Strange objects in the complex plane

Julia sets are examined as examples of strange objects which arise in the study of long time properties of simple dynamical systems. Technically they are the closure of the set of unstable cycles of analytic maps. Physically, they are sets of points which lead to chaotic behavior. The mapf(z)=z²+p is analyzed for smallp where the Julia set is a closed curve, and for largep where the Julia set is completely disconnected. In both cases the Hausdorff dimension is calculated in perturbation theory and numerically. An expression for the rate at which points escape from the neighborhood of the Julia set is derived and tested in a numerical simulation of the escape.     

Vector meson dominance model in γ+γ*(Q 2)→f(1270,2+)→ππ

Constructed is the vector dominance model for the γ+γ^*(Q ²)→f(1270,2⁺) process which is in good agreement with the experiment. We note that the experimental data do not require the consideration of the generalized vector dominance model. It is also shown that the vector dominance model (non-generalized) with the natural assumption of the vector meson-hadron scattering total cross section independence of vector meson helicity state describes well the total cross section of γγ^*→hardons process in the Regge region.     

The exact solution of an elimination problem in kinetic theory: I. The one-dimensional hard sphere gas

A class of initial value problems for a one-dimensional hard sphere gas is considered where a specified particle has a given distribution f⁽¹⁾(z₁; 0) and the rest are in equilibrium at t=0. An exact expansion is obtained for a certain n-particle reduced distribution function f⁽ⁿ⁾(z₁;…;z_n; t) in terms of the 1-particle reduced distribution function f⁽¹⁾(z₁; t) for the specified particle by starting with separate expressions for these functions in terms of f⁽¹⁾(z₁; 0). Expansions for the corresponding cluster functions are first obtained and then graph theoretic methods applied to obtain a solution.     

Study of inclusive production of boson resonances in\bar pp-interactions at 32 GeV/c

The inclusive production ofK ^*±(890), ρ^ρ(770), ω⁰(780) andf ⁰(1,270) mesons is studied in \(\bar pp\) experiment at 32GeV/c in the MIRABELLE bubble chamber by the analysis of (K ⁰π^±) and (π⁺π^?) invariant mass distributions. The inclusive cross sections are σ(K ^*±)=2.45±0.30 mb, σ(ρ⁰)=8.8 ±0.9 mb, σ(ω⁺)?6.2 mb, and σ(f ⁰)=2.2±0.7 mb. The C.M. longitudinal distribution of theK ^*±(890) is slightly shifted to the primary particles fragmentation regions; the ρ⁰ is produced more centrally. The studied mesons alone are responsible for 40% of final state poins. The production features of various resonances are compared with quark models predictions.     

Statistical analysis of24Mg+24Mg elastic and inelastic scattering

The data on the excitation functions of²⁴Mg+²⁴Mg elastic and inelastic (²⁴Mg +²⁴Mg^*(2⁺),²⁴Mg^*(2⁺)+²⁴Mg^*(2⁺),²⁴Mg+²⁴Mg^*(4⁺),²⁴Mg^*(4⁺)+²⁴Mg^*(2⁺),²⁴Mg+²⁴Mg^*(6⁺)) scattering fromE _c.m=42 to 56 MeV have been subjected to a statistical analysis consisting of calculations of deviation function, cross-correlation function, cross-channel correlation coefficients, coherence widths, and the distribution of cross sections. On the basis of the analysis resonant structures atE _c.m=45.70, 46.65, 47.35 and 47.75 MeV have been confirmed. Two new resonant structures atE _c.m=44.55 and 50.50 MeV have been identified.     

InclusiveK *+(1430),K *0(1430), andf(1270) production inK + p interactions at 32 GeV/c

Results are presented on total and semiinclusive cross sections, longitudinal and transverse momentum distributions of the tensor mesonsK ^*+(1430),K ^*0(1430), andf(1270) observed inK ⁺ p interactions at 32 GeV/c. The data are compared withK ^*+(892).K ^*0(896) and ?⁰ vector meson production in the same experiment. The shapes of inclusive invariantx-spectra for the strange vector and tensor mesons are found to be rather similar, after removal of diffractive and quasi-two-body channels, in agreement with quark-recombination ideas.     

