adaptive hybrid grid methods an adaptive refinement and redistribution method is presented for use on hybrid grids composed of prismatic pyramidal and tetrahedral elements the combined refinement and redistribution scheme minimizes the computational time and memory required for the solver and maximizes the numerical accuracy grid refinement is achieved via a priori source placement and also solution based splitting of the edges of the grid elements the prismatic and pyramidal elements are divided directionally and the tetrahedral elements may be refined isotropically or directionally the prismatic grid redistribution scheme attracts or repels points that are close to the wall surface so as to better capture boundary layers a method is developed to place newly created boundary nodes on the actual splined surface instead of the previous discretized surface this ensures that the grid refinement always represents the geometric model accurately several examples are presented to demonstrate the a priori source placement the solution based grid refinement and the prismatic grid redistribution

non linear rough heat equations this article is devoted to define and solve an evolution equation of the form dyt yt dt dx t yt wherestands for the laplace operator on a space of the form l p r n and x is a finite dimensional noisy nonlinearity whose typical form is given by xt n i 1 x i fi where each x x 1 x n is a holder function generating a rough path and each fi is a smooth enough function defined on l p r n the generalization of the usual rough path theory allowing to cope with such

prismatic grid generation for three dimensional complex geometries the paper describes generation of semiunstructu red prismatic grids for three dimensional complex geometries a new method of generation is presented which couples both algebraic and elliptic steps this combination yields low computation cost as well as smoothness of the grid main advantages of the developed grid generator are its direct control of grid orthogonality and spacing as well as its generality for treatment of three dimensional geometries the method marches a surface consisting of triangular faces away from the body surface employment of prismatic elements is a relatively new approach toward three dimensional complex geometry numerical simulations of viscous flows a body surface is covered with triangles which gives the geometric flexibility whereas the structure of the mesh in the direction normal to the surface provides better resolution of the viscous regions an f 16a aircraft geometry was included in the applications to investigate efficiency and to demonstrate robustness of the method in handling relatively complex topologies grid generation time for the entire aircraft domain was of the order of minutes on a workstation

integration with respect to fractal functions and stochastic calculus i the classical lebesgue stieltjes integral b a fdg of real or complex valued functions on a finite interval a b is extended to a large class of integrands f and integrators g of unbounded variation the key is to use composition formulas and integration by part rules for fractional integrals and weyl derivatives in the special case of holder continuous functions f and g of summed order greater than 1 convergence of the corresponding riemann stieltjes sums is proved the results are applied to stochastic integrals where g is replaced by the wiener process and f by adapted as well as anticipating random functions in the anticipating case we work within slobodeckij spaces and introduce a stochastic integral for which the classical ito formula remains valid moreover this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional brownian motion

on perturbations of quasiperiodic schrodinger operators abstract using relative oscillation theory and the reducibility result of eliasson we study perturbations of quasiperiodic schrodinger operators in particular we derive relative oscillation criteria and eigenvalue asymptotics for critical potentials

almost periodic solutions of delay difference systems in the present paper we study the properties of almost periodic solutions for delay difference systems and establish some theorems on the existence of almost periodic solutions for delay difference systems by means of liapunov functionals the obtained results confirm not only the existence but also the uniqueness as well as the uniform asymptotic stability of the almost periodic solution

oscillation constants for second order ordinary differential equations related to elliptic equations with p laplacian abstract in this paper we consider the second order nonlinear differential equation t 1 x t 1 p f x 0 x x p 2 x p 1 r with f satisfying x f x 0 x 0 we analyze the difference between the cases p p and p in each case we give a condition on the function f which guarantees that solutions of eq are non oscillatory the principal methods used in this paper are the riccati technique and its modifications the results of our paper complement and extend several previously obtained results on the subject

hybrid prismatic tetrahedral grid generation for viscous flow applications a method for the automatic generation of unstructured grids composed of tetrahedra and prisms is proposed the prismatic semistructured grid is generated around viscous boundary surfaces and covers viscous regions whereas the tetrahedral grid covers the rest of the computational domain the delaunay approach for tetrahedral grid generation is used the proposed prismatic grid is structured in directions normal to the boundary faces but the number of prisms generated from one boundary face is variable from face to face unlike conventional prismatic grid generators this technique works well even in regions of cavities and gaps the delaunay background grid generated for surface nodes serves as an efficient data structure to check possible intersections of prisms particular attention is given to the boundary constraining problem a robust algorithm for the boundary recovery by edge swapping followed by a direct subdivision of tetrahedra is used grid examples for internal and external flow problems of complex shapes demonstrate the efficiency of the method

