The block numerical range of analytic operator functions

We introduce the block numerical range Wn(L) of an operator function L with respect to a decomposition H = H1⊕. . .⊕Hn of the underlying Hilbert space. Our main results include the spectral inclusion property and estimates of the norm of the resolvent for analytic L . They generalise, and improve, the corresponding results for the numerical range (which is the case n = 1) since the block numerical range is contained in, and may be much smaller than, the usual numerical range. We show that refinements of the decomposition entail inclusions between the corresponding block numerical ranges and that the block numerical range of the operator matrix function L contains those of its principal subminors. For the special case of operator polynomials, we investigate the boundedness of Wn(L) and we prove a Perron-Frobenius type result for the block numerical radius of monic operator polynomials with coefficients that are positive in Hilbert lattice sense.

An Eigen Approach to Transmit Beamforming in Wireless Networks With Single Antenna Receivers

In this paper, we propose an eigen framework for transmit beamforming for single-hop and dual-hop network models with single antenna receivers. In cases where number of receivers is not more than three, the proposed Eigen approach is vastly superior in terms of ease of implementation and computational complexity compared with the existing convex-relaxation-based approaches. The essential premise is that the precoding problems can be posed as equivalent optimization problems of searching for an optimal vector in the joint numerical range of Hermitian matrices. We show that the latter problem has two convex approximations: the first one is a semi-definite program that yields a lower bound on the solution, and the second one is a linear matrix inequality that yields an upper bound on the solution. We study the performance of the proposed and existing techniques using numerical simulations.

Paranormal operator on a Hilbert space

In this thesis an extensive study is made of the set P of all paranormal operators in B(H), the set of all bounded endomorphisms on the complex Hilbert space H. T ϵ B(H) is paranormal if for each z contained in the resolvent set of T, d(z, σ(T))//(T-zI)^-1 = 1 where d(z, σ(T)) is the distance from z to σ(T), the spectrum of T. P contains the set N of normal operators and P contains the set of hyponormal operators. However, P is contained in L, the set of all T ϵ B(H) such that the convex hull of the spectrum of T is equal to the closure of the numerical range of T. Thus, N≤P≤L.

If the uniform operator (norm) topology is placed on B(H), then the relative topological properties of N, P, L can be discussed. In Section IV, it is shown that: 1) N P and L are arc-wise connected and closed, 2) N, P, and L are nowhere dense subsets of B(H) when dim H ≥ 2, 3) N = P when dimH ˂ ∞ , 4) N is a nowhere dense subset of P when dimH ˂ ∞ , 5) P is not a nowhere dense subset of L when dimH ˂ ∞ , and 6) it is not known if P is a nowhere dense subset of L when dimH ˂ ∞.

The spectral properties of paranormal operators are of current interest in the literature. Putnam [22, 23] has shown that certain points on the boundary of the spectrum of a paranormal operator are either normal eigenvalues or normal approximate eigenvalues. Stampfli [26] has shown that a hyponormal operator with countable spectrum is normal. However, in Theorem 3.3, it is shown that a paranormal operator T with countable spectrum can be written as the direct sum, N ⊕ A, of a normal operator N with σ(N) = σ(T) and of an operator A with σ(A) a subset of the derived set of σ(T). It is then shown that A need not be normal. If we restrict the countable spectrum of T ϵ P to lie on a C²-smooth rectifiable Jordan curve G_o, then T must be normal [see Theorem 3.5 and its Corollary]. If T is a scalar paranormal operator with countable spectrum, then in order to conclude that T is normal the condition of σ(T) ≤ G_o can be relaxed [see Theorem 3.6]. In Theorem 3.7 it is then shown that the above result is not true when T is not assumed to be scalar. It was then conjectured that if T ϵ P with σ(T) ≤ G_o, then T is normal. The proof of Theorem 3.5 relies heavily on the assumption that T has countable spectrum and cannot be generalized. However, the corollary to Theorem 3.9 states that if T ϵ P with σ(T) ≤ G_o, then T has a non-trivial lattice of invariant subspaces. After the completion of most of the work on this thesis, Stampfli [30, 31] published a proof that a paranormal operator T with σ(T) ≤ G_o is normal. His proof uses some rather deep results concerning numerical ranges whereas the proof of Theorem 3.5 uses relatively elementary methods.

