poincare bendixson theory for leaves of codimension one the level of a local minimal set of a c2 codimension one foliation of a compact manifold is a nonnegative integer defined inductively level zero corresponding to the minimal sets in the usual sense each leaf of a local minimal set at level k is at level k the authors develop a theory of local minimal sets level and how leaves at level k asymptotically approach leaves at lower level this last generalizes the classical poincare bendixson theorem and provides information relating growth topological type and level e g if l is a totally proper leaf at level k then l has exactly polynomial growth of degree k and topological type k

surfaces et cohomologie bornee ce travail comprend deux parties dans la premiere nous etudions le genre minimum d une surface qui represente une classe d homologie donnee de degre 2 la deuxieme partie est consacree a une analyse du second groupe de cohomologie bornee d une surface

green s function for an unbounded biaxial medium in cylindrical coordinates the dyadic green s function for an unbounded biaxial medium is treated analytically in the fourier domain the green s function is initially expressed as a triple fourier integral which is next reduced to a double one by performing the integration over the longitudinal fourier variable a delta type source term is extracted which is dependent on the particular coordinate system

boolean functions projection operators and quantum error correcting codes this paper describes a fundamental correspondence between boolean functions and projection operators in hilbert space the correspondence is widely applicable and it is used in this paper to provide a common mathematical framework for the design of both additive and nonadditive quantum error correcting codes the new framework leads to the construction of a variety of codes including an infinite class of codes that extend the original 5 6 2 code found by rains it also extends to operator quantum error correcting codes

mellin transform methods applied to integral evaluation taylor series and asymptotic approximations mellin transform methods have been used to obtain taylor series or asymptotic approximations to definite integrals whose integrands consist of the product of two generalized hypergeometric functions this approach has proved extremely useful in many applied physics problems including the analysis of electromagnetic propagation in a turbulent medium the main result in this article is a generalization of the method to integrands containing the product of an arbitrary number of hypergeometric functions providing a very powerful and general technique for integral evaluation with many applications in physics

asymptotic expansion of a integral involving a bessel function abstract an asymptotic expansion is derived for a quadruple integral involving the bessel function j0 xy zw where each of the integration variables x y z and w belongs to the interval 0 1 and the large variable tends to infinity corresponding results are also obtained for similar integrals in two and three dimensions

nonbinary stabilizer codes over finite fields one formidable difficulty in quantum communication and computation is to protect information carrying quantum states against undesired interactions with the environment to address this difficulty many good quantum error correcting codes have been derived as binary stabilizer codes fault tolerant quantum computation prompted the study of nonbinary quantum codes but the theory of such codes is not as advanced as that of binary quantum codes this paper describes the basic theory of stabilizer codes over finite fields the relation between stabilizer codes and general quantum codes is clarified by introducing a galois theory for these objects a characterization of nonbinary stabilizer codes over fq in terms of classical codes over fq 2 is provided that generalizes the well known notion of additive codes over f4 of the binary case this paper also derives lower and upper bounds on the minimum distance of stabilizer codes gives several code constructions and derives numerous families of stabilizer codes including quantum hamming codes quadratic residue codes quantum melas codes quantum bose chaudhuri hocquenghem bch codes and quantum character codes the puncturing theory by rains is generalized to additive codes that are not necessarily pure bounds on the maximal length of maximum distance separable stabilizer codes are given a discussion of open problems concludes this paper

asymptotic expansions of integrals of two bessel functions via the generalized hypergeometric and meijer functions abstract asymptotic expansions of certain finite and infinite integrals involving products of two bessel functions of the first kind are obtained by using the generalized hypergeometric and meijer functions the bessel functions involved are of arbitrary generally different orders but of the same argument containing a parameter which tends to infinity these types of integrals arise in various contexts including wave scattering and crystallography and are of general mathematical interest being related to the riemann liouville and hankel integrals the results complete the asymptotic expansions derived previously by two different methods a straightforward approach and the mellin transform technique these asymptotic expansions supply practical algorithms for computing the integrals the leading terms explicitly provide valuable analytical insight into the high frequency behavior of the solutions to the wave scattering problems

