critical exponents for semilinear equations of mixed elliptic hyperbolic and degenerate types for semilinear gellerstedt equations with tricomi goursat or dirichlet boundary conditions we prove poho aev type identities and derive nonexistence results that exploit an invariance of the linear part with respect to certain nonhomogeneous dilations a critical exponent phenomenon of power type in the nonlinearity is exhibited in these mixed elliptic hyperbolic or degenerate settings where the power is 1 less than the critical exponent in a relevant sobolev embedding c 2003 wiley periodicals inc

global existence for the one dimensional second order semilinear hyperbolic equations abstract the aim of the paper is to study necessary and sufficient conditions for the existence of the global solution of the one dimensional semilinear equation appearing in the boundary value problems of gas dynamics we investigate the cauchy problem for such equation in the domain where the operator is weakly hyperbolic we obtain the necessary condition for the existence of the self similar solutions for the semilinear gellerstedt type equation the approach used in the paper is based on the fundamental solution of the linear gellerstedt operator and the l p l q estimates

the self similar solutions of the one dimensional semilinear tricomi type equations abstract in this article we investigate the issue of existence of global in time solutions of semilinear tricomi type equations we give conditions that relate the nonlinearity the speed of propagation and the order of singularities of initial data these conditions guarantee existence of global in time solutions in particular we prove existence of solutions invariant under dilation by solving the cauchy problem with initial data which are homogeneous functions

linear prediction methods for blind fractionally spaced equalization we describe adaptive methods for estimating fir zero forcing blind equalizers with arbitrary delay directly from the linear predictions of the observations while most current methods require inversion or singular value decomposition svd of the correlation matrix our methods need only to solve two linear prediction problems they can be implemented as rls or lms algorithms to recursively update the equalizer estimation they are computationally efficient the computational complexity in each recursion can be less than 15 ln sup 2 in the rls case where ln equals the equalizer length and 3l ln in the lms case where l is the number of subchannels the performance of the proposed methods and comparisons with existing approaches are shown by simulation to demonstrate their effectiveness

qr factorization based blind channel identification with second order statistics most eigenstructure based blind channel identification and equalization algorithms with second order statistics need svd or evd of the correlation matrix of the received signal in this paper we address new algorithms based on qr factorization of the received signal directly without calculating the correlation matrix this renders the qr factorization based algorithms more robust against ill conditioned channels i e those channels with almost common zeros among the subchannels first we present a block algorithm that performs the qr factorization of the received data matrix as a whole then a recursive algorithm is developed based on the qr factorization by updating a rank revealing ulv decomposition compared with existing algorithms in the same category our algorithms are computationally more efficient the computation in each recursion of the recursive algorithm is on the order of o m sup 2 if only equalization is required where m is the dimension of the received signal vector our recursive algorithm preserves the fast convergence property of the subspace algorithms thus converging faster than other adaptive algorithms such as the super exponential algorithm with comparable computational complexities moreover our proposed algorithms do not require noise variance estimation numerical simulations demonstrate the good performance of the proposed algorithms

a robust past algorithm for subspace tracking in impulsive noise the past algorithm is an effective and low complexity method for adaptive subspace tracking however due to the use of the recursive least squares rls algorithm in estimating the conventional correlation matrix like other rls algorithms it is very sensitive to impulsive noise and the performance can be degraded substantially to overcome this problem a new robust correlation matrix estimate based on robust statistics concept is proposed in this paper it is derived from the maximum likelihood ml estimate of a multivariate gaussian process in contaminated gaussian noise cg similar to the m estimates in robust statistics this new estimator is incorporated into the past algorithm for robust subspace tracking in impulsive noise furthermore a new restoring mechanism is proposed to combat the hostile effect of long burst of impulses which sporadically occur in communications systems the convergence of this new algorithm is analyzed by extending a previous ordinary differential equation ode based method for past both theoretical and simulation results show that the proposed algorithm offers improved robustness against impulsive noise over the past algorithm the performance of the new algorithm in nominal gaussian noise is very close to that of the past algorithm