Large scale heterogeneous networks, the Davis-Wielandt shell, and graph separation

We consider a large scale network of interconnected heterogeneous dynamical components. Scalable stability conditions are derived that involve the input/output properties of individual subsystems and the interconnection matrix. The analysis is based on the Davis-Wielandt shell, a higher dimensional version of the numerical range with important convexity properties. This can be used to allow heterogeneity in the agent dynamics while relaxing normality and symmetry assumptions on the interconnection matrix. The results include small gain and passivity approaches as special cases, with the three dimensional shell shown to be inherently connected with corresponding graph separation arguments. © 2012 Society for Industrial and Applied Mathematics.

Regularity of Wiener-Hopf plus Hankel operators

In this thesis we consider Wiener-Hopf-Hankel operators with Fourier symbols in the class of almost periodic, semi-almost periodic and piecewise almost periodic functions. In the first place, we consider Wiener-Hopf-Hankel operators acting between L2 Lebesgue spaces with possibly different Fourier matrix symbols in the Wiener-Hopf and in the Hankel operators. In the second place, we consider these operators with equal Fourier symbols and acting between weighted Lebesgue spaces Lp(R;w), where 1 < p < 1 and w belongs to a subclass of Muckenhoupt weights. In addition, singular integral operators with Carleman shift and almost periodic coefficients are also object of study. The main purpose of this thesis is to obtain regularity properties characterizations of those classes of operators. By regularity properties we mean those that depend on the kernel and cokernel of the operator. The main techniques used are the equivalence relations between operators and the factorization theory. An invertibility characterization for the Wiener-Hopf-Hankel operators with symbols belonging to the Wiener subclass of almost periodic functions APW is obtained, assuming that a particular matrix function admits a numerical range bounded away from zero and based on the values of a certain mean motion. For Wiener-Hopf-Hankel operators acting between L2-spaces and with possibly different AP symbols, criteria for the semi-Fredholm property and for one-sided and both-sided invertibility are obtained and the inverses for all possible cases are exhibited. For such results, a new type of AP factorization is introduced. Singular integral operators with Carleman shift and scalar almost periodic coefficients are also studied. Considering an auxiliar and simpler operator, and using appropriate factorizations, the dimensions of the kernels and cokernels of those operators are obtained. For Wiener-Hopf-Hankel operators with (possibly different) SAP and PAP matrix symbols and acting between L2-spaces, criteria for the Fredholm property are presented as well as the sum of the Fredholm indices of the Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operators. By studying dependencies between different matrix Fourier symbols of Wiener-Hopf plus Hankel operators acting between L2-spaces, results about the kernel and cokernel of those operators are derived. For Wiener-Hopf-Hankel operators acting between weighted Lebesgue spaces, Lp(R;w), a study is made considering equal scalar Fourier symbols in the Wiener-Hopf and in the Hankel operators and belonging to the classes of APp;w, SAPp;w and PAPp;w. It is obtained an invertibility characterization for Wiener-Hopf plus Hankel operators with APp;w symbols. In the cases for which the Fourier symbols of the operators belong to SAPp;w and PAPp;w, it is obtained semi-Fredholm criteria for Wiener-Hopf-Hankel operators as well as formulas for the Fredholm indices of those operators.

Communicating risk of medication side effects: an empirical evaluation of EU recommended terminology

