Solution of a System of Generalized Abel Integral Equations Using Fractional Calculus

A method is presented for obtaining useful closed form solution of a system of generalized Abel integral equations by using the ideas of fractional integral operators and their applications. This system appears in solving certain mixed boundary value problems arising in the classical theory of elasticity.

A computational algorithm for the verification of tautologies in propositional calculus

A computational algorithm (based on Smullyan's analytic tableau method) that varifies whether a given well-formed formula in propositional calculus is a tautology or not has been implemented on a DEC system 10. The stepwise refinement approch of program development used for this implementation forms the subject matter of this paper. The top-down design has resulted in a modular and reliable program package. This computational algoritlhm compares favourably with the algorithm based on the well-known resolution principle used in theorem provers.

An interpreter for slips-An applicative language based on LAMBDA-Calculus

An applicative language based on the LAMBDA-Calculus is presented. The language, SLIPS (Small Language for Instruction Purposes), is described using the LAMBDA-Calculus as a metalanguage. A call-by-need mechanism of function invocation eliminates the drawbacks of both call-by-name and call-by-value. The system has been implemented in PASCAL.

Quantum stochastic calculus associated with quadratic quantum noises

We first study a class of fundamental quantum stochastic processes induced by the generators of a six dimensional non-solvable Lie dagger-algebra consisting of all linear combinations of the generalized Gross Laplacian and its adjoint, annihilation operator, creation operator, conservation, and time, and then we study the quantum stochastic integrals associated with the class of fundamental quantum stochastic processes, and the quantum Ito formula is revisited. The existence and uniqueness of solution of a quantum stochastic differential equation is proved. The unitarity conditions of solutions of quantum stochastic differential equations associated with the fundamental processes are examined. The quantum stochastic calculus extends the Hudson-Parthasarathy quantum stochastic calculus. (C) 2016 AIP Publishing LLC.

Some remarks on the symplectic pairing on the moduli space of representations of the fundamental group of surfaces

This work grew out of an attempt to understand a conjectural remark made by Professor Kyoji Saito to the author about a possible link between the Fox-calculus description of the symplectic structure on the moduli space of representations of the fundamental group of surfaces into a Lie group and pairs of mutually dual sets of generators of the fundamental group. In fact in his paper [3] , Prof. Kyoji Saito gives an explicit description of the system of dual generators of the fundamental group.

Paraxial Maxwell beams: transformation by general linear optical systems

Extending the work of earlier papers on the relativistic-front description of paraxial optics and the formulation of Fourier optics for vector waves consistent with the Maxwell equations, we generalize the Jones calculus of axial plane waves to describe the action of the most general linear optical system on paraxial Maxwell fields. Several examples are worked out, and in each case it is shown that the formalism leads to physically correct results. The importance of retaining the small components of the field vectors along the axis of the system for a consistent description is emphasized.

Complete Pick positivity and unitary invariance

The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foias. Just as a contraction is related to the Szego kernel k(S) (z, w) = (1 - z (w) over tilde)(-1) for |z|, |w| < 1, by means of (1/k(S))(T,T*) >= 0, we consider an arbitrary open connected domain Omega in C-n, a complete Pick kernel k on Omega and a tuple T = (T-1, ..., T-n) of commuting bounded operators on a complex separable Hilbert space H such that (1/k)(T,T*) >= 0. For a complete Pick kernel the 1/k functional calculus makes sense in a beautiful way. It turns out that the model theory works very well and a characteristic function can be associated with T. Moreover, the characteristic function is then a complete unitary invariant for a suitable class of tuples T.

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.

Damping and synchronising torques contributed by a voltage regulator to a synchronous generator: a quantitative assessment

In the paper, the total damping and synchronising torques, which determine the dynamic stability of a synchronous generator in a power system, have been traced to their origin. The positive and negative components released or consumed by the voltage regulator, and by the various windings of the machine, have been isolated, with the object of making a quantitative assessment of the effects of various gains and time constants on the dynamic stability of a synchronous machine under different operating conditions. The analysis is based on the properties of quadratic invariance in tensor calculus. An alternative solution by network analysis has also been provided to establish the validity of the tensor approach.

First passage distributions for long memory processes

We study the distribution of first passage time for Levy type anomalous diffusion. A fractional Fokker-Planck equation framework is introduced.For the zero drift case, using fractional calculus an explicit analytic solution for the first passage time density function in terms of Fox or H-functions is given. The asymptotic behaviour of the density function is discussed. For the nonzero drift case, we obtain an expression for the Laplace transform of the first passage time density function, from which the mean first passage time and variance are derived.

