a general mass conservative numerical solution for the unsaturated flow equation numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems numerical solution based on the standard h based form of richards equation generally yields poor results characterized by large mass balance errors and erroneous estimates of infiltration depth conversely numerical solutions based on the mixed form of richards equation can be shown to possess the conservative property so that mass is perfectly conserved this leads to significant improvement in numerical solution performance while requiring no additional computational effort however use of the mass conservative method does not guarantee good solutions accurate solution of the unsaturated flow equation also requires use of a diagonal time or mass matrix only when diagonal time matrices are used can the solution be shown to obey a maximum principle which guarantees smooth nonoscillatory infiltration profiles this highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow

multichannel interpolation of nonuniform samples with application to image recovery abstract the multichannel trigonometric reconstruction from uniform samples was proposed recently it not only makes use of multichannel information about the signal but is also capable to generate various kinds of interpolation formulas according to the types and amounts of the collected samples the paper presents the theory of multichannel interpolation from nonuniform samples two distinct models of nonuniform sampling patterns are considered namely recurrent and generic nonuniform sampling each model involves two types of samples nonuniform samples of the observed signal and its derivatives numerical examples and quantitative error analysis are provided to demonstrate the effectiveness of the proposed algorithms additionally the proposed algorithms for recovering highly corrupted images are also investigated in comparison with the median filter and the shepard interpolation the proposed algorithms produce superior results with lower errors

on the rhs description of resonances and virtual states we discuss the role of the m ller wave operators in a rigged hilbert space rhs model for resonances we also construct a new rhs of analytic functions valid for the description of resonances and virtual states these become generalized state vectors in a general physically realizable representation

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

an irregular sampling theorem for functions bandlimited in a generalized sense it is shown that a well known irregular sampling formula for ordinary bandlimited functions is also valid for functions bandlimited in a generalized sense except for some necessary modifications

the rigged hilbert space approach to the lippmann schwinger equation ii the analytic continuation of the lippmann schwinger bras and kets the analytic continuation of the lippmann schwinger bras and kets is obtained and characterized it is shown that the natural mathematical setting for the analytic continuation of the solutions of the lippmann schwinger equation is the rigged hilbert space rather than just the hilbert space it is also argued that this analytic continuation entails the imposition of a time asymmetric boundary condition upon the group time evolution resulting in a semigroup time evolution physically the semigroup time evolution is simply a retarded or advanced propagator

a note on sampling of bandlimited stochastic processes it is pointed out that an irregular sampling theorem of the author siam j appl math vol 47 no 5 p 1112 16 1987 can be used to generalize the results of a paper by z a piranashvili theory prob appl vol 12 p 647 57 1967 on interpolation of stochastic processes only the important case of wide sense stationary processes is considered here

reply to comment on on the inconsistency of the bohm gadella theory with quantum mechanics in this reply we show that when we apply standard distribution theory to the lippmann schwinger equation the resulting spaces of test functions would comply with the hardy axiom only if classic results of paley and wiener of gelfand and shilov and of the theory of ultradistributions were wrong also we point out several differences between the standard method of constructing rigged hilbert spaces in quantum mechanics and the method used in time asymmetric quantum theory

the rigged hilbert space of the algebra of the one dimensional rectangular barrier potential the rigged hilbert space of the algebra of the one dimensional rectangular barrier potential is constructed the one dimensional rectangular potential provides another opportunity to show that the rigged hilbert space fully accounts for dirac s bra ket formalism the analogy between dirac s formalism and fourier methods is pointed out

innovative modelling of 3d unsaturated flow in porous media by coupling independent models for vertical and lateral flows unsaturated groundwater flows are mathematically represented by the richards equation hitherto in hydrology solutions of this equation mainly serve as an alimentation of the source term for the surface runoff modelling therefore the complete resolution of the 3d model looks surplus to requirements and the infiltration is dealt either thanks to 1d vertical modelling of the richards equation or through derivate models like e g the green ampt infiltration model or the horton law thus ignoring eventual horizontal transfers nowadays the request for more detailed information is real and the physics of groundwater unsaturated flow needs to be represented more reliably this information could be furnished by the resolution of the complete 3d model but although numerically mastered and well documented it is very costly for large scale both in time and space real applications climate change adaptation of watersheds the authors propose an original solution decoupling the 3d equations into 1d vertical equations and a 2d depth integrated horizontal equation the aim is to consider independent vertical columns of infiltration coupled with lateral transfer of mass through the boundary conditions on this basis they postulate that the mass transfers in the three dimensions are correctly represented this way problematic like the supply of the aquifers the re emergence of groundwater to surface water or especially the capability of memorization of past rainy events could be reliably depicted the two coupled models are solved on a unique numerical frame a cell centred finite volume method is used to solve the parabolic partial differential equations the spatial derivatives are approximate by a second order central difference scheme while the time splitting follows an implicit backward euler scheme coupled with picard iteration the method has been tested and its reliability assessed on different theoretical two dimensional cross sectional test cases representing infiltration phenomena

on the inconsistency of the bohm gadella theory with quantum mechanics the bohm gadella theory sometimes referred to as the time asymmetric quantum theory of scattering and decay is based on the hardy axiom the hardy axiom asserts that the solutions of the lippmann schwinger equation are functionals over spaces of hardy functions the preparation registration arrow of time provides the physical justification for the hardy axiom in this paper it is shown that the hardy axiom is incorrect because the solutions of the lippmann schwinger equation do not act on spaces of hardy functions it is also shown that the derivation of the preparation registration arrow of time is flawed thus hardy functions neither appear when we solve the lippmann schwinger equation nor should they appear it is also shown that the bohm gadella theory does not rest on the same physical principles as quantum mechanics and that it does not solve any problem that quantum mechanics cannot solve the bohm gadella theory must therefore be abandoned

