A Note on Minimal Reed-Muller Canonical Forms of Switching Functions

A nonexhaustive procedure for obtaining minimal Reed-Muller canonical (RMC) forms of switching functions is presented. This procedure is a modification of a procedure presented earlier in the literature and enables derivation of an upper bound on the number of RMC forms to be derived to choose a minimal one. It is shown that the task of obtaining minimal RMC forms is simplified in the case of symmetric functions and self-dual functions.

Automata and logics over finitely varying functions

We extend some of the classical connections between automata and logic due to Büchi (1960) [5] and McNaughton and Papert (1971) [12] to languages of finitely varying functions or “signals”. In particular, we introduce a natural class of automata for generating finitely varying functions called View the MathML source’s, and show that it coincides in terms of language definability with a natural monadic second-order logic interpreted over finitely varying functions Rabinovich (2002) [15]. We also identify a “counter-free” subclass of View the MathML source’s which characterise the first-order definable languages of finitely varying functions. Our proofs mainly factor through the classical results for word languages. These results have applications in automata characterisations for continuously interpreted real-time logics like Metric Temporal Logic (MTL) Chevalier et al. (2006, 2007) [6] and [7].

Post-liquefaction undrained monotonic behaviour of sands: experiments and DEM simulations

This paper presents the results of a series of servo-controlled cyclic triaxial tests and numerical simulations using the three- dimensional discrete element method (DEM) on post-liquefaction undrained monotonic strength of granular materials. In a first test series,undrained monotonic tests were carried out after dissipating the excess pore water pressure developed during liquefaction. The influence of different parameters such as amplitude of axial strain,relative density and confining pressure prior to liquefaction on the post-liquefaction undrained response have been investigated.The results obtained highlight an insignificant influence of amplitude of axial strain, confining pressure and a significant influence of relative density on the post-liquefaction undrained monotonic stress-strain response.In the second series, undrained monotonic tests were carried out on similar triaxial samples without dissipating the excess pore water pressure developed during liquefaction. The results highlight that the amplitude of axial strain prior to liquefaction has a significant influence on the post-liquefaction undrained monotonic response.In addition,DEM simulations have been carried out on an assembly of spheres to simulate post-liquefaction behaviour.The simulations were very similar to the experiments with an objective to understand the behaviour of monotonic strength of liquefied samples from the grain scale. The numerical simulations using DEM have captured qualitatively all the features of the post-liquefaction undrained monotonic response in a manner similar to that of the experiments.In addition,a detailed study on the evolution of micromechanical parameters such as the average coordination number and induced anisotropic coefficients has been reported during the post-liquefaction undrained monotonic loading.

A Legendre spectral element model for sloshing and acoustic analysis in nearly incompressible fluids

A new spectral finite element formulation is presented for modeling the sloshing and the acoustic waves in nearly incompressible fluids. The formulation makes use of the Legendre polynomials in deriving the finite element interpolation shape functions in the Lagrangian frame of reference. The formulated element uses Gauss-Lobatto-Legendre quadrature scheme for integrating the volumetric stiffness and the mass matrices while the conventional Gauss-Legendre quadrature scheme is used on the rotational stiffness matrix to completely eliminate the zero energy modes, which are normally associated with the Lagrangian FE formulation. The numerical performance of the spectral element formulated here is examined by doing the inf-sup test oil a standard rectangular rigid tank partially filled with liquid The eigenvalues obtained from the formulated spectral element are compared with the conventional equally spaced node locations of the h-type Lagrangian finite element and the predicted results show that these spectral elements are more accurate and give superior convergence The efficiency and robustness of the formulated elements are demonstrated by solving few standard problems involving free vibration and dynamic response analysis with undistorted and distorted spectral elements. and the obtained results are compared with available results in the published literature (C) 2009 Elsevier Inc All rights reserved

Guessing based on length functions

Close relationships between guessing functions and length functions are established. Good length functions lead to good guessing functions. In particular, guessing in the increasing order of Lempel-Ziv lengths has certain universality properties for finite-state sources. As an application, these results show that hiding the parameters of the key-stream generating source in a private key crypto-system may not enhance the privacy of the system, the privacy level being measured by the difficulty in brute-force guessing of the key stream.