Two experiments compared people's interpretation of verbal and numerical descriptions of the risk of medication side effects occurring. The verbal descriptors were selected from those recommended for use by the European Union (very common, common, uncommon, rare, very rare). Both experiments used a controlled empirical methodology, in which nearly 500 members of the general population were presented with a fictitious (but realistic) scenario about visiting the doctor and being prescribed medication, together with information about the medicine's side effects and their probability of occurrence. Experiment 1 found that, in all three age groups tested (18 - 40, 41 - 60 and over 60), participants given a verbal descriptor (very common) estimated side effect risk to be considerably higher than those given a comparable numerical description. Furthermore, the differences in interpretation were reflected in their judgements of side effect severity, risk to health, and intention to comply. Experiment 2 confirmed these findings using two different verbal descriptors (common and rare) and in scenarios which described either relatively severe or relatively mild side effects. Strikingly, only 7 out of 180 participants in this study gave a probability estimate which fell within the EU assigned numerical range. Thus, large scale use of the descriptors could have serious negative consequences for individual and public health. We therefore recommend that the EU and National authorities suspend their recommendations regarding these descriptors until a more substantial evidence base is available to support their appropriate use.

On the spectra and pseudospectra of a class of non-self-adjoint random matrices and operators

In this paper we develop and apply methods for the spectral analysis of non-selfadjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudo-ergodic in the sense of E. B. Davies (Commun. Math. Phys. 216 (2001), 687–704). As a major application to illustrate our methods we focus on the “hopping sign model” introduced by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433–6443), in which the main objects of study are random tridiagonal matrices which have zeros on the main diagonal and random ±1’s as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semi-infinite and bi-infinite matrix cases, for example showing that the numerical range and p-norm ε - pseudospectra (ε > 0, p ∈ [1,∞] ) of the random finite matrices converge almost surely to their infinite matrix counterparts, and that the finite matrix spectra are contained in the infinite matrix spectrum Σ. We also propose a sequence of inclusion sets for Σ which we show is convergent to Σ, with the nth element of the sequence computable by calculating smallest singular values of (large numbers of) n×n matrices. We propose similar convergent approximations for the 2-norm ε -pseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below.

Domaine Numérique du produit AB avec A normal

2000 Mathematics Subject Classification: 18B30, 47A12.

Generalized Derivations and Norm Equality in Normed Ideals

2000 Mathematics Subject Classification: 47A10, 47A12, 47A30, 47B10, 47B20, 47B37, 47B47, 47D50.

Comparison of numerical models for vibro-acoustic analysis of structural panels in low modal density range engaging air layers

During launch, satellite and their equipment are subjected to loads of random nature and with a wide frequency range. Their vibro-acoustic response is an important issue to be analysed, for example for folded solar arrays and antennas. The main issue at low modal density is the modelling combinations engaging air layers, structures and external fluid. Depending on the modal density different methodologies, as FEM, BEM and SEA should be considered. This work focuses on the analysis of different combinations of the methodologies previously stated used in order to characterise the vibro-acoustic response of two rectangular sandwich structure panels isolated and engaging an air layer between them under a diffuse acoustic field. Focusing on the modelling of air layers, different models are proposed. To illustrate the phenomenology described and studied, experimental results from an acoustic test on an ARA-MKIII solar array in folded configuration are presented along with numerical results.

Passive control of a bi-ventricular assist device : an experimental and numerical investigation

