index and stability of symmetric periodic orbits in hamiltonian systems with application to figure eight orbit in this paper using the maslov index theory in symplectic geometry we build up some stability criteria for symmetric periodic orbits in a hamiltonian system which is motivated by the recent discoveries in the n body problem the key ingredient is a generalized bott type iteration formula for periodic solution in the presence of finite group action on the orbit for second order system we prove under general boundary conditions the close formula for the relationship between the morse index of an orbit in a lagrangian system and the maslov index of the fundamental solution for the corresponding orbit in its hamiltonian system counterpart and the boundary conditions cover the cases which appeared in the n body problem as an application we consider the stability problem of the celebrated figure eight orbit due to chenciner and montgomery in the planar three body problem with equal masses and we clarify the relationship between linear stability and its variational nature on various loop spaces the basic idea is as follows the variational characterization of the figure eight orbit provides information about its morse index based on its relation to the maslov index our stability criteria come into play

a multiple level set method for 3d inversion of magnetic data abstractwe have developed a multiple level set method for inverting magnetic data produced by weak induced magnetization only the method is designed to deal with a specific class of 3d magnetic inverse problems in which the magnetic susceptibility is known and the objective of the inversion is to find the boundary or geometric shape of the causative bodies we adopt the conceptual representation of the subsurface geologic structure by a set of magnetic bodies each having a uniform magnetic susceptibility embedded in a nonmagnetic background this representation enables us to reformulate the magnetic inverse problem into a domain inverse problem for those unknown domains defining the supports of the magnetic causative bodies because each body may take on a variety of shapes and we may not know the number of bodies a priori either we use multiple level set functions to parameterize these domains so that the domain inverse problem can be further reduced to an optimization problem of multiple level set f

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

covering radii of ternary linear codes of small dimensions and codimensions the covering radii and the least covering radii of all two and three dimensional codes over gf 3 are given in the present work the exact values of the least covering radii of some ternary linear codes of codimensions up to 8 are also determined and a table of values of t sub 3 n k for n spl les 27 is presented

locally optimal nonshortening linear covering codes and minimal saturating sets in projective spaces a concept of locally optimal lo linear covering codes is introduced in accordance with the concept of minimal saturating sets in projective spaces over finite fields an lo code is nonshortening in the sense that one cannot remove any column from a parity check matrix without increasing the code covering radius several q sup m concatenating constructions of lo covering codes are described taking a starting lo code as a seed such constructions produce an infinite family of lo codes with the same covering radius the infinite families of lo codes are designed using minimal saturating sets as starting codes new upper bounds on the length function are given new extremal and classification problems for linear covering codes are formulated and investigated in particular the spectrum of possible lengths of lo codes including the greatest possible length the complete computer classification of the minimal saturating sets in small geometries and of the corresponding lo codes is obtained

existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four body problem abstract in this paper we study the existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four body problem the hamiltonian of the differential system is regularized by a levi civita type transformation and an appropriate scaling of time the initial condition of this orbit is shown to be the infimum of some well chosen set this existence proof is direct and surprisingly simple further a careful study shows that this orbit has a symmetry group isomorphic to the dihedral group d 4 then roberts symmetry reduction method is applied to show the linear stability it turns out that the rhomboidal periodic orbit in the planar equal mass four body problem is linearly stable

short codes with a given covering radius the covering radius r of a code is the maximum distance from any vector in the space containing the code to the nearest codeword the authors introduce a new function l m r called the length function which equals the smallest length of a binary code of codimension m and covering radius r they investigate basic properties of the length function projective geometries over larger fields are used to construct families of codes which improve significantly the upper bound for l m 2 obtained by amalgamation of hamming codes general methods are developed for ruling out the existence of codes of covering radius 2 with a given codimension and length resulting in lower bounds for l m 2 a table is presented which gives the best results now known for l m r with m

linear stability of double double orbits in the parallelogram four body problem abstract we study the linear stability of a two parameter family of periodic orbits in the parallelogram four body problem this family was numerically found and named as double double orbits by vanderbei a demonstration of such an orbit is shown by introducing new transformations and applying roberts symmetry reduction method the linear stability can be simplified to the calculation of two eigenvalues a global picture of linear stability of this set with respect to the two parameters is given for the first time actually most of the double double orbits are numerically proved to be linearly stable

