numerical estimation of coercivity constants for boundary integral operators in acoustic scattering coercivity is an important concept for proving existence and uniqueness of solutions to variational problems in hilbert spaces but while coercivity estimates are well known for many variational problems arising from partial differential equations they are still an open problem in the context of boundary integral operators arising from acoustic scattering problems where rigorous coercivity results have so far only been established for combined integral operators on the unit circle and sphere the fact that coercivity holds even in these special cases is perhaps surprising as formulations of helmholtz problems are generally thought to be indefinite the main motivation for investigating coercivity in this context is that it has the potential to give error estimates for the galerkin method which are both explicit in the wavenumber k and valid regardless of the approximation space used thus they apply to hybrid asymptotic numerical methods recently developed for the high frequency case one way to interpret coercivity is by considering the numerical range of the operator the numerical range is a well established tool in spectral theory and algorithms exist to approximate the numerical range of finite dimensional matrices we can therefore use galerkin projections of the boundary integral operators to approximate the numerical range of the original operator we prove convergence estimates for the numerical range of galerkin projections of a general bounded linear operator on a hilbert space to justify this approach by computing the numerical range of the combined integral operator in acoustic scattering for several interesting convex nonconvex smooth and polygonal domains we numerically study coercivity estimates for varying wavenumbers we find that coercivity holds uniformly in the wavenumber k for a wide variety of domains finally we consider a trapping domain for which there exist resonances also called scattering poles very close to the real line to demonstrate that coercivity for a certain wavenumber k seems to be strongly dependent on the distance to the nearest resonance

analysis of multiple scattering iterations for high frequency scattering problems i the two dimensional case we present an analysis of a recently proposed integral equation method for the solution of high frequency electromagnetic and acoustic scattering problems that delivers error controllable solutions in frequency independent computational times within single scattering configurations the method is based on the use of an appropriate ansatz for the unknown surface densities and on suitable extensions of the method of stationary phase the extension to multiple scattering configurations in turn is attained through consideration of an iterative neumann series that successively accounts for further geometrical wave reflections as we show for a collection of two dimensional cylindrical convex obstacles this series can be rearranged into a sum of periodic orbits of increasing period each corresponding to contributions arising from waves that reflect off a fixed subset of scatterers when these are transversed sequentially in a periodic manner here we analyze the properties of these periodic orbits in the high frequency regime by deriving precise asymptotic expansions for the currents i e the normal derivative of the fields that they induce on the surface of the obstacles as we demonstrate these expansions can be used to provide accurate estimates of the rate at which their magnitude decreases with increasing number of reflections which defines the overall rate of convergence of the multiple scattering series moreover we show that the detailed asymptotic knowledge of these currents can be used to accelerate this convergence and thus to reduce the number of iterations necessary to attain a prescribed accuracy

optical orthogonal codes with unequal auto and cross correlation constraints an optical orthogonal code ooc is a collection of binary sequences with good auto and cross correlation properties they were defined by salehi and others as a means of obtaining code division multiple access on optical networks up to now all work on oocs have assumed that the constraint placed on the autocorrelation and that placed on the cross correlation are the same we consider codes for which the two constraints are not equal specifically we develop bounds on the size of such oocs and demonstrate constriction techniques for building them the results demonstrate that a significant increase in the code size is possible by letting the autocorrelation constraint exceed the cross correlation constraint these results suggest that for a given performance requirement the optimal ooc may be one with unequal constraints this paper also views oocs with unequal auto and cross correlation constraints as constant weight unequal error protection uep codes with two levels of protection the bounds derived are interpreted from this viewpoint

analysis of multiple scattering iterations for high frequency scattering problems ii the three dimensional scalar case in this paper we continue our analysis of the treatment of multiple scattering effects within a recently proposed methodology based on integral equations for the numerical solution of scattering problems at high frequencies in more detail here we extend the two dimensional results in part i of this work to fully three dimensional geometries as in the former case our concern here is the determination of the rate of convergence of the multiple scattering iterations for a collection of three dimensional convex obstacles that are inherent in the aforementioned high frequency schemes to this end we follow a similar strategy to that we devised in part i first we recast the iterated neumann multiple scattering series in the form of a sum of periodic orbits of increasing period corresponding to multiple reflections that periodically bounce off a series of scattering sub structures then we proceed to derive a high frequency recurrence that relates the normal derivatives of the fields induced on these structures as the waves reflect periodically and finally we analyze this recurrence to provide an explicit rate of convergence associated with each orbit while the procedure is analogous to its two dimensional counterpart the actual analysis is significantly more involved and perhaps more interestingly it uncovers new phenomena that cannot be distinguished in two dimensional configurations e g the further dependence of the convergence rate on the relative orientation of interacting structures as in the two dimensional case and beyond their intrinsic interest we also explain here how the results of our analysis can be used to accelerate the convergence of the multiple scattering series and thus to provide significant savings in computational times