irregular sampling theorems and series expansions of band limited functions abstract we present a new approach to the problem of irregular sampling of band limited functions that is based on the approximation and factorization of convolution operators a special case of our main result is the following theorem if r n is compact g l 1 rn a band limited function with g t 0 on and xi i i and yj j j are two discrete sets in rn which are dense enough then every band limited function f lwp 1 p our methods work a for a very general class of function spaces not only weighted lwp spaces b in all dimensions even in general locally compact abelian groups c to provide the correct behavior of the coefficients in contrast to the traditional estimates currently used by engineers furthermore the method is constructive and stable with respect to input errors such as round off errors truncation effors jitter errors or aliasing errors

nonuniform sampling of bandlimited functions in this correspondence we show that a function in the bernstein class bpi p 1 les p les infin can be reconstructed in the lagrange interpolation form with sampling points tn satisfying tn n les d and 4d additional points where d is any nonnegative real number that is not necessarily less than 1 4 here x denotes the integer part of x

numerical approximations of darcian flow through unsaturated soil the choice of appropriate intrablock approximations for the unsaturated hydraulic conductivity is a critical step for the numerical solution of richards equation forms commonly used include arithmetic geometric and harmonic averaging the choice is most often based on convenience and custom this study presents a framework for comparing any scheme for weighting the unsaturated hydraulic conductivity against the correct darcian flow across a block for steady conditions such comparisons are carried out for a number of commonly used averages and errors are shown to be alarmingly high in some cases a scheme for approximately matching the correct darcian flux is illustrated by two infiltration examples in the first the arithmetic and weighted schemes are superior to the geometric average when compared to a quasi analytical solution for the second example the weighted solution is slightly better than either the arithmetic or geometric averaging schemes these examples also illustrate that an acceptable mass balance should not be used as a sole criterion for success as it can probably be attained with most schemes the conclusions are believed relevant for algorithms which are pressure head based water content based or other choices and for multidimensional as well as one dimensional flow problems

a comparison of picard and newton iteration in the numerical solution of multidimensional variably saturated flow problems picard iteration is a widely used procedure for solving the nonlinear equation governing flow in variably saturated porous media the method is simple to code and computationally cheap but has been known to fail or converge slowly the newton method is more complex and expensive on a per iteration basis than picard and as such has not received very much attention its robustness and higher rate of convergence however make it an attractive alternative to the picard method particularly for strongly nonlinear problems in this paper the picard and newton schemes are implemented and compared in one two and three dimensional finite element simulations involving both steady state and transient flow the eight test cases presented highlight different aspects of the performance of the two iterative methods and the different factors that can affect their convergence and efficiency including problem size spatial and temporal discretization initial solution estimates convergence error norm mass lumping time weighting conductivity and moisture content characteristics boundary conditions seepage faces and the extent of fully saturated zones in the soil previous strategies for enhancing the performance of the picard and newton schemes are revisited and new ones are suggested the strategies include chord slope approximations for the derivatives of the characteristic equations relaxing convergence requirements along seepage faces dynamic time step control nonlinear relaxation and a mixed picard newton approach the tests show that the picard or relaxed picard schemes are often adequate for solving richards equation but that in cases where these fail to converge or converge slowly the newton method should be used the mixed picard newton approach can effectively overcome the newton scheme s sensitivity to initial solution estimates while comparatively poor performance is reported for the various chord slope approximations finally given the reliability and efficiency of current conjugate gradient like methods for solving linear nonsymmetric systems the only real drawback of using newton rather than picard iteration is the algebraic complexity and computational cost of assembling the derivative terms of the jacobian matrix and it is suggested that both methods can be effectively implemented and used in numerical models of richards equation

comment on on the inconsistency of the bohm gadella theory with quantum mechanics rigged hilbert spaces of hardy functions lead to a consistent theory of resonance scattering and decay contrary to the claims of a recent article 8 the theory holds for a wide range of potentials and rigorously describes the asymmetric time evolution of resonances

some new procedures for numerical solution of variably saturated flow problems persistent difficulties that arise in forming numerical solutions of variably saturated flow problems include controlling the stability of the nonlinear equation solvers and devising a reliable yet efficient method for determining the positions of seepage surfaces new techniques for addressing these problems are applied to a subdomain finite element discretization of the governing flow equations a series of test problems demonstrates that the techniques are reliable and efficient for a wide variety of problems

the rigged hilbert space approach to the gamow states we use the resonances of the spherical shell potential to present a thorough description of the gamow quasinormal states within the rigged hilbert space it will be concluded that the natural setting for the gamow states is a rigged hilbert space whose test functions fall off at infinity faster than gaussians

on computing the trace of the kernel of the homogeneous fredholm s equation a method for computing the trace of the kernel of the homogeneous fredholm s equation for resonant states arising from nonlocal potentials is proposed we show that this integral formulation is convergent

the lippmann schwinger equations in the rigged hilbert space we give a rigorous meaning to the lippmann schwinger equations relating generalized eigenvectors of the free and perturbed hamiltonians in a scattering process some related formulae are also presented a discussion is made on the context of rigged hilbert space formulation of quantum mechanics

the rigged hilbert space approach to the lippmann schwinger equation part i we exemplify the way the rigged hilbert space deals with the lippmann schwinger equation by way of the spherical shell potential we explicitly construct the lippmann schwinger bras and kets along with their energy representation their time evolution and the rigged hilbert spaces to which they belong it will be concluded that the natural setting for the solutions of the lippmann schwinger equation and therefore for scattering theory is the rigged hilbert space rather than just the hilbert space