properties of the dyadic green s function for an unbounded anisotropic medium radiation in an unbounded anisotropic medium is treated analytically by studying the dyadic green s function of the problem initially expressed as a triple fourier integral which is next reduced to a double one under certain conditions the existence of incoming waves is verified it is also found that exponentially decaying waves are possible in such media finally the existence of branch points in the remaining integrand function is investigated and appropriate branch cuts are proposed

field patterns of resonant noncircular closed loop arrays further analysis previous papers have introduced the continuous current model for resonant closed loop arrays of cylindrical dipoles and for a certain class of array shapes have determined the resulting field patterns analytically those analytical calculations involve several approximations in the present paper we first study the aforementioned model numerically for the case of elliptical closed loop arrays the results which involve fewer approximations than those of the previous papers but which refer to different array shapes imply that elliptical arrays have a bidirectional highly directive field and unlike conventional arrays have no sidelobes combining analytical and numerical methods we then investigate the relationship between the aforementioned continuous current model and what we call the discrete current model which in a certain sense is more realistic the results indicate that subject to suitable conditions the two models yield virtually identical results in our analytical investigations an important tool is the poisson summation formula for finite sums we have devoted a stand alone section to this formula which supplements recent unrelated publications in antenna theory

evaluating infinite integrals involving products of bessel functions of arbitrary order the difficulties involved with evaluating infinite integrals involving products of bessel functions are considered and a method for evaluating these integrals is outlined the method makes use of extrapolation on a sequence of partial sums and requires rewriting the product of bessel functions as the sum of two more well behaved functions numerical results are presented to demonstrate the efficiency of this method where it is shown to be significantly superior to standard infinite integration routines

enlargement of calderbank shor steane quantum codes it is shown that a classical error correcting code c n k d which contains its dual c sup spl perp spl sube c and which can be enlarged to c n k k 1 d can be converted into a quantum code of parameters n k k n min d 3d 2 this is a generalization of a previous construction it enables many new codes of good efficiency to be discovered examples based on classical bose chaudhuri hocquenghem bch codes are discussed

salvatore pincherle the pioneer of the mellin barnes integrals the 1888 paper by salvatore pincherle professor of mathematics at the university of bologna on generalized hypergeometric functions is revisited we point out the pioneering contribution of the italian mathematician towards the mellin barnes integrals based on the duality principle between linear differential equations and linear difference equation with rational coefficients by extending the original arguments used by pincherle we also show how to formally derive the linear differential equation and the mellin barnes integral representation of the meijer g functions

on an integral related to biaxially anisotropic media an integral arising in certain studies of theoretical electromagnetics is evaluated and its properties are discussed in some detail the integral has three integer two real and one complex parameter the integrand involves a product of bessel functions of different argument and order several generalizations are discussed

foliations with good geometry the goal of this article is to show that there is a large class of closed hyperbolic 3 manifolds admitting codimension one foliations with good large scale geometric properties we obtain results in two directions first there is an internal result a possibly singular foliation in a manifold is quasi isometric if when lifted to the universal cover distance along leaves is efficienit up to a bounded multiplicative distortion in measuring distance in the universal cover this means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight this is the best geometric behavior we previously proved that nonsingular codimension one foliations in closed hyperbolic 3 manifolds can never be quasi isometric in this article we produce a large class of singular quasi isometric codimension one foliations in closed hyperbolic 3 manifolds the foliations are stable and unstable foliations of pseudo anosov flows our second result is an external result relating nonsingular foliations in hyperbolic 3 manifolds with their limit sets in the universal cover that is showing that leaves in the universal cover have good asymptotic behavior let g be a reebless finite depth foliation in a closed hyperbolic 3 manifold then g is not quasi isometric but suppose that g is transverse to a quasigeodesic pseudo anosov flow with quasi isometric stable and unstable foliations which are given by the internal result we then show that the lifts of leaves of g to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis the main tools used to prove these results are first a link between geometric properties of stable unstable foliations of pseudo anosov flows and the topology of these foliations in the tuniversal cover and second a topological theory of the joint structure of the pseudo anosov foliations in the universal cover reebless codimension one foliations are extremely useful for understanding the topology of 3 manifolds for instance they imply that the manifold is irreducible ro its universal cover is homeomorphic to r3 pa leaves are 7rl injective no and transversals are never null homotopic no hence they reflect topological properties of the manifold as for which manifolds have reebless foliations gabai gal ga2 ga3 proved that any compact oriented irreducible 3 manifold with nonzero first betti number has many reebless finite depth foliations roughly a