half linear equations and characteristic properties of the principal solution abstract the characteristic properties of the principal solution for half linear differential equation a t x b t x 0 where the functions a b are positive and continuous for t 0 and u u p 2u p 1 are investigated in the linear case it is well known that the principal solution is the smallest one in a neighbourhood of infinity we show that this property continues to hold in the half linear case in addition it is proved that the principal solutions can be fully characterized by means of two different integral criteria which reduce to that one well known in the linear case

perturbations of the half linear euler weber type differential equation abstract we investigate oscillatory properties of the half linear second order differential equation r t x c t x 0 x x p 2 x p 1 viewed as a perturbation of another half linear differential equation of the same form r t x c t x 0 the obtained oscillation and nonoscillation criteria are formulated in terms of the integral c t c t h p t d t where h is a function which is close to the principal solution of in a certain sense a typical model of in applications is the half linear euler weber differential equation with the critical coefficients x p t p p t p log 2 t x 0 p p 1 p p p 1 2 p 1 p p 1 we establish oscillation and nonoscillation criteria for perturbations of this equation some open problems and perspectives of the further research along this line are also formulated

effective prufer angles and relative oscillation criteria abstract we present a streamlined approach to relative oscillation criteria based on effective prufer angles adapted to the use at the edges of the essential spectrum based on this we provided a new scale of oscillation criteria for general sturm liouville operators which answer the question whether a perturbation inserts a finite or an infinite number of eigenvalues into an essential spectral gap as a special case we recover and generalize the gesztesy unal criterion which works below the spectrum and contains classical criteria by kneser hartman hille and weber and the well known results by rofe beketov including the extensions by schmidt

relative oscillation theory for sturm liouville operators extended we extend relative oscillation theory to the case of sturm liouville operators hu r 1 pu qu with different p s we show that the weighted number of zeros of wronskians of certain solutions equals the value of krein s spectral shift function inside essential spectral gaps

exponential separation exponential dichotomy and almost periodicity of linear difference equations abstract in this paper we study the relation between the concept of exponential separation and those of reducibility exponential dichotomy and almost periodicity of linear difference equations

nonoscillatory half linear differential equations and generalized karamata functions abstract we introduce a natural generalization of the concept of regularly varying functions in the sense of karamata and show that the class of generalized karamata functions is a well suited framework for the study of the asymptotic behavior of nonoscillatory solutions of the half linear differential equation of the type a p t y 1 y q t y 1 y 0

growth conditions for uniform asymptotic stability of damped oscillators abstract the present paper is devoted to an investigation on the uniform asymptotic stability for the linear differential equation with a damping term x h t x 2 x 0 and its generalization p x h t p x p p x 0 where 0 and p z z p 2 z with p 1 sufficient conditions are obtained for the equilibrium x x 0 0 to be uniformly asymptotically stable under the assumption that the damping coefficient h t is integrally positive the obtained condition for the damped linear differential equation is given by the form of a certain uniform growth condition on h t another representation which is equivalent to this uniform growth condition is also given our results assert that the equilibrium can be uniformly asymptotically stable even if h t is unbounded an example is attached to show this fact in addition easy to use conditions are given to guarantee that the uniform growth condition is satisfied moreover a sufficient condition expressed by an infinite series is presented the relation between the representation of an infinite series and the uniform growth condition is also clarified finally our results are extended to be able to apply to the above mentioned nonlinear differential equation

almost periodic and almost automorphic solutions of linear differential difference equations without favard s separation condition ii abstract the well known favard s theorem states that the linear differential equation 1 x a t x f t with bohr almost periodic coefficients admits at least one bohr almost periodic solution if it has a bounded solution the main assumption in this theorem is the separation among bounded solutions of homogeneous equations 2 x b t x where b h a b b t lim n a t t n if there are bounded solutions which are non separated sometimes almost periodic solutions do not exist r johnson r ortega and m tarallo v zhikov and b levitan in this paper we prove that linear differential equation 1 with levitan almost periodic coefficients has a unique levitan almost periodic solution if it has at least one bounded solution and the bounded solutions of the homogeneous equation 3 x a t x are homoclinic to zero i e lim t t 0 for all bounded solutions of 3 if the coefficients of 1 are bohr almost periodic and all bounded solutions of all limiting equations 2 are homoclinic to zero then eq 1 admits a unique almost automorphic solution the analogue of this result for difference equations is also given we study the problem of existence of bohr levitan almost periodic solutions of eq 1 in the framework of general non autonomous dynamical systems cocycles

the transformation operator for schrodinger operators on almost periodic infinite gap backgrounds abstract we investigate the kernels of the transformation operators for one dimensional schrodinger operators with potentials which are asymptotically close to bohr almost periodic infinite gap potentials

