A geometrical approach to bifurcation for nonlinear boundary value problems

Summary In this note we use a new averaging method, which was introduced in [2], to explain the geometrical behaviour of systems governed by nonlinear boundary value problems of the formy+g(y)=K sin(t),y(0)=y(/)=0. We show by numerical computations that global features of the solutions (such as the number of solutions, their magnitude, bifurcation behaviour, etc.) agree in both the original and averaged model. As an example, the pendulum equation is discussed in detail.

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Zusammenfassung In dieser Arbeit benutzen wir eine neue, in [2] eingeführte Mittelwertmethode, um das geometrische Verhalten nichtlinearer Randwertprobleme der Formy+g(y)=K sin(t),y(0)=y(/)=0. zu erklären. Wir belegen durch numerische Untersuchungen, daß globale Eigenschaften der Lösungen (wie z. B. die Anzahl der Lösungen, ihre Größenordnung, das Verzweigungsverhalten usw.) in der originalen und genäherten Gleichung übereinstimmen. Als Beispiel wird die Pendelgleichung ausführlich diskutiert.

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Supported by the Deutsche Forschungsgemeinschaft under grant No. BA 735/3-1     

Applying a fuzzy relation to describe influence of flux behaviour in a printed coil

Summary Being a subject of expectedly fuzzy character, an attempt is made to apply fuzzy functions, more precisely fuzzy relations to investigate the problem of flux distortion in a printed coil.

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Zusammenfassung Da die Vorgänge in einer gedruckten Spule sich aus einer Vielzahl schwerer erfaßbarer Einzeleinflüsse zusammensetzen, die insbesondere den Magnetfluß verzerren, lag es nahe, den Versuch zu machen, die Fuzzy Functions (die logische Algebra, gemäß [1]) darauf anzuwenden. Die Fuzzy Functions sind gedacht für das Beschreiben von Vorgängen, oder Ereignissen, die im Ansatz ein breit gefächertes (fuzzy) Verhalten, ohne scharfe Grenzen aufweisen; siehe Fig. 4.Dieser gefächerten logischen Funktion entspricht eine reguläre mathematische Kurvenschar, die meßtechnisch nachgewiesen werden muß.Im vorliegenden Falle ergab sich eine Korrekturfunktion (5), die es gestattet, die Induktivität gedruckter Spulen, in einem weiten Bereich von Abmessungen und Windungen, mit einer Genauigkeit von –2% bis +5% zu bestimmen.

     

k-convexn-person games and their cores

For any natural numbersk andn, the subclass ofk-convexn-person games is introduced. In casek=n, the subclass consists of the convexn-person games. Ak-convexn-person game is characterized in several ways in terms of the core and certain marginal worth vectors. The marginal worth vectors of a game are described in terms of an upper bound for the core and the corresponding gap function. It is shown that thek-convexity of ann-person gamev is equivalent to

1. all marginal worth vectors ofv belong to the core ofv; or
2. the core ofv is the convex hull of the set consisting of all marginal worth vectors ofv; or
3. the extreme points of the core ofv are exactly the marginal worth vectors ofv.

Examples ofk-convexn-person games are also treated.     

Improved limits on the mass of the tau neutrino

Several methods to determine (or set an upper bound on) the mass of the tau neutrino are described. The decay spectrum of \(\tau \to e\bar \nu _e \nu _\tau \) near the high center-of-mass electron-energy end is particularly sensitive to the tau-neutrino mass. It is shown that an upper bound of 20 MeV or lower may be feasible. In addition, improved limits on the mass and mixing angle of a heavy component of the tau-neutrino may be readily obtained. We also discuss the purely leptonic decays of theD andF mesons, and show how measurements of the ratiosB(D→τν_τ/B(D→μν_μ,B(F→τν_τ/B(D→μν_μ can be sensitive to tau-neutrino masses of 30 MeV or greater.     

Joint production of ϕ mesons andπ ±,π 0,p,\bar p,K s 0 andK ± in hadronic interactions

The joint production of ? mesons andπ ^±,π ⁰,p, \(\bar p\) ,K _s ⁰ andK ^± is investigated using a sample of 600,000 inclusive ? meson events obtained in hadron Be interactions with incidentπ ^±,p, \(\bar p\) andK ^± beams. Evidence is presented for the joint production of ? mesons and strange particles produced with non-strange incident beams. With incidentK ^± beam the number of additional strange particles is suppressed. The results are found to be in agreement with the qualitative predictions of a parton fusion model. The comparison with the Lund model for lowp _T processes is fair.     