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

projection approximation subspace tracking subspace estimation plays an important role in a variety of modern signal processing applications we present a new approach for tracking the signal subspace recursively it is based on a novel interpretation of the signal subspace as the solution of a projection like unconstrained minimization problem we show that recursive least squares techniques can be applied to solve this problem by making an appropriate projection approximation the resulting algorithms have a computational complexity of o nr where n is the input vector dimension and r is the number of desired eigencomponents simulation results demonstrate that the tracking capability of these algorithms is similar to and in some cases more robust than the computationally expensive batch eigenvalue decomposition relations of the new algorithms to other subspace tracking methods and numerical issues are also discussed

plane rotation based evd updating schemes for efficient subspace tracking we present new algorithms based on plane rotations for tracking the eigenvalue decomposition evd of a time varying data covariance matrix these algorithms directly produce eigenvectors in orthonormal form and are well suited for the application of subspace methods to nonstationary data after recasting evd tracking as a simplified rank one evd update problem computationally efficient solutions are obtained in two steps first a new kind of parametric perturbation approach is used to express the eigenvector update as an unimodular orthogonal transform which is represented in exponential matrix form in terms of a reduced set of small unconstrained parameters second two approximate decompositions of this exponential matrix into products of plane or givens rotations are derived one of which being previously unknown these decompositions lead to new plane rotation based evd updating schemes proteus whose main feature is the use of plane rotations for updating the eigenvectors thereby preserving orthonormality finally the proteus schemes are used to derive new evd trackers whose convergence and numerical stability are investigated via simulations one algorithm can track all the signal subspace evd components in only o lm operations where l and m respectively denote the data vector and signal subspace dimensions while achieving a performance comparable to an exact evd approach and maintaining perfect orthonormality of the eigenvectors the new algorithms show no signs of error buildup

an updating algorithm for subspace tracking in certain signal processing applications it is required to compute the null space of a matrix whose rows are samples of a signal with p components the usual tool for doing this is the singular value decomposition however the singular value decomposition has the drawback that it requires o p sup 3 operations to recompute when a new sample arrives it is shown that a different decomposition called the urv decomposition is equally effective in exhibiting the null space and can be updated in o p sup 2 time the updating technique can be run on a linear array of p processors in o p time

blind adaptive multiuser detection in multipath cdma channels based on subspace tracking a blind adaptive technique for signal demodulation in multipath code division multiple access cdma communication channels is proposed this technique is based on signal subspace estimation the receiver employs a bank of linear filters decorrelating filters or linear mmse filters at the front end to mitigate the multiple access interference and the multipath interference a channel estimator is used to estimate the channel state for diversity combining it is shown that through the use of signal subspace estimation both the decorrelating filterbank and the linear mmse filterbank can be obtained blindly i e they can be estimated from the received signal with the prior knowledge of only the signature waveform of the desired user two forms of the subspace based linear filterbanks are developed and their equivalence in terms of the interference suppression capability is established these subspace based blind adaptive interference suppression techniques require at each symbol epoch the eigenvalues and the eigenvectors of an appropriate signal subspace which ran be obtained using computationally efficient sequential adaptive eigendecomposition subspace tracking algorithms moreover a blind adaptive method for estimating the channel state is developed which also produces the postcombining decision statistic as a byproduct

a note on the fundamental solution for the tricomi type equation in the hyperbolic domain abstract barros neto and gelfand duke math j 98 3 1999 465 duke math j 117 2 2003 561 constructed for the tricomi operator y x 2 y 2 on the plane the fundamental solutions with the supports in the regions related to the geometry of the characteristics of the tricomi operator we give for the tricomi type operator t 2 t m x a fundamental solution relative to an arbitrary point of r n 1 with the support in the region t 0 where the operator is hyperbolic our key observation is that the fundamental solution for the tricomi type operator can be written like an integral of the distributions generated by the fundamental solution of the cauchy problem for the wave equation the application of that fundamental solution to the l p l q estimate for the forced tricomi type equation is given as well

