Some Stability and Instability Criteria for Ideal Plane Flows

We investigate stability and instability of steady ideal plane flows for an arbitrary bounded domain. First, we obtain some general criteria for linear and nonlinear stability. Second, we find a sufficient condition for the existence of a growing mode to the linearized equation. Third, we construct a steady flow which is nonlinearly and linearly stable in the L² norm of vorticity but linearly unstable in the L² norm of velocity.     

L_2-L_∞ learning of dynamic neural networks

This paper proposes an L2 -L∞ learning law as a new learning method for dynamic neural networks with external disturbance. Based on linear matrix inequality (LMI) formulation, the L2-L∞ learning law is presented to not only guarantee asymptotical stability of dynamic neural networks but also reduce the effect of external disturbance to an L2-L∞ induced norm constraint. It is shown that the design of the L2-L∞ learning law for such neural networks can be achieved by solving LMIs, which can be easily facilitated by using some standard numerical packages. A numerical example is presented to demonstrate the validity of the proposed learning law.     

Algebraic Rate of Decay for the Excess Free Energy and Stability of Fronts for a Nonlocal Phase Kinetics Equation with a Conservation Law. I

This is the first of two papers devoted to the study of a nonlocal evolution equation that describes the evolution of the local magnetization in a continuum limit of an Ising spin system with Kawasaki dynamics and Kac potentials. We consider subcritical temperatures, for which there are two local equilibria, and begin the proof of a local nonlinear stability result for the minimum free energy profiles for the magnetization at the interface between regions of these two different local equilibria; i.e., the fronts. We shall show in the second paper that an initial perturbation v ₀ of a front that is sufficiently small in L ² norm, and sufficiently localized that x ² v ₀(x)² dx<, yields a solution that relaxes to another front, selected by a conservation law, in the L ¹ norm at an algebraic rate that we explicitly estimate. There we also obtain rates for the relaxation in the L ² norm and the rate of decrease of the excess free energy. Here we prove a number of estimates essential for this result. Moreover, the estimates proved here suffice to establish the main result in an important special case.on leave from     

On the roles of minimization and linearization in least-squares finite element models of nonlinear boundary-value problems

In this paper we examine the roles of minimization and linearization in the least-squares finite element formulations of nonlinear boundary-values problems. The least-squares principle is based upon the minimization of the least-squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations (in the present case we consider L₂ norms). Since the least-squares method is independent of the discretization procedure and the solution scheme, the least-squares principle suggests that minimization should be performed prior to linearization, where linearization is employed in the context of either the Picard or Newton iterative solution procedures. However, in the least-squares finite element analysis of nonlinear boundary-value problems, it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations. The main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged. The assessment is performed mathematically, through an examination of the variational setting for the least-squares formulation of an abstract nonlinear boundary-value problem, and also computationally, through the numerical simulation of the least-squares finite element solutions of both a nonlinear form of the Poisson equation and also the incompressible Navier–Stokes equations. The assessment suggests that although the least-squares principle indicates that minimization should be performed prior to linearization, such an approach is often impractical and not necessary.     

On conservation integrals in micropolar elasticity

Two conservation laws of nonlinear micropolar elasticity (J_k = 0 and L_k = 0) are derived within the framework of the Noether theorem on invariant variational principles, thereby extending the earlier authors' results from the couple stress elasticity. Two non-conserved M-type integrals of linear micropolar elasticity are then derived and their values discussed. A comparison with related work is also given.     

A Maximum Entropy Method Based on Piecewise Linear Functions for the Recovery of a Stationary Density of Interval Mappings

Let S:[0,1]→[0,1] be a nonsingular transformation such that the corresponding Frobenius-Perron operator P _S :L ¹(0,1)→L ¹(0,1) has a stationary density f ^∗. We propose a maximum entropy method based on piecewise linear functions for the numerical recovery of f ^∗. An advantage of this new approximation approach over the maximum entropy method based on polynomial basis functions is that the system of nonlinear equations can be solved efficiently because when we apply Newton’s method, the Jacobian matrices are positive-definite and tri-diagonal. The numerical experiments show that the new maximum entropy method is more accurate than the Markov finite approximation method, which also uses piecewise linear functions, provided that the involved moments are known. This is supported by the convergence rate analysis of the method.     

Uniqueness and a priori bounds for certain homoclinic orbits of a Boussinesq system modelling solitary water waves

This paper establishes surprisingly precise a priori bounds on theL -norm of certain singular solutions of a system of two nonlinear Sturm-Liouville equations which model solitary water waves.These solutions can be interpreted as homoclinic orbits for a system of four first order ordinary differential equations. The uniqueness of these homoclinic orbits is established for certain choices of a parameterc, the phase speed of the waves. These observations do not result from perturbation of linear theory, but are global.     