surface wave and superdirectivity aspects of effective current for linear antennas previous papers have defined the concept of effective current for linear antennas as a method of postprocessing the unphysical solutions obtained by applying discretization galerkin methods to the usual integral equations that employ the approximate or reduced kernel such an unphysical solution can be thought of as an auxiliary rapidly oscillating current located on the axis of the linear antenna while the effective current is the magnetic field due to the oscillating current at a distance equal to the antenna radius multiplied by a factor equal to the antenna circumference in the present paper we focus on the rapid decrease of the aforementioned magnetic field and point out similarities to surface waves to the well known phenomenon of superdirectivity and to the method of auxiliary sources mas our study sheds light on the unusual nature of electromagnetic waves generated by currents with very rapid spatial variations subject to the condition that the discretization is fine we treat th

codeword stabilized quantum codes we present a unifying approach to quantum error correcting code design that encompasses additive stabilizer codes as well as all known examples of nonadditive codes with good parameters we use this framework to generate new codes with superior parameters to any previously known in particular we find 10 18 3 and 10 20 3 codes we also show how to construct encoding circuits for all codes within our framework

a frequency domain approach to eigenvalue based detection with diversity reception and spectrum estimation in this paper we investigate a frequency domain approach for eigenvalue based detection of a primary user based on equal gain combining egc and spectrum estimation with bartlett s method this paper considers two techniques for eigenvalue detection which are maximum eigenvalue detection med and the maximum minimum eigenvalue mme detector we exploit the eigenvalues that are associated with the hermitian form representation of bartlett s estimate to assess the performance of the aforementioned eigenvalue techniques in the frequency domain for each case we quantify the performance based on the probabilities of false alarm and missed detection over rayleigh and rician fading a bivariate mellin transform approach is employed to obtain the probability distribution function for the ratio of the extreme eigenvalues under each hypothesis all obtained formulas are validated via monte carlo simulations and the results give a clear insight into the performance of the investigated methods in frequency domain med outperforms both the mme detector and periodogram based energy detection even in a worst case scenario of noise uncertainty while the mme detector exhibits heavy tailed statistical characteristics and thus its receiver operating characteristics tend to stay on the line of no discrimination the performance of med is further enhanced by careful choice of combinations of the total length of the sensing frame and number of sub slots

codeword stabilized quantum codes algorithm and structure the codeword stabilized cws quantum code formalism presents a unifying approach to both additive and nonadditive quantum error correcting codes ieee trans inf theory 55 433 2009 this formalism reduces the problem of constructing such quantum codes to finding a binary classical code correcting an error pattern induced by a graph state finding such a classical code can be very difficult here we consider an algorithm which maps the search for cws codes to a problem of identifying maximum cliques in a graph while solving this problem is in general very hard we provide three structure theorems which reduce the search space specifying certain admissible and optimal n k d additive codes in particular we find that the re does not exist any 7 3 3 cws code though the linear programming bound does not rule it out the complexity of the cws search algorithm is compared with the contrasting method introduced by aggarwal and calderbank ieee trans inf theory 54 1700 2008

on functions defined by sums of products of bessel functions various functions defined as infinite series of products of bessel functions of the first kind are studied integral representations are obtained and then used to deduce asymptotic approximations although several methods have been investigated including power series expansions and integral transforms methods based on fourier series emerge as the most useful

a nonadditive quantum code every good quantum error correcting code discovered thus far such as those known as stabilizer or additive codes has had the structure of an eigenspace of an abelian group generated by tensor products of pauli matrices in this letter we present the first example of a code that is better than any code of this type it encodes six states in five qubits and can correct the erasure of any single qubit