similarity solutions of f 3 x f xx f y d y d with applications to transonic aerodynamics of den se gases similarity solutions of the nonlinear small disturbance equation phi x 3 phi xx phi tilde y tilde y which describes a two dimensional near sonic potential flow of dense gases are studied the analysis is based on a coordinate transformation from the cartesian physical plane to the hodograph velocity plane at which the governing equation becomes linear the relations between the similarity parameters in these two planes are constructed the first order similarity solution of the velocity potential phi in the physical plane is described analytically by a parametric representation in terms of the hodograph similarity variable the new solutions are applied to the problem of a near sonic small disturbance flow of dense gases in the surroundings of two dimensional slender semi infinite bodies with x n generators where 2 7 le n 1 and x is the distance along the body axis in this problem the similarity term describes a symmetric flow about the body axis in the case when

fundamental solutions for the tricomi operator in this paper we explicitly calculate fundamental solutions for the tricomi operator relative to an arbitrary point in the plane and show that all such fundamental solutions originate from the hypergeometric function f 1 6 1 6 1 that is obtained when we look for homogeneous solutions to the reduced hyperbolic tricomi equation

stochastic homogenisation of free discontinuity problems in this paper we study the stochastic homogenisation of free discontinuity functionals assuming stationarity for the random volume and surface integrands we prove the existence of a homogenised random free discontinuity functional which is deterministic in the ergodic case moreover by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise subadditive ergodic theorem by akcoglou and krengel we characterise the limit volume and surface integrands in terms of asymptotic cell formulas

direct estimation of blind zero forcing equalizers based on second order statistics most existing zero forcing equalization algorithms rely either on higher than second order statistics or on partial or complete channel identification we describe methods for computing fractionally spaced zero forcing blind equalizers with arbitrary delay directly from second order statistics of the observations without channel identification we first develop a batch type algorithm then adaptive algorithms are obtained by linear prediction and gradient descent optimization our adaptive algorithms do not require channel order estimation nor rank estimation compared with other second order statistics based approaches ours do not require channel identification at all on the other hand compared with the cma type algorithms ours use only second order statistics thus no local convergence problem exists and faster convergence can be achieved simulations show that our algorithms outperform most typical existing algorithms

sequences of non local functionals which approximate free discontinuity problems we prove by means of a generalized version of the well known fundamental estimate that the limit in the sense of convergence of a wide class of families of non local integral functionals is a functional related to a free discontinuity problem

blind equalization and multiuser detection in dispersive cdma channels the problem of blind demodulation of multiuser information symbols in a high rate code division multiple access cdma network in the presence of both multiple access interference mai and intersymbol interference isi is considered the dispersive cdma channel is first cast into a multiple input multiple output mimo signal model framework by applying the theory of blind mimo channel identification and equalization it is then shown that under certain conditions the multiuser information symbols can be recovered without any prior knowledge of the channel or the users signature waveforms including the desired user s signature waveform although the algorithmic complexity of such an approach is prohibitively high however in practice the signature waveform of the user of interest is always available at the receiver it is shown that by incorporating this knowledge the impulse response of each user s dispersive channel can be identified using a subspace method it is further shown that based on the identified signal subspace parameters and the channel response two linear detectors that are capable of suppressing both mai and isi i e a zero forcing detector and a minimum mean square error mmse detector can be constructed in closed form at almost no extra computational cost data detection can then be furnished by applying these linear detectors obtained blindly to the received signal the major contribution of this paper is the development of these subspace based blind techniques for joint suppression of mai and isi in the dispersive cdma channels