the periodic boundary value problem for semilinear elastic beam equations the resonance case this paper discusses the existence of generalized solutions to periodic boundary value problems for semilinear elastic beam equations under a resonance condition the argument presented makes use of the global inverse theorem and galerkin s method

corner treatments for high order local absorbing boundary conditions in high frequency acoustic scattering this paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal computational domains for the finite element solution of high frequency acoustic scattering problems while high order absorbing boundary conditions habcs are accurate for smooth fictitious boundaries the precision of the solution drops in the presence of corners if no specific treatment is applied we present and analyze two strategies to preserve the accuracy of pade type habcs at corners first by using compatibility relations derived for right angle corners and second by regularizing the boundary at the corner we show that the former strategy is well adapted to right corners and efficient for nearly right corners while the later is better for very obtuse corners numerical results are proposed to analyze and compare the approaches for two and three dimensional problems

optical orthogonal codes new bounds and an optimal construction a technique for constructing optimal oocs optical orthogonal codes is presented it provides the only known family of optimal with respect to family size oocs having lambda 2 the parameters n omega lambda are respectively p sup 2m 1 p sup m 1 2 where p is any prime and the family size is p sup m 2 three distinct upper bounds on the size of an ooc are presented that for many values of the parameter set n omega lambda improve upon the tightest previously known bound

asymptotic bifurcation results for coupled nonlinear wave equations with variable coefficients abstract this paper is concerned with the bifurcation phenomenon of large amplitude periodic solutions of coupled nonlinear wave equations with either constant or variable coefficients such model arises when two waves propagate simultaneously in nonisotropic media the change of degree argument is used to obtain necessary and sufficient conditions for the existence of asymptotic bifurcation points with respect to the matrix spectrum the results we get are also valid for uncoupled systems by changing one of our assumptions on the nonlinear term we also obtain the necessary and sufficient conditions for the existence of the standard bifurcation points

frequency adapted galerkin boundary element methods for convex scattering problems we introduce a class of hybrid boundary element methods for the solution of sound soft scattering problems in the exterior of two dimensional smooth convex obstacles to facilitate the applicability of our algorithms throughout the entire frequency spectrum we have enriched our galerkin approximation spaces through incorporation of oscillations in the incident field of radiation into the algebraic and trigonometric polynomial approximation spaces the resulting methodologies have three distinctive properties indeed from a theoretical point of view 1 they can be tuned to demand only an mathcal o k varepsilon o k increase for any varepsilon 0 0 in the number of degrees of freedom to maintain a fixed accuracy with increasing wavenumberk owing to the optimal adaptation of approximation spaces to asymptotic stretching shrinking of illuminated and deep shadow regions shadow boundaries and 2 they are convergent for each fixed wavenumberk thanks to the additional approximation spaces in the deep shadow region perhaps more importantly from a practical point of view 3 they give rise to linear systems with significantly enhanced condition numbers and this in turn allows for more accurate solutions if desired

gauss jacobi quadratures for weakly strongly hyper and nearly singular integrals in boundary integral equation methods for domains with sharp edges and corners abstract we present gauss jacobi quadrature rules in terms of hypergeometric functions for the discretization of weakly singular strongly singular hypersingular and nearly singular integrals that arise in integral equation formulations of potential problems for domains with sharp edges and corners the rules are tailored to weight functions with algebraic endpoint singularities of a fairly general form thus allowing one to easily incorporate a wide class of domains into the analysis numerical examples illustrate the accuracy and stability of the proposed algorithms it is shown that the same level of high accuracy can be achieved for any choice of the external variable the usefulness of the method is exemplified by application to the solution of a singular integral equation that arises in time harmonic electromagnetic scattering by either closed or open perfectly conducting cylindrical objects with edges and corners such as polygon cylinders and bent strips some practical aspects concerning the role of nearby singularities in achieving a highly accurate solution of singular integral equations are also discussed

bifurcation problems for the laplacian in ⁿ in this paper we consider the bifurcation problem div ivulp 2vu ag x julp 2u f a x u in rn with p 1 we show that a continuum of positive solutions bifurcates out from the principal eigenvalue a1 of the problem div ivuip 2vu ag x ulp 2u here both functions f and g may change sign