quantum codes of minimum distance two it is reasonable to expect the theory of quantum codes to be simplified in the case of codes of minimum distance 2 thus it makes sense to examine such codes in the hopes that techniques that prove effective there will generalize with this in mind we present a number of results on codes of minimum distance 2 we first compute the linear programming bound on the dimension of such a code then show that this bound can only be attained when the code either is of even length or is of length 3 or 5 we next consider questions of uniqueness showing that the optimal code of length 2 or 1 is unique implying that the well known one qubit in five single error correcting code is unique and presenting nonadditive optimal codes of all greater even lengths finally we compute the full automorphism group of the more important distance 2 codes allowing us to determine the full automorphism group of any gf 4 linear code

on the use of nonsingular kernels in certain integral equations for thin wire circular loop antennas fundamental properties of certain popular integral equations for thin wire circular loop antennas with nonsingular kernels are studied the cornerstone of our study is a large asymptotic formula for the n th fourier coefficient of the nonsingular kernel four different methods of driving the circular loop are considered namely the delta function generator the finite gap generator a certain type of frill generator and the case of plane wave incidence the excitation model is crucial to our discussions since it determines the behavior of the solution convergence also discussed are associated difficulties that may arise when moment methods are applied to the aforementioned equations as well as a simple improvement to the fourier series method for determining the current in the case of the frill generator the main results herein closely parallel recent results for the case of the straight antenna and similarities and differences between the straight and circular cases are discussed

dehn surgery on knots in d dehn considered the following method for constructing 3 manifolds remove a solid torus neighborhood n k of some knot x in the 3 sphere s and sew it back differently in particular he showed that taking x to be the trefoil one could obtain infinitely many non simply connected homology spheres o in this way let mx s n k then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus omk that bounds a meridional disk in the re attached solid torus we denote the resulting closed oriented 3 manifold by mk and say that it is obtained by r dehn surgery on x more generally one can consider the manifolds ml obtained by r dehn surgery on a fc component link l xi u u kk in s where r r it turns out that every closed oriented 3 manifold can be constructed in this way wal lie thus a good understanding of dehn surgery might lead to progress on general questions about the structure of 3 manifolds starting with the case of knots it is natural to extend the context a little and consider the manifolds m r obtained by attaching a solid torus v to an arbitrary compact oriented irreducible every 2 sphere bounds a 3 ball 3 manifold m with dm an incompressible torus where r is the isotopy class slope on dm of the boundary of a meridional disk of v we say that m r is the result of r dehn filling on m an observed feature of this construction is that

comments on asymptotic expansion of a bessel function integral using hypergeometric functions by l j landau and n j luswili in a recent paper landau and luswili j comput appl math 132 2001 387 used generalized hypergeometric functions to obtain a complete asymptotic expansion for the integral 0 2j sin jv sin d where j is the th order bessel function of the first kind and is a large parameter tending to infinity the purpose of this note is to point out that the same complete asymptotic expansion for this integral as well as another one for a hankel type integral has previously been obtained by stoyanov et al j comput appl math 50 1994 533 by using the same method in addition an alternative simpler representation of the algebraic series contribution to the asymptotic expansion is provided a few errors are also corrected and additional relevant references indicated

original and modified kernels in method of moments analyses of resonant circular arrays of cylindrical dipoles properly dimensioned circular arrays of cylindrical dipoles are known to possess very narrow resonances it is also known that analyzing such arrays using moment methods presents unique and particular difficulties as application of such methods to the usual hallen type integral equations can yield meaningless results from which no further conclusions should be drawn in the present paper we apply moment methods to properly modified integral equations and obtain much more reliable results we also observe that certain difficulties still remain and discuss them in detail

formulas for the force dipole interaction of surface line defects in homoepitaxy using asymptotics we derive explicit simplified formulas for integrals representing the force dipole interaction energy per unit length between line defects steps of the same sign that form perturbations of circles in homoepitaxy our starting point is continuum linear elasticity in accordance with the classic model by marchenko and parshin 1981 sov phys jetp 52 129 in the case with concentric circular steps we define a small geometric parameter 2 which expresses the smallness of interstep distance relative to the circle radii we invoke the mellin transform with respect to 2 and derive systematically an approximation for the requisite integral this technique offers an alternative to an exact evaluation in terms of elliptic integrals we then demonstrate the use of the mellin transform when calculating the force dipole interaction energy between smoothly slowly varying steps that form perturbations of circles we discuss the implications of our results for small amplitude modulations of circular step profiles

