the asymptotic form of the titchmarsh weyl m function associated with a second order differential equation with locally integrable coefficient on deduit un developpement asymptotique pour la m fonction de titchmarsh weyl associee a l equation differentielle lineaire d ordre 2 y qy y sur 0 dans le cas ou la seule restriction sur la fonction a valeur reelle q est q l 1 0 e pour e 0

the boundary control approach to the titchmarsh weyl m function i the response operator and the a amplitude we link the boundary control theory and the titchmarsh weyl theory this provides a natural interpretation of the a amplitude due to simon and yields a new efficient method to evaluate the titchmarsh weyl m function associated with the schrodinger operator h x 2 q x on l 2 0 with dirichlet boundary condition at x 0

an exact method for the calculation of certain titchmarsh weyl m functions we consider the second order linear differential equation where q is real valued in the case we calculate the titchmarsh weyl m function associated with

optimal average sample number of the sprt chart for monitoring fraction nonconforming abstract the sequential probability ratio test sprt control chart is a powerful tool for monitoring manufacturing processes it is highly suitable for the applications where testing is destructive or very expensive such as the automobile airbags test this article studies the effect of the average sample number asn i e the average sample size on the chart s performance a design algorithm is proposed to develop the optimal sprt chart for monitoring the fraction nonconforming p of bernoulli processes by optimizing the asn and other charting parameters the average detection speed of the sprt chart is almost doubled it is also found that the optimal sprt chart significantly outperforms the optimal np and binomial cusum charts in terms of average number of defectives and under different combinations of the design specifications it is observed that the sprt chart using a relatively smaller asn and a shorter sampling interval h has a higher overall detection effectiveness

the interpolation of the titchmarsh weyl function by using the idea of interpolation of functions in hardy spaces we obtain a new representation of the titchmarsh weyl function known also as the m function in the case of short range potentials the key idea is to use the gelfand levitan marchenko theory which allows us to represent jost solutions by the fourier transform

optimal np control chart with curtailment this article presents an algorithm for the optimization design of the np control chart with curtailment the objective is to maximize the effectiveness of the np chart while holding the false alarm rate at a specified level and maintaining its simplicity for understanding and operation to a large degree the optimization algorithm can be applied to a 100 inspection as well as a general random sampling inspection the effectiveness of the chart is measured by the out of control average time to signal ats under the steady state mode this mode allows a shift in fraction nonconforming p to occur at any time during the inspection of a sample and therefore provides more realistic results than the zero state mode according to several performance studies and an illustrative example that are conducted in this research the optimal np chart with curtailment may reduce the out of control ats by nearly half on average compared to the conventional np charts

the classification of compact simply connected biquotients in dimension 6 and 7 we classify all compact simply connected biquotients of dimension 6 and 7 for each 6 dimensional biquotient all pairs of groups g h and homomorphisms h rightarrow g times g giving rise to it are classified

curvature and symmetry of milnor spheres since milnor s discovery of exotic spheres mi one of the most intriguing problems in riemannian geometry has been whether there are exotic spheres with positive curvature it is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature hi on the other hand there are many examples of exotic spheres with positive ricci curvature cf ch1 he po and na and this work recently culminated in wr where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive ricci curvature this includes all exotic spheres in dimension 7 so far however no example of an exotic sphere with positive sectional curvature has been found in fact until now only one example of an exotic sphere with nonnegative sectional curvature was known the so called gromoll meyer sphere gm in dimension 7 as one of our main results we prove

a property of the asymptotic series for a class of titchmarsh weyl m functions in an earlier paper 6 we showed that if q c n 0 e for some e 0 then the titchmarsh weyl m function associated with the second order linear differential equation has the asymptotic expansion as a in a sector of the form 0 we show that if the real valued function q admits the expansion in a neighbourhood of 0 then