High-p T αα andαp interactions

Properties of secondaries associated with a high-p _T charged trigger particle (3<p _T <5 GeV/c) were studied for αp and αα interactions at c.m. energies \(\sqrt s = 88\) GeV and 125 GeV, respectively. Thep _T distributions of secondaries in the away-jet and trigger-jet regions were compared with those for high-p _{T pp} interactions. No statistically significant differences were seen, except at lowp _T . Momentum and angular distributions of spectator and leading protons were studied as a function of charge and rapidity of the trigger hadron. The observed correlations between trigger charge and number of spectator protons provide evidence of valence quark contributions to the trigger jet.     

Signals for compositeness atep colliders: “Leptogluons” and exotic colored vectorbosons

Production rates and signals for coloroctet leptonsl ₈ atep machines like Hera and a \(\sqrt s = 2\) TeV collider are presented. In case of very heavy leptogluons:m ₈≈0(100 GeV) we argue that rates based on direct electron-gluon fusion are possibly strongly overestimated in the literature. In composite models the appearance of leptogluons always implies the existence of color exotic vectorbosons. Therefore we considerl ₈ production in electron-quark scatterings. We find that even at HERA energies such reactions should be detectable if there is nearby compositeness (Λ _H ～ several TeV) and the mass of the coloroctet neutrinom _v8<180 GeV. From such events it would be possible to extract information on the deeper structure of the preon model.     

Phenomenology of composite colored weak bosons

In composite models of quarks, leptons and weak bosons whereW-constituents are colored objects, color octet partners ofW ^± andZ ⁰ are predicted. We study in detail the phenomenology of these particles. Independent of the specific model one expects a color octet isotriplet of vector bosons (W ₈ ^± ,Z ₈ ⁰ ) with mass in the range of 100–200 GeV, and a color octet isosinglet vector bosonV ₈ ⁰ with substantially larger mass, due to mixing with the gluon. Moreover, relatively light color octet excitations of the leptons appear, while the existence of “color exotic” partners of the quarks is model dependent. These particles decay mainly into a lepton (quark) and a gluon. We construct the couplings ofW ₈ ^± ,Z ₈ ⁰ andV ₈ ⁰ to ordinary and “color exotic” fermions. The signals of color octet weak bosons in low energy weak reactions are explored in detail. The production cross section ofW ₈ ^± (Z ₈ ⁰ ) in hadron-hadron collisions is calculated for \(0.54TeV \leqq \sqrt s \leqq 20TeV\) . Various decay modes of colored weak bosons are studied. The most prominent decay signatures ofW ₈ ^± andZ ₈ ⁰ are events of the type (l ^+-: charged lepton;j: hadronic jet; : missing transverse momentum). The present CERN \(p\bar p\) collider data on such events are discussed in the light ofW ₈ ^± andZ ₈ ⁰ decays. If colored weak bosons are not found with a mass less than ～250 GeV composite model building will be strongly restricted.     

Site-specific core level spectroscopy of CO and NO adsorption on Pt(110)(1×2) and (1×1) surfaces

The adsorption of CO and NO on the (1×2) and (1×1) modifications of the Pt(110) surface was studied by x-ray photoemission spectroscopy, LEED and work-function change measurements. The O(1s) binding energy of adsorbed CO is site-specific and differentiates between on-top and bridge adsorbed species. CO adsorption on Pt(110)(1×2) at 120 K occurred sequentially into on-top and bridge sites yielding an orderedc(8×4) layer at the maximum coverage. At 300 K only on-top bonded CO was present after CO adsorption on the (1×2) surface. CO adsorption on the (1×1) surface at 120 K showed a transient bridge adsorbed CO and on-top CO at saturation, with an ordered (2×1)p1g1 LEED pattern. Heating the (2×1)p1g1 CO layer to 400 K also showed this transient bridge CO species. Work function changes generally correlated with the appearance of different CO species but were complex in detail. The findings for CO adsorption are consistent with the missing row model of the (1×2) surface.Parallel data for NO adsorption on (1×2) and (1×1) surfaces at 120 K were less informative than those for CO because O(1s) spectra showed single broad peaks. Peak contributions due to bridge and on-top bonded NO could be estimated.