the rem zeros in the complex temperature and magnetic field planes abstract the distribution of zeros of the partition function of the random energy model is described both for the complex temperature and magnetic field planes there are regions where these zeros become dense it is shown that for t t c a dense region of zeros reaches the real magnetic field axis as the thermodynamic limit is taken for a fixed sample this manifests itself as jumps in the equilibrium magnetization m whose position is correlated with the closest zeros

the probability distribution of the partition function of the random energy model the authors give the expression for both integer and non integer moments of the partition function z of the random energy model in the thermodynamic limit they find that the probability distribution p z can be decomposed into two parts for log z log z finite the distribution is independent of n the size of the system whereas for log z log z positive and of order n the distribution is gaussian these two parts match in the region 1 log z log z n where the distribution is exponential

logarithmic laws for compressible turbulent boundary layers dimensional similarity arguments proposed by millikan are used with the morkovin hypothesis to deduce logarithmic laws for compressible turbulent boundary layers as an alternative to the traditional van driest analysis it is shown that an overlap exists between the wall layer and the defect layer and this leads to logarithmic behavior in the overlap region the von karman constant is found to depend parametrically on the mach number based on the friction velocity the dimensionless total heat flux and the specific heat ratio even though it remains constant at approximately 0 41 for a freestream mach number range of 0 to 4 544 with adiabatic wall boundary conditions it rises sharply as the mach number increases significantly beyond 4 544 the intercept of the logarithmic law of the wall is found to depend on the mach number based on the friction velocity the dimensionless total heat flux the prandtl number evaluated at the wall and the specific heat ratio on the other hand the intercept of the logarithmic defect law is parametric in the pressure gradient parameter and all of the aforementioned dimensionless variables except the prandtl number a skin friction law is also deduced for compressible boundary layers the skin friction coefficientmore is shown to depend on the momentum thickness reynolds number the wall temperature ratio and all of the other parameters already mentioned 26 refs less

nonlinear equations involving m accretive operators abstract let x be a banach space and t an m accretive operator defined on a subset d t of x and taking values in 2x for the class of spaces whose bounded closed and convex subsets have the fixed point property for nonexpansive self mappings it is shown here that two boundary conditions which imply existence of zeroes for t appear to be equivalent this fact is then used to prove that if there exists x0 d t and a bounded open neighborhood u of x0 such that t x 0 for all x u d t then the open ball b 0 r is contained in the range of t

the law of the wall in turbulent flow the law of the wall for the inner part of a turbulent shear flow over a solid surface is one of the cornerstones of fluid dynamics and one of the very few pieces of turbulence theory whose results include a simple analytic function for the mean velocity distribution the logarithmic law various aspects of the law have recently been questioned and this paper is a summary of the present position although the law of the wall for velocity has apparently been confirmed by experiment well outside its original range the law of the wall for temperature seems to apply only to very simple flows since the two laws are derived by closely analogous arguments this throws suspicion on the law of the wall for velocity analysis of simulation data for all the reynolds stresses including the shear stress shows that law of the wall scaling fails spectacularly in the viscous wall region even when the logarithmic law is relatively well behaved virtually all turbulence models are calibrated to reproduce the law of the wall in simple flows and we discuss whether in practice or in principle their range of validity is larger than that of the law of the wall itself the present answer is that it is not so that when the law of the wall or the mixing length formula fails current reynolds averaged turbulence models are likely to fail too

a defect stream function law of the wall wake method for compressible turbulent boundary layers the application of the defect stream function to the solution of the two dimensional compressible boundary layer is examined a law of the wall law of the wake formulation for the inner part of the boundary layer is presented which greatly simplifies the computational task near the wall and eliminates the need for an eddy viscosity model in this region the eddy viscosity model in the outer region is arbitrary the modified crocco temperature velocity relationship is used as a simplification of the differential energy equation formulations for both equilibrium and nonequilibrium boundary layers are presented including a constrained zero order form which significantly reduces the computational workload while retaining the significant physics of the flow a formulation for primitive variables is also presented results are given for the constrained zero order and second order equilibrium formulations and are compared with experimental data a compressible wake function valid near the wall has been developed from the present results

