The K-Quotient

The Rayleigh Quotient and a recently proposed Timoshenko Quotient [1] are upper bounds to the fundamental eigenvalue of a discrete dynamic system. The closeness of these upper bounds to the fundamental eigenvalue depends on the closeness of the trial vectors to the eigenmode used in the calculation. In the present paper, a new quotient is presented. This quotient does not require the closeness of the trial vector to the eigenmode and its accuracy is improvable by raising the numerical value of the parameter p.     

Perturbation theory and the Rayleigh quotient

The characteristic frequencies ω of the vibrations of an elastic solid subject to boundary conditions of either zero displacement or zero traction are given by the Rayleigh quotient expressed in terms of the corresponding exact eigenfunctions. In problems that can be analytically expanded in a small parameter ε, it is shown that when an approximate eigenfunction is known with an error O(ε^N), the Rayleigh quotient gives the frequency with an error O(ε^2N), a gain of N orders. This result generalizes a well-known theorem for N=1. A non-trivial example is presented for N=4, whereby knowledge of the 3rd-order eigenfunction (error being 4th order) gives the eigenvalue with an error that is 8th order; the 6th-order term thus determined provides an unambiguous derivation of the shear coefficient in Timoshenko beam theory.     

Critical value computation for H∞-Riccati difference equations

In designing finite horizon discrete time H_∞ controllers, the associated H_∞-Riccati difference equations must be solved. But the Riccati equation has a non-negative solution only when γ⁻² is small enough. So it is important to get the upper bound of the parameter, i.e., the critical value that ensures the existence of the solution to the Riccati equation. The solution sequence of the Riccati difference equation can be constructed by the conjoined basis of an associated linear Hamiltonian difference system. Based on this expression and the Hamiltonian difference system eigenvalue theorems, the equivalence between the critical value and the first order eigenvalue of the linear Hamiltonian difference system is presented. Since the critical value is also shown to be the fundamental eigenvalue of a generalized Rayleigh quotient, an extended form of Wittrick-Williams algorithm is presented to search this value.     

Longitudinal vibrations in circular rods: A systematic approach

A systematic method is developed for expressing the frequency squared ω² and the corresponding displacement fields of harmonic waves in a long thin rod as an even power series in qa, where q is the wavenumber along the rod and a is a representative transverse dimension. For longitudinal waves in a circular rod, the evaluation is reduced to algebraic recursion, giving coefficients analytically in terms of Poisson's ratio v, to many orders. The second nontrivial coefficient, corresponding to Rayleigh–Love theory in the present longitudinal case and Timoshenko theory in the flexural case, is thus put on a firm footing without reliance on ad hoc physical assumptions. The results are compared to available exact predictions, and shown to be accurate for moderate values of qa (5% accuracy for qa≤1.5) with just two terms. Improvements based on the Rayleigh quotient guarantee positivity and the correct asymptotic power, and the variational principle further ensures that the accuracy improves monotonically with the order of approximation. With these features, accurate results are obtained for larger qa (5% accuracy for qa≤3), so that results are valid for rods that are by no means thin. Application of these methods to the flexural case has been presented separately.     

Circuit bounds on stochastic transport in the Lorenz equations

In turbulent Rayleigh–Bénard convection one seeks the relationship between the heat transport, captured by the Nusselt number, and the temperature drop across the convecting layer, captured by the Rayleigh number. In experiments, one measures the Nusselt number for a given Rayleigh number, and the question of how close that value is to the maximal transport is a key prediction of variational fluid mechanics in the form of an upper bound. The Lorenz equations have traditionally been studied as a simplified model of turbulent Rayleigh–Bénard convection, and hence it is natural to investigate their upper bounds, which has previously been done numerically and analytically, but they are not as easily accessible in an experimental context. Here we describe a specially built circuit that is the experimental analogue of the Lorenz equations and compare its output to the recently determined upper bounds of the stochastic Lorenz equations [1]. The circuit is substantially more efficient than computational solutions, and hence we can more easily examine the system. Because of offsets that appear naturally in the circuit, we are motivated to study unique bifurcation phenomena that arise as a result. Namely, for a given Rayleigh number, we find a reentrant behavior of the transport on noise amplitude and this varies with Rayleigh number passing from the homoclinic to the Hopf bifurcation.     

Schrödinger uncertainty relation and squeezed state

In a quantum optical model, we demonstrate both analytically and numerically that if the measurement of physical observables corresponds to non-canonical operators, the Schrödinger uncertainty relation may be used to define the squeezing, where the Schrödinger lower limit sets a higher bound on quantum fluctuations than the Heisenberg one does. The effect of the second-order correction to Rayleigh scattering on the squeezing is also discussed.     