on the oscillation of hill s equations under periodic forcing abstract we use the floquet theory of the hill s equation to prove the conjecture that all solutions of the second order forced linear differential equation y c sin t y cos t are oscillatory on 0 for all c 0

hybrid prismatic tetrahedral grid generation for viscous flows around complex geometries a method for the generation of hybrid prismatic tetrahedral grids is described for complex three dimensional geometries including multibody domains and employed for viscous flows around an aircraft the prisms cover the region close to each body s surface while tetrahedra are created elsewhere a special method is presented that allows the generation of single block nonoverlapping prismatic meshes even in complex geometries that contain narrow gaps and cavities examples of cases treated by this method include wing engine configurations as well as multi element wings the second development is a combined octree advancing front method for the generation of the tetrahedra of the hybrid mesh the main feature of this octree based tetrahedra generator is that it does not require the creation of a background mesh by the user for the determination of the grid spacing and stretching parameters these are determined via an automatically generated octree the hybrid grid generator is robust geometry independent and requires no user intervention the suitability of hybrid meshes for capturing viscous flow phenomena is demonstrated by simulation of viscous flows around a high speed civil transport type of aircraft configuration reduction in computer resources has been substantial allowing flow simulations to be performed on workstations rather than supercomputers

nonoscillation of mathieu equations with two frequencies abstract as is well known mathieu s equation is a representative of mathematical models describing parametric excitation phenomena this paper deals with the oscillation problem for mathieu s equation with two frequencies the ratio of these two frequencies is not necessarily a rational number when the ratio is an irrational number the coefficient of mathieu s equation is quasi periodic but not periodic for this reason the basic knowledge for linear periodic systems such as floquet theory is not useful whether all solutions of mathieu s equation oscillate or not is determined by parameters and frequencies our results provide parametric conditions to guarantee that all solutions are nonoscillatory the advantage of the obtained parametric conditions is that it can be easily checked parametric nonoscillation region is drawn to understand these results easily finally several simulations are carried out to clarify the remaining problems

evolution equations driven by a fractional brownian motion abstract in this paper we study nonlinear stochastic evolution equations in a hilbert space driven by a cylindrical fractional brownian motion with hurst parameter h 1 2 and nuclear covariance operator we establish the existence and uniqueness of a mild solution under some regularity and boundedness conditions on the coefficients and for some values of the parameter h this result is applied to stochastic parabolic equation perturbed by a fractional white noise in this case if the coefficients are lipschitz continuous and bounded the existence and uniqueness of a solution holds if h d 4 the proofs of our results combine techniques of fractional calculus with semigroup estimates

rough path analysis via fractional calculus using fractional calculus we define integrals of the form b a f x t dy t where x and y are vector valued holder continuous functions of order p 1 3 1 2 and f is a continuously differentiable function such that f is a holder continuous for some a 1 2 under some further smooth conditions on f the integral is a continuous functional of x y and the tensor product x y with respect to the holder norms we derive some estimates for these integrals and we solve differential equations driven by the function y we discuss some applications to stochastic integrals and stochastic differential equations

on favard s theorems abstract in this paper we improve and extend the classical favard s theorems i e favard s theorem of the module containment favard s theorem of linear differential equations we study favard s theory of linear differential equations with piecewise constant argument an example shows that the new module containment is necessary in the study of differential equations with piecewise constant argument the equivalence between almost automorphic functions and n almost periodic ones is studied

zeros of the wronskian and renormalized oscillation theory for general sturm liouville operators with separated boundary conditions we prove the following if e 1 2 r and if u 1 2 solve the differential equation hu j e ju j j 1 2 and respectively satisfy the boundary condition on the left right then the dimension of the spectral projection p e 1 e 2 h of h equals the number of zeros of the wronskian of u 1 and u 2

almost periodic and almost automorphic solutions of linear differential equations the paper is dedicated to the study problem of existence of almost periodic respectively almost automorphic recurrent solutions of linear differential equations u t a t y u t f t y y y in banach space with almost periodic respectively almost automorphic recurrent coefficients where y r is a dynamical system on the metric space y in particular we prove that if the operator a y is positive and auto adjoint then for the equation 1 u t a t y u t 0 one of the following alternative is fulfilled i there exists a complete trajectory of 1 with constant positive norm ii the trivial solution of equation 1 is uniformly asymptotically stable this problem we investigate in the framework of general linear non autonomous dynamical system we apply our general results also for the functional differential equations and difference equations

