Nielsen numbers in topological coincidence theory

We discuss coincidences of pairs (f_1, f_2) of maps between manifolds. We recall briefly the definition of four types of Nielsen numbers which arise naturally from the geometry of generic coincidences. They are lower bounds for the minimum numbers MCC and MC which measure to some extend the 'essential' size of a coincidence phenomenon. In the setting of fixed point theory these Nielsen numbers all coincide with the classical notion but in general they are distinct invariants. We illustrate this by many examples involving maps from spheres to the real, complex or quaternionic projective space KP(n'). In particular, when n' is odd and K = R or C or when n' = 23 mod 24 and K = H, we compute the minimum number MCC and all four Nielsen numbers for every pair of these maps, and we establish a 'Wecken theorem' in this context (in the process we correct also a mistake in previous work concerning the quaternionic case). However, when n' is even, counterexamples can occur, detected e.g. by Kervaire invariants.Comment: Coincidence, minimum number, Nielsen number, Reidemeister number, Wecken theorem, projective spac

Optimal interval clustering: Application to Bregman clustering and statistical mixture learning

We present a generic dynamic programming method to compute the optimal clustering of $n$ scalar elements into $k$ pairwise disjoint intervals. This case includes 1D Euclidean $k$-means, $k$-medoids, $k$-medians, $k$-centers, etc. We extend the method to incorporate cluster size constraints and show how to choose the appropriate $k$ by model selection. Finally, we illustrate and refine the method on two case studies: Bregman clustering and statistical mixture learning maximizing the complete likelihood.Comment: 10 pages, 3 figure

Nielsen-Olesen strings in Supersymmetric models

We investigate the behaviour of a model with two oppositely charged scalar fields. In the Bogomol'nyi limit this may be seen as the scalar sector of N=1 supersymmetric QED, and it has been shown that cosmic strings form. We examine numerically the model out of the Bogomol'nyi limit, and show that this remains the case. We then add supersymmetry-breaking mass terms to the supersymmetric model, and show that strings still survive. Finally we consider the extension to N=2 supersymmetry with supersymmetry-breaking mass terms, and show that this leads to the formation of stable cosmic strings, unlike in the unbroken case.Comment: 7 pages, 2 figues, uses revtex4; minor typos corrected; references adde

Out of equilibrium dynamics of coherent non-abelian gauge fields

We study out-of-equilibrium dynamics of intense non-abelian gauge fields. Generalizing the well-known Nielsen-Olesen instabilities for constant initial color-magnetic fields, we investigate the impact of temporal modulations and fluctuations in the initial conditions. This leads to a remarkable coexistence of the original Nielsen-Olesen instability and the subdominant phenomenon of parametric resonance. Taking into account that the fields may be correlated only over a limited transverse size, we model characteristic aspects of the dynamics of color flux tubes relevant in the context of heavy-ion collisions.Comment: 12 pages, 10 figures; PRD version, minor change

Analysis of DsDK∗D_s D K^* and DsD∗K∗D_s D^{*} K^* vertices and branching ratio of B+→K∗0π+B^+\to {K^*}^0 \pi^+

In this paper, the strong form factors and coupling constants of $D_sDK^*$ and $D_sD^*K^*$ vertices are investigated within the three-point QCD sum rules method with and without the $SU_{f}(3)$ symmetry. In this calculation, the contributions of the quark-quark, quark-gluon, and gluon-gluon condensate corrections are considered. As an example of specific application of these coupling constants, the branching ratio of the hadronic decay $B^+\to {K^*}^0 \pi^+$ is analyzed based on the one-particle-exchange which is one of the phenomenological models. In this model, $B$ decays into a $D_s D^*$ intermediate state, and then these two particles exchange a $D (D^*)$ producing the final $K^*$ and $\pi$ mesons. In order to compute the effect of these interactions, the $D_s D K^*$ and $D_s D^* K^*$ form factors are needed.Comment: 14 pages, 8 figures. arXiv admin note: substantial text overlap with arXiv:1509.0171

Geometric and homotopy theoretic methods in Nielsen coincidence theory

In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. Here we extend it to pairs (f_1, f_2) of maps between manifolds of arbitrary dimensions. This leads to estimates of the minimum numbers MCC(f_1, f_2) (and MC(f_1, f_2), resp.) of pathcomponents (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to (f_1, f_2). Furthermore we deduce finiteness conditions for MC(f_1, f_2). As an application we compute both minimum numbers explicitly in four concrete geometric sample situations. The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space E(f_1, f_2) into pathcomponents. Its higher dimensional topology captures further crucial geometric coincidence data. An analoguous approach can be used to define also Nielsen numbers of certain link maps