On estimates for short wave stability and long wave instability in three-layer Hele-Shaw flows

We consider the linear stability of three-layer Hele-Shaw flows with each layer having constant viscosity and viscosity increasing in the direction of a basic uniform flow. While the upper bound results on the growth rate of long waves are well known from our earlier works, lower bound results on the growth rate of short stable waves are not known to date. In this paper, we obtain such a lower bound. In particular, we show the following results: (i) the lower bound for stable short waves is also a lower bound for all stable waves, and the exact dispersion curve for the most stable eigenvalue intersects the dispersion curve based on the lower bound at a wavenumber where the most stable eigenvalue is zero; (ii) the upper bound for unstable long waves is also an upper bound for all unstable waves, and the exact dispersion curve for the most unstable eigenvalue intersects the dispersion curve based on the upper bound at a wavenumber where the most unstable eigenvalue is zero. Numerical results are provided which support these findings.     

Upper bounds on the convective heat transport in a rotating fluid layer of infinite Prandtl number: case of large Taylor numbers

By means of the Howard-Busse method of the optimum theory of turbulence we obtain upper bounds on the convective heat transport in a horizontal fluid layer heated from below and rotating about a vertical axis. We consider the interval of large Taylor numbers where the intermediate layers of the optimum fields expand in the direction of the corresponding internal layers. We consider the 1 - α-solution of the arising variational problem for the cases of rigid-stress-free, stress-free, and rigid boundary conditions. For each kind of boundary condition we discuss four cases: two cases where the boundary layers are thinner than the Ekman layers of the optimum field and two cases where the boundary layers are thicker than the Ekman layers. In most cases we use an improved solution of the Euler-Lagrange equations of the variational problem for the intermediate layers of the optimum fields. This solution leads to corrections of the thicknesses of the boundary layers of the optimum fields and to lower upper bounds on the convective heat transport in comparison to the bounds obtained by Chan [J. Fluid Mech. 64, 477 (1974)] and Hunter and Riahi [J. Fluid Mech. 72, 433 (1975)]. Compared to the existing experimental data for the case of a fluid layer with rigid boundaries the corresponding upper bounds on the convective heat transport is less than two times larger than the experimental results, the corresponding upper bound on the convective heat transport, obtained by Hunter and Riahi is about 10% higher than the bound obtained in this article. When Rayleigh number and Taylor number are high enough the upper bound on the convective heat transport ceases to depend on the boundary conditions. Received 30 January 2001 and Received in final form 28 May 2001     

Bounds to the wave-vector dependent susceptibility of the ising model

It is shown that rigorous upper and lower bounds to the wave-vector dependent susceptibility of the Ising Model are obtained either when an upper bound to the spin pair correlation function and a lower bound to the susceptibility at zero wave-vector are given, or when a lower bound to the former and an upper bound to the latter are given. An example of the numerical computation of the bounds is presented for the Ising model on the sc lattice.     

Vibration and stability of rectangular strip-plates

The problem considered is that of free vibration and stability of a simply supported rectangular strip-plate subjected to constant in-plane forces. The relevant continuity conditions at the interface between the adjacent regions, which play a significant role in this type of problem, are derived. The eigenfrequencies and the buckling load are estimated by the method of the new quotient which is based on a variational statement proposed by Nemat-Nasser. The results are compared with those obtained by means of the usual Rayleigh quotient and the exact solution. The good accuracy obtained by the application of the method of the new quotient is demonstrated by means of numerical examples.     

Error thresholds for quasispecies on dynamic fitness landscapes

In this paper we investigate error thresholds on dynamic fitness landscapes. We show that there exists both a lower and an upper threshold, representing limits to the copying fidelity of simple replicators. The lower bound can be expressed as a correction term to the error threshold present on a static landscape. The upper error threshold is a new limit that only exists on dynamic fitness landscapes. We also show that for long genomes and/or highly dynamic fitness landscapes there exists a lower bound on the selection pressure required for the effective selection of genomes with superior fitness independent of mutation rates, i.e. there are distinct nontrivial limits to evolutionary parameters in dynamic environments.     

Dispersion curves for a viscoelastic Timoshenko beam with fractional derivatives

The Kramers–Kronig dispersion relation, often used as a viscoelastic constitutive law for polymeric materials, is based on purely mathematical properties of linearity, convergence of improper integrals, and causality; thus, it may also be valid as a viscoelastic constitutive law for general structural materials. Accordingly, the motion equation of a Timoshenko beam composed of conventional elastic structural materials is extended to one composed of viscoelastic materials. From the derived governing equation, a dispersive equation is derived for a viscoelastic Timoshenko beam. By plotting phase velocity curves and group velocity curves for a beam of solid circular cross-section composed of a viscoelastic material (polyvinyl chloride foam), the influence of the fractional order of viscoelasticity is examined. As a result, it is found that, in the high frequency range, only the first mode of a Timoshenko beam converged to the propagation velocity of the Rayleigh wave, which takes account of the fractional order of viscoelasticity. In addition, the phase velocity and the group velocity were found to increase as the fractional order approaches 0, and to decrease as the fractional order approaches 1. Furthermore, the rate of velocity change becomes greater as the fractional order approaches 0, and becomes smaller as the fractional order approaches 1.     