relative oscillation theory for dirac operators we develop relative oscillation theory for one dimensional dirac operators which rather than measuring the spectrum of one single operator measures the difference between the spectra of two different operators this is done by replacing zeros of solutions of one operator by weighted zeros of wronskians of solutions of two different operators in particular we show that a sturm type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the number of eigenvalues in essential spectral gaps furthermore the connection with krein s spectral shift function is established as an application we extend a result by k m schmidt on the finiteness infiniteness of the number of eigenvalues in essential spectral gaps of perturbed periodic dirac operators

almost periodic transformable difference systems abstract we analyse almost periodic homogeneous linear difference systems whose coefficient matrices belong to the so called transformable groups we introduce the notion of weakly transformable groups and we use this notion to obtain generalizations of known results about non almost periodic solutions of considered systems

rough evolution equations we generalize lyons rough paths theory in order to give a pathwise meaning to some nonlinear infinite dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise as an illustration we discuss a class of linear and nonlinear 1d spdes driven by a space time gaussian noise with singular space covariance and brownian time dependence

controlling rough paths abstract we formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints this allows us to define a good notion of integral with respect to irregular paths with holder exponent greater than 1 3 e g samples of brownian motion and study the problem of the existence uniqueness and continuity of solution of differential equations driven by such paths we recover young s theory of integration and the main results of lyons theory of rough paths in holder topology

global existence and nonexistence of solutions for second order nonlinear differential equations abstract this paper deals with the global existence and nonexistence of solutions of the second order nonlinear differential equation x x 0 satisfying x 0 x 0 and x 0 x 1 where is a positive parameter and with 0 and 0 is strictly increasing odd bijective and continuous on necessary and sufficient conditions are obtained for the initial value problem to have a unique global solution which is oscillatory and periodic examples are given to illustrate our main result finally a nonexistence result for the equation with a damping term is discussed as an application to our result

even order half linear advanced differential equations we establish some new criteria for oscillation and asymptotic behavior of solutions of even order half linear advanced differential equations we study the case of canonical and the case of noncanonical equations subject to various conditions

oscillation criteria for second order nonlinear differential equations of euler type abstract the purpose of this paper is to solve the oscillation problem for the nonlinear euler differential equation t 2 x g x 0 and the extended equation x a t g x 0 here g x satisfies the sign condition xg x 0 if x 0 but is not assumed to be monotone we give necessary and sufficient conditions for all nontrivial solutions to be oscillatory to this end we use phase plane analysis of the lienard system and the oscillation result on the riemann weber version of the linear euler differential equation t 2 y 1 4 log t 2 y 0 our results are a negative answer to a conjecture which was given by wong finally we illustrate our results by two examples

local mild solutions for rough stochastic partial differential equations abstract we investigate mild solutions for stochastic evolution equations driven by a fractional brownian motion fbm with hurst parameter h 1 3 1 2 in infinite dimensional banach spaces using elements from rough paths theory we introduce an appropriate integral with respect to the fbm this allows us to solve pathwise our stochastic evolution equation in a suitable function space

overset adaptive cartesian prism grid method for stationary and moving boundary flow problems t he use of unstructured grids in computational fluid dynamics cfd has become widespread during the last two decades due to their ability to discretize arbitrarily complex geometries and the flexibility in supporting solution based grid adaptations to enhance the solution accuracy and efficiency 1 7 in the early days of unstructured grid development triangular tetrahedral grids were employed the most in dealing with complex geometries recently mixed or hybrid grids including many different cell types have gained popularity because of the improved efficiency and accuracy over pure tetrahedral grids for example hybrid prism tetrahedral grids 8 mixed grids including tetrahedral prism pyramid hexahedral cells 9 and adaptive cartesian grid methods 10 15 have been used in many applications with complex configurations the success demonstrated by unstructured grids for steady flow problems has prompted their applications to unsteady movingboundary flow problems a powerful approach for moving boundary flow problems is the overset chimera grid method 16 originally the chimera grid method was used to simplify domain decomposition for complex geometries using structured grids the method is particularly useful for moving boundary flow simulations because grid remeshing can be avoided 17 however frequent hole cutting and donor cell searching may be necessary to facilitate communications between the moving chimera grids with continuous improvement over the last decade and a half the chimera grid method has achieved tremendous success in handling very complex moving boundary flow problems more recently to further simplify the grid generation process unstructured grids are also used in achimera grid system for moving boundary flow computations making the approach even more flexible in handling complex geometries 18 in this technical note we advocate the use of an overset adaptive cartesian prism grid method for moving boundary flow computations themethod combines the advantage of the adaptive cartesian prism grid in geometry flexibility with that of the chimera approach in tacklingmoving boundary flowwithout grid remeshing there are several reasons why an adaptive cartesian grid is used for movingboundary problems 1 cartesian cells are more efficient at filling space given a certain length scale than triangular tetrahedral cells 2 searching operations can be performed very efficiently with the octree data structure 3 solution based and geometry based grid adaptations are straightforward to carry out the grid generation process is as follows body fitted prism grids are generated first near solid bodies to resolve viscous boundary layers using the advancing layer approach by marching the prism layer grid in the approximate surface normal directions the prism grid generation approaches and their limitations have been well researched in the literature in the past two decades for example in 6 8 18 19 and the current implementation follows similar ideas therefore interested readers should consult these references for prism grid generation methods and their limitations an adaptive cartesian grid is then generated to cover the outer domain and serve as the background grid for bridging the gaps between the prism grids the outer boundaries of the prism grids are used to generate holes in the adaptive cartesian grid to facilitate data communication if the bodies move the prism grids move with the bodies whereas the cartesian grid remains stationary after a few tens of time steps new holes are cut out of the cartesian grids and new donor cells are also identified solution fields are interpolated from the old cartesian grid to the new grid using a cellwise linear reconstruction technique this technical note is organized as follows in the next section the overset adaptive cartesian prism grid generation and holecutting approaches will be presented together with illustrative examples in sec iii several steady and unsteady moving boundary flow problems are computed grid refinement studies are performed to ensure the computational solutions are grid independent computational results are compared with experimental data and other simulations whenever possible finally conclusions from this study are summarized in sec iv

