Delay-dependent exponential stabilization for uncertain linear systems with interval non-differentiable time-varying delays

In this paper, the problem of exponential stabilization for a class of linear systems with time-varying delay is studied. The time delay is a continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, but the delay function is not necessary to be differentiable. Based on the construction of improved Lyapunov-Krasovskii functionals combined with Leibniz-Newton’s formula, new delay-dependent sufficient conditions for the exponential stabilization of the systems are first established in terms of LMIs. Numerical examples are given to demonstrate that the derived conditions are much less conservative than those given in the literature.     

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.     

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.     

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.     

Generation of entanglement via atomic coherence in a two-mode three-level cascade atomic system

The generation of continuous variable entanglement via atomic coherence in a two-mode three-level cascade atomic system is discussed according to the entanglement criterion proposed by Duan et al. [Phys. Rev. Lett. 84, 2722 (2000)]. Atomic coherence between the top and bottom levels is induced with two photons of a strong external pump field. It shows that entanglement for the two-mode field in the cavity can be generated under certain conditions. Moreover, by means of the input-output theory, we show that the two-mode entanglement could also be approached at the output.     

Effects of phase decoherence on the entanglement of a two-qubit anisotropic Heisenberg XYZ chain with an in-plane magnetic field

We investigate the phase decoherence effects on the entanglement of a two-qubit anisotropic Heisenberg model with a nonuniform magnetic field in the x–z-plane. As a measure of the entanglement, the concurrence of the system is calculated. It is shown that when the magnetic field is along the z-axis, the nonuniform and uniform components of the field have no influence on the entanglement for the cases of and , respectively. But when the magnetic field is not along the z-axis, both the uniform and the nonuniform components of the field will introduce the decoherence effects. It is found that the effects of the Heisenberg chain's anisotropy in the Z-direction on the entanglement are dependent on the direction of the field. Moreover, the larger the initial concurrence is, the higher value it will exhibit during the time evolution of the system for a proper set of the parameters ν, Δ, θ, γ , B and b.     

Multi-way dual Cheeger constants and spectral bounds of graphs

We introduce a set of multi-way dual Cheeger constants and prove universal higher-order dual Cheeger inequalities for eigenvalues of normalized Laplace operators on weighted finite graphs. Our proof proposes a new spectral clustering phenomenon deduced from metrics on real projective spaces. We further extend those results to a general reversible Markov operator and find applications in characterizing its essential spectrum.