On an Inequality of Pual Turan Concerning Polynomials-Ⅱ

Let P(z) be a polynomial of degree n and for any complex number α, let D_αP(z) = nP(z)+(α- z) P′(z) denote the polar derivative of the polynomial P(z) with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all zeros inside a circle. Our results shall generalize and sharpen some well-known results of Turan, Govil, Dewan et al. and others.     

Photoinduced alignment of polymer liquid crystals containing azobenzene moieties in the side group VII. On He-Ne laser beam irradiation

Photoinduced alignment in a polymer liquid crystal prepared from 6-{1-[4-(2-cyano-4-nitrophenylazo)phenyl]piperazino}hexyl acrylate and 4'-[6-(methacryloyloxy)hexyloxy]-4-cyanobiphenyl was investigated for the first time on irradiation with a polarized He-Ne laser beam at 633 nm. The azobenzene moieties as well as the inert cyanobiphenyl mesogenic units were aligned with the molecular long axis perpendicular to the polarization direction of the irradiation light. Alignment induced on short irradiation was reversible, while that induced under prolonged irradiation was irreversible due to the occurrence of crosslinking which might be caused by photoinduced decomposition of the azobenzene moieties during the photoirradiation process.     

A Minty variational principle for set optimization

Extremal problems are studied involving an objective function with values in (order) complete lattices of sets generated by so-called set relations. Contrary to the popular paradigm in vector optimization, the solution concept for such problems, introduced by F. Heyde and A. Löhne, comprises the attainment of the infimum as well as a minimality property. The main result is a Minty type variational inequality for set optimization problems which provides a sufficient optimality condition under lower semicontinuity assumptions and a necessary condition under appropriate generalized convexity assumptions. The variational inequality is based on a new Dini directional derivative for set-valued functions which is defined in terms of a “lattice difference quotient.” A residual operation in a lattice of sets replaces the inverse addition in linear spaces. Relationships to families of scalar problems are pointed out and used for proofs. The appearance of improper scalarizations poses a major difficulty which is dealt with by extending known scalar results such as Diewert's theorem to improper functions.     

A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces

On doubling metric measure spaces endowed with a strongly local regular Dirichlet form, we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale-invariant inequalities that correspond respectively to the Nash inequality and to a Gagliardo–Nirenberg type inequality when the volume growth is polynomial. This yields a new proof and a generalisation of the well-known equivalence between classical heat kernel upper bounds and relative Faber–Krahn inequalities or localised Sobolev or Nash inequalities. We are able to treat more general pointwise estimates, where the heat kernel rate of decay is not necessarily governed by the volume growth. A crucial role is played by the finite propagation speed property for the associated wave equation, and our main result holds for an abstract semigroup of operators satisfying the Davies–Gaffney estimates.     

A simultaneous approach to inverse source problem by Green's function

In this paper, we consider an inverse source problem of identification of F(t) function in the linear parabolic equation u_t = u_xx + F(t) and u₀(x) function as the initial condition from the measured final data and local boundary data. Based on the optimal control framework by Green's function, we construct Fréchet derivative of Tikhonov functional. The stability of the minimizer is established from the necessary condition. The CG algorithm based on the Fréchet derivative is applied to the inverse problem, and results are presented for a test example. Copyright © 2014 John Wiley & Sons, Ltd.     

An inverse coefficient problem for a quasilinear parabolic equation with periodic boundary and integral overdetermination condition

In this paper, the inverse problem of finding the time‐dependent coefficient of heat capacity together with the solution periodic boundary and integral overdetermination conditions is considered. Under some natural regularity and consistency conditions on the input data, the existence, uniqueness, and continuous dependence upon the data of the solution are shown. Some considerations on the numerical solution for this inverse problem are presented with an example. Copyright © 2014 John Wiley & Sons, Ltd.     

Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs

This paper addresses the solution of parabolic evolution equations simultaneously in space and time as may be of interest in, for example, optimal control problems constrained by such equations. As a model problem, we consider the heat equation posed on the unit cube in Euclidean space of moderately high dimension. An a priori stable minimal residual Petrov–Galerkin variational formulation of the heat equation in space–time results in a generalized least squares problem. This formulation admits a unique, quasi‐optimal solution in the natural space–time Hilbert space and serves as a basis for the development of space–time compressive solution algorithms. The solution of the heat equation is obtained by applying the conjugate gradient method to the normal equations of the generalized least squares problem. Starting from stable subspace splittings in space and in time, multilevel space–time preconditioners for the normal equations are derived. In order to reduce the complexity of the full space–time problem, all computations are performed in a compressed or sparse format called the hierarchical Tucker format, supposing that the input data are available in this format. In order to maintain sparsity, compression of the iterates within the hierarchical Tucker format is performed in each conjugate gradient iteration. Its application to vectors in the hierarchical Tucker format is detailed. Finally, numerical results in up to five spatial dimensions based on the recently developed htucker toolbox for MATLAB are presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Approximate controllability for a class of hemivariational inequalities

In this paper, we deal with the approximate controllability for control systems described by a class of hemivariational inequalities. Firstly, we introduce the concept of mild solutions for hemivariational inequalities. Then the approximate controllability is formulated and proved by utilizing a fixed-point theorem of multivalued maps and properties of generalized Clarke subdifferential.     

Metric inequalities for routings on direct connections with application to line planning

We consider multi-commodity flow problems in which capacities are installed on paths. In this setting, it is often important to distinguish between flows on direct connection routes, using single paths, and flows that include path switching. We derive a feasibility condition for path capacities supporting such direct connection flows similar to the well-known feasibility condition for arc capacities in ordinary multi-commodity flows. The condition can be expressed in terms of a class of metric inequalities for routings on direct connections. We illustrate the concept on the example of the line planning problem in public transport and present an application to large-scale real-world problems.