Contractive Hilbert modules and their dilations

In this note, we show that a quasi-free Hilbert module R defined over the polydisk algebra with kernel function k(z,w) admits a unique minimal dilation (actually an isometric co-extension) to the Hardy module over the polydisk if and only if S (-1)(z, w)k(z, w) is a positive kernel function, where S(z,w) is the Szego kernel for the polydisk. Moreover, we establish the equivalence of such a factorization of the kernel function and a positivity condition, defined using the hereditary functional calculus, which was introduced earlier by Athavale [8] and Ambrozie, Englis and Muller [2]. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. The proof works for a wider class of Hilbert modules in which the Hardy module is replaced by more general quasi-free Hilbert modules such as the classical spaces on the polydisk or the unit ball in a'', (m) . Some consequences of this more general result are then explored in the case of several natural function algebras.

Simulation of length-preserving motions of flexible one dimensional oObjects using optimization

During the motion of one dimensional flexible objects such as ropes, chains, etc., the assumption of constant length is realistic. Moreover,their motion appears to be naturally minimizing some abstract distance measure, wherein the disturbance at one end gradually dies down along the curve defining the object. This paper presents purely kinematic strategies for deriving length-preserving transformations of flexible objects that minimize appropriate ‘motion’. The strategies involve sequential and overall optimization of the motion derived using variational calculus. Numerical simulations are performed for the motion of a planar curve and results show stable converging behavior for single-step infinitesimal and finite perturbations 1 as well as multi-step perturbations. Additionally, our generalized approach provides different intuitive motions for various problem-specific measures of motion, one of which is shown to converge to the conventional tractrix-based solution. Simulation results for arbitrary shapes and excitations are also included.

Natural motion of one-dimensional flexible objects using minimization approaches

For one-dimensional flexible objects such as ropes, chains, hair, the assumption of constant length is realistic for large-scale 3D motion. Moreover, when the motion or disturbance at one end gradually dies down along the curve defining the one-dimensional flexible objects, the motion appears ``natural''. This paper presents a purely geometric and kinematic approach for deriving more natural and length-preserving transformations of planar and spatial curves. Techniques from variational calculus are used to determine analytical conditions and it is shown that the velocity at any point on the curve must be along the tangent at that point for preserving the length and to yield the feature of diminishing motion. It is shown that for the special case of a straight line, the analytical conditions lead to the classical tractrix curve solution. Since analytical solutions exist for a tractrix curve, the motion of a piecewise linear curve can be solved in closed-form and thus can be applied for the resolution of redundancy in hyper-redundant robots. Simulation results for several planar and spatial curves and various input motions of one end are used to illustrate the features of motion damping and eventual alignment with the perturbation vector.

A fractional order proportional integral controller for path following and trajectory tracking of miniature air vehicles

In this paper, a fractional order proportional-integral controller is developed for a miniature air vehicle for rectilinear path following and trajectory tracking. The controller is implemented by constructing a vector field surrounding the path to be followed, which is then used to generate course commands for the miniature air vehicle. The fractional order proportional-integral controller is simulated using the fundamentals of fractional calculus, and the results for this controller are compared with those obtained for a proportional controller and a proportional integral controller. In order to analyze the performance of the controllers, four performance metrics, namely (maximum) overshoot, control effort, settling time and integral of the timed absolute error cost, have been selected. A comparison of the nominal as well as the robust performances of these controllers indicates that the fractional order proportional-integral controller exhibits the best performance in terms of ITAE while showing comparable performances in all other aspects.

Globally coupled stochastic two-state oscillators: Fluctuations due to finite numbers

Infinite arrays of coupled two-state stochastic oscillators exhibit well-defined steady states. We study the fluctuations that occur when the number N of oscillators in the array is finite. We choose a particular form of global coupling that in the infinite array leads to a pitchfork bifurcation from a monostable to a bistable steady state, the latter with two equally probable stationary states. The control parameter for this bifurcation is the coupling strength. In finite arrays these states become metastable: The fluctuations lead to distributions around the most probable states, with one maximum in the monostable regime and two maxima in the bistable regime. In the latter regime, the fluctuations lead to transitions between the two peak regions of the distribution. Also, we find that the fluctuations break the symmetry in the bimodal regime, that is, one metastable state becomes more probable than the other, increasingly so with increasing array size. To arrive at these results, we start from microscopic dynamical evolution equations from which we derive a Langevin equation that exhibits an interesting multiplicative noise structure. We also present a master equation description of the dynamics. Both of these equations lead to the same Fokker-Planck equation, the master equation via a 1/N expansion and the Langevin equation via standard methods of Ito calculus for multiplicative noise. From the Fokker-Planck equation we obtain an effective potential that reflects the transition from the monomodal to the bimodal distribution as a function of a control parameter. We present a variety of numerical and analytic results that illustrate the strong effects of the fluctuations. We also show that the limits N -> infinity and t -> infinity(t is the time) do not commute. In fact, the two orders of implementation lead to drastically different results.