Group law computations on Jacobians of hyperelliptic Curves

We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form.

A finite volume method with linearisation in time for the solution of advection-reaction-diffusion systems

The numerical solution in one space dimension of advection--reaction--diffusion systems with nonlinear source terms may invoke a high computational cost when the presently available methods are used. Numerous examples of finite volume schemes with high order spatial discretisations together with various techniques for the approximation of the advection term can be found in the literature. Almost all such techniques result in a nonlinear system of equations as a consequence of the finite volume discretisation especially when there are nonlinear source terms in the associated partial differential equation models. This work introduces a new technique that avoids having such nonlinear systems of equations generated by the spatial discretisation process when nonlinear source terms in the model equations can be expanded in positive powers of the dependent function of interest. The basis of this method is a new linearisation technique for the temporal integration of the nonlinear source terms as a supplementation of a more typical finite volume method. The resulting linear system of equations is shown to be both accurate and significantly faster than methods that necessitate the use of solvers for nonlinear system of equations.

Hybrid frequency–time domain models for dynamic response analysis of marine structures

Time-domain models of marine structures based on frequency domain data are usually built upon the Cummins equation. This type of model is a vector integro-differential equation which involves convolution terms. These convolution terms are not convenient for analysis and design of motion control systems. In addition, these models are not efficient with respect to simulation time, and ease of implementation in standard simulation packages. For these reasons, different methods have been proposed in the literature as approximate alternative representations of the convolutions. Because the convolution is a linear operation, different approaches can be followed to obtain an approximately equivalent linear system in the form of either transfer function or state-space models. This process involves the use of system identification, and several options are available depending on how the identification problem is posed. This raises the question whether one method is better than the others. This paper therefore has three objectives. The first objective is to revisit some of the methods for replacing the convolutions, which have been reported in different areas of analysis of marine systems: hydrodynamics, wave energy conversion, and motion control systems. The second objective is to compare the different methods in terms of complexity and performance. For this purpose, a model for the response in the vertical plane of a modern containership is considered. The third objective is to describe the implementation of the resulting model in the standard simulation environment Matlab/Simulink.

High order unconditionally stable difference schemes for the Riesz space-fractional telegraph equation

In this paper, a class of unconditionally stable difference schemes based on the Pad´e approximation is presented for the Riesz space-fractional telegraph equation. Firstly, we introduce a new variable to transform the original dfferential equation to an equivalent differential equation system. Then, we apply a second order fractional central difference scheme to discretise the Riesz space-fractional operator. Finally, we use (1, 1), (2, 2) and (3, 3) Pad´e approximations to give a fully discrete difference scheme for the resulting linear system of ordinary differential equations. Matrix analysis is used to show the unconditional stability of the proposed algorithms. Two examples with known exact solutions are chosen to assess the proposed difference schemes. Numerical results demonstrate that these schemes provide accurate and efficient methods for solving a space-fractional hyperbolic equation.

Tracking adversarial targets

We study linear control problems with quadratic losses and adversarially chosen tracking targets. We present an efficient algorithm for this problem and show that, under standard conditions on the linear system, its regret with respect to an optimal linear policy grows as O(log^2 T), where T is the number of rounds of the game. We also study a problem with adversarially chosen transition dynamics; we present an exponentiallyweighted average algorithm for this problem, and we give regret bounds that grow as O(sqtr p T).

Nonstationary Random Critical Seismic Excitations

A method is presented to find nonstationary random seismic excitations with a constraint on mean square value such that the response variance of a given linear system is maximized. It is also possible to incorporate the dominant input frequency into the analysis. The excitation is taken to be the product of a deterministic enveloping function and a zero mean Gaussian stationary random process. The power spectral density function of this process is determined such that the response variance is maximized. Numerical results are presented for a single-degree system and an earth embankment modeled as shear beam.

Implementation of parallel tridiagonal solvers for a heterogeneous computing environment

Tridiagonal diagonally dominant linear systems arise in many scientific and engineering applications. The standard Thomas algorithm for solving such systems is inherently serial forming a bottleneck in computation. Algorithms such as cyclic reduction and SPIKE reduce a single large tridiagonal system into multiple small independent systems which can be solved in parallel. We have developed portable cyclic reduction and SPIKE algorithm OpenCL implementations with the intent to target a range of co-processors in a heterogeneous computing environment including Field Programmable Gate Arrays (FPGAs), Graphics Processing Units (GPUs) and other multi-core processors. In this paper, we evaluate these designs in the context of solver performance, resource efficiency and numerical accuracy.

Multi-time scale analysis applied to long time simulation of power systems

The results are presented of applying multi-time scale analysis using the singular perturbation technique for long time simulation of power system problems. A linear system represented in state-space form can be decoupled into slow and fast subsystems. These subsystems can be simulated with different time steps and then recombined to obtain the system response. Simulation results with a two-time scale analysis of a power system show a large saving in computational costs.

The Noether-Lefschetz theorem for the divisor class group

Let X be a normal projective threefold over a field of characteristic zero and vertical bar L vertical bar be a base-point free, ample linear system on X. Under suitable hypotheses on (X, vertical bar L vertical bar), we prove that for a very general member Y is an element of vertical bar L vertical bar, the restriction map on divisor class groups Cl(X) -> Cl(Y) is an isomorphism. In particular, we are able to recover the classical Noether-Lefschetz theorem, that a very general hypersurface X subset of P-C(3) of degree >= 4 has Pic(X) congruent to Z.

