More maximal arcs in Desarguesian projective planes and their geometric structure

In a previous paper R. Mathon gave a new construction method for maximal arcs in finite Desarguesian projective planes via closed sets of conics, as well as giving many new examples of maximal arcs. In the current paper, new classes of maximal arcs are constructed, and it is shown that every maximal arc so constructed gives rise to an infinite class of maximal arcs. Apart from when they are of Denniston type or dual hyperovals, closed sets of conics are shown to give maximal arcs that are not isomorphic to the known constructions. An easy characterisation of when a closed set of conics is of Denniston type is given. Results on the geometric structure of the maximal arcs and their duals are proved, as well as on elements of their collineation stabilisers.

Semi-analytical solution for nonlinear vibration of laminated FGM plates with geometric imperfections

This paper investigates the nonlinear vibration of imperfect shear deformable laminated rectangular plates comprising a homogeneous substrate and two layers of functionally graded materials (FGMs). A theoretical formulation based on Reddy's higher-order shear deformation plate theory is presented in terms of deflection, mid-plane rotations, and the stress function. A semi-analytical method, which makes use of the one-dimensional differential quadrature method, the Galerkin technique, and an iteration process, is used to obtain the vibration frequencies for plates with various boundary conditions. Material properties are assumed to be temperature-dependent. Special attention is given to the effects of sine type imperfection, localized imperfection, and global imperfection on linear and nonlinear vibration behavior. Numerical results are presented in both dimensionless tabular and graphical forms for laminated plates with graded silicon nitride/stainless steel layers. It is shown that the vibration frequencies are very much dependent on the vibration amplitude and the imperfection mode and its magnitude. While most of the imperfect laminated plates show the well-known hard-spring vibration, those with free edges can display soft-spring vibration behavior at certain imperfection levels. The influences of material composition, temperature-dependence of material properties and side-to-thickness ratio are also discussed. (C) 2004 Elsevier Ltd. All rights reserved.

A novel phantom and method for comprehensive 3-dimensional measurement and correction of geometric distortion in magnetic resonance imaging

A phantom that can be used for mapping geometric distortion in magnetic resonance imaging (MRI) is described. This phantom provides an array of densely distributed control points in three-dimensional (3D) space. These points form the basis of a comprehensive measurement method to correct for geometric distortion in MR images arising principally from gradient field non-linearity and magnet field inhomogeneity. The phantom was designed based on the concept that a point in space can be defined using three orthogonal planes. This novel design approach allows for as many control points as desired. Employing this novel design, a highly accurate method has been developed that enables the positions of the control points to be measured to sub-voxel accuracy. The phantom described in this paper was constructed to fit into a body coil of a MRI scanner, (external dimensions of the phantom were: 310 mm x 310 mm x 310 mm), and it contained 10,830 control points. With this phantom, the mean errors in the measured coordinates of the control points were on the order of 0.1 mm or less, which were less than one tenth of the voxel's dimensions of the phantom image. The calculated three-dimensional distortion map, i.e., the differences between the image positions and true positions of the control points, can then be used to compensate for geometric distortion for a full image restoration. It is anticipated that this novel method will have an impact on the applicability of MRI in both clinical and research settings. especially in areas where geometric accuracy is highly required, such as in MR neuro-imaging. (C) 2004 Elsevier Inc. All rights reserved.

A proposed scheme for comprehensive characterization of the measured geometric distortion in magnetic resonance imaging using a three-dimensional phantom

Recently, a 3-dimensional phantom that can provide a comprehensive, accurate and complete measurement of the geometric distortion in MRI has been developed. In this paper, a scheme for characterizing the measured geometric distortion using the 3-D phantom is described. In the proposed scheme, a number of quantitative measures are developed and used to characterize the geometric distortion. These measures encompass the overall and spatial aspects of the geometric distortion. Two specific types of volume of interest, rectangular parallelepipeds (including cubes) and spheres are considered in the proposed scheme. As an illustration, characterization of the geometric distortion in a Siemens 1.5T Sonata MRI system using the proposed scheme is presented. As shown, the proposed scheme provides a comprehensive assessment of the geometric distortion. The scheme can be potentially used as a standard procedure for the assessment of geometric distortion in MRI. (C) 2004 American Association of Physicists in Medicine.

Geometric distortion in clinical MRI systems - Part II: correction using a 3D phantom

In this paper, we present the correction of the geometric distortion measured in the clinical magnetic resonance imaging (MRI) systems reported in the preceding paper (Part 1) using a 3D method based on the phantom-mapped geometric distortion data. This method allows the correction to be made on phantom images acquired without or with the vendor correction applied. With the vendor's 2D correction applied, the method corrects for both the residual geometric distortion still present in the plane in which the correction method was applied (the axial plane) and the uncorrected geometric distortion along the axis non-nal to the plane. The evaluation of the effectiveness of the correction using this new method was carried out through analyzing the residual geometric distortion in the corrected phantom images. The results show that the new method can restore the distorted images in 3D nearly to perfection. For all the MRI systems investigated, the mean absolute deviations in the positions of the control points (along x-, y- and z-axes) measured on the corrected phantom images were all less than 0.2 mm. The maximum absolute deviations were all below similar to0.8 mm. As expected, the correction of the phantom images acquired with the vendor's correction applied in the axial plane performed equally well. Both the geometric distortion still present in the axial plane after applying the vendor's correction and the uncorrected distortion along the z-axis have all been restored. (C) 2004 Elsevier Inc. All rights reserved.