widely linear versus linear blind multiuser detection with subspace based channel estimation finite sample size effects in a recent paper a s cacciapuoti et al ldquofinite sample performance analysis of widely linear multiuser receivers in ds cdma systems ieee transactions on signal processing vol 56 no 4 pp 1572 1588 apr 2008 we presented the finite sample theoretical performance comparison between linear l and widely linear wl minimum output energy moe receivers for direct sequence code division multiple access ds cdma systems worked out under the assumption that the channel impulse response of the desired user is exactly known the main scope of this paper is to extend such an analysis taking into account not only autocorrelation matrix acm estimation effects but also the accuracy of subspace based blind channel estimation ce we aim to answer the two following questions which of the two estimation processes acm or ce is the main source of degradation when implementing the receivers on the basis of a finite sample size compared with the l moe one is the finite sample wl moe receiver with blind ce capable of achieving the performance gains predicted by the theory to this goal simple and easily interpretable formulas are developed for the signal to interference plus noise ratio sinr at the output of the l and wl moe receivers with blind ce when they are implemented using either the sample acm or its eigendecomposition in addition the derived formulas which are validated by simulations allow one to recognize and discuss interesting tradeoffs between the main parameters of the ds cdma system

prediction error method for second order blind identification blind channel identification methods based on the oversampled channel output are a problem of current theoretical and practical interest in this paper we introduce a second order blind identification technique based on a linear prediction approach in contrast to eigenstructure based methods it will be shown that the linear prediction error method is robust to order overdetermination an asymptotic performance analysis of the proposed estimation method is carried out consistency and asymptotic normality of the estimates is established a closed form expression for the asymptotic covariance of the estimates is given numerical simulations and investigations are finally presented to demonstrate the potential and the robustness of the proposed method

a subspace algorithm for certain blind identification problems the problem of blind identification of p inputs q outputs fir transfer functions is addressed existing subspace identification methods derived for p 1 are first reformulated in particular the links between the noise subspace of a certain covariance matrix of the output signals on which subspace methods build on and certain rational subspaces associated with the transfer function to be identified are elucidated based on these relations we study the behavior of the subspace method in the case where the order of the transfer function is overestimated next an asymptotic performance analysis of this estimation method is carried out consistency and asymptotical normality of the estimates is established a closed form expression for the asymptotic covariance of the estimates is given numerical simulations and investigations are presented to demonstrate the potential of the subspace method finally we take advantage of our new reformulation to discuss the extension of the subspace method to the case p 1 we show where the difficulties lie and we briefly indicate how to solve the corresponding problems the possible connections with classical approaches for ma model estimations are also outlined

existence theorem for a minimum problem with free discontinuity set we study the variational problem where is an open set in n n 2g l q l 1 q o andh n 1 is the n 1 dimensional hausdorff measure

blind fractionally spaced equalization of noisy fir channels direct and adaptive solutions blind fractionally spaced equalizers reduce intersymbol interference using second order statistics without the need for training sequences methods for finding fir zero forcing blind equalizers directly from the observations are described and adaptive versions are developed in contrast most current methods require channel estimation as a first step to estimating the equalizer the direct methods can be zero forcing minimum mean square error or even minimum mean square error mmse within the class of zero forcing equalizers performance of the proposed methods and comparisons with existing approaches are shown for a variety of channels including an empirically measured digital microwave channel

noniterative subspace tracking a rank one spherical subspace update that is appropriate for subspace based methods like music and minimum norm is introduced this noniterative highly parallel numerically stabilized subspace update is closely related to rank one eigenstructure updating however a rank one subspace update involves less computation than simple rank one correlation accumulation moreover the frequency tracking capabilities of the noniterative subspace update are virtually identical to and in some case more robust than the more computationally expensive eigen based methods

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

two algorithms for fast approximate subspace tracking new fast algorithms are presented for tracking singular values singular vectors and the dimension of a signal subspace through an overlapping sequence of data matrices the basic algorithm is called fast approximate subspace tracking fast the algorithm is derived for the special case in which the matrix is changed by deleting the oldest column shifting the remaining columns to the left and adding a new column on the right a second algorithm fast2 is specified by modifying fast to trade reduced accuracy for higher speed the speed and accuracy are compared with the pl algorithm the past and pastd algorithms and the fst algorithm an extension to multicolumn updates for the fast algorithm is also discussed

a γ convergence approach to stability of unilateral minimality properties in fracture mechanics and applications we prove a stability result for a large class of unilateral minimality properties which arise naturally in the theory of crack propagation proposed by francfort marigo in 14 then we give an application to the quasistatic evolution of cracks in composite materials the main tool in the analysis is a convergence result for energies of the form

