on the soliton resolution for equivariant wave maps to the sphere we consider finite energy corotational wave maps with target manifold s2 we prove that for a sequence of times they decompose as a sum of decoupled harmonic maps in the light cone and a smooth wave map in the blowup case or a linear scattering term in the global case up to an error which tends to 0 in the energy space 2015 wiley periodicals inc

two bubble dynamics for threshold solutions to the wave maps equation we consider the energy critical wave maps equation mathbb r 1 2 rightarrow mathbb s 2 in the equivariant case with equivariance degree k ge 2 it is known that initial data of energy 8 pi k and topological degree zero leads to global solutions that scatter in both time directions we consider the threshold case of energy 8 pi k we prove that the solution is defined for all time and either scatters in both time directions or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction in the latter case we describe the asymptotic behavior of the scales of the two harmonic maps the proof combines the classical concentration compactness techniques of kenig merle with a modulation analysis of interactions of two harmonic maps in the absence of excess radiation

multipath clustering and cluster tracking for geometry based stochastic channel modeling this paper presents a clustering and tracking method that exploits the geometry of the scattering points sps obtained from the measurement based ray tracer the multipath components mpcs were categorized into clusters by applying the kpowermeans kpm framework to those sps the clusters were tracked by comparing the cluster centroid sps of the adjacent snapshots the clusters were estimated based on the indoor environment geometry at 11 ghz and their physical mechanisms were interpreted the complexity and performance of this method was assessed and compared with that of conventional kpm by comparing the number of floating point operations flops and the channel eigenvalues obtained from the reconstructed channel matrices which were calculated by superposing the mpcs randomly generated from intracluster parameters the verification of this method showed that most clusters were estimated and tracked according to the physical location of the scatterers in the environment with acceptable error moreover the eigenvalues reconstructed from the proposed method were closer to the measured ones with less number of flops which indicates both accuracy and complexity improvement the results also imply that multiple input multiple output performance is highly dependent on the radio propagation channel therefore it is imperative that clusters in the channel be determined accurately

factorization of j unitary matrix polynomials on the line and a schur algorithm for generalized nevanlinna functions we prove that a 2 2 matrix polynomial which is j unitary on the real line can be written as a product of normalized elementary j unitary factors and a j unitary constant in the second part we give an algorithm for this factorization using an analog of the schur transformation

on the formation of singularities in the critical o 3 sigma model we study the phenomena of energy concentration for the critical o 3 sigma model also known as the wave map flow from 2 1 minkowski space into the sphere s 2 we establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities the construction and analysis are done in the context of the k equivariant symmetry reduction and we restrict to maps with homotopy class k 4 the concentration mechanism we uncover is essentially due to a resonant self focusing shrinking of a corresponding harmonic map we show that the phenomenon is generic e g in certain sobolev spaces in that it persists under small perturbations of initial data while the resulting blowup is bounded by a log modified self similar asymptotic

characterization of large energy solutions of the equivariant wave map problem ii we consider 1 equivariant wave maps from bbb r 1 2 to bbb s 2 of finite energy we establish a classification of all degree one global solutions whose energies are less than three times the energy of the harmonic map q in particular for each global energy solution of topological degree one we show that the solution asymptotically decouples into a rescaled harmonic map plus a radiation term together with a companion article part i where we consider the case of finite time blow up this gives a characterization of all 1 equivariant degree 1 wave maps in the energy regime e q 3e q

on collapse of wave maps abstract we derive the universal collapse law of degree 1 equivariant wave maps solutions of the sigma model from the 2 1 minkowski space time to the 2 sphere to this end we introduce a nonlinear transformation from original variables to blowup ones our formal derivations are confirmed by numerical simulations

regularization of the indefinite stieltjes moment problem abstract a not regular indefinite stieltjes moment problem is studied in the generalized stieltjes class n k the new class of the sequence s s i i 0 l is defined which is called regular class regularization of the indefinite stieltjes moment problem is obtained the set of solutions of this problem is found in terms of the generalized stieltjes polynomials and s fractions with the shift

a validity measure for fuzzy clustering the authors present a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c partitions without assumptions as to the number of substructures inherent in the data this function depends on the data set geometric distance measure distance between cluster centroids and more importantly on the fuzzy partition generated by any fuzzy algorithm used the function is mathematically justified via its relationship to a well defined hard clustering validity function the separation index for which the condition of uniqueness has already been established the performance of this validity function compares favorably to that of several others the application of this validity function to color image segmentation in a computer color vision system for recognition of ic wafer defects which are otherwise impossible to detect using gray scale image processing is discussed

jl unitary factorization and the schur algorithm for nevanlinna functions in an indefinite setting abstract we introduce a schur transformation for generalized nevanlinna functions and show that it can be used in obtaining the unique minimal factorization of a class of rational j l unitary 2 2 matrix functions into elementary factors from the same class

silhouettes a graphical aid to the interpretation and validation of cluster analysis a new graphical display is proposed for partitioning techniques each cluster is represented by a so called silhouette which is based on the comparison of its tightness and separation this silhouette shows which objects lie well within their cluster and which ones are merely somewhere in between clusters the entire clustering is displayed by combining the silhouettes into a single plot allowing an appreciation of the relative quality of the clusters and an overview of the data configuration the average silhouette width provides an evaluation of clustering validity and might be used to select an appropriate number of clusters

