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Leading Term (leading + term)
Selected AbstractsUniform asymptotic Green's functions for efficient modeling of cracks in elastic layers with relative shear deformation controlled by linear springsINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 3 2009Anthony P. Peirce Abstract We present a uniform asymptotic solution (UAS) for a displacement discontinuity (DD) that lies within the middle layer of a three-layer elastic medium in which relative shear deformation between parallel interfaces is controlled by linear springs. The DD is assumed to be normal to the two interfaces between the elastic media. Using the Fourier transform method we construct a leading term in the asymptotic expansion for the spectral coefficient functions for a DD in a three-layer-spring medium. Although a closed-form solution will require a solution in terms of an infinite series, we demonstrate how this UAS can be used to construct highly efficient and accurate solutions even in the case in which the DD actually touches the interface. We compare the results using the Green's function UAS solution for a crack crossing a soft interface with results obtained using a multi-layer boundary element method. We also present results from an implementation of the UAS Green's function approach in a pseudo-3D hydraulic fracturing simulator to analyze the effect of interface shear deformation on the fracture propagation process. These results are compared with field measurements. Copyright © 2008 John Wiley & Sons, Ltd. [source] On the inverse of generalized ,-matrices with singular leading termINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2006N. A. Dumont Abstract An algorithm is introduced for the inverse of a ,-matrix given as the truncated series A0,i,A1,,2A2+i,3A3+,4A4+···+O(,n+1) with square coefficient matrices and singular leading term A0. Moreover, A1 may be conditionally singular and no restrictions are made for the remaining terms. The result is a ,-matrix given as a unique, truncated series of the same error order. Motivation for this problem is the evaluation of the frequency-dependent stiffness matrix of general boundary or macro-finite elements in the frame of a hybrid variational formulation that is based on a flexibility matrix F expressed as a truncated power series of the circular frequency ,. Copyright © 2005 John Wiley & Sons, Ltd. [source] Iterated Neumann problem for the higher order Poisson equationMATHEMATISCHE NACHRICHTEN, Issue 1-2 2006H. Begehr Abstract Rewriting the higher order Poisson equation ,nu = f in a plane domain as a system of Poisson equations it is immediately clear what boundary conditions may be prescribed in order to get (unique) solutions. Neumann conditions for the Poisson equation lead to higher-order Neumann (Neumann- n ) problems for ,nu = f . Extending the concept of Neumann functions for the Laplacian to Neumann functions for powers of the Laplacian leads to an explicit representation of the solution to the Neumann- n problem for ,nu = f . The representation formula provides the tool to treat more general partial differential equations with leading term ,nu in reducing them into some singular integral equations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Some bounds on the coupon collector problemRANDOM STRUCTURES AND ALGORITHMS, Issue 2 2004Servet Martinez Abstract This work addresses on the coupon collector problem and its generalization introduced by Flajolet, Gardy, and Thimonier. In our main results, we show a ratio limit theorem for the random time of the generalized coupon collector problem, and, further, we give the leading term and the geometric rate for the distribution of this random time, when the number of throws is large. For the classical coupon collector problem, we give a bound on the conditional second moment for the number of visits to the coupons, relying strongly on a result of Holst on extremal distributions. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004 [source] Semiclassical determination of exponentially small intermode transitions for 1 + 1 spacetime scattering systems,COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 8 2007Alain Joye We consider the semiclassical limit of systems of autonomous PDEs in 1 + 1 spacetime dimensions in a scattering regime. We assume the matrix-valued coefficients are analytic in the space variable, and we further suppose that the corresponding dispersion relation admits real-valued modes only with one-dimensional polarization subspaces. Hence a BKW-type analysis of the solutions is possible. We typically consider time-dependent solutions to the PDE that are carried asymptotically in the past and as x , ,, along one mode only and determine the piece of the solution that is carried for x , +, along some other mode in the future. Because of the assumed nondegeneracy of the modes, such transitions between modes are exponentially small in the semiclassical parameter; this is an expression of the Landau-Zener mechanism. We completely elucidate the spacetime properties of the leading term of this exponentially small wave, when the semiclassical parameter is small, for large values of x and t, when some avoided crossing of finite width takes place between the involved modes. © 2006 Wiley Periodicals, Inc. [source] Sum rules and exact relations for quantal Coulomb systemsCONTRIBUTIONS TO PLASMA PHYSICS, Issue 5-6 2003V.M. Adamyan Abstract A complex response function describing a reaction of a multi-particle system to a weak alternating external field is the boundary value of a Nevanlinna class function (i.e. a holomorphic function with non-negative imaginary part in the upper half-plane). Attempts of direct calculations of response functions based on standard approximations of the kinetic theory for real Coulomb condensed systems often result in considerable discrepancies with experiments and computer simulations. At the same time a relatively simple approach using only the exact values of leading asymptotic terms of the response function permits to restrict essentially a subset of Nevanlinna class functions containing this response function, and in this way to obtain sufficient data to explain and predict experimental results. Mathematical details of this approach are demonstrated on an example with the response function being the (external) dynamic electrical conductivity of cold dense hydrogen-like plasmas. In particular, the exact values of the leading terms of asymptotic expansions of the conductivity are calculated. (© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] A method for analysing the transient and the steady-state oscillations in third-order oscillators with shifting biasINTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS, Issue 5 2001A. Buonomo Abstract We provide an asymptotic method for systematically analysing the transient and the steady-state oscillations in third-order oscillators with shifting bias. The method allows us to construct the general solution of the weakly non-linear differential equation describing these oscillators through an iteration procedure of successive approximations typical of perturbation methods. The approximation to first order is obtained solving a system of two first-order non-linear differential equations in the leading terms of solution (dc component and fundamental harmonic), whereby the dominant dynamics, the stationary states and their stability can be easily analysed. Unlike existing approaches, our method also enables us to determine the higher harmonics as well as the frequency shift from the system's natural frequency in the exact solution through analytical formulae. In addition, formulae for higher-order approximations of the above quantities are determined. The proposed method is applied to a practical circuit to show its usefulness in both analysis and design problems. Copyright © 2001 John Wiley & Sons, Ltd. [source] Weak asymptotics of the spectral shift functionMATHEMATISCHE NACHRICHTEN, Issue 11 2007Vincent Bruneau Abstract We consider the three-dimensional Schrödinger operator with constant magnetic field of strength b > 0, and with smooth electric potential. The weak asymptotics of the spectral shift function with respect to b , +, is studied. First, we fix the distance to the Landau levels, then the distance to Landau levels tends to infinity as b , +,. In particular we give explicitly the leading terms in the asymptotics and in some case we obtain full asymptotics expansions. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] | |||||||||||||