global solutions and self similar solutions of semilinear wave equation we prove existence uniqueness and regularity results for the global solutions of the semilinear wave equations in particular we show existence of regular self similar solutions also we build some finite energy asymptotically self similar solutions

an l p a priori estimate for the tricomi equation in the upper half space we establish an lp a priori estimate for the tricomi equation our main tool is mihlin s multiplier theorem combined with well known estimates of the newtonian potential 0 introduction the purpose of this paper is to establish an lp a priori estimate for the tricomi equation in the upper half space the tricomi equation arises in transonic gas dynamics and is a typical model equation of changing type it has been extensively investigated from the various viewpoints the tricomi equation can be interpreted as an elliptic equation which degenerates on the boundary which is our viewpoint in this paper we can formulate the dirichlet boundary value problem in the upper half space as follows 02 yacuz f in r 0 2 u 0 x x on mt where r y x y 0 x e rn and x is the laplacian in the x variable an l2 a priori estimate can be obtained very easily following the presentation in 8 we suppose that u e co rn and multiply the equation 0 1 by 92u then we integrate over r r to obtain 0 3 ir dy2x 2jj v dydx rn 1 o 9y2 rn 1 o a y 2irn1l vq dx j 1 f 3 dydx where vx stands for the gradient in x e r n this yields 0 4 2u f 1 yx 4y21 l2 rn dy l2 rn l2 rn m f 1 l2 rn 11vx01il2 rn 1 7 received by the editors december 30 1996 and in revised form february 10 1998 1991 mathematics subject classification primary 35j70 35b45

performance analysis of an adaptive algorithm for tracking dominant subspaces this paper provides a performance analysis of a least mean square lms dominant invariant subspace algorithm based on an unconstrained minimization problem this algorithm is a stochastic gradient algorithm driving the columns of a matrix w to an orthonormal basis of a dominant invariant subspace of a correlation matrix we consider the stochastic algorithm governing the evolution of ww sup h to the projection matrix onto this dominant invariant subspace and study its asymptotic distribution a closed form expression of its asymptotic covariance is given in the case of independent observations and is further analyzed to provide some insights into the behavior of this lms type algorithm in particular it is shown that even though the algorithm does not constrain w to have orthonormal columns there is deviation from orthonormality of the first order we also give a closed form expression of the asymptotic covariance of the doas estimated by the music algorithm applied to the adaptive estimate of the projector it is found that the asymptotic distributions have a structure that is very similar to those describing batch estimation techniques because both algorithms are obtained from the minimization of the same criterion finally the accuracy of the asymptotic analysis is checked by numerical simulations and is found to be valid not only for a small step size but in a very large domain

approximation of functional depending on jumps by elliptic functional via t convergence we show how it is possible to approximate the mumford shah see 29 image segmentation functional by elliptic functionals defined on sobolev spaces the heuristic idea is to consider functionals h u z with z ranging between 0 and 1 and related to the set k the minimizing zh are near to 1 in a neighborhood of the set k and far from the neighborhood they are very small the neighborhood shrinks as h for a similar approach to the problem compare kulkarni see 25 the approximation of h to takes place in a variational sense the de giorgi f convergence

on updating signal subspaces the authors develop an algorithm for adaptively estimating the noise subspace of a data matrix as is required in signal processing applications employing the signal subspace approach the noise subspace is estimated using a rank revealing qr factorization instead of the more expensive singular value or eigenvalue decompositions using incremental condition estimation to monitor the smallest singular values of triangular matrices the authors can update the rank revealing triangular factorization inexpensively when new rows are added and old rows are deleted experiments demonstrate that the new approach usually requires o n sup 2 work to update an n n matrix and that it accurately tracks the noise subspace

on estimating the condition of eigenvalues and eigenvectors abstract a method is developed for estimating the accuracy of computed eigenvalues and eigenvectors that are obtained via certain eispack subroutines it does this at a cost of o n 2 flops per eigenpair assuming that the eigenpair is known and assuming that the original matrix has been reduced to hessenberg form the heart of the technique involves estimating the smallest singular value of a certain nearly triangular submatrix this is accomplished by some standard zero chasing with givens transformations and with a 2 norm version of the linpack condition estimator an eispack compatible code has been developed and its performance is discussed