cheeger manifolds and the classification of biquotients a closed manifold is called a biquotient if it is diffeomorphic to k g h for some compact lie group g with closed subgroups k and h such that k acts freely on g h every biquotient has a riemannian metric of nonnegative sectional curvature in fact almost all known manifolds of nonnegative curvature are biquotients the only known closed manifolds of nonnegative sectional curvature which were not defined in this way are those found by cheeger 6 in 1973 and grove and ziller 16 in 2000 in order to understand these constructions better this paper analyzes which of the cheeger and grove ziller manifolds are actually diffeomorphic to biquotients in the process we develop a general approach to the classification of biquotient manifolds a priori it is hard to determine whether a given manifold is a biquotient because there is no obvious upper bound on the dimension of the groups involved we give a procedure which allows us to reduce the dimension of the groups needed to describe a given manifold as a biquotient a surprising application is that gromoll meyer s example of an exotic 7 sphere which is a biquotient 15 is the only exotic sphere of any dimension which is a biquotient more generally we classify all biquotients which are simply connected rational homology spheres of any dimension theorem 6 1 the classification of biquotients which are simply connected rational homology spheres has been given independently by kapovitch and ziller 17 more generally they classify all simply connected biquotients whose rational cohomology ring is generated by one element another application of our general classification theory for biquotients is the precise determination of which cheeger manifolds the connected sums of two rank one symmetric spaces with any orientations are diffeomorphic to biquotients theorem 2 1 some of the cheeger manifolds were known to be biquotients such as cp2 cp2 the nontrivial s2 bundle over s2 but we find that many of the other cheeger manifolds are also biquotients such as cp2 cp2 on the other hand the cheeger manifold cp4 hp2 is not diffeomorphic or even homotopy equivalent to a biquotient the positive result that some cheeger manifolds such as cp2 cp2 are biquotients implies that they have nonnegatively curved riemannian metrics with various good properties that were not clear from cheeger s construction first the new metrics are real analytic further the new metrics determine a natural complex analytic structure on the whole tangent bundle since aguilar 2 showed that biquotients have this property my paper 26 has a weaker result which applies to all the cheeger manifolds finally the geodesic flow for the new metrics on the cheeger manifolds has zero topological entropy paternain 23 proved the latter property for cheeger s metric on cp2 cp2 but now we know it for some metric on a large class of the cheeger manifolds finally our general classification results imply that there are only finitely many

on the location of the weyl circles the paper deals with explicit estimates concerning certain circles in the complex plane which were associated with sturm liouville problems by h weyl by the use of riccati equations instead of linear integral equations improvements are obtained for results of everitt and halvorsen concerning the behaviour of the titchmarsh weyl m coefficient

three triplet np control charts abstract an algorithm is presented for designing the np control charts on the basis of their statistical performance the algorithm gives the sample size and control limits for an np chart the 3 triplet np chart based on the specified values of a the in control fraction nonconforming value p 0 a downward shifted value p and an upward shifted value p b the corresponding design run lengths rl and c the corresponding probabilities w specifically whenever the process fraction nonconforming equals one of the three specified values the 3 triplet np chart will produce a signal in a predetermined run length with a predetermined probability this feature makes the operating characteristics of the np chart highly specifiable and provides the quality assurance engineers with better perception and more control in the design and operation of the np charts the 3 triplet np chart is designed by seeking an optimal combination of the sample size and the control limits

quasi positive curvature on homogeneous bundles we provide new examples of manifolds which admit a riemannian metric with sectional curvature nonnegative and strictly positive at one point our examples include the unit tangent bundles of n n and mathbb o n the cayley plane and a family of lens space bundles over n

the form of the spectral functions associated with a class of sturm liouville equations with integrable coefficient we consider the second order linear differential equation in the case where roughly q l 1 0 we devise a representation for the spectral function t associated with which is valid for t sufficiently large our results are the best possible

a homotopy classification of certain 7 manifolds this paper gives a homotopy classification of wallach spaces and a partial homotopy classification of closely related spaces obtained by free sl actions on su 3 and on s3 x s5

asymptotic phase asymptotic modulus and titchmarsh weyl coefficient for a dirac system abstract this paper studies a series representation for the jost function associated with a dirac system under suitable hypotheses the series yields the asymptotic form of the titchmarsh weyl coefficient valid not only in sectors of the upper halfplane but also the real axis the asymptotic form of the spectral function higher derivatives of the jost function and the limiting form of the asymptotic phase are also considered extensions of certain scattering formulas known in the schrodinger case are given for the dirac system

cohomogeneity two torus actions on non negatively curved manifolds of low dimension let mathrm m n n in 4 5 6 be a compact simply connected n manifold which admits some riemannian metric with non negative curvature and an isometry group of maximal possible rank then any smooth effective action on mathrm m n by a torus mathrm t n 2 is equivariantly diffeomorphic to an isometric action on a normal biquotient furthermore it follows that any effective isometric circle action on a compact simply connected non negatively curved four dimensional manifold is equivariantly diffeomorphic to an effective isometric action on a normal biquotient

on the curvature of biquotients as a means to better understanding manifolds with positive curvature there has been much recent interest in the study of non negatively curved manifolds which contain either a point or an open dense set of points at which all 2 planes have positive curvature we study infinite families of biquotients defined by eschenburg and bazaikin from this viewpoint together with torus quotients of s3 s3