lyapunov exponents of large sparse random matrices and the problem of directed polymers with complex random weights we present results on two different problems the lyapunov exponent of large sparse random matrices and the problem of polymers on a cayley tree with random complex weights we give an analytic expression for the largest lyapunov exponent of products of random sparse matrices with random elements located at random positions in the matrix this expression is obtained through an analogy with the problem of random directed polymers on a cayley tree i e in the mean field limit which itself can be solved using its relationship with random energy models rem and grem for the random polymer problem with complex weights we find that in addition to the high and the low temperature phases which were already known in the case of positive weights the mean field theory predicts a new phase phase iii which is dominated by interference effects

the zeroes of the partition function of the random energy model abstract the random energy model is one of the simplest disordered models containing some of the physics of spin glasses the zeroes of the partition function in the complex temperature plane are determined for this model they either lie on simple lines or are dense in some regions of the complex plane

statistical mechanics of conducting phase transitions the critical behavior of the effective conductivity of the random resistor network in zd near its percolation threshold is considered the network has bonds assigned the conductivities 1 and e 0 in the volume fractions p and 1 p motivated by the statistical mechanics of an ising ferromagnet at temperature t in a field h we introduce a partition function and free energy for the resistor network which establishes a direct correspondence between the two problems in particular we show that the free energies for the resistor network and the ising model both have the same type of integral representation which has the interpretation of the complex potential due to a charge distribution on 0 1 in the s 1 1 e plane for the resistor network and on the unit circle in the z exp 2 h plane for the ferromagnet based on this correspondence we develop a yang lee picture of the onset of nonanalytic behavior of the effective conductivity so that the percolation threshold p pc is characterized as a

morkovin hypothesis and the modeling of wall bounded compressible turbulent flows modelingofcompressiblewall boundedturbulente owsreliesonthehypothesisofmorkovin who suggestedthat compressibility effects on turbulence could be accounted for by the mean density variations alone this hypothesis has been shown to yield good results for the mean velocity and mean temperature e elds when the incompressible turbulence models are extended directly to calculate compressible turbulent boundary layers however its applicability for the turbulence e eld has been less closely scrutinized the reason is the lack of sufe ciently detailed compressible turbulence data for comparison such data are now becoming available therefore the purpose here is to assess the applicability of the morkovin hypothesis to the turbulence e eld using direct numerical simulation data of a supersonic e at plate boundary layer a near wall reynolds stress closure based on a quasi linear pressure strain model is used to calculatethis supersonic boundary layer e ow comparisons between calculations and direct numerical simulation data show that the morkovin hypothesis is just as applicable for the turbulence e eld and there is a dynamic similarity between the near wall turbulence e eld of an incompressible and a compressible wall bounded turbulent e ow in addition the validation of this model is reported for compressible e ow calculations covering a wide range of mach numbers with adiabatic and constant temperature wall boundary conditions these results show that the model yields good predictions of e at plate turbulent boundary layers up to a mach number of 10 31

ranges of perturbed maximal monotone and m accretive operators in banach spaces a more comprehensive and unified theory is developed for the solvability of the inclusions s c r a b ints c r a b where a x d d a 2y b x d d b y and s c x here x is a real banach space and y x or y x mainly a is either maximal monotone or maccretive and b is either pseudo monotone or compact cases are also considered where a has compact resolvents and b is continuous and bounded these results are then used to obtain more concrete sets in the ranges of sums of such operators a and b various results of browder calvert and gupta gupta gupta and hess and kartsatos are improved and or extended the methods involve the application of a basic result of browder concerning pseudo monotone perturbations of maximal monotone operators and the leray schauder degree theory