For the last two decades heart disease has been the highest single cause of death for the human population. With an alarming number of patients requiring heart transplant, and donations not able to satisfy the demand, treatment looks to mechanical alternatives. Rotary Ventricular Assist Devices, VADs, are miniature pumps which can be implanted alongside the heart to assist its pumping function. These constant flow devices are smaller, more efficient and promise a longer operational life than more traditional pulsatile VADs. The development of rotary VADs has focused on single pumps assisting the left ventricle only to supply blood for the body. In many patients however, failure of both ventricles demands that an additional pulsatile device be used to support the failing right ventricle. This condition renders them hospital bound while they wait for an unlikely heart donation. Reported attempts to use two rotary pumps to support both ventricles concurrently have warned of inherent haemodynamic instability. Poor balancing of the pumps’ flow rates quickly leads to vascular congestion increasing the risk of oedema and ventricular ‘suckdown’ occluding the inlet to the pump. This thesis introduces a novel Bi-Ventricular Assist Device (BiVAD) configuration where the pump outputs are passively balanced by vascular pressure. The BiVAD consists of two rotary pumps straddling the mechanical passive controller. Fluctuations in vascular pressure induce small deflections within both pumps adjusting their outputs allowing them to maintain arterial pressure. To optimise the passive controller’s interaction with the circulation, the controller’s dynamic response is optimised with a spring, mass, damper arrangement. This two part study presents a comprehensive assessment of the prototype’s ‘viability’ as a support device. Its ‘viability’ was considered based on its sensitivity to pathogenic haemodynamics and the ability of the passive response to maintain healthy circulation. The first part of the study is an experimental investigation where a prototype device was designed and built, and then tested in a pulsatile mock circulation loop. The BiVAD was subjected to a range of haemodynamic imbalances as well as a dynamic analysis to assess the functionality of the mechanical damper. The second part introduces the development of a numerical program to simulate human circulation supported by the passively controlled BiVAD. Both investigations showed that the prototype was able to mimic the native baroreceptor response. Simulating hypertension, poor flow balancing and subsequent ventricular failure during BiVAD support allowed the passive controller’s response to be assessed. Triggered by the resulting pressure imbalance, the controller responded by passively adjusting the VAD outputs in order to maintain healthy arterial pressures. This baroreceptor-like response demonstrated the inherent stability of the auto regulating BiVAD prototype. Simulating pulmonary hypertension in the more observable numerical model, however, revealed a serious issue with the passive response. The subsequent decrease in venous return into the left heart went unnoticed by the passive controller. Meanwhile the coupled nature of the passive response not only decreased RVAD output to reduce pulmonary arterial pressure, but it also increased LVAD output. Consequently, the LVAD increased fluid evacuation from the left ventricle, LV, and so actually accelerated the onset of LV collapse. It was concluded that despite the inherently stable baroreceptor-like response of the passive controller, its lack of sensitivity to venous return made it unviable in its present configuration. The study revealed a number of other important findings. Perhaps the most significant was that the reduced pulse experienced during constant flow support unbalanced the ratio of effective resistances of both vascular circuits. Even during steady rotary support therefore, the resulting ventricle volume imbalance increased the likelihood of suckdown. Additionally, mechanical damping of the passive controller’s response successfully filtered out pressure fluctuations from residual ventricular function. Finally, the importance of recognising inertial contributions to blood flow in the atria and ventricles in a numerical simulation were highlighted. This thesis documents the first attempt to create a fully auto regulated rotary cardiac assist device. Initial results encourage development of an inlet configuration sensitive to low flow such as collapsible inlet cannulae. Combining this with the existing baroreceptor-like response of the passive controller will render a highly stable passively controlled BiVAD configuration. The prototype controller’s passive interaction with the vasculature is a significant step towards a highly stable new generation of artificial heart.

Numerical analyses of crack spacing due to shrinkage in reinforced concrete slabs

Shrinkage cracking is commonly observed in concrete flat structures such as highway pavements, slabs, and bridge decks. Crack spacing due to shrinkage has received considerable attention for many years [1-3]. However, some aspects concerning the mechanism of crack spacing still remain un-clear. Though it is well known that the interval of the cracks generally falls with a range, no satisfactory explanation has been put forward as to why the minimum spacing exists.

Numerical and experimental investigation of wedge tip radius effect on wedge plasmons

We report numerical analysis and experimental observation of strongly localized plasmons guided by triangular metal wedges and pay special attention to the effect of smooth (nonzero radius) tips. Dispersion, dissipation, and field structure of such wedge plasmons are analyzed using the compact two-dimensional finite-difference time-domain algorithm. Experimental observation is conducted by the end-fire excitation and near-field scanning optical microscope detection of the predicted plasmons on 40°silver nanowedges with the wedge tip radii of 20, 85, and 125 nm that were fabricated by the focused-ion beam method. The effect of smoothing wedge tips is shown to be similar to that of increasing wedge angle. Increasing wedge angle or wedge tip radius results in increasing propagation distance at the same time as decreasing field localization (decreasing wave number). Quantitative differences between the theoretical and experimental propagation distances are suggested to be due to a contribution of scattered bulk and surface waves near the excitation region as well as the addition of losses due to surface roughness. The theoretical and measured propagation distances are several plasmon wavelengths and are useful for a range of nano-optical applications