New rational interpolation functions for finite element analysis of rotating beams

A rotating beam finite element in which the interpolating shape functions are obtained by satisfying the governing static homogenous differential equation of Euler–Bernoulli rotating beams is developed in this work. The shape functions turn out to be rational functions which also depend on rotation speed and element position along the beam and account for the centrifugal stiffening effect. These rational functions yield the Hermite cubic when rotation speed becomes zero. The new element is applied for static and dynamic analysis of rotating beams. In the static case, a cantilever beam having a tip load is considered, with a radially varying axial force. It is found that this new element gives a very good approximation of the tip deflection to the analytical series solution value, as compared to the classical finite element given by the Hermite cubic shape functions. In the dynamic analysis, the new element is applied for uniform, and tapered rotating beams with cantilever and hinged boundary conditions to determine the natural frequencies, and the results compare very well with the published results given in the literature.

Novel window functions for digital filters

Window technique is one of the simplest methods to design Finite Impulse Response (FIR) filters. It uses special functions to truncate an infinite sequence to a finite one. In this paper, we propose window techniques based on integer sequences. The striking feature of the proposed work is that it overcomes all the problems posed by floating point numbers and inaccuracy, as the sequences are made of only integers. Some of these integer window sequences, yield sharp transition, while some of them result in zero ripple in passband and stopband.

Stiff-String Basis Functions for Vibration Analysis of High Speed Rotating Beams

A new rotating beam finite element is developed in which the basis functions are obtained by the exact solution of the governing static homogenous differential equation of a stiff string, which results from an approximation in the rotating beam equation. These shape functions depend on rotation speed and element position along the beam and account for the centrifugal stiffening effect. Using this new element and the Hermite cubic finite element, a convergence study of natural frequencies is performed, and it is found that the new element converges much more rapidly than the conventional Hermite cubic element for the first two modes at higher rotation speeds. The new element is also applied for uniform and tapered rotating beams to determine the natural frequencies, and the results compare very well with the published results given in the literature.

Time and Energy Complexity of Distributed Computation of a Class of Functions in Wireless Sensor Networks

We consider a scenario in which a wireless sensor network is formed by randomly deploying n sensors to measure some spatial function over a field, with the objective of computing a function of the measurements and communicating it to an operator station. We restrict ourselves to the class of type-threshold functions (as defined in the work of Giridhar and Kumar, 2005), of which max, min, and indicator functions are important examples: our discussions are couched in terms of the max function. We view the problem as one of message-passing distributed computation over a geometric random graph. The network is assumed to be synchronous, and the sensors synchronously measure values and then collaborate to compute and deliver the function computed with these values to the operator station. Computation algorithms differ in (1) the communication topology assumed and (2) the messages that the nodes need to exchange in order to carry out the computation. The focus of our paper is to establish (in probability) scaling laws for the time and energy complexity of the distributed function computation over random wireless networks, under the assumption of centralized contention-free scheduling of packet transmissions. First, without any constraint on the computation algorithm, we establish scaling laws for the computation time and energy expenditure for one-time maximum computation. We show that for an optimal algorithm, the computation time and energy expenditure scale, respectively, as Theta(radicn/log n) and Theta(n) asymptotically as the number of sensors n rarr infin. Second, we analyze the performance of three specific computation algorithms that may be used in specific practical situations, namely, the tree algorithm, multihop transmission, and the Ripple algorithm (a type of gossip algorithm), and obtain scaling laws for the computation time and energy expenditure as n rarr infin. In particular, we show that the computation time for these algorithms scales as Theta(radicn/lo- g n), Theta(n), and Theta(radicn log n), respectively, whereas the energy expended scales as , Theta(n), Theta(radicn/log n), and Theta(radicn log n), respectively. Finally, simulation results are provided to show that our analysis indeed captures the correct scaling. The simulations also yield estimates of the constant multipliers in the scaling laws. Our analyses throughout assume a centralized optimal scheduler, and hence, our results can be viewed as providing bounds for the performance with practical distributed schedulers.