embedded function approach for turbulent flow prediction in conventional prediction methods for turbulent flow influence on surface properties very small mesh sizes and considerable computational effort is required to adequately resolve the intense velocity and temperature profile variations that occur in the wall layer region in this study an approach is described wherein an outer region numerical solution is smoothly matched to a set of embedded analytic profile functions in the near wall region these wall layer profile functions having been derived through consideration of the coherent structure of the time dependent near wall flow provide analytic expressions for the mean velocity and enthalpy profiles in the wall layer the concept is demonstrated through example applications to turbulent boundary layer flows in two dimensions the technique is shown to be very efficient and it is demonstrated that a reduction of approximately half the mesh points across the 2 d layer may be realized as compared to conventional methods with no degradation in accuracy

location of zeros in the complex temperature plane absence of lee yang theorem in the yang lee theory 1952 the occurrence of a phase transition at real physical values of the magnetic field is related to the behaviour of the density of zeros of the partition function in the complex magnetic field plane as shown by lee and yang these zeros fall on lines for a wide class of ferromagnetic models in extensions of the yang lee theory to the complex temperature plane it has often been assumed that the zeros of the partition function would again fall on lines rather than in areas though this happens to be true for the isotropic ising model the authors point out that there is in general no basis for this assumption they discuss the location of the zeros of the partition function of the anisotropic ising model in the complex temperature plane and show that they indeed always fall in areas

on the semi monotone operator theory and applications abstract in this paper eis a real banach space a e e e is a semi monotone operator we first study the variational inequality problem findu k such that a u u v u 0 wherek e is a closed convex subset then the operator equationa u u p and we also construct a degree theory for demi continuous semi monotone operators in reflexive banach spaces

skin friction and velocity profile family for compressible turbulent boundary layers the paper presents a general approach to constructing mean velocity profiles for compressible turbulent boundary layers with isothermal or adiabatic walls the theory is based on a density weighted transformation that allows the extension of the incompressible similarity laws of the wall to the compressible regions the velocity profile family is compared to a range of experimental data and excellent agreement is obtained a self consistent skin friction law which satisfies the proposed velocity profile family is derived and compared with the well known van driest ii theory for boundary layers in zero pressure gradient the results are found to be at least as good as those obtained by using the van driest ii transformation

defect stream function law of the wall wake method for turbulent boundary layers a law of the wall law of the wake formulation for the inner part of the boundary layer is presented that greatly simplifies the computational task near the wall and eliminates the need for an eddy viscosity model in this region the eddy viscosity model in the outer part of the boundary layer is arbitrary formulations for both equilibrium and nonequilibrium boundary layers are presented and results are compared with previous methods for equilibrium boundary layers

analyticity of the partition function of the random energy model we consider the random energy model with complex temperature for beta lying in a suitable set the function zn beta ezn beta is at the thermodynamic limit an analytic function of beta analytic properties of the high temperature free energy follow directly

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

numerical complex zeros of the random energy model abstract the distribution of zeros of the partition function of the random energy model rem is numerically estimated in order to do this we devise a discrete version of the model drem for which the partition function is a polynomial and whose roots can therefore be numerically obtained a continuous limit transforms drem into rem so the zeros of the latter can be obtained from those of the former

on the density of partition function temperature zeros the density of partition function temperature zeros for the two dimensional spatially anisotropic ising model in the absence of a magnetic field satisfies a linear homogeneous partial differential equation a generalised form of this equation for a non zero specific heat exponent can be derived by applying non linear scaling to the expression for the specific heat as an integral over the density of zeros

