Equivariant Birch-Swinnerton-Dyer Conjecture for the Base Change of Elliptic Curves: An Example

Let E be an elliptic curve defined over Q and let K/Q be a finite Galois extension with Galois group G. The equivariant Birch-Swinnerton-Dyer conjecture for h(1)(E x(Q) K)(1) viewed as amotive over Q with coefficients in Q[G] relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper I prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S-3-extension of Q.

Numerical studies on quasilinear and linear elliptic equations

We explore here the acceleration of convergence of iterative methods for the solution of a class of quasilinear and linear algebraic equations. The specific systems are the finite difference form of the Navier-Stokes equations and the energy equation for recirculating flows. The acceleration procedures considered are: the successive over relaxation scheme; several implicit methods; and a second-order procedure. A new implicit method—the alternating direction line iterative method—is proposed in this paper. The method combines the advantages of the line successive over relaxation and alternating direction implicit methods. The various methods are tested for their computational economy and accuracy on a typical recirculating flow situation. The numerical experiments show that the alternating direction line iterative method is the most economical method of solving the Navier-Stokes equations for all Reynolds numbers in the laminar regime. The usual ADI method is shown to be not so attractive for large Reynolds numbers because of the loss of diagonal dominance. This loss can however be restored by a suitable choice of the relaxation parameter, but at the cost of accuracy. The accuracy of the new procedure is comparable to that of the well-tested successive overrelaxation method and to the available results in the literature. The second-order procedure turns out to be the most efficient method for the solution of the linear energy equation.

PIV studies of large scale structures in the near field of small aspect ratio elliptic jets

The near flow field of small aspect ratio elliptic turbulent free jets (issuing from nozzle and orifice) was experimentally studied using a 2D PIV. Two point velocity correlations in these jets revealed the extent and orientation of the large scale structures in the major and minor planes. The spatial filtering of the instantaneous velocity field using Gaussian convolution kernel shows that while a single large vortex ring circumscribing the jet seems to be present at the exit of nozzle, the orifice jet exhibited a number of smaller vortex ring pairs close to jet exit. The smaller length scale observed in the case of the orifice jet is representative of the smaller azimuthal vortex rings that generate axial vortex field as they are convected. This results in the axis-switching in the case of orifice jet and may have a mechanism different from the self induction process as observed in the case of contoured nozzle jet flow.

Variational calculation for the spin-(1/2 Heisenberg antiferromagnet on a square lattice

The ground-state properties of the spin-(1/2 Heisenberg antiferromagnet on a square lattice are studied by using a simple variational wave function that interpolates continuously between the Néel state and short-range resonating-valence-bond states. Exact calculations of the variational energy for small systems show that the state with the lowest energy has long-range antiferromagnetic order. The staggered magnetization in this state is approximately 70% of its maximum possible value. The variational estimate of the ground-state energy is substantially lower than the value obtained for the nearest-neighbor resonating-valence-bond wave function.

Some nonstandard error analysis of discontinuous Galerkin methods for elliptic problems

An a priori error analysis of discontinuous Galerkin methods for a general elliptic problem is derived under a mild elliptic regularity assumption on the solution. This is accomplished by using some techniques from a posteriori error analysis. The model problem is assumed to satisfy a GAyenrding type inequality. Optimal order L (2) norm a priori error estimates are derived for an adjoint consistent interior penalty method.

Free convection about cylinders of elliptic cross section embedded in a porous medium

Numerical solutions are presented for the free convection boundary layers over cylinders of elliptic cross section embedded in a fluid-saturated porous medium. The transformed conservation equations of the nonsimilar boundary layers are solved numerically by an efficient finite-difference method. The theory was applied to a number of cylinders and the results compared very well with published analytical solutions. The results are of use in the design of underground electrical cables, power plant steam, and water distribution lines, among others.

Variational principles in evolution

For a one-locus selection model, Svirezhev introduced an integral variational principle by defining a Lagrangian which remained stationary on the trajectory followed by the population undergoing selection. It is shown here (i) that this principle can be extended to multiple loci in some simple cases and (ii) that the Lagrangian is defined by a straightforward generalization of the one-locus case, but (iii) that in two-locus or more general models there is no straightforward extension of this principle if linkage and epistasis are present. The population trajectories can be constructed as trajectories of steepest ascent in a Riemannian metric space. A general method is formulated to find the metric tensor and the surface-in the metric space on which the trajectories, which characterize the variations in the gene structure of the population, lie. The local optimality principle holds good in such a space. In the special case when all possible linkage disequilibria are zero, the phase point of the n-locus genetic system moves on the surface of the product space of n higher dimensional unit spheres in a certain Riemannian metric space of gene frequencies so that the rate of change of mean fitness is maximum along the trajectory. In the two-locus case the corresponding surface is a hyper-torus.