oscillation and non oscillation of euler type half linear differential equations abstract we investigate oscillatory properties of second order euler type half linear differential equations whose coefficients are given by periodic functions and functions having mean values we prove the conditional oscillation of these equations in addition we prove that the known oscillation constants for the corresponding equations with only periodic coefficients do not change in the studied more general case the presented results are new for linear equations as well

oscillation and nonoscillation of hill s equation with periodic damping we prove new results on the oscillation and nonoscillation of the hill s equation with periodic damping y p t y q t y 0 t 0 where p t and q t are continuous and periodic the results show that the equation y sint y cost y 0 is nonoscillatory whilst the equation y cost y sint y 0 is oscillatory

oscillation criterion for half linear differential equations with periodic coefficients abstract in this paper we present an oscillation criterion for second order half linear differential equations with periodic coefficients the method is based on the nonexistence of a proper solution of the related modified riccati equation our result can be regarded as an oscillatory counterpart to the nonoscillation criterion by sugie and matsumura 2008 these two theorems provide a complete half linear extension of the oscillation criterion of kwong and wong 2003 dealing with the hill s equation

modified prufer and efgp transforms and deterministic models with dense point spectrum we provide a new proof of the theorem of simon and zhu that in the region e for a e energies d 2 dx 2 cos x 0 1 has lyapunov behavior with a quasi classical formula for the lyapunov exponent we also prove lyapunov behavior for a e e 2 2 for the discrete model with v j 2 e j v n 0 if n 1 4 9 the arguments depend on a direct analysis of the equations for the norm of a solution

limit periodic homogeneous linear difference systems we study limit periodic homogeneous linear difference systems where the coefficient matrices belong to a bounded group we find groups of matrices with the property that the systems which do not possess any non zero asymptotically almost periodic solution form a dense subset in the space of all considered systems analogously we analyse almost periodic systems as well

eigenvalue excluding for perturbed periodic one dimensional schrodinger operators subject of investigation in this paper is a one dimensional schrodinger equation where the potential is a sum of a periodic function and a perturbation decaying at graphic 1 it is well known that the essential spectrum consists of spectral bands and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands while enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations e g by d weinstein s bounds there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps i e which identifies subregions of a gap which contain no eigenvalues here we propose such a method it makes heavy use of computer assistance nevertheless the results are completely rigorous in the strict mathematical sense because all computational errors are taken into account 1 embed inline graphic 1 gif

a theory of regularity structures we introduce a new notion of regularity structure that provides an algebraic framework allowing to describe functions and or distributions via a kind of jet or local taylor expansion around each point the main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose built for the problem at hand in particular this allows to describe the local behaviour not only of functions but also of large classes of distributions we then build a calculus allowing to perform the various operations multiplication composition with smooth functions integration against singular kernels necessary to formulate fixed point equations for a very large class of semilinear pdes driven by some very singular typically random input this allows for the first time to give a mathematically rigorous meaning to many interesting stochastic pdes arising in physics the theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems possibly modified by the addition of diverging counterterms these counterterms arise naturally through the action of a renormalisation group which is defined canonically in terms of the regularity structure associated to the given class of pdes our theory also allows to easily recover many existing results on singular stochastic pdes kpz equation stochastic quantisation equations burgers type equations and to understand them as particular instances of a unified framework one surprising insight is that in all of these instances local solutions are actually smooth in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions distributions that play the role of polynomials in the theory as an example of a novel application we solve the long standing problem of building a natural markov process that is symmetric with respect to the finite volume measure describing the phi 4 3 euclidean quantum field theory it is natural to conjecture that the markov process built in this way describes the glauber dynamic of 3 dimensional ferromagnets near their critical temperature