acceleration of an iterative method for the evaluation of high frequency multiple scattering effects high frequency integral equation methodologies display the capability of reproducing single scattering returns in frequency independent computational times and employ a neumann series formulation to handle multiple scattering effects this requires the solution of an enormously large number of single scattering problems to attain a reasonable numerical accuracy in geometrically challenging configurations here we propose a novel and effective krylov subspace method suitable for the use of high frequency integral equation techniques that significantly accelerates the convergence of neumann series we additionally complement this strategy utilizing a preconditioner based upon kirchhoff approximations that provides a further reduction in the overall computational cost

computing fresnel integrals via modified trapezium rules in this paper we propose methods for computing fresnel integrals based on truncated trapezium rule approximations to integrals on the real line these trapezium rules modified to take into account poles of the integrand near the real axis our starting point is a method for computation of the error function of complex argument due to matta and reichel j math phys 34 298 307 1956 and hunter and regan math comp 26 539 541 1972 we construct approximations which we prove are exponentially convergent as a function of n n the number of quadrature points obtaining explicit error bounds which show that accuracies of 10 15 10 15 uniformly on the real line are achieved with n 12 n 12 this confirmed by computations the approximations we obtain are attractive additionally in that they maintain small relative errors for small and large argument are analytic on the real axis echoing the analyticity of the fresnel integrals and are straightforward to implement

code division multiple access techniques in optical fiber networks part iii optical and logic gate receiver structure with generalized optical orthogonal codes in this paper we present a deep insight into the behavior of optical code division multiple access cdma systems based on an incoherent intensity encoding decoding technique using a well known class of codes namely optical orthogonal codes oocs as opposed to parts i and ii of this paper where oocs with cross correlation lambda 1 were considered we consider generalized oocs with 1 leq lambda leq w where w is the weight of the corresponding codes to enhance the performance of such systems we propose that use of optical and gate receiver which in an ideal case e g in the absence of any noise source except the optical multiple access noise is optimum using some basic laws on probability we present direct and exact solutions for oocs with lambda 1 2 3 ldots w with optical and gate as receiver using the exact solution we obtain empirical solutions that can be easily used in optimizing the design criteria of such systems from our optimization scheme we obtain some fresh insight into the performance of oocs with lambdageq 1 in particular we can obtain some simple relations between p e min minimum error rate l min minimum required ooc length and n max maximum number of interfering users to be supported which are the most desired parameters for any optical cdma system design furthermore we show that in most practical cases oocs with lambda 2 or 3 perform better than oocs with lambda 1 while having a much bigger cardinality finally we show that an upper bound on the maximum weight of oocs are on the order of sqrt2lambda l where l is the length of the oocs used in systems

a high frequency boundary element method for scattering by a class of nonconvex obstacles in this paper we propose and analyse a hybrid numerical asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound soft nonconvex polygons the approximation space is enriched with carefully chosen oscillatory basis functions these are selected via a study of the high frequency asymptotic behaviour of the solution we demonstrate via a rigorous error analysis supported by numerical examples that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases in contrast to the at least linear growth required by conventional methods this appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle our analysis is based on new frequency explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane

existence and multiplicity results of positive doubly periodic solutions for nonlinear telegraph system abstract in this paper the existence of positive doubly periodic solutions for nonlinear telegraph system is discussed using the method of upper and lower solutions

wavenumber explicit hp bem for high frequency scattering for the helmholtz equation with wavenumber k and analytic curves or surfaces gamma we analyze the galerkin discretization of classical combined field integral equations in an l 2 setting we give abstract conditions on the approximation properties of the ansatz space that ensure stability and quasi optimality of the galerkin method special attention is paid to the hp version of the boundary element method hp bem under the assumption of polynomial growth of the solution operator we show stability and quasi optimality of the hp bem if the following scale resolution condition is satisfied the polynomial degree p is at least o log k and kh p is bounded by a number that is sufficiently small but independent of k under this assumption the constant in the quasi optimality estimate is independent of k numerical examples in two dimensions illustrate the theoretical results and even suggest that in many cases quasi optimality is given under the weaker condition that kh p is sufficiently small

optical orthogonal codes design analysis and applications an optical orthogonal code ooc is a family of 0 1 sequences with good auto and cross correlation properties i e the autocorrelation of each sequence exhibits the thumbtack shape and the cross correlation between any two sequences remains low throughout the use of optical orthogonal codes enables a large number of asynchronous users to transmit information efficiently and reliably the thumbtack shaped autocorrelation facilitates the detection of the desired signal and low profile cross correlation reduces interference from unwanted signals theoretical upper and lower bounds on the maximum possible size of oocs are derived methods for the design and analysis of oocs using tools from projective geometry the greedy algorithm iterative constructions algebraic coding theory block design and various other combinational disciplines are discussed