Peristaltic transport of a biofluid in a pipe of elliptic cross section

Peristaltic transport of two fluids occupying the peripheral layer and the core in an elliptic tube is, investigated in elliptic cylindrical co-ordinate system, under long wavelength and low Reynolds number approximations. The effect of peripheral-layer viscosity on the flow rate and the frictional force for a slightly elliptic tube is discussed. The limiting results for the one-fluid model are obtained for different eccentricities of the undisturbed tube cross sections with the same area. As a result of non-uniformity of the peristaltic wave, two different amplitude ratios are defined and the time-averaged flux and mechanical efficiency are studied for different eccentricities. It is observed that the time-averaged flux is not affected significantly by the pressure drop when the eccentricity is large. For the peristaltic waves with same area variation, the pumping seems to improve with the eccentricity.

Superconductivity from repulsion: a variational view

This letter presents a new class of variational wavefunctions for Fermi systems in any dimension. These wavefunctions introduce correlations between Cooper pairs in different momentum states and the relevant correlations can be computed analytically. At half filling we have a ground state with critical superconducting correlations, that causes negligible increase of the kinetic energy. We find large enhancements in a Cooper-pair correlation function caused purely by the interplay between the uncertainty principle, repulsion and the proximity of half filling. This is surprising since there is no accompanying signature in usual charge and spin response functions, and typifies a novel kind of many-body cooperative behaviour.

Elliptic slot antenna for broadband wireless communications

This paper presents the design of a broadband antenna suitable for wireless communications operating over the frequency range of 3.1-10.6 GHz. Parametric studies on the effect of stub and elliptic slot have been carried out to arrive at optimum dimensions to achieve enhanced bandwidth of the proposed antenna. An experimental antenna has been designed and tested to validate the proposed design. Measured return loss characteristics have been compared against the simulation results. Simulated radiation patterns at 3.1 GHz, 6.85 GHz and 10.6 GHz have also been presented in this paper.

Two-Parameter Uniformly Elliptic Sturm–Liouville Problems With Eigenparameter-Dependent Boundary Conditions

We consider the two-parameter Sturm–Liouville system $$ -y_1''+q_1y_1=(\lambda r_{11}+\mu r_{12})y_1\quad\text{on }[0,1], $$ with the boundary conditions $$ \frac{y_1'(0)}{y_1(0)}=\cot\alpha_1\quad\text{and}\quad\frac{y_1'(1)}{y_1(1)}=\frac{a_1\lambda+b_1}{c_1\lambda+d_1}, $$ and $$ -y_2''+q_2y_2=(\lambda r_{21}+\mu r_{22})y_2\quad\text{on }[0,1], $$ with the boundary conditions $$ \frac{y_2'(0)}{y_2(0)} =\cot\alpha_2\quad\text{and}\quad\frac{y_2'(1)}{y_2(1)}=\frac{a_2\mu+b_2}{c_2\mu+d_2}, $$ subject to the uniform-left-definite and uniform-ellipticity conditions; where $q_{i}$ and $r_{ij}$ are continuous real valued functions on $[0,1]$, the angle $\alpha_{i}$ is in $[0,\pi)$ and $a_{i}$, $b_{i}$, $c_{i}$, $d_{i}$ are real numbers with $\delta_{i}=a_{i}d_{i}-b_{i}c_{i}>0$ and $c_{i}\neq0$ for $i,j=1,2$. Results are given on asymptotics, oscillation of eigenfunctions and location of eigenvalues.

Variational Asymptotic Analysis of a Self-healing SMA Composite

A Shape Memory Alloy (SMA) wire reinforced composite shell structure is analyzed for self-healing characteristic using Variational Asymptotic Method (VAM). SMA behavior is modeled using a onedimensional constitutive model. A pre-notched specimen is loaded longitudinally to simulate crack propagation. The loading process is accompanied by martensitic phase transformation in pre-strained SMA wires, bridging the crack. To heal the composite, uniform heating is required to initiate reverse transformation in the wires and bringing the crack faces back into contact. The pre-strain in the SMA wires used for reinforcement, causes a closure force across the crack during reverse transformation of the wires under heating. The simulation can be useful in design of self-healing composite structures using SMA. Effect of various parameters, like composite and SMA material properties and the geometry of the specimen, on the cracking and self-healing can also be studied.