Stability of a cylindrical plasma in the presence of a monotonic field-II

The stability of an incompressible inviscid, perfectly conducting cylindrical plasma against azimuthal disturbances in the presence of a monotonic decreasing magnetic field having a constant pitch is discussed by using energy principle. The results obtained by this principle are compared for m = 1 mode (which is a dangerous mode in which there is a lateral shift of the entire column) with that obtained by normal mode analysis. It is found that m = 1 mode is always unstable. Further, an axial line current, external axial field and the surface tension tend to stabilise m ≠ modes.

Force constants and thermodynamic functions of iodine heptafluoride

Normal coordinate analysis of a molecule of the type XY7 (point group D5h) has been carried out using Wilson's FG, matrix method and the results have been utilized to calculate the force constants of IF7 from the available Raman and infrared data. Some of the assignments made previously by Lord and others have been revised and with the revised assignments the thermodynamic quantities of IF7 have been computed from 300°K to 1000°K under rigid rotator and harmonic oscillator approximation.

Potential functions for hydrogen bond interactions - III. Empirical Potential Function for the Peptide N-H...O=C Hydrogen Bond

A careful comparison of the distribution in the (R, θ)-plane of all NH ... O hydrogen bonds with that for bonds between neutral NH and neutral C=O groups indicated that the latter has a larger mean R and a wider range of θ and that the distribution was also broader than for the average case. Therefore, the potential function developed earlier for an average NH ... O hydrogen bond was modified to suit the peptide case. A three-parameter expression of the form {Mathematical expression}, with △ = R - Rmin, was found to be satisfactory. By comparing the theoretically expected distribution in R and θ with observed data (although limited), the best values were found to be p1 = 25, p3 = - 2 and q1 = 1 × 10-3, with Rmin = 2·95 Å and Vmin = - 4·5 kcal/mole. The procedure for obtaining a smooth transition from Vhb to the non-bonded potential Vnb for large R and θ is described, along with a flow chart useful for programming the formulae. Calculated values of ΔH, the enthalpy of formation of the hydrogen bond, using this function are in reasonable agreement with observation. When the atoms involved in the hydrogen bond occur in a five-membered ring as in the sequence[Figure not available: see fulltext.] a different formula for the potential function is needed, which is of the form Vhb = Vmin +p1△2 +q1x2 where x = θ - 50° for θ ≥ 50°, with p1 = 15, q1 = 0·002, Rmin = 2· Å and Vmin = - 2·5 kcal/mole. © 1971 Indian Academy of Sciences.

Potential functions for hydrogen bond interactions - IV. Minimum Energy Conformation of the α-Helical Structure of PoIy-L-Alanine

Making use of the empirical potential functions for peptide NH .. O bonds, developed in this laboratory, the relative stabilities of the rightand left-handed α-helical structures of poly-L-alanine have been investigated, by calculating their conformational energies (V). The value of Vmin of the right-handed helix (αP) is about - 10.4 kcal/mole, and that of the left-handed helix (αM) is about - 9.6 kcal/mole, showing that the former is lower in energy by 0.8 kcal/mole. The helical parameters of the stable conformation of αP are n ∼ 3.6 and h ∼ 1.5 Å. The hydrogen bond of length 2.85 Å and nonlinearity of about 10° adds about 4.0 kcal/ mole to the stabilising energy of the helix in the minimum enregy region. The energy minimum is not sharply defined, but occurs over a long valley, suggesting that a distribution of conformations (φ{symbol}, ψ) of nearly the same energy may occur for the individual residues in a helix. The experimental data of a-helical fibres of poly-L-alanine are in good agreement with the theoretical results for αP. In the case of proteins, the mean values of (φ{symbol}, ψ) for different helices are distributed, but they invariably occur within the contour for V = Vmin + 2 kcal/mole for αP.

On some integral properties of hypergeometric functions

In this note certain integrals involving hypergeometric functions have been evaluated in convenient and elegant forms. © 1971 Indian Academy of Sciences.