On the Spectral Theory of Operator Pencils in a Hilbert Space

Abstract

Consider the operator pencil L _λ = A ? λB ? λ ² C, where A, B, and C are linear, in general unbounded and nonsymmetric, operators densely defined in a Hilbert space H. Sufficient conditions for the existence of the eigenvalues of L _λ are investigated in the case when A, B and C are K-positive and K-symmetric operators in H, and a method to bracket the eigenvalues of L _λ is developed by using a variational characterization of the problem (i) L _λ u = 0. The method generates a sequence of lower and upper bounds converging to the eigenvalues of L _λ and can be considered an extension of the Temple-Lehman method to quadratic eigenvalue problems (i).     

Smoothly Clipped Absolute Deviation (SCAD) regularization for compressed sensing MRI Using an augmented Lagrangian scheme

PurposeCompressed sensing (CS) provides a promising framework for MR image reconstruction from highly undersampled data, thus reducing data acquisition time. In this context, sparsity-promoting regularization techniques exploit the prior knowledge that MR images are sparse or compressible in a given transform domain. In this work, a new regularization technique was introduced by iterative linearization of the non-convex smoothly clipped absolute deviation (SCAD) norm with the aim of reducing the sampling rate even lower than it is required by the conventional l₁ norm while approaching an l₀ norm.Materials and MethodsThe CS-MR image reconstruction was formulated as an equality-constrained optimization problem using a variable splitting technique and solved using an augmented Lagrangian (AL) method developed to accelerate the optimization of constrained problems. The performance of the resulting SCAD-based algorithm was evaluated for discrete gradients and wavelet sparsifying transforms and compared with its l₁-based counterpart using phantom and clinical studies. The k-spaces of the datasets were retrospectively undersampled using different sampling trajectories. In the AL framework, the CS-MRI problem was decomposed into two simpler sub-problems, wherein the linearization of the SCAD norm resulted in an adaptively weighted soft thresholding rule with a sparsity enhancing effect.ResultsIt was demonstrated that the proposed regularization technique adaptively assigns lower weights on the thresholding of gradient fields and wavelet coefficients, and as such, is more efficient in reducing aliasing artifacts arising from k-space undersampling, when compared to its l₁-based counterpart.ConclusionThe SCAD regularization improves the performance of l₁-based regularization technique, especially at reduced sampling rates, and thus might be a good candidate for some applications in CS-MRI.     

Improving Visible Light Sensitization of Luminescent Europium Complexes

The synthesis and characterization of the new ligands L ₁ , L ₂ and L ₄ are described with the series of four europium complexes of formula [EuL _n (TTA)₃] in which TTA refers to 2-thenoyltrifluoroacetonate and L _n to tridentate ligands with nitrogen containing heterocyclic structure, such as a 2,6-bis(3-methyl-pyrazolyl)-4-(p-toluyl-ethynyl)-triazine for L ₁ , or terpyridines functionalized at the 4′ position by a phenyl-vinylene for L ₂ , a p-dimethylamino-phenylene for L ₃ , or a p-aminophenyl-ethynylene for L ₄ . The spectroscopic properties of the ligands and of the complexes are studied by means of UV–Vis absorption spectroscopy, as well as steady-state and time-resolved luminescence spectroscopy. All complexes display europium centred luminescence upon ligand excitation. Careful examination of the excitation spectra revealed differences in the ligand based sensitization efficiencies. For complexes of L ₁ and L ₂ , excitation of europium is mainly achieved through the TTA moieties and the photo-physical studies on [EuL ₁ (TTA)₃] evidenced a weaker coordination of the bispyrazolyltriazine tridentate ligand, resulting from a partial decomplexation upon dilution. Complexes of L ₃ and L ₄ display intense excitation through the tridentate units, which extend down to 460 nm in the visible region. In the case of L ₃ , selective excitation reveals the presence of a ligand-centred emission band at 520 nm which is likely ascribed to a L ₃ centred charge transfer state.     

Mimetic finite difference method for the Stokes problem on polygonal meshes

Various approaches to extend finite element methods to non-traditional elements (general polygons, pyramids, polyhedra, etc.) have been developed over the last decade. The construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis. The mimetic finite difference (MFD) method works on general polygonal meshes and has many similarities with low-order finite element methods. Both schemes try to preserve the fundamental properties of the underlying physical and mathematical models. The essential difference between the two schemes is that the MFD method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices. Since no extension of basis functions inside the mesh elements is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we present a new MFD method for the Stokes problem on arbitrary polygonal meshes and analyze its stability. The method is developed for the general case of tensor coefficients, which allows us to apply it to a linear elasticity problem, as well. Numerical experiments show, for the velocity variable, second-order convergence in a discrete L² norm and first-order convergence in a discrete H¹ norm. For the pressure variable, first-order convergence is shown in the L² norm.     

Stochastic operators,information, and entropy

For a stochastic operatorU on andL ₁-space, i.eU is linear, positive, and norm preserving on the positive cone ofL ₁, it is shown thatU decreases relative information between two nonnegativeL ₁-functions. Furthermore it is shown that the following properties ofU are closely related:U is energy decreasing (energy preserving),U isH-decreasing, whereH is Boltzmann'sH-functional, and the Maxwell distributions are fixed points ofU.     