simple choreographies of the planar newtonian n body problem in the n body problem a simple choreography is a periodic solution where all masses chase each other on a single loop in this paper we prove that for the planar newtonian n body problem with equal masses n 3 there are at least 2n 3 2 n 3 2 different main simple choreographies this confirms a conjecture given by chenciner et al geometry mechanics and dynamics springer new york pp 287 308 2002 all the simple choreoagraphies we prove belong to the linear chain family

nonlinear magnetic inversion using a random search method a robust magnetic interpretation technique is described it is based on the minimization of an objective function by a random search algorithm instead of just one estimate this algorithm yields several points in parameter space all producing objective function values below an assumed noise level the centroid of the search points is in general a better estimate than each individual search point especially when data are corrupted by noise since no derivatives are employed even nondifferentiable objective functions such as the l1 norm of residuals can be used computation of parameter covariance matrices and confidence ellipses is possible using simple formulas the usefulness of r and q mode factor analysis applied to the correlation matrix is demonstrated the r mode factor analysis can be used in parameter variance control and the q mode factor analysis provides means to find models which are extreme in some sense the technique was applied to theoretical data and to field data from three diffe

morse index and stability of elliptic lagrangian solutions in the planar three body problem abstract we illustrate a new way to study the stability problem in celestial mechanics in this paper using the variational nature of elliptic lagrangian solutions in the planar three body problem we study the relation between morse index and its stability via maslov type index theory of periodic solutions of hamiltonian system for elliptic lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the morse index which is the key to understand the stability problem as a special case we provide a criterion to spectral stability of relative equilibrium

constructions and families of nonbinary linear codes with covering radius 2 new constructions of linear nonbinary codes with covering radius r 2 are proposed they are in part modifications of earlier constructions by the author and in part are new using a starting code with r 2 as a seed these constructions yield an infinite family of codes with the same covering radius new infinite families of codes with r 2 are obtained for all alphabets of size q spl ges 4 and all codimensions r spl ges 3 with the help of the constructions described the parameters obtained are better than those of known codes new estimates for some partition parameters in earlier known constructions are used to design new code families complete caps and other saturated sets of points in projective geometry are applied as starting codes a table of new upper bounds on the length function for q 4 5 7 r 2 and r spl les 24 is included

adaptive sampling of potential field data a direct approach to compressive inversion abstractcompressive inversion uses computational algorithms that decrease the time and storage needs of a traditional inverse problem most compression approaches focus on the model domain and very few other than traditional downsampling focus on the data domain for potential field applications to further the compression in the data domain a direct and practical approach to the adaptive downsampling of potential field data for large inversion problems has been developed the approach is formulated to significantly reduce the quantity of data in relatively smooth or quiet regions of the data set while preserving the signal anomalies that contain the relevant target information two major benefits arise from this form of compressive inversion first because the approach compresses the problem in the data domain it can be applied immediately without the addition of or modification to existing inversion software second as most industry software use some form of model or sensitivity compression the

quasi perfect linear codes with minimum distance 4 some new infinite families of short quasi perfect linear codes are described such codes provide improvements on the currently known upper bounds on the minimal length of a quasi perfect n n m 4 q code when either 1 q 16 m ges 5 m odd or 2 q 2i 7 les i les 15 m ges 4 or 3 q 22lscr lscr ges 8 m ges 5 m odd as quasi perfect n n m 4 q codes and complete n caps in projective spaces pg m 1 q are equivalent objects new upper bounds on the size of the smallest complete cap in pg m 1 q are obtained

simultaneous inversion of hydrogeologic and thermal data 1 theory and application using hydraulic head data one goal in solving the inverse problem in groundwater hydrology is to obtain parameter estimates for use in a simulation model a philosophy about the inverse is adopted here that takes the form of constructing a variety of models under different norms large differences in parameter estimates obtained using different norms suggests the possibility of an incorrect interpretational model an optimization procedure is used that is based on the constrained simplex method this algorithm easily handles constraints does not require derivatives of the functional with respect to the parameters and can be employed to minimize complicated functionals the algorithm produces a cluster of search points in parameter space about the global minimum of the objective function each of these points generates data that matches observed values within a prescribed tolerance the centroid of the search points is taken as the best estimate of the parameters the covariances of the estimated parameters and estimated data can also be computed this inverse technique is applied to a sparse inaccurate hydraulic head data set observed at downie slide british columbia canada for this cross sectional problem with noisy data and a strong drift component interpolation of hydraulic head values using kriging techniques was not possible the domain is represented as a homogeneous isotropic medium four parameters are estimated the free surface position of the water table and three boundary conditions further simulations using a synthetic data set with assumed properties similar to that of downie slide show that with noise free data and an adequate spacing between measurement points it is possible to interpolate an unbiased estimate of hydraulic head data at all nodes in the equivalent discretized domain when the inverse technique is applied the hydraulic conductivity structure is correctly identified the implications for downie slide are that in order to resolve more than a simple hydraulic conductivity structure accurate field measurements of hydraulic head a better spacing between measurements and more detailed knowledge of the boundary conditions are required