Depolarized light scattering in a 2.6-lutidine-water solution near the critical point

The spectra of depolarized Rayleigh light scattering, i.e., the Rayleigh line wing (RLW), and Raman scattering in a solution with the lower critical point in the phase separation diagram is experimentally studied. The Rayleigh line wing is studied to 30 cm^?1 from the exciting light frequency; the Raman scattering line (RSL) is studied in the near profile region. In the vicinity of the lower critical point, as well as in solutions with the upper critical point, strong narrowing of the RLW and RSL is observed. In this case, the conditions arise for the manifestation of the profile in the Rayleigh line spectrum, which is caused by the interaction of concentration and anisotropy fluctuations. Previously, this spectral line was theoretically predicted by I. A. Chaban in [1].     

On universal quantum cloning and state estimation

Both perfect cloning and perfect state estimation of an unknown pure quantum state are impossible, due to principles of quantum mechanics. Nevertheless, they can be performed imperfectly. A link between these two scenarios allows us to derive an upper bound for the fidelity in one of them, given an upper bound is known in the other. Furthermore, it is shown that also a lower bound on cloning is related to an upper bound on state estimation. Received: 15 June 1999 / Revised version: 23 September 1999 / Published online: 10 November 1999     

The pressure in the Huang-Yang-Luttinger model of an interacting boson gas

This completes our study of the equilibrium thermodynamics of the Huang-Yang-Luttinger model of a boson gas with a hard-sphere repulsion. In an earlier paper we obtained a lower bound on the pressure, but our proof of an upper bound held only for a truncated version of the model. In this paper we establish an upper bound on the pressure in the full model; the upper and lower bounds coincide and provide a variational formula for the pressure. The proof relies on recent second-level large deviation results for the occupation measure of the free boson gas.     

Variational Bounds for the Generalized Random Energy Model

We compute the pressure of the random energy model (REM) and generalized random energy model (GREM) by establishing variational upper and lower bounds. For the upper bound, we generalize Guerra’s “broken replica symmetry bounds,” and identify the random probability cascade as the appropriate random overlap structure for the model. For the REM the lower bound is obtained, in the high temperature regime using Talagrand’s concentration of measure inequality, and in the low temperature regime using convexity and the high temperature formula. The lower bound for the GREM follows from the lower bound for the REM by induction. While the argument for the lower bound is fairly standard, our proof of the upper bound is new.     

On the width of resonances

A method to get both upper and lower bounds on real and imaginary parts of resonance eigenvalues is extended to Schrödinger operators with exterior dilation analytic potential. We apply it to a simple model potential where the bound states and resonances are exactly known.     

Practical SARG04 quantum key distribution

We have presented a method to estimate parameters of the decoy state method based on one decoy state protocol for SARG04. This method has given lower bound of the fraction of single-photon counts (y ₁), the fraction of two-photon counts (y ₂), the upper bound QBER of single-photon pulses (e ₁), the upper bound QBER of two-photon pulses (e ₂), and the lower bound of key generation rate for both BB84 and SARG04. The numerical simulation has shown that the fiber based QKD and free space QKD systems using the proposed method for BB84 are able to achieve both a higher secret key rate and greater secure distance than that of SARG04. Also, it is shown that bidirectional ground to satellite and inter-satellite communications are possible with our protocol.     

Non λ−4 wavelength dependence of rayleigh-scattering loss in waveguides

We show that, in small-core waveguides, where a considerable fraction of radiation power is guided in the evanescent field outside the fiber core, the wavelength dependence of losses induced by Rayleigh scattering can substantially deviate from the λ^?4 scaling law. In the limiting case of an ultrathin waveguiding wire surrounded by air, the ultimate lower bound level of losses, controlled by Rayleigh scattering, asymptotically scales as exp(?κλ²) with radiation wavelength λ, with κ being a wavelength-independent constant.     

Sub-Rayleigh imaging via N-photon detection

The Rayleigh diffraction bound sets the minimum separation for two point objects to be distinguishable in a conventional imaging system. We demonstrate sub-Rayleigh resolution by scanning a focused beam--in an arbitrary, object-covering pattern that is unknown to the imager--and using N-photon photodetection implemented with a single-photon avalanche detector array. Experiments show resolution improvement by a factor ～(N-N(max))(?) beyond the Rayleigh bound, where N(max) is the maximum average detected photon number in the image, in good agreement with theory.