Production of meson resonances as leading particles in jets in proton-proton collisions at\sqrt s = 62 GeV at the CERN ISR

The production of ρ(770)⁰,K ^*(892)⁰, \(\bar K^* (892)^0 \) andf ₂(1270) mesons as leading particles in hadron jets (<z>～0.7) has been measured in deep inelasticpp interactions at the CERN ISR at \(\sqrt s = 62\) GeV. The comparison of vector and pseudoscalar meson production at the same transverse momentum provides a rather model independent way to determine the fragmentation parameterV/P. We determine(V/P) _u,d =1.66±0.18±0.63 from the ρ/π ratio and(V/P) _s =0.90±0.13±0.27 from theK ^*/K ratio. For thef ₂(1270) production we findf ₂/ρ⁰=0.30±0.08.     

Inclusive meson resonance production inK + p interactions at 250 GeV/c

A detailed analysis of the inclusive production of the vector mesonsφ,K ^*+,0(892) \(\bar K*^0 (892),\rho ^{ + ,0} ,\omega \) and the tensor mesonsK ₂ ^*0 )(1430) andf ₂(1270) inK ⁺ p interactions at 250 GeV/c is presented The data are compared with results at lower energies and with various quark-parton models. The production ofρ ⁰,K ^*0(892) and \(\bar K*^0 (892)\) increases at the same rate as a function ofs, is concentrated in the central region and is not reproduced by the models. Production of the tensor mesonsf ₂(1270) andK ₂ ^*0 (1340) is suppressed relative toρ ⁰ andK ^*0(892) by a factor of about 3.     

An uncertainty principle for fermions with generalized kinetic energy

We derive semiclassical upper bounds for the number of bound states and the sum of negative eigenvalues of the one-particle Hamiltoniansh=f(?i?)+V(x) acting onL ²(?ⁿ). These bounds are then used to derive a lower bound on the kinetic energy \(\sum\limits_{j = 1}^N {\left\langle {\psi ,f( - i\nabla _j )\psi } \right\rangle }\) for anN-fermion wavefunction ψ. We discuss two examples in more detail:f(p)=|p| andf(p)=(p ²+m ²)^1/2?m, both in three dimensions.     

Julia Sets in Parameter Spaces

Given a complex number λ of modulus 1, we show that the bifurcation locus of the one parameter family {f _b (z)=λz+b z ²+z ³} _{b ∈ ℂ} contains quasi-conformal copies of the quadratic Julia set J(λz+z ²). As a corollary, we show that when the Julia set J(λz+z ²) is not locally connected (for example when z↦λz+z ² has a Cremer point at 0), the bifurcation locus is not locally connected. To our knowledge, this is the first example of complex analytic parameter space of dimension 1, with connected but non-locally connected bifurcation locus. We also show that the set of complex numbers λ of modulus 1, for which at least one of the parameter rays has a non-trivial accumulation set, contains a dense G _δ subset of S ¹. Received: 22 September 2000 / Accepted: 16 January 2001     

Tc级数解的收敛性质

本文证明了,当|z_ph|c级数解的收敛圆是由奇点决定的,但是只要1/λ<[0.63+3μ⁺(1/Λ₁)] 1/(Λ₁),T_c级数解在收敛圆外仍然是个很好的渐近展开式,其中1/Λ₁=min(|z|,1);μ⁺(x)=(1-2μ^*x)μ^*。 关键词：     

The reaction Pomeron-Pomeron →π + π − and an unusual production mechanism for thef 2 (1270)

Data are presented on Pomeron-Pomeron interactions which produce a centralπ ⁺ π ^? system in proton-proton collisions at \(\sqrt s = 62 GeV\) at the CERN Intersecting Storage Rings. This process may favor the production of gluonic bound states. A partial-wave analysis of theπ ⁺ π ^? system shows evidence for the production of the statesf ₀(975),f ₀(1400), andf ₂(1270). The fitted mass for thef ₂(1270) is about 50 MeV below the world average. In addition, the production mechanism for thef ₂(1270) is uniquely different from that for the other final states in that there is a correlation between the outgoing protons. this is consistent with a picture of two-gluon exchange with thef ₂(1270) produced by gluon fusion, and could indicate that thef ₂(1270) has a glueball component.     

