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

Modules of Abelian integrals and Picard-Fuchs systems

We give a simple proof of an isomorphism between the two $\mathbb{C}[t]$-modules: the module of relative cohomologies $\Lambda^2/dH\land \Lambda^1$ and the module of Abelian integrals corresponding to a regular at infinity polynomial $H$ in two variables. Using this isomorphism, we prove existence and deduce some properties of the corresponding Picard-Fuchs system.Comment: A separate section discusses Fuchsian properties of the Picard-Fuchs system, Morse condition exterminated. Few errors were correcte

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

Leading-order QCD Analysis of Neutrino-Induced Dimuon Events

The results of a leading-order QCD analysis of neutrino-induced charm production are presented. They are based on a sample of 4111 \numu- and 871 \anumu-induced opposite-sign dimuon events with $E_{\mu 1},E_{\mu 2} > 6~{\rm GeV}$, $35 5.5\,{\rm GeV^2}$, observed in the CHARM~II detector exposed to the CERN wideband neutrino and antineutrino beams. The analysis yields the value of \linebreak the charm quark mass $m_c=1.79\pm0.38\,{\rm GeV}/c^2$ and the Cabibbo--Kobayashi--Maskawa matrix element $|V_{cd}|=0.219\pm0.016$. The strange quark content of the nucleon is found to be suppressed with respect to non-strange sea quarks by a factor $\kappa =0.39\pm0.09$

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)}