Solutions and applications of tridiagonal vector recurrence relations

The applications of infinite systems of linear first order differential equations with 2L+1-term recursion formulas are discussed. It is shown that such systems can be written as a system of linear tridiagonal vector equations of dimensionL. A general method is presented by which the initial value problem can be solved by iteration. For special but physically important initial conditions the solution is given by a matrix continued fraction. The eigenvalues of the tridiagonal vector recurrence relations are obtained as the roots of aL×L determinant the elements of which are determined by a matrix continued fraction. The applicability of the method is demonstrated by calculating the eigenvalues of the laser Fokker-Planck operator.     

A Family of Linearizations of Autonomous Ordinary Differential Equations with Scalar Nonlinearity

Abstract

This paper deals with a method for the linearization of nonlinear autonomous differential equations with a scalar nonlinearity. The method consists of a family of approximations which are time independent, but depend on the initial state. The family of linearizations can be used to approximate the derivative of the nonlinear vector field, especially at equilibrium points, which are of particular interest, it can be used also to determine the asymptotic stability of equilibrium point, especially in the non-hyperbolic case. Using numerical experiments, we show that the method presents good agreement with the nonlinear system even in the case of highly nonlinear systems.     

Sparse hyperspectral unmixing using an approximate L0 norm

Sparse unmixing aims at finding an optimal subset of spectral signatures in a large spectral library to effectively model each pixel in the hyperspectral image and compute their fractional abundances. In most previous work concerned with the sparse unmixing, L₂ norm is used to measure the error tolerance and the L₁ norm is added as the sparsity regularization. However, in some applications, using L₁ norm to measure the error tolerance has significant robustness advantages over the L₂ norm. Besides, in some cases, using a smooth function to approximate the L₀ norm can obtain more accurate results than the L₁ norm in the field of sparse regression. Thus, in this paper, we consider the two alternative choices for sparse unmixing. A reweighted iteration algorithm is also proposed so that the unconvex regularizer (smoothed L₀ norm) can be efficiently solved through transforming it into a series of weighted L₁ regularizer problems. Experimental results on both synthetic and real hyperspectral data demonstrate the efficacy of the new models.     

Zygmund Spaces,Inviscid Limit and Uniqueness of Euler Flows

The paper improves the classical uniqueness result for the incompressible Euler system in the n dimensional case assuming that , only. Moreover the rate of the convergence for the inviscid limit of solutions to the Navier-Stokes equations is obtained, under the same regularity of the limit Eulerian flow. A key element of the proof is a logarithmic inequality between the Hardy and L ₁ spaces which is a consequence of the basic properties of the Zygmund space L ln L.     

Even the first iterate of a Markov operator is contracting in anL 2 norm

A weightedL ₂ norm is introduced in which Markov operators, e.g., associated with noisy maps, are contracting provided the kernel (i.e., the transitional distribution) is smooth enough. This results in strong relaxational properties of noisy maps. Similar to this norm, integral functionals appear useful when studying spatiotemporal chaos and random fields.     

Force-torque correlations

The equilibrium force-torque correlation ?Nf? is studied, first for a general fluid in which the intermolecular potential is expanded in generalized spherical harmonics, and then for the specific case of homonuclear diatomic molecules. If coordinate axes are taken with z axis parallel to the 12 intermolecular vector, it is shown that two (and only two) elements of the tensor are non-zero : ?n_x (12)f_y (12)? = -?n_y (12)f_x (12)?. For the linear molecule, these elements are evaluated in the limit of zero density for the atom-atom model of the potential. Example calculations are done, numerically for finite L (where L is the length of the diatomic molecule) and analytically in the limit of small L.     

Linear and nonlinear optical properties of semiconductor particles

Abstract

The field of nonlinear optics is briefly introduced, followed by a review of quantum size effects in small semiconductor particles, with special reference to the linear and nonlinear optical properties. Experimental methods, especially the “laser induced grating” technique are described. Semiconductor (CdS _x Se_1?x ) doped glasses are examined, both the commercially available niters and a specially made experimental CdSe doped glass. Both linear and nonlinear optical properties are discussed. Colloidal and polymers systems are also briefly discussed.     

On the logical foundations of the Jauch-Piron approach to quantum physics

We make a critical analysis of the basic concepts of the Jauch-Piron (JP) approach to quantum physics. Then, we exhibit a formalized presentation of the mathematical structure of the JP theory by introducing it as a completely formalized syntactic system, i.e., we construct a formalized languageL _e and formally state the logical-deductive structure of the JP theory by means ofL _e . Finally, we show that the JP syntactic system can be endowed with an intended interpretation, which yields a physical model of the system. A mathematical model endowed with a physical interpretation is given which establishes (in the usual sense of the model theory) the coherence of the JP syntactic system.