fixed point theory and nonlinear problems introduction among the most original and far reaching of the contributions made by henri poincare to mathematics was his introduction of the use of topological or qualitative methods in the study of nonlinear problems in analysis his starting point was the study of the differential equations of celestial mechanics and in particular of their periodic solutions his work on this topic began with his thesis in 1879 and was developed in detail in his great three volume work methodes nouvelles de la mechanique celeste which appeared in the early 1890s and summarized his many memoirs of the intervening period it continued until his memoir shortly before his death in 1912 in which he put forward the unproved fixed point result usually referred to as poincare s last geometric theorem the ideas introduced by poincare include the use of fixed point theorems the continuation method and the general concept of global analysis the writer s acquaintance with poincare s influence came through contact with solomon lefschetz and marston morse both of whom were very explicit as to the role of poincare as an initiator in this direction of mathematical development in 1934 in the foreword to his colloquium volume on the calculus of variations in the large morse had put this forward very forcefully in the first paragraph

aeroacoustics of t junction merging flow this paper reports a numerical study of the aeroacoustics of merging flow at t junction the primary focus is to elucidate the acoustic generation by the flow unsteadiness the study is conducted by performing direct aeroacoustic simulation approach which solves the unsteady compressible navier stokes equations and the perfect gas equation of state simultaneously using the conservation element and solution element method for practical flows the reynolds number based on duct width is usually quite high 105 in order to properly account for the effects of flow turbulence a large eddy simulation methodology together with a wall modeling derived from the classical logarithm wall law is adopted the numerical simulations are performed in two dimensions and the acoustic generation physics at different ratios of side branch to main duct flow velocities vr 0 5 0 67 1 0 2 0 are studied both the levels of unsteady interactions of merging flow structures and the efficiency of acoustic generation are obser

van driest transformation and compressible wall bounded flows the transformation validity question utilizing resulting data from direct numerical simulations dns of supersonic isothermal cold wall channel flow was investigated the dns results stood for a wide scope of parameter and were suitable for the purpose of examining the generality of van driest transformation the van driest law of the wall can be obtained from the inner layer similarity arguments it was demonstrated that the van driest transformation cannot be incorporated to collapse the sublayer and log layer velocity profiles simultaneously velocity and temperature predictions according to the preceding composite mixing length model were presented despite satisfactory congruity with the dns data the model must be perceived as an engineering guide and not as a rigorous analysis

ranges of densely defined generalized pseudomonotone perturbations of maximal monotone operators abstract let x be a real reflexive banach space and a x 2 x be maximal monotone let b x 2 x be quasibounded finitely continuous and generalized pseudomonotone with x d b where x is a dense subspace of x such that x d a let s x conditions are given under which s r a b and int s int r a b results of browder concerning everywhere defined continuous and bounded operators b are improved extensions of this theory are also given using the degree theory of the last two authors concerning densely defined perturbations of nonlinear maximal monotone operators which satisfy a generalized s condition applications of this extended theory are given involving nonlinear parabolic problems on cylindrical domains

on the ranges of nonlinear set valued accretive operators in banach spaces abstract let e be a real banach space and a d a e 2 e b d b e 2 e be two set valued accretive operators we study the range problems of a i and a b i for 0 the results obtained in this paper generalize some other results

finite difference outer layer analytic inner layer method for turbulent boundary layers a new turbulent boundary layer method is developed which models the inner region with the law of the wall whereas the outer region uses clauser s eddy viscosity in matsuno s finite difference method the match point between the inner and outer regions as well as the wall shear stress is determined at each marching step during the computation results obtained for incompressible two dimensional flow over flat plates and ellipses are compared with solutions from a baseline method that uses a finite difference method for the entire boundary layer since the present method used the finite difference method in the outer region only the number of grid points required was about half that needed for the baseline method accurate displacement and momentum thicknesses were predicted for all cases skin friction was predicted well for the flat plate but the accuracy decreased significantly for the ellipses adding a wake function to the law of the wall allows some of the pressure gradient effect to be taken into account thereby increasing the accuracy of the method

quantum chaotic dynamics and random polynomials we investigate the distribution of roots of polynomials of high degree with random coefficients which among others appear naturally in the context of quantum chaotic dynamics it is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane in order to further increase this tendency we study in detail the particular case of self inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle correlation functions of these roots are also computed analytically and compared to the correlations of eigenvalues of random matrices the problem of ergodicity of chaotic wavefunctions is also considered for that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space while these results are consistent with random matrix theory predictions they provide a new and different insight into the problem of quantum ergodicity special attention is devoted to the role of symmetries in the distribution of roots of random polynomials