integral evaluation using the mellin transform and generalized hypergeometric functions tutorial and applications to antenna problems this is a tutorial presentation of the mellin transform mt method for the exact calculation of one dimensional definite integrals and an illustration of the application of this method to antenna electromagnetics problems once the basics have been mastered one quickly realizes that the mt method is extremely powerful often yielding closed form expressions very difficult to come up with other methods or to deduce from the usual tables of integrals yet as opposed to other methods the mt method is very straightforward to apply it usually requires laborious calculations but little ingenuity in fact the mt method is used by mathematica to symbolically calculate definite integrals the first part of this paper is a step by step tutorial proceeding from first principles it includes basic information on mellin barnes integrals and generalized hypergeometric functions and summarizes the key ideas of the mt method in the remaining parts the mt method is applied to three examples from the antenna area the results here are believed to be new at least in the antenna electromagnetics literature in our first example we obtain a closed form expression as a generalized hypergeometric function for the power radiated by a constant current circular loop antenna this quantity has been extensively discussed recently our second example concerns the admittance of a 2 d slot antenna in both these examples the exact closed form expressions are applied to improve upon existing formulas in standard antenna textbooks in our third example finally we obtain a very simple expression for an integral arising in recent unpublished studies of unbounded biaxially anisotropic media

two dimensional highly directive currents on large circular loops properties of idealized two dimensional current distributions on circular loops are investigated analytically via the solution of a constrained optimization problem the directivity in the far field is maximized under a fixed c n t where n is the integral of the square of the current magnitude and t is the total radiated power c enters the ensuing fourier series for the current implicitly through a lagrange multiplier for non negative and large electrical radius ka the directivity and the current are evaluated approximately via combined use of the poisson summation formula and the mellin transform technique as a result a geometrical ray representation for the current is derived for the case of directivities that are slightly larger than that of the uniform distribution the analysis indicates certain advantages of large radiating structures for moderate values of the constraint c in the limit c of oseen s einstein needle radiation an asymptotic formula for the directivity is obtained pos

comment on two dimensional highly directive currents on large circular loops j math phys 41 6130 2000 margetis and fikioris apparently unaware of the earlier results available in the literature used an elaborate mellin transform technique to obtain an asymptotic approximation for a particular case of a bessel function integral here the earlier more general results are pointed out

highly directive current distributions general theory a theoretical scheme for studying the properties of localized monochromatic and highly directive classical current distributions in two and three dimensions is formulated and analyzed for continuous current distributions it is shown that maximizing the directivity d in the far field while constraining c5n t where n is the integral of the square of the magnitude of the current density and t is proportional to the total radiated power leads to a fredholm integral equation of the second kind for the optimum current this equation is a useful analytical tool for studying currents that produce optimum directivities above the directivity of a uniform distribution various consequences of the present formulation are examined analytically for essentially arbitrary geometries of the current carrying region in particular certain properties of the optimum directivity are derived and differences between the continuous and discrete cases are pointed out when c the directivity tends to infinity monotonically in accord with oseen s einstein needle radiation s1063 651x 98 02108 4

asymptotic expansion of a bessel function integral using hypergeometric functions abstract generalized hypergeometric functions are used to extend simplify and complete the analysis of stoyanov and farrel math comput 49 1987 275 279 and of wong math comput 50 1998 229 234 as well as putting their considerations within a wider framework the integral 0 2 sin a cos b j sin j sin d where j is the bessel function of order is analyzed in terms of generalized hypergeometric functions and a complete asymptotic expansion is given for 0 2 j sin j sin d

knot floer homology detects genus one fibred knots ozsv ath and szab o conjectured that knot floer homology detects fibred knots we propose a strategy to approach this conjecture based on contact topology and gabai s theory of sutured manifold decomposition we implement this strategy for genus one knots obtaining as a corollary that if rational surgery on a knot k gives the poincar e homology sphere sigma 2 3 5 then k is the left handed trefoil knot