inverse problems for periodic generalized jacobi matrices abstract some inverse problems for semi infinite periodic generalized jacobi matrices are considered in particular a generalization of the abel criterion is presented the approach is based on the fact that the solvability of the pell abel equation is equivalent to the existence of a certainly normalized j unitary 2 2 matrix polynomial the monodromy matrix

asymptotic decomposition for semilinear wave and equivariant wave map equations in this paper we give a unified proof to the soliton resolution conjecture along a sequence of times for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations including the four dimensional radial yang mills equation without using outer energy type inequalities such inequalities have played a crucial role in previous works with similar results roughly speaking we prove that along a sequence of times t n to t the maximal time of existence the solution decouples to a sum of rescaled solitons and a term vanishing in the energy space plus a free radiation term in the global case or a regular part in the finite time blow up case the main difficulty is that in general for instance for the radial four dimensional yang mills case and the radial six dimensional semilinear wave case we do not have a favorable outer energy inequality for the associated linear wave equations our main new input is the simultaneous use of two virial identities

free versus constrained evolution of the 2 1 equivariant wave map we compare the numerical solutions of the 2 1 equivariant wave map problem computed with the symplectic constraint respecting rattle algorithm and the well known fourth order runge kutta method we show the advantages of the rattle algorithm for the constrained system compared to the free evolution with the runge kutta method we also present an expression which represents the energy loss due to constraint violation taking this expression into account we can achieve energy conservation for the runge kutta scheme which is better than with the rattle method using the symplectic scheme with constraint enforcement we can reproduce previous calculations of the equivariant case without imposing the symmetry explicitly thereby confirming that the critical behaviour is stable

the schur algorithm for generalized schur functions iii j unitary matrix polynomials on the circle the main result is that for j 1 0 0 1 every j unitary 2 x 2 matrix polynomial on the unit circle is an essentially unique product of elementary j unitary 2 x 2 matrix polynomials which are either of degree 1 or 2k this is shown by means of the generalized schur transformation introduced in ann inst fourier 8 1958 211 ann acad sci fenn ser a i 250 9 1958 1 7 and studied in pisot and salem numbers birkhauser verlag basel 1992 philips j res 41 1 1986 1 54 and also in the first two parts operator theory adv appl 129 birkhauser verlag basel 2000 p 1 monatshefte fur mathematik in press of this series the essential tool in this paper are the reproducing kernel pontryagin spaces associated with generalized schur functions c 2003 elsevier science inc all fights reserved

validity index for crisp and fuzzy clusters abstract in this article a cluster validity index and its fuzzification is described which can provide a measure of goodness of clustering on different partitions of a data set the maximum value of this index called the pbm index across the hierarchy provides the best partitioning the index is defined as a product of three factors maximization of which ensures the formation of a small number of compact clusters with large separation between at least two clusters we have used both the k means and the expectation maximization algorithms as underlying crisp clustering techniques for fuzzy clustering we have utilized the well known fuzzy c means algorithm results demonstrating the superiority of the pbm index in appropriately determining the number of clusters as compared to three other well known measures the davies bouldin index dunn s index and the xie beni index are provided for several artificial and real life data sets

existence of constant mean curvature foliations in spacetimes with two dimensional local symmetry it is shown that in a class of maximal globally hyperbolic spacetimes ad mitting two local killing vectors the past defined with respect to an appropriate time orientation of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces moreover the mean cur vature of the leaves of this foliation takes on arbitrarily negative values and so the initial singularity in these spacetimes is a crushing singularity the simplest examples occur when the spatial topology is that of a torus with the standard global killing vectors but more exotic topologies are also covered in the course of the proof it is shown that in this class of spacetimes a kind of positive mass theorem holds the symmetry singles out a compact surface passing through any given point of spacetime and the hawking mass of any such surface is non negative if the hawking mass of any one of these surfaces is zero then the entire spacetime is flat

on the regularity of spherically symmetric wave maps wave maps are critical points u m n of the lagrangian u m du 2 where m is an einsteinian manifold and n a riemannian one for the case m 2 1 and u a spherically symmetric map it is shown that the solution to the cauchy problem for u with smooth initial data of arbitrary size is smooth for all time provided the target manifold n satisfies the two conditions that 1 it is either compact or there exists an orthonormal frame of smooth vectorfields on n whose structure functions are bounded and 2 there are two constants c and c such that the smallest eigenvalue and the largest eigenvalue of the second fundamental form kab of any geodesic sphere p s of radius s centered at p n satisfy s c and s a c 1 s this is proved by first analyzing the energy momentum tensor and using the second condition to show that near the first possible singularity the energy of the solution cannot concentrate and hence is small one then proves that for targets satisfying the first condition initial data of small energy imply global regularity of the solution 1993 john wiley sons inc

on stable self similar blowup for equivariant wave maps reference epfl article 166023doi 10 1002 cpa 20366 url http arxiv org abs 1003 0707 record created on 2011 05 23 modified on 2017 05 12