complex random energy model zeros and fluctuations the partition function of the random energy model at inverse temperature beta is a sum of random exponentials mathcal z n beta sum k 1 n exp beta sqrt n x k where x 1 x 2 ldots are independent real standard normal random variables random energies and n log n we study the large n limit of the partition function viewed as an analytic function of the complex variable beta we identify the asymptotic structure of complex zeros of the partition function confirming and extending predictions made in the theoretical physics literature we prove limit theorems for the random partition function at complex beta both on the logarithmic scale and on the level of limiting distributions our results cover also the case of the sums of independent identically distributed random exponentials with any given correlations between the real and imaginary parts of the random exponent

mean field theory of directed polymers with random complex weights we show that for the problem of directed polymers on a tree with i i d random complex weights on each bond three possible phases can exist the phase of a particular system is determined by the distribution of the random weights for each of these three phases we give the expression of the free energy per unit length in the limit of infinitely long polymers our proofs require several hypotheses on the distribution most importantly that the amplitude and the phase of each complex weight be statistically independent the main steps of our proofs use bounds on noninteger moments of the partition function and self averaging properties of the free energy we illustrate our results by some examples and discuss possible generalizations to a larger class of distributions to random energy models and to the finite dimensional case we note that our results are not in agreement with the predictions of a recent replica approach to a similar problem

on the connection between the existence of zeros and the asymptotic behavior of resolvents of maximal monotone operators in reflexive banach spaces a more systematic approach is introduced in the theory of zeros of maximal monotone operators t x d d t 2x where x is a real banach space a basic pair of necessary and sufficient boundary conditions is given for the existence of a zero of such an operator t these conditions are then shown to be equivalent to a certain asymptotic behavior of the resolvents or the yosida resolvents of t furthermore several interesting corollaries are given a nd the extendability of the necessary and sufficient conditions to the existence of zeros of locally defined demicontinuous monotone mappings is demonstrated a result of guan about a pathwise connected set lying in the range of a morlotone operator is improved by including non convex domains a partial answer to nirenberg s problem is also given namely it is shown that a continzuous expansive mapping t on a real hilbert space 1t is surjective if there exists a constant ae g 0 1 such that ktx ty x y cax y 112 x y e h the methods for these results do not involve explicit use of any degree theory

the generalized degree for compact perturbations of m accretive operators and applications soit x un espace de banach reel ou complexe a un operateur m accretif a une seule valeur ou multivalue de domaine d a et de portee r a dans x et c un operateur compact sur x on definit une fonction degre generalisee pour a c on demontre que cette fonction a les proprietes essentielles du degre topologique

mapping theorems involving compact perturbations and compact resolvents of nonlinear operators in banach spaces where it is assumed that the resolvent l t l n l i ii 1 2 exists and is defined at least on the range of the operator f c such a class of operators t includes all m accretive operators cf kato 5 i the equations g can be solved by a degree theory argument if we assume among other things that l is compact and c is continuous and bounded or that l is continuous and c compact in most of our results we do not explicitly assume the strong continuity of the operator t due to thi s fact our theorems complement and extend various results of browder 2 petryshyn 6 81 petryshyn and tucker 9 as well as kartsatos 4 and ward 9 0 ur results are particularly related to theorem 2 of petryshyn 7 where operators t are studied with d t x and range a banach space y however these spaces are assumed to possess

remarks on perturbation theory of maximal monotone and m accretive operators in banach spaces in this paper we get and improve some results in the perturbation theory of maximal monotone and m accretive operators having compact resolvents in banach spaces in which the composition of resolvent and perturbation is assumed compact

