Upper bounds for the number of orbital topological types of planar polynomial vector fields "modulo limit cycles"

The paper deals with planar polynomial vector fields. We aim to estimate the number of orbital topological equivalence classes for the fields of degree n. An evident obstacle for this is the second part of Hilbert's 16th problem. To circumvent this obstacle we introduce the notion of equivalence modulo limit cycles. This paper is the continuation of the author's paper in [Mosc. Math. J. 1 (2001), no. 4] where the lower bound of the form 2^{cn^2} has been obtained. Here we obtain the upper bound of the same form. We also associate an equipped planar graph to every planar polynomial vector field, this graph is a complete invariant for orbital topological classification of such fields.Comment: 23 pages, 5 figure

Parametric Polyhedra with at least kk Lattice Points: Their Semigroup Structure and the k-Frobenius Problem

Given an integral $d \times n$ matrix $A$, the well-studied affine semigroup \mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\{x: Ax=b, x\geq0\}$. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{ Sg}(A) such that $P_A(b) \cap {\mathbb Z}^n$ has at least $k$ solutions. We demonstrate that this set is finitely generated, it is a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computations. Related results can be derived for those right-hand-side vectors $b$ for which $P_A(b) \cap {\mathbb Z}^n$ has exactly $k$ solutions or fewer than $k$ solutions. (2) A computational complexity theory. We show that, when $n$, $k$ are fixed natural numbers, one can compute in polynomial time an encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors of bounded norm that have at least $k$ solutions. (3) Applications and computation for the $k$-Frobenius numbers. Using Generating functions we prove that for fixed $n,k$ the $k$-Frobenius number can be computed in polynomial time. This generalizes a well-known result for $k=1$ by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of $k$-Frobenius numbers and their relatives

Experimental search for muonic photons

We report new limits on the production of muonic photons in the CERN neutrino beam. The results are based on the analysis of neutrino production of dimuons in the CHARM II detector. A $90\%$ CL limit on the coupling constant of muonic photons, $\alpha_{\mu} / \alpha < (1.5 \div 3.2) \times10^{-6}$ is derived for a muon neutrino mass in the range $m_{\nu_{\mu}} = (10^{-20} \div 10^5)$ eV. This improves the limit obtained from a precision measurement of the anomalous magnetic moment of the muon $(g-2)_\mu$ by a factor from 8 to 4

Observation of weak neutral current neutrino production of J/ψJ/\psi

Observation of \jpsi production by neutrinos in the calorimeter of the CHORUS detector exposed to the CERN SPS wide-band \numu beam is reported. A spectrum-averaged cross-section $\sigma^{\mathrm{J/\psi}}$ = (6.3 $\pm$ 3.0) $\times \mathrm{10^{-41}~cm^{2}}$ is obtained for 20 GeV $\leq E_{\nu} \leq$ 200 GeV. The data are compared with the theoretical model based on the QCD Z-gluon fusion mechanism

The CHORUS neutrino oscillation search experiment

The CHORUS experiment has successfully finished run I (320~000 recorded \numu\ CC in 94/95) and performed half of run II (225~000 \numu\ CC in 96). The analysis chain was exercised on a small data sample for the muonic \tdecay\ search using for the first time fully automatic emulsion scanning. This pilot analysis, resulting in a limit \sintth \leq 3 \cdot 10^{-2}, confirms that the CHORUS proposal sensitivity (\sintth \leq 3 \cdot 10^{-4}) is within reach in two years

Measurement of inclusive π0\pi^{0} production in hadronic Z0Z^{0} decays

An analysis is presented of inclusive \pi^0 production in Z^0 decays measured with the DELPHI detector. At low energies, \pi^0 decays are reconstructed by \linebreak using pairs of converted photons and combinations of converted photons and photons reconstructed in the barrel electromagnetic calorimeter (HPC). At high energies (up to x_p = 2 \cdot p_{\pi}/\sqrt{s} = 0.75) the excellent granularity of the HPC is exploited to search for two-photon substructures in single showers. The inclusive differential cross section is measured as a function of energy for {q\overline q} and {b \bar b} events. The number of \pi^0's per hadronic Z^0 event is N(\pi^0)/ Z_{had}^0 = 9.2 \pm 0.2 \mbox{(stat)} \pm 1.0 \mbox{(syst)} and for {b \bar b}~events the number of \pi^0's is {\mathrm N(\pi^0)/ b \overline b} = 10.1 \pm 0.4 \mbox{(stat)} \pm 1.1 \mbox{(syst)} . The ratio of the number of \pi^0's in b \overline b events to hadronic Z^0 events is less affected by the systematic errors and is found to be 1.09 \pm 0.05 \pm 0.01. The measured \pi^0 cross sections are compared with the predictions of different parton shower models. For hadronic events, the peak position in the \mathrm \xi_p = \ln(1/x_p) distribution is \xi_p^{\star} = 3.90^{+0.24}_{-0.14}. The average number of \pi^0's from the decay of primary \mathrm B hadrons is found to be {\mathrm N} (B \rightarrow \pi^0 \, X)/\mbox{B hadron} = 2.78 \pm 0.15 \mbox{(stat)} \pm 0.60 \mbox{(syst)}

Search for Neutral Heavy Leptons Produced in Z Decays

Weak isosinglet Neutral Heavy Leptons ($\nu_m$) have been searched for using data collected by the DELPHI detector corresponding to $3.3\times 10^{6}$ hadronic~Z$^{0}$ decays at LEP1. Four separate searches have been performed, for short-lived $\nu_m$ production giving monojet or acollinear jet topologies, and for long-lived $\nu_m$ giving detectable secondary vertices or calorimeter clusters. No indication of the existence of these particles has been found, leading to an upper limit for the branching ratio $BR($Z$^0\rightarrow \nu_m \overline{\nu})$ of about $1.3\times10^{-6}$ at 95\% confidence level for $\nu_m$ masses between 3.5 and 50 GeV/$c^2$. Outside this range the limit weakens rapidly with the $\nu_m$ mass. %Special emphasis has been given to the search for monojet--like topologies. One event %has passed the selection, in agreement with the expectation from the reaction: %$e^+e^- \rightarrow\ell \bar\ell \nu\bar\nu$. The results are also interpreted in terms of limits for the single production of excited neutrinos

Investigation of the splitting of quark and gluon jets

The splitting processes in identified quark and gluon jets are investigated using longitudinal and transverse observables. The jets are selected from symmetric three-jet events measured in Z decays with the Delphi detector in 1991-1994. Gluon jets are identified using heavy quark anti-tagging. Scaling violations in identified gluon jets are observed for the first time. The scale energy dependence of the gluon fragmentation function is found to be about two times larger than for the corresponding quark jets, consistent with the QCD expectation TeX . The primary splitting of gluons and quarks into subjets agrees with fragmentation models and, for specific regions of the jet resolution TeX , with NLLA calculations. The maximum of the ratio of the primary subjet splittings in quark and gluon jets is TeX . Due to non-perturbative effects, the data are below the expectation at small TeX . The transition from the perturbative to the non-perturbative domain appears at smaller TeX for quark jets than for gluon jets. Combined with the observed behaviour of the higher rank splittings, this explains the relatively small multiplicity ratio between gluon and quark jets