesimal longitudinal homogeneous plane wave may propagate in every direction when the material is maintained in a state of arbitrary finite static homogeneous deformation. Here, as regards the wave, 208261q/xxlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">homogeneous208261q/xxlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0"> means that the direction of propagation of the wave is parallel to the direction of eventual attenuation; and 208261q/xxlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">longitudinal208261q/xxlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0"> means that the wave is linearly polarized in a direction parallel to the direction of propagation. In other words, the displacement is of the form u = ncos k(n · x – ct), where n is a real vector. It is seen that the Hadamard material is the most general one for which a 208261q/xxlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">longitudinal208261q/xxlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0"> inhomogeneous plane wave may also propagate in any direction of a predeformed body. Here, 208261q/xxlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">inhomogeneous208261q/xxlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0"> means that the wave is attenuated, in a direction distinct from the direction of propagation; and 208261q/xxlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">longitudinal208261q/xxlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0"> means that the wave is elliptically polarized in the plane containing these two directions, and that the ellipse of polarization is similar and similarly situated to the ellipse for which the real and imaginary parts of the complex wave vector are conjugate semi-diameters. In other words, the displacement is of the form u = 208261q/xxlarge8476.gif" alt="real" align="BASELINE" BORDER="0">{S exp i208261q/xxlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">(S · x – ct)}, where S is a complex vector (or bivector). Then a Generalized Hadamard material is introduced. It is the most general homogeneous isotropic compressible material which allows the propagation of infinitesimal 208261q/xxlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">longitudinal208261q/xxlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0"> inhomogeneous plane circularly polarized waves for all choices of the isotropic directional bivector. Finally, the most general forms of response functions are found for homogeneously deformed isotropic elastic materials in which 208261q/xxlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">longitudinal208261q/xxlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0"> inhomogeneous plane waves may propagate with a circular polarization in each of the two planes of central circular section of the 208261q/10659_2005_Article_4399_TeX2GIFIE1.gif" alt=" $$\mathbb{B}$$ " align="middle" border="0"> ⁿ -ellipsoid, where 208261q/10659_2005_Article_4399_TeX2GIFIE2.gif" alt=" $$\mathbb{B}$$ " align="middle" border="0"> is the left Cauchy-Green strain tensor corresponding to the primary pure homogeneous deformation.     

On Estimates of the Boltzmann Collision Operator with Cut-off

We present new estimates of the Boltzmann collision operator in weighted Lebesgue and Bessel potential spaces. The main focus is put on hard potentials under the assumption that the angular part of the collision kernel fulfills some weighted integrability condition. In addition, the proofs for some previously known 20m/21_2004_Article_144_TeX2GIFIEq1.gif" alt="$$\mathbb{L}_p $$" align="middle" border="0"> -estimates have been considerably shortened and carried out by elementary methods. For a class of metric spaces, the collision integral is seen to be a continuous operator into the same space. Furthermore, we give a new pointwise lower bound as well as asymptotic estimates for the loss term without requiring that the entropy is finite.     

Epitaxy of Binary Compounds and Alloys

The present work aims at constructing a theoretical framework within which to address the issues of morphological instabilities (one-dimensional step bunching and two-dimensional step meandering), alloying, and phase segregation in binary systems in the context of (physical or chemical) vapor deposition. The length scale of interest, although nanoscopic, is sufficiently large that the steps on a vicinal surface can be viewed as smooth curves and, correspondingly, the theory is a continuum one. In a departure from theories inaugurated by Burton, Cabrera, and Frank [The growth of crystals and the equilibrium structure of their surfaces. Phil. Trans. Roy. Soc. A 243 (1951) 299–358] the steps are endowed with a thermodynamic structure whose main ingredients are a step free-energy density and edge species chemical potentials. Moreover, crystal anisotropy, with its altering of the dynamics of steps and the associated morphological instabilities, is accounted for – in a manner consistent with the second law – both in the thermodynamic and kinetic properties of terraces and, more importantly, of steps. Additionally, in contrast with most of the literature on the subject (cf. [J. Krug, Introduction to step dynamics and step instabilities. In: A. Voigt (ed.) Multiscale Modeling in Epitaxial Growth. Birkhäusser, Berlin (2005)]), adsorption–desorption along the steps, bulk atomic diffusion, and chemical reactions (both on the terraces and along the step edges) are incorporated and coupled to the other mechanisms, e.g., terrace adatom diffusion and step attachment–detachment kinetics, whose interplay governs the evolution of steps on vicinal surfaces. Importantly, aided by the concept of configurational forces for which a separate balance law is postulated Configurational Forces as Basic Concepts of Continuum Physics. Springer, Berlin Heidelberg New York (2000)]), the proposed theory allows the steps to evolve away from local equilibrium thus contributing to a general treatment of the dynamics of steps. Finally, a specialization to the epitaxy of binary compounds and alloys is afforded, yielding a generalization of the classical Gibbs–Thomson relation in the former and novel evolution equations for an individual step in the latter.     

Delay equations with fluctuating delay related to the regenerative chatter

The mathematical models representing machine tool chatter dynamics have been cast as differential equations with delay. In this paper, non-linear delay differential equations with periodic delays which model the machine tool chatter with continuously modulated spindle speed are studied. The explicit time-dependent delay terms, due to spindle speed modulation, are replaced by state-dependent delay terms by augmenting the original equations. The augmented system of equations is autonomous and has two pairs of pure imaginary eigenvalues without resonance. The reduced bifurcation equation is obtained by making use of Lyapunov-Schmidt Reduction method. By using the reduced bifurcation equations, the periodic solutions are determined to analyze the tool motion. Analytical results show both modest increase of stability and existence of periodic solutions near the new stability boundary.