modified prufer angle and conditional oscillation of perturbed linear and half linear differential equations abstract the research and results described in this paper belong to the qualitative theory of differential equations more precisely the partial differential equations with the one dimensional p laplacian using a method whose core is formed by the prufer technique we identify a borderline case between oscillatory and non oscillatory equations moreover we are able to decide whether the studied equations are oscillatory or not even in the so called critical i e the borderline case the advantage of our approach is the fact that we obtain new and strong results for linear and half linear equations i e the equations with the one dimensional p laplacian at the same time in addition we are able to work with equations whose coefficients are non constant and non periodic the novelty of our results is documented by examples and corollaries

algebraic turbulence modeling for adaptive unstructured grids algebraic turbulence models are quite common in flow numerical simulations their simplicity and low computation cost as well as their applicability to specific classes of flows have made them attractive compared to the more expensive multiple equation models adaptive unstructured grid algorithms have become quite popular over the last years due to their flexibility in treating complex geometries as well their high resolution capability for local flow phenomena however application of algebraic turbulence models with unstructured grids proves to be nontrivial the difficulties associated with implementation of those models with such grids have generally impeded use of unstructured grids for turbulent flow simulations this paper presents a novel method of implementing an algebraic turbulence model with embedded unstructured grids the baldwinlomax model was chosen in order to test the implementation modifications to the application of the original model on structured grids are suggested as well comparisons with both experiments and other numerical simulations are employed in order to evaluate accuracy of the implementation method onsiderable progress has been made in the development of numerical methods for the solution of the navier stokes equations however most of those methods are impractical for complicated flows in a design environment the primary reason is poor efficiency which makes it difficult to obtain accurate results very fine resolution is needed which results in long computation times even with the use of available supercomputers in general the selection of the equations to be solved the scheme and the grid all are determined a priori by the user before starting the solution procedure and quite often some and even all of these factors must be modified by the user in order to improve the results the robustness of current numerical schemes as well as present computer capabilities has recently allowed a dramatic change in this philosophy general algorithms have been developed that are flexible enough to adaptively adjust those parameters during the solution procedure without intervention by the user one of the most efficient adaptive methods consists of local refinement by dividing grid cells an initially coarse grid is embedded in regions with large flow gradients e g boundary layers shocks wakes etc the algorithm senses high gradient regions and automatically divides grid cells in such regions this approach has been used for the resolution of shocks in flowfields described by euler equations 1 2 the embedded scheme was used as part of an adaptive algorithm for turbulent flows 3 6 algebraic turbulence models are quite common in aerodynamics numerical simulations their simplicity and low computation cost as well as their accuracy for specific classes of flows have made them attractive compared to the more expensive multiple equation models however such models require nonlocal information that characterizes an entire shearlayer profile the adaptive algorithm results in an unstructured mesh which is a major issue in the implementation of an algebraic turbulence model grid interfaces frequently interrupt normal to wall meshlines and therefore models that employ information from an entire shear layer profile cannot be applied similarly in the case of triangular meshes such normal grid lines generally do not exist a new method of

half linear oscillation criteria perturbation in term involving derivative abstract we consider the nonoscillatory half linear differential equation r t x c t x 0 x x p 2 x p 1 and we study oscillatory properties of its perturbation r t r t x c t c t x 0 we use the riccati technique and relationship between and a certain associated linear equation the results are applied to the perturbed euler type equation

global attractivity for half linear differential systems with periodic coefficients abstract the system to be considered in this paper is x y a t x y b t p x p y here a t is a 2 2 diagonal matrix and b t is a 2 2 anti diagonal matrix and q z z q 2 z with q 1 the coefficients of b t are assumed to be periodic but the coefficients of a t are not necessarily periodic the system is of half linear type sufficient conditions are given for the zero solution of the half linear system to be globally asymptotically stable the zero solution of the system x y b t p x p y is stable but is not attractive our concern is to clarify a positive effect of the diagonal part a t x y on the global asymptotic stability for the half linear system some simple examples are included to illustrate the main result