constrained topological degree and positive solutions of fully nonlinear boundary value problems abstract in the first part of the paper we provide a construction of an abstract homotopy invariant detecting zeros of maps of the form a f where a d a e is a densely defined m accretive operator in a banach space e and f u e is a tangent field defined on an open subset u of a neighborhood retract m being invariant with respect to the resolvents of a the construction is performed under the assumption that resolvents of a are completely continuous in the second part we derive index formulae for isolated zeros and apply them to show the existence of nontrivial positive steady state solutions for a class of nonlinear reaction diffusion equations and equations with one dimensional p laplacian with possibly non positive perturbations as well as some controlled neumann like problems

reply by the authors to f motallebi m rightly pointed out the misprints in eqs 8 and 10 ref 1 in addition to these errors introduced in printing there are others that have been reported in the errata that appeared in aiaa journal vol 32 no 11 1993 p 2192 these errors do not change our results and conclusions since our program and analysis are correct the horizontal axis of fig la is labeled correctly although in step 2 of the skin friction algorithm it is referred to as re 2 when discussing fig 1 we considered only incompressible flows and hence the parameter req was used however as we extended the algorithm from incompressible flows to compressible flows an important step was to change the parameter in the n correlation from rcq viscosity evaluated in the freestream to re 2 viscosity evaluated at the wall following fernholz and finley see the discussion following eq 8 this change was stated clearly in the beginning of sec ii we disagree with motallebi s statement that the algorithm is limited to hypersonic flows the current method can be considered as an extension of coles method which was proposed mainly for low speed flows for m 0 the method reproduces the karman schoenherr skin friction formula in the published paper we gave examples only for high mach number flows because this is where the model is most likely to break down supersonic flows are less severe test cases and the model works well figure 1 shows an example where the comparison of the present algorithm with the data of moore and harkness was made for a flow over an

new results in the perturbation theory of maximal monotone and accretive operators in banach spaces let x be a real banach space and g a bounded open and convex subset of x the solvability of the fixed point problem tx cx 3 x in d t n g is considered where t x d d t 2x is a possibly discontinuous m dissipative operator and c g x is completely continuous it is assumed that x is uniformly convex d t n g 0 and t c d t n og c g a result of browder concerning single valued operators t that are either uniformly continuous or continuous with x uniformly convex is extended to the present case browder s method cannot be applied in this setting even in the single valued case because there is no class of permissible homeomorphisms let if 7z r 3 r o 0 as r oo the effect of a weak boundary condition of the type u cx x 3 11xfl flxl12 on the range of operators t c is studied for m accretive and maximal monotone operators t here b e f x c d t with sufficiently large norm and u c tx various new eigenvalue results are given involving the solvability of tx acx 3 0 with respect to a x e 0 oo x d t several results do not require the continuity of the operator c four open problems are also given the solution of which would improve upon certain results of the paper

the closedness property and the pseudo a properness of accretive operators on demontre un theoreme sur la propriete de fermeture des operateurs accretifs sans les hypotheses de separabilite et de convexite uniforme de l espace de banach e

on 1 set contraction perturbations of accretive operators in cones of banach spaces abstract letpbe a cone in a banach spacee in this paper we show the existence of solutions of the operator equationy ax txfory p wheretis a 1 set contraction operator inpandais an accretive operator inpsatisfyingr i a pfor all 0 further a sufficient condition forr i a pis also given

on the solvability of abstract operator equations involving compact perturbations of m accretive operators on considere des equations de la forme tx cx f ou t est un operateur m accretif et c est compact

skin friction law for compressible turbulent flow an algebraic skin friction law is derived for adiabatic compressible equilibrium turbulent boundary layer flow an outer solution in terms of the clauser defect stream function is matched to an inner empirical expression composed of compressible laws of the wall and wake the modified crocco temperature velocity relationship and the clauser eddy viscousity model are used in the outer solution the skin friction law pertains for all pressure gradients in the incompressible through supersonic range and for small pressure gradients in the hypersonic range excellent comparisons with experiment are obtained in the appropriate parameter ranges the application to numerical computation is discussed