Geometric distortion in clinical MRI systems - Part I: evaluation using a 3D phantom

Recently, a 3D phantom that can provide a comprehensive and accurate measurement of the geometric distortion in MRI has been developed. Using this phantom, a full assessment of the geometric distortion in a number of clinical MRI systems (GE and Siemens) has been carried out and detailed results are presented in this paper. As expected, the main source of geometric distortion in modern superconducting MRI systems arises from the gradient field nonlinearity. Significantly large distortions with maximum absolute geometric errors ranged between 10 and 25 mm within a volume of 240 x 240 x 240 mm(3) were observed when imaging with the new generation of gradient systems that employs shorter coils. By comparison, the geometric distortion was much less in the older-generation gradient systems. With the vendor's correction method, the geometric distortion measured was significantly reduced but only within the plane in which these 2D correction methods were applied. Distortion along the axis normal to the plane was, as expected, virtually unchanged. Two-dimensional correction methods are a convenient approach and in principle they are the only methods that can be applied to correct geometric distortion in a single slice or in multiple noncontiguous slices. However, these methods only provide an incomplete solution to the problem and their value can be significantly reduced if the distortion along the normal of the correction plane is not small. (C) 2004 Elsevier Inc. All rights reserved.

Measuring geometric phases of scattering states in nanoscale electronic devices

We show how a quantum property, a geometric phase, associated with scattering states can be exhibited in nanoscale electronic devices. We propose an experiment to use interference to directly measure the effect of this geometric phase. The setup involves a double-path interferometer, adapted from that used to measure the phase evolution of electrons as they traverse a quantum dot (QD). Gate voltages on the QD could be varied cyclically and adiabatically, in a manner similar to that used to observe quantum adiabatic charge pumping. The interference due to the geometric phase results in oscillations in the current collected in the drain when a small bias across the device is applied. We illustrate the effect with examples of geometric phases resulting from both Abelian and non-Abelian gauge potentials.

Geometric phases of scattering states in a ring geometry: adiabatic pumping in mesoscopic devices

Geometric phases of scattering states in a ring geometry are studied on the basis of a variant of the adiabatic theorem. Three timescales, i.e., the adiabatic period, the system time and the dwell time, associated with adiabatic scattering in a ring geometry play a crucial role in determining geometric phases, in contrast to only two timescales, i.e., the adiabatic period and the dwell time, in an open system. We derive a formula connecting the gauge invariant geometric phases acquired by time-reversed scattering states and the circulating (pumping) current. A numerical calculation shows that the effect of the geometric phases is observable in a nanoscale electronic device.

A geometric approach to quantum circuit lower bounds

What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on the manifold SU(2(n)). The geodesic curves on these manifolds have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size. For each Finsler metric we give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call Pauli geodesics, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.

Local geometric changes in left ventricular remodeling post-infarction can lead to apparent reduction in scar thickness

Serial reduction in scar thickness has been shown in animal models. We sought whether this reduction in scar thickness may be a result of dilatation of the left ventricle (LV) with stretching and thinning of the wall. Contrast enhanced magnetic resonance imaging (CMRI) was performed to delineate radial scar thickness in 25 patients (age 63±10, 21 men) after myocardial infarction. The LV was divided into 16 segts and the absolute radial scar thickness (ST) and percentage scar to total wall thickness (%ST) were measured. Regional end diastolic (EDV) and end systolic volumes (ESV) of corresponding segments were measured on CMRI. All patients underwent revascularization and serial changes in ST, %ST, and regional volumes were assessed with a mean follow up of 15±5 months. CMRI identified a total of 93 scar segments. An increase in EDV or ESV was associated with a serial reduction inST(versusEDV, r =−0.3, p = 0.01; versusESV, r =−0.3, p = 0.005) and%ST(versusEDV, r =−0.2, p = 0.04; versus ESV, r =−0.3, p = 0.001). For segts associated with a positive increase in EDV (group I) or ESV (group II) there was a significant decrease in ST and %ST, but in those segts with stable EDV (group III) or ESV (group IV) there were no significant changes in ST and %ST (Table).

The Hierarchical Structure of the Firm: A Geometric Perspective

This paper incorporates hierarchical structure into the neoclassical theory of the firm. Firms are hierarchical in two respects: the organization of workers in production and the wage structure. The firm’s hierarchy is represented as the sector of a circle, where the radius represents the hierarchy’s height, the width of the sector represents the breadth of the hierarchy at a given height, and the angle of the sector represents span of control for any given supervisor. A perfectly competitive firm then chooses height and width, as well as capital inputs, in order to maximize profit. We analyze the short run and long run impact of changes in scale economies, input substitutability and input and output prices on the firm’s hierarchical structure. We find that the firm unambiguously becomes more hierarchical as the specialization of its workers increases or as its output price increases relative to input prices. The effect of changes in scale economies is contingent on the output price. The model also brings forth an analysis of wage inequality within the firm, which is found to be independent of technological considerations, and only depends on the firm’s wage schedule.