wavenumber explicit analysis for the helmholtz h bem error estimates and iteration counts for the dirichlet problem we consider solving the exterior dirichlet problem for the helmholtz equation with the h version of the boundary element method bem using the standard second kind combined field integral equations we prove a new sharp bound on how the number of gmres iterations must grow with the wavenumber k to have the error in the iterative solution bounded independently of k as k rightarrow infty when the boundary of the obstacle is analytic and has strictly positive curvature to our knowledge this result is the first ever sharp bound on how the number of gmres iterations depends on the wavenumber for an integral equation used to solve a scattering problem we also prove new bounds on how h must decrease with k to maintain k independent quasi optimality of the galerkin solutions as k rightarrow infty when the obstacle is nontrapping

global results concerning bifurcation for fredholm maps of index zero with a transversality condition on considere l equation f x ax bx r x 0 ou a b est la derivee de frechet de f a l origine dans x et r un terme d ordre eleve x est un espace de banach et f x r y est un operateur non lineaire continu et y un espace de banach on etudie comment l ensemble solution varie avec le parametre reel

on the bifurcation of large amplitude solutions for a system of wave and beam equations we consider the existence of solutions for linearly coupled system of wave and beam equation with a sublinear perturbation the main concept is the matrix spectrum which is a natural extension of the usual spectrum using the standard change of degree argument we shall find necessary and sufficient conditions for the existence of asymptotic bifurcation with respect to the matrix spectrum both in one parameter and four parameter cases obviously the results can be modified for more general systems of partial differential equations

schaefer type theorem and periodic solutions of evolution equations abstract two fixed point theorems for the sum of contractive and compact operators are obtained in this paper which generalize and improve the corresponding results in h schaefer uber die methode der a priori schranken math ann 129 1955 415 416 t a burton integral equations implicit functions and fixed points proc amer math soc 124 1996 2383 2390 v i istr tescu fixed point theory an introduction reidel dordrecht 1981 t a burton k colleen a fixed point theorem of krasnoselskii schaefer type math nachr 189 1998 23 31 d r smart fixed point theorems cambridge univ press cambridge 1980 as the applications for the results we obtain the existence of periodic solutions for some evolution equations with delay which extend the corresponding results in t a burton b zhang periodic solutions of abstract differential equations with infinite delay j differential equations 90 1991 357 396

nonnegative doubly periodic solutions for nonlinear telegraph system with twin parameters in this paper by using krasnosel skii fixed point theorem and under suitable conditions we present the existence and multiplicity of nonnegative doubly periodic solutions for the following system u t t u x x c 1u t a 1 1 t x u a 1 2 t x v b 1 t x f t x u v v t t v x x c 2v t a 2 1 t x u a 2 2 t x v b 2 t x g t x u v where c i 0 is a constant a 1 1 t x a 2 2 t x b 1 t x b 2 t x c r 2 r a 1 2 t x a 2 1 t x c r 2 r f t x u v g t x u v c r 2xr xr r and a i j b i f g are email protected in t and x we derive two explicit intervals of b 1 t x and b 2 t x such that for any b 1 t x and b 2 t x in the two intervals respectively the existence of at least one solution for the system is guaranteed and the existence of at least two solutions for b 1 t x and b 2 t x in appropriate intervals is also discussed

on generalized gaussian quadrature rules for singular and nearly singular integrals we construct and analyze generalized gaussian quadrature rules for integrands with endpoint singularities or near endpoint singularities the rules have quadrature points inside the interval of integration and the weights are all strictly positive such rules date back to the study of chebyshev sets but their use in applications has only recently been appreciated we provide error estimates and we show that the convergence rate is unaffected by the singularity of the integrand we characterize the quadrature rules in terms of two families of functions that share many properties with orthogonal polynomials but that are orthogonal with respect to a discrete scalar product that in most cases is not known a priori

a new family of optical code sequences for use in spread spectrum fiber optic local area networks the problem of the algebraic construction of a particular family of optical codes for use in code division multiple access cdma fiber optic local area networks lans is treated the conditions that the code families have to satisfy when used in such systems are reviewed the new codes are called quadratic congruence codes and the construction of the corresponding sequences is based on the number theoretic concept of quadratic congruences it is shown that p 1 codes exist for every odd prime p and can serve as many as p 1 different users in the cdma fiber optic system the codes belong to the family of optical orthogonal codes their auto and cross correlation properties are established and their performance is compared to that of the previous optical codes examples of the codes and examples of their auto and cross correlation functions are given