oscillation and non oscillation criteria for linear and half linear difference equations abstract we find very general classes of linear and half linear difference equations which are conditionally oscillatory we identify the critical oscillation constant whose value implies the oscillation or non oscillation of studied equations our results are divided into oscillatory and non oscillatory theorems which determine the critical oscillation constant for coefficients given by sequences having mean values in addition our approach enables to analyse also the oscillatory properties of equations whose coefficients are not given by sequences with mean values we point out that the obtained results are new even for linear equations with periodic coefficients such consequences are formulated at the end of this paper

a nonoscillation theorem for half linear differential equations with periodic coefficients abstract the half linear differential equation p x a t p x b t p x 0 is considered under the assumption that the coefficient a t and an indefinite integral b t of b t are periodic functions with period t 0 it is proved that p 1 p b t a t b t 0 0 t t is sufficient for all nontrivial solutions to be nonoscillatory here p 1 and q y y q 2 y for q p or q p p p 1 the proof is given by means of riccati technique the condition is shown to be sharp sufficient conditions are also presented for all nontrivial solutions are oscillatory in the linear case p 2 some examples and simulations are included to illustrate our results

variational technique and principal solution in half linear oscillation criteria we investigate oscillatory properties of half linear differential equations the investigated half linear equation is viewed as a perturbation of another nonoscillatory equation of the same form and perturbation is allowed both in the absolute term and in the derivative term first the sturmian type theorem for solutions of the associated riccati equation is established in the second part of the paper we prove new half linear oscillation criteria using the variational technique

critical coupling constants and eigenvalue asymptotics of perturbed periodic sturm liouville operators perturbations of asymptotic decay c r 2 arise in the partial wave analysis of rotationally symmetric partial differential operators we show that for each end point 0 of the spectral bands of a perturbed periodic sturm liouville operator there is a critical coupling constant c crit such that eigenvalues in the spectral gap accumulate at 0 if and only if c c crit 1 the oscillation theoretic method used in the proof also yields the asymptotic distribution of the eigenvalues near 0

simple conditions for parametrically excited oscillations of generalized mathieu equations abstract the following equation is considered in this paper x cos t x 0 where and are real parameters and 0 this equation is referred to as mathieu s equation when 2 the parameters determine whether all solutions of this equation are oscillatory or nonoscillatory our results provide parametric conditions for oscillation and nonoscillation there is a feature in which it is very easy to check whether these conditions are satisfied or not parametric oscillation and nonoscillation regions are drawn to help understand the obtained results

sample paths estimates for stochastic fast slow systems driven by fractional brownian motion we analyze the effect of additive fractional noise with hurst parameter h 1 2 on fast slow systems our strategy is based on sample paths estimates similar to the approach by berglund and gentz in the brownian motion case yet the setting of fractional brownian motion does not allow us to use the martingale methods from fast slow systems with brownian motion we thoroughly investigate the case where the deterministic system permits a uniformly hyperbolic stable slow manifold in this setting we provide a neighborhood tailored to the fast slow structure of the system that contains the process with high probability we prove this assertion by providing exponential error estimates on the probability that the system leaves this neighborhood we also illustrate our results in an example arising in climate modeling where time correlated noise processes have become of greater relevance recently

critical oscillation constant for euler type dynamic equations on time scales in this paper we study the second order dynamic equation on the time scale t of the form r t y d email protected t email protected t y s 0 wherer qare positive rd continuous periodic functions with inf r t email protected t 0 and c is an arbitrary real constant this equation corresponds to euler type differential resp euler type difference equation for continuous resp discrete case our aim is to prove that this equation is conditionally oscillatory i e there exists a constant c 0 such that studied equation is oscillatory for c c and non oscillatory for c c

hermitian symplectic geometry and extension theory here we give brief account of hermitian symplectic spaces showing that they are intimately connected to symmetric as well as self adjoint extensions of a symmetric operator furthermore we find an explicit parametrization of the lagrange grassmannian in terms of the unitary matrices this allows us to explicitly describe all self adjoint boundary conditions for the schrodinger operator on the graph in terms of a unitary matrix we show that the asymptotics of the scattering matrix can be expressed simply in terms of this unitary matrix

octree advancing front method for generation of unstructured surface and volume meshes amethodisdescribedforthegeneration ofunstructuredgridsforcomplexthree dimensionalgeometriesincluding multibodydomains acombinedoctree advancing frontmethod ispresentedforgenerationoftheunstructured surface mesh as well as the tetrahedral volume mesh the unique feature of this generator is that a special octree provides the local length scales of the mesh which can vary signi cantly this octree is automatically created and eliminates the need for a user constructed background mesh to determine the grid spacing and stretching parameters the grid generator isrobust isgeometry independent and requires minimal userinteraction a techniqueis also presented for generation of anisotropic surface meshes that result in a signi cant reduction in the number of triangular faces automatic local remeshing is also employed to improve the quality of the tetrahedral mesh the geometry independence of the octree advancing front method is demonstrated through both surface and volume