Search for e + e − → f 0(600)γ, f 0(980)γ, f 0(1350)γ, and f 2(1270)γ processes in the energy range from 1.05 to 1.38 GeV

In the energy range from 1.05 to 1.38 GeV, the upper limits for the cross sections of e ⁺ e ^? → f ₀(600)γ, f ₀(980)γ, f ₀(1350)γ, and f ₂(1270)γ → π⁰π⁰γ processes have been established. Measurements are performed in accordance with the statistics gathered in experiment with a spherical neutral detector (SND) on the VEPP-2M e ⁺ e ^? collider. The resultant upper limits vary from 6 to 42 pb depending on the energy and the model.     

A study of the centrally producedK *0 K^{*0} \bar K^{*0} final state in the reactionpp→p f (K + K -π+π-)p s at 300 GeV/c

The reactionpp→p _f (K ⁺ K ^-π⁺π^-)p _s , where theK ⁺ K ^? π⁺π^- system is centrally produced, has been studied at 300 GeV/c. TheK ^*0 \(K^{*0} \bar K^{*0} \) final state has been observed and the cross sections for its central production are found to be the same at 300 and 85 GeV/c. TheK ^*0 \(K^{*0} \bar K^{*0} \) final state appears to be produced as a non-resonant threshold enhancement.     

Nonstatistical structures in the excitation functions of20Ne(16O,12C)24Mg and20Ne(16O,20Ne)16O reactions

The data on the excitation functions of²⁰Ne(¹⁶O,¹²C)²⁴Mg,²⁰Ne(¹⁶O,¹²C)²⁴Mg^*(1.37, 2⁺),²⁰Ne(¹⁶O,¹²C)²⁴Mg^*(4.12, 4⁺+4.24, 2⁺) +²⁰Ne(¹⁶O,¹²C^*(4.44, 2⁺))²⁴Mg,²⁰Ne(¹⁶O,¹²C)²⁴Mg^*(6.01, 4⁺+6.43, 0⁺),²⁰Ne(¹⁶O,²⁰Ne)¹⁶O,²⁰Ne(¹⁶O,²⁰Ne^*(1.63, 2⁺))¹⁶O, and²⁰Ne(¹⁶O,²⁰Ne^*(4.25, 4⁺))¹⁶O reactions atθ _lab=13° fromE _c.m.=22.8 to 38.6 MeV have been subjected to a statistical analysis comprising of the calculations of the distribution of cross sections, deviation functions, cross-correlation functions, summed excitation functions, cross-channel correlation coefficients and coherence widths. The analysis confirms the existence of nonstatistical structures atE _c.m.=24.6, 27.8, 31.7 and 35.5 MeV, and identifies a new structure of the same nature atE _c.m. =25.6 MeV.     

Oscillator strengths for SiH and SiH deduced from the solar spectrum

Recent identifications of SiH (A²Δ-X²Π) and SiH⁺ (A¹Π-X¹Σ⁺) in the solar photospheric spectrum have enabled us to derive absolute oscillator strengths for the (0, 0) bands of these transitions: f₀₀(SiH) = 0.0033 and f₀₀(SiH⁺) = 0.0005. Our result for SiH is compared with other values.     

Characterization of polycrystalline (Na,K)β/β′- alumina ceramics

K β′-alumina is unstable at >1300°C. Mixed alkali β′-alumina has a variable stability depending on the alkali ratio, [K⁺]/([Na⁺]+[K⁺]). For f(β)<[K⁺]/([Na