nonadiabatic and three dimensional effects in compressible turbulent boundary layers a defect stream function formulation for nonadiabatic flow with small crossflow is developed the first inte gral property of this formulation provides for the removal of the streamline curvature term in the governing equation so that the form of the reduced equation for small crossflow is the same as that for two dimensional flow the combined law of the wall and wake is used in place of the no slip boundary condition a three dimensional law of the wall formulation is used to develop a closed form solution for the crossflow angle near the wall analytic solutions for nonadiabatic compressible equilibrium flow with a clauser outer region eddy viscosity model are obtained and excellent agreement with experimental skin friction and velocity profile data for nonadiabatic compressible flat plate flow is achieved an analytic solution for a linear inner region eddy viscosity model is also obtained the wake function part of this solution is found to be inconsistent with the empirically established law of the wake all comparisons with experiment are for two dimensional equi librium flow ecently barnwell and wahls1 developed an analytic solution for adiabatic compressible turbulent bound ary layer flows for which the profile of the deficit in velocity from the edge value normalized with the shear stress velocity is invariant to displacement in the streamwise direction in the outer region such flows which are said to be in equilibrium were originally studied experimentally by clauser 2 3 flat plate flow is an example the equation that the analytic solution of ref 1 satisfies is an approximate form of the defect stream function equation for compressible boundary layer flow a principal advantage of the approximate equation is that it can be reduced to a second order differential equation by means of a first integral wahls4 and wahls et al 5 have shown with extensive numerical comparisons that solutions to the approx imate form of the defect stream function equation agree re markably well with solutions to the full equation form wall boundary conditions are imposed by joining the analytic solu tion for the outer region to an empirical expression for the inner region composed of the laws of the wall and wake a similar simpler approach was used by clauser 3 for incom pressible flat plate flow the application of the defect stream function formulation to the numerical solution of compressible flow is a generaliza tion of a similar treatment for incompressible flow6 that is based on the earlier analysis of mellor and gibson 7 the principal difference in the methods of refs 6 and 7 is the manner in which the surface boundary conditions are im posed rather than use an empirical analytic expression in the inner region mellor and gibson solved that region numeri cally using an inner region eddy viscosity model the resulting solutions were difficult to implement and deviated from the experimentally observed laws of the wall and wake but agreed closely with experiment on skin friction the method of ref 6 successfully overcame these disadvantages while maintaining the advantages the present paper presents results on three topics in turbu lent boundary layer analysis these topics include the law of the wall for three dimensional flow the solution of nonadia batic compressible flow and the effectiveness of linear eddy viscosity models in the inner region the boussinesq approximation and a three dimensio nal law of the wall concept are used to derive a closed form ex pression for the crossflow angle near the wall in addition the approximate second order defect layer governing equations for two dimensions and three dimensions with small crossflow are found to be the same the analytic solution of ref 1 for compressible turbulent equilibrium flow is extended to include nonadiabatic effects excellent agreement with experiment is demonstrated for sev eral nonadiabatic compressible flat plate flows the author has long been concerned about the redundancy and possible inconsistency of using both an eddy viscosity and the laws of the wall and wake in the inner region an analytic solution is obtained for a linear inner region eddy viscosity model this solution is inconsistent with the law of the wake and not completely compatible with the law of the wall

ishikawa iterative processes with mixed errors for nonlinear equations with perturbed m accretive mappings in banach spaces abstract the purpose of this paper is to study the convergence of the ishikawa and mann iterative sequences with mixed errors to approximate solutions of nonlinear operator equations with perturbed m accretive mappings in arbitrary banach spaces the results presented in this paper extend and improve some recent results in jung and morales nonlinear anal tma 46 2001 231 243 jung et al applied math comput 133 2002 389 406 and kartsatos and liu nonlinear anal tma 24 1995 469 492