renormalized oscillation theory for hamiltonian systems abstract we extend a result on renormalized oscillation theory originally derived for sturm liouville and dirac type operators on arbitrary intervals in the context of scalar coefficients to the case of general hamiltonian systems with block matrix coefficients in particular this contains the cases of general sturm liouville and dirac type operators with block matrix valued coefficients as special cases the principal feature of these renormalized oscillation theory results consists in the fact that by replacing solutions by appropriate wronskians of solutions oscillation theory now applies to intervals in essential spectral gaps where traditional oscillation theory typically fails

curvature based wall boundary condition for the euler equations on unstructured grids a curvature corrected symmetry technique developed by dadone and grossman aiaa j vol 32 1994 for a structured grid euler solver has been extended to unstructured grids and to arbitrary curved boundaries in this paper the local curvature is estimated numerically with a robust procedure and is used successfully in the boundary treatment numerical results for 2d inviscid flows around circular and elliptic cylinders and the naca0012 airfoil with the new boundary condition showed dramatic improvements over those with a conventional non curvature corrected approach in all cases spurious entropy productions with the new boundary treatment are significantly reduced and sometimes by several orders of magnitude

relative oscillation theory weighted zeros of the wronskian and the spectral shift function we develop an analog of classical oscillation theory for sturm liouville operators which rather than measuring the spectrum of one single operator measures the difference between the spectra of two different operators

on certain riccati integral equations and second order linear oscillation are closely related is well known many important results in the oscillation theory of 1 1 are in fact established by studying either 1 2 or 1 3 recent studies suggest that a great deal can still be said about 1 2 and l 3 thus leading to new results on 1 1 see 7 91 particularly useful in these studies is the theory of differential and integral inequalities the present work further supports this view point in this paper we first give a comparison theorem section 2 theorem 1 for riccati integral equations of the form 1 3 involving monotonic rearrangements of the function q t ji q s ds as a corollary we establish an extension of the well known oscillation criteria of hille s 6 to more general potentials section 3 theorem 4 in section 4 we elaborate on a method of hartman to derive a comparison type result involving the square of the integral j h q s ds and then deduce a lyapunov type inequality for disfocality the preliminaries will be introduced in the appropriate sections 315 0022 247x 82 020315 16 02 00 o

oscillation problems for hill s equation with periodic damping abstract this paper deals with the second order linear differential equation x a t x b t x 0 where a and b are periodic coefficients the main purpose is to present new criteria which guarantee that all nontrivial solutions are nonoscillatory and that those are oscillatory our nonoscillation theorem and oscillation theorem are proved by using the riccati technique in our theorem the composite function of an indefinite integral of b and a suitable multiple valued continuously differentiable function are focused and the composite function of them plays an important role the results obtained here include a result by kwong and wong 15 and a result by sugie and matsumura 26 an application to a equation of whittaker hill type is given to show the usefulness of our results finally simulations are also attached to illustrate that our oscillation criterion is sharp

globally dynamical behaviors for a class of nonlinear functional difference equation with almost periodic coefficients abstract a new approach will be used to obtained the existence and uniqueness of positive almost periodic solution x n of the following nonlinear functional difference equation x n a n x n f n x n n also some sufficient conditions are established for global attractivity of x n for the proof of existence and uniqueness of x n the method used here is better than contraction mapping principle meanwhile when our main results applied to some famous discrete models lasota wazewska model and hematopoiesis model some new statements will be obtained and these results complement existing ones

scattering theory for schrodinger operators on steplike almost periodic infinite gap backgrounds abstract we develop direct scattering theory for one dimensional schrodinger operators with steplike potentials which are asymptotically close to different bohr almost periodic infinite gap potentials on different half axes

generalized prufer variables for perturbations of jacobi and cmv matrices abstract prufer variables are a standard tool in spectral theory developed originally for perturbations of the free schrodinger operator we adapt the generalized prufer variables introduced by kiselev remling simon for perturbations of an arbitrary schrodinger operator to the setting of jacobi and szeg recursions we present an application to random l 2 perturbations of jacobi and cmv matrices and an application to decaying oscillatory perturbations of periodic jacobi and cmv matrices

pseudo almost periodic solutions for equation with piecewise constant argument abstract by using the roughness theory of exponential dichotomies and the contraction mapping some sufficient conditions are obtained for the existence and uniqueness of pseudo almost periodic solution of the above differential equation with piecewise constant argument d x d t a t x t j 0 r a j t x t j g t x t x t x t r