extended reed solomon codes for optical cdma in this paper the extended reed solomon codes are modified to construct a new family of 2 d codes for asynchronous optical code division multiple access o cdma in addition of having expanded and asymptotically optimal cardinality these 2 d asynchronous optical codes can be partitioned into multiple tree structures of code subsets in which code cardinality is a function of the periodic cross correlation value assigned to the subset the performance of these 2 d optical codes is analyzed and compared with that of the multilevel prime codes our results show that the unique partition property of the new optical codes supports a trade off between code cardinality and performance for meeting different system requirements such as user capacity and throughput in addition the multiple tree structures of the new codes potentially support applications that require rapid switching of many codewords such as in o cdma network gateway or in strategic environments where code obscurity is essential

multiple wavelength optical orthogonal codes under prime sequence permutations for optical cdma a new family of two dimensional 2 d wavelength hopping time spreading codes which employs wavelength hopping algebraically under prime sequence permutations on top of time spreading optical orthogonal codes is studied and analyzed different from other 2 d codes our new codes allow the number of wavelengths and code length to be chosen independently and at the same time the code cardinality is a quadratic function of the number of wavelengths without sacrificing the maximum cross correlation value i e still at most one they are particularly suitable for high bit rate optical code division multiple access systems with broadband mode locked lasers in which the number of time slots is very limited and system capacity can only be grown by increasing the number of wavelengths rather than code length finally a novel wavelength aware detector for wavelength hopping time spreading codes is discussed and shown to provide improved code performance

construction of optimal 2d optical codes using n w 2 2 optical orthogonal codes two dimensional 2d wavelength time coding schemes have recently been studied for optical code division multiple access in this paper we propose a new family of 2d codes which use synchronized quadratic congruence sequences for wavelength hopping and n w 2 2 optical orthogonal codes for time spreading the new 2d codes provide larger asymptotically optimal code cardinality than other 2d codes by relaxing the maximum cross correlation function from one to two our study shows that the heavier code weight supported by our 2d codes results in better code performance than some of previously proposed 2d codes of the maximum cross correlation functions of one and two the new codes potentially support more subscribers and heavier bursty asynchronous traffic with gradual performance degradation as traffic load increases

a high frequency hp boundary element method for scattering by convex polygons in this paper we propose and analyze a hybrid hp boundary element method for the solution of problems of high frequency acoustic scattering by sound soft convex polygons in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution we demonstrate both theoretically and via numerical examples exponential convergence with respect to the order of the polynomials moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern in which the dependence on the frequency of all constants is explicit importantly these estimates prove that to achieve any desired accuracy in the computation of these quantities it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases in contrast to the at least linear growth required by conventional methods

nonnegative doubly periodic solutions for nonlinear telegraph system abstract this paper deals with the nonnegative doubly periodic solutions for nonlinear telegraph system u t t u x x c 1 u t a 11 t x u a 12 t x v b 1 t x f t x u v v t t v x x c 2 v t a 21 t x u a 22 t x v b 2 t x g t x u v where c i 0 is a constant a 11 a 22 b 1 b 2 c r 2 r a 12 a 21 c r 2 r f g c r 2 r r r and a i j b i f g are 2 periodic in t and x we show the existence and multiplicity results when 0 a i i t x c i 2 4 and f g are superlinear or sublinear on u v by using the fixed point theorem in cones

bifurcations associated with a double zero and a pair of pure imaginary eigenvalues interactive static and dynamic bifurcations associated with a nonlinear autonomous system and the stability properties of various solutions are explored attention is focused on the vicinity of a compound critical point where the jacobian of the system exhibits a double zero eigenvalue of index one and a pair of pure imaginary eigenvalues the system under consideration has three independent parameters and depending on the route followed in the parameter space the system may exhibit static bifurcations hopf bifurcations secondary hopf bifurcations and bifurcations into two or three dimensional tori the general analysis is based on a perturbation method which combines an intrinsic harmonic balancing technique with a certain unification procedure this approach leads to bifurcation equations and simplified differential equations governing the local dynamics in the vicinity of the compound critical point a control system is analyzed for illustration

a galerkin boundary element method for high frequency scattering by convex polygons in this paper we consider the problem of time harmonic acoustic scattering in two dimensions by convex polygons standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave here we present a novel galerkin boundary element method which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon we prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency numerical results demonstrate the same logarithmic dependence on the frequency for the galerkin method solution our boundary element method is a discretization of a well known second kind combined layer potential integral equation we provide a proof that this equation and its adjoint are well posed and equivalent to the boundary value problem in a sobolev space setting for general lipschitz domains