Composite scheme using localized relaxation with non-standard finite difference method for hyperbolic conservation laws

Non-standard finite difference methods (NSFDM) introduced by Mickens [Non-standard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994] are interesting alternatives to the traditional finite difference and finite volume methods. When applied to linear hyperbolic conservation laws, these methods reproduce exact solutions. In this paper, the NSFDM is first extended to hyperbolic systems of conservation laws, by a novel utilization of the decoupled equations using characteristic variables. In the second part of this paper, the NSFDM is studied for its efficacy in application to nonlinear scalar hyperbolic conservation laws. The original NSFDMs introduced by Mickens (1994) were not in conservation form, which is an important feature in capturing discontinuities at the right locations. Mickens [Construction and analysis of a non-standard finite difference scheme for the Burgers–Fisher equations, Journal of Sound and Vibration 257 (4) (2002) 791–797] recently introduced a NSFDM in conservative form. This method captures the shock waves exactly, without any numerical dissipation. In this paper, this algorithm is tested for the case of expansion waves with sonic points and is found to generate unphysical expansion shocks. As a remedy to this defect, we use the strategy of composite schemes [R. Liska, B. Wendroff, Composite schemes for conservation laws, SIAM Journal of Numerical Analysis 35 (6) (1998) 2250–2271] in which the accurate NSFDM is used as the basic scheme and localized relaxation NSFDM is used as the supporting scheme which acts like a filter. Relaxation schemes introduced by Jin and Xin [The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications in Pure and Applied Mathematics 48 (1995) 235–276] are based on relaxation systems which replace the nonlinear hyperbolic conservation laws by a semi-linear system with a stiff relaxation term. The relaxation parameter (λ) is chosen locally on the three point stencil of grid which makes the proposed method more efficient. This composite scheme overcomes the problem of unphysical expansion shocks and captures the shock waves with an accuracy better than the upwind relaxation scheme, as demonstrated by the test cases, together with comparisons with popular numerical methods like Roe scheme and ENO schemes.

Formal Verification of Full-Wave Rectifier: A Case Study

We present a case study of formal verification of full-wave rectifier for analog and mixed signal designs. We have used the Checkmate tool from CMU [1], which is a public domain formal verification tool for hybrid systems. Due to the restriction imposed by Checkmate it necessitates to make the changes in the Checkmate implementation to implement the complex and non-linear system. Full-wave rectifier has been implemented by using the Checkmate custom blocks and the Simulink blocks from MATLAB from Math works. After establishing the required changes in the Checkmate implementation we are able to efficiently verify, the safety properties of the full-wave rectifier.

An ab-initio study of the proton affinity of conjugated schiff-base and related nitrogen compounds: an analysis of the triggering site of bacteriorhodopsin

The chemical groups which take part in the proton transfer reaction in bacteriorhodopsin have been studied by ab initio quantum chemical methods. The various factors such as conjugation with a linear system, electron delocalization of the guanidine type, cis-trans isomerism, geometry distortion and hydrogen bonding with charged groups can influence the properties of a given chemical group. Several systems are studied at 4-31G and STO-3G levels. Some of the Schiff-base analogues and guanidine type molecules are characterized by their molecular orbital diagrams, energy levels and the nature of charge distribution. Also, the effects of the above-mentioned factors on proton affinity are studied. It is hoped that the values thus obtained can be helpful in evaluating various structural models for proton transfer.

Some new results regarding spikes and a heuristic for spike construction

his paper addresses the problem of minimizing the number of columns with superdiagonal nonzeroes (viz., spiked columns) in a square, nonsingular linear system of equations which is to be solved by Gaussian elimination. The exact focus is on a class of min-spike heuristics in which the rows and columns of the coefficient matrix are first permuted to block lower-triangular form. Subsequently, the number of spiked columns in each irreducible block and their heights above the diagonal are minimized heuristically. We show that ifevery column in an irreducible block has exactly two nonzeroes, i.e., is a doubleton, then there is exactly one spiked column. Further, if there is at least one non-doubleton column, there isalways an optimal permutation of rows and columns under whichnone of the doubleton columns are spiked. An analysis of a few benchmark linear programs suggests that singleton and doubleton columns can abound in practice. Hence, it appears that the results of this paper can be practically useful. In the rest of the paper, we develop a polynomial-time min-spike heuristic based on the above results and on a graph-theoretic interpretation of doubleton columns.

Critical earthquake input power spectral density function models for engineering structures

The problem of determining optimal power spectral density models for earthquake excitation which satisfy constraints on total average power, zero crossing rate and which produce the highest response variance in a given linear system is considered. The solution to this problem is obtained using linear programming methods. The resulting solutions are shown to display a highly deterministic structure and, therefore, fail to capture the stochastic nature of the input. A modification to the definition of critical excitation is proposed which takes into account the entropy rate as a measure of uncertainty in the earthquake loads. The resulting problem is solved using calculus of variations and also within linear programming framework. Illustrative examples on specifying seismic inputs for a nuclear power plant and a tall earth dam are considered and the resulting solutions are shown to be realistic.

A nonlinear spectral finite element model for analysis of wave propagation in solid with internal friction and dissipation

A geometrically non-linear Spectral Finite Flement Model (SFEM) including hysteresis, internal friction and viscous dissipation in the material is developed and is used to study non-linear dissipative wave propagation in elementary rod under high amplitude pulse loading. The solution to non-linear dispersive dissipative equation constitutes one of the most difficult problems in contemporary mathematical physics. Although intensive research towards analytical developments are on, a general purpose cumputational discretization technique for complex applications, such as finite element, but with all the features of travelling wave (TW) solutions is not available. The present effort is aimed towards development of such computational framework. Fast Fourier Transform (FFT) is used for transformation between temporal and frequency domain. SFEM for the associated linear system is used as initial state for vector iteration. General purpose procedure involving matrix computation and frequency domain convolution operators are used and implemented in a finite element code. Convergnence of the spectral residual force vector ensures the solution accuracy. Important conclusions are drawn from the numerical simulations. Future course of developments are highlighted.