blind channel identification subspace tracking method without rank estimation subspace ss methods are an effective approach for blind channel identification however these methods also have two major disadvantages 1 they require accurate channel length estimation and or rank estimation of the correlation matrix which is difficult with noisy channels and 2 they require a large amount of computation for the singular value decomposition svd which makes it inconvenient for adaptive implementation although many adaptive subspace tracking algorithms can be applied the computational complexity is still o m sup 3 where m is the data vector length in this paper we introduce new recursive subspace algorithms using ulv updating and successive cancellation techniques the new algorithms do not need to estimate the rank of the correlation matrix furthermore the channel length can be overestimated initially and be recovered at the end by a successive cancellation procedure which leads to more convenient implementations the adaptive algorithm has computations of o m sup 2 in each recursion the new methods can be applied to either the single user or the multiuser cases simulations demonstrate their good performance

spectral bounds for tricomi problems and application to semilinear existence and existence with uniqueness results abstract for the linear tricomi problem it is shown that real eigenvalues corresponding to generalized eigenfunctions must be positive and that the energy integral methods used to prove solvability results can give lower bounds on the spectrum exploiting the linear solvability theory and spectral information standard nonlinear analysis tools are employed to yield results on existence and uniqueness for semilinear problems in particular using the leray schauder principle existence of generalized solutions with sublinear nonlinearities is established for sublinear or asymptotically linear nonlinearities that satisfy a lipschitz condition the contraction mapping principle is employed to give results on existence with uniqueness the lipschitz constant depends on lower bounds for the spectrum of the linear problem for certain superlinear problems maximum principles for the linear problem are used via the method of upper and lower solutions to give results on existence

interior regularity of the dirichlet problem for the tricomi equation for the tricomi equation with dirichlet boundary conditions we study the relationship between singularites at the boundary and singularities in the interior of a bounded planar region with smooth non characteristic boundary necessary and sufficient conditions for interior smoothness are stated in terms of microlocal regularity at the boundary and are proven via known microlocal propagation of singularities results along the generalized bicharacteristic flow in particular a trapped gliding ray phenomenon at parabolic boundary points is demonstrated under a sharp geometric hypothesis which provides a microlocal explanation for the possibility of having only isolated singularities at the boundary which is a question left open in the work of morawetz

novel similarity solutions of the sonic small disturbance equation with applications to airfoil transonic aerodynamics similarity solutions of the small disturbance equation for a two dimensional near sonic potential flow are studied the analysis uses basic similarity solutions of the problem in the hodograph plane the first order similarity solution of the velocity potential in the physical plane is described analytically by a parametric representation in terms of the hodograph similarity variable and function the second order similarity term is governed by a linear equation and is described analytically as a product of two functions one is the power of the basic hodograph similarity function and the other is a solution of the basic hodograph hypergeometric equation with a different constant the application of the present solutions to the far field approximation of a sonic flow about a thin airfoil results in a relation between frankl s special similarity parameter and the hodograph similarity variable the new solutions are also applied to the problem of a near sonic small disturbance flow in the surroundings of a

transonic flow around optimum critical airfoils optimum critical nonlifting airfoils that have the highest free stream mach number m infty for a given thickness ratio delta have recently been presented by schwendeman kropinski and cole z angew math phys 44 1993 pp 556 571 the asymptotic shape of these airfoils as m infty to 1 and delta to 0 consists of a sonic arc with a x 2 5 nose in the present paper the transonic potential flow around these asymptotic shapes is analyzed asymptotic expansions of the velocity potential function are constructed in terms of delta in an outer region around the airfoil and in an inner region near the nose the outer expansion consists of the transonic small disturbance theory where a leading edge singularity appears the inner expansion accounts for the flow around the nose where a stagnation point exists the matching between the expansions results in a boundary value problem in the inner region for the solution of a uniform sonic flow about a two dimensional x 2 5