minimizing periodic orbits with regularizable collisions in the n body problem in the newtonian n body problem there are various subsystems with two degrees of freedom such as the collinear three body problem and the isosceles three body problem after we determine a normal form of the lagrangians of these subsystems we prove the existence of periodic solutions with regularizable collisions for these systems our result includes several examples such as schubart s orbit with or without equal masses among others

binary and ternary linear quasi perfect codes with small dimensions the aim of this work is a systematic investigation of the possible parameters of quasi perfect qp binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes first we give a list of infinite families of qp codes which includes all binary ternary and quaternary codes known to us we continue further with a list of sporadic examples of binary and ternary qp codes later we present the results of our investigation where binary qp codes of dimensions up to 14 and ternary qp codes of dimensions up to 13 are classified

linear codes with covering radius r 2 3 and codimension tr let n n r sub q r denote a linear code over f sub q with length n codimension r and covering radius r we use a modification of constructions of 2q 1 2q 3 sub q 2 and 3q 1 3q 5 sub q 3 codes q spl ges 5 to produce infinite families of good codes with covering radius 2 and 3 and codimension tr

variational property of periodic kepler orbits in constant curvature spaces abstract we study the variational property of the periodic kepler orbits on the sphere the plane and the hyperbolic plane we first classify the orbits by the two constants of motion the energy and the angular momentum then we characterize the local variational property of the closed orbits by computing the maslov type indices finally we study the global variational property of the closed orbits we prove that the closed orbits on the hyperbolic plane minimizes the action among all loops which encircle the attracting center

constructions and families of covering codes and saturated sets of points in projective geometry in davydov 1990 constructions of linear binary covering codes were considered in the present paper constructions and techniques of the earlier paper are developed and modified for q ary linear nonbinary covering codes q spl ges 3 and new constructions are proposed the described constructions design an infinite family of codes with covering radius r based on a starting code of the same covering radius for arbitrary r spl ges 2 q spl ges 3 new infinite families of nonbinary covering codes with good parameters are obtained with the help of an iterative process when constructed codes are the starting codes for the following steps the table of upper bounds on the length function for codes with q 3 r 2 3 and codimension up to 24 is given the author proposes to use saturated sets of points in projective geometries over finite fields as parity check matrices of starting codes new saturated sets are obtained

linear codes with covering radius 2 3 and saturating sets in projective geometry infinite families of linear codes with covering radius r 2 3 and codimension tr 1 are constructed on the base of starting codes with codimension 3 and 4 parity check matrices of the starting codes are treated as saturating sets in projective geometry that are obtained by computer search using projective properties of objects upper bounds on the length function and on the smallest sizes of saturating sets are given

linear stability analysis of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four body problem abstract we apply the symmetry reduction method of roberts to numerically analyze the linear stability of a one parameter family of symmetric periodic orbits with regularizable simultaneous binary collisions in the planar pairwise symmetric four body problem with a mass m 0 1 as the parameter this reduces the linear stability analysis to the computation of two eigenvalues of a 3 3 matrix for each m 0 1 obtained from numerical integration of the linearized regularized equations along only the first one eighth of each regularized periodic orbit the results are that the family of symmetric periodic orbits with regularizable simultaneous binary collisions changes its linear stability type several times as m varies over 0 1 with linear instability for m close or equal to 0 01 and linear stability for m close or equal to 1

a separating surface for sitnikov like n 1 body problems abstract we consider the generalized sitnikov problem of newtonian mechanics for periodic planar configurations of n bodies which are symmetric under rotation by a fixed angle the z axis is invariant we consider the effect of placing a massless particle on the z axis the study of the motion of this particle can then be modeled as a time dependent hamiltonian system we give a geometric construction of a surface in the three dimensional phase space separating orbits for which the massless particle escapes to infinity from those for which it does not a procedure for removing the periodicity condition of the planar configuration is outlined the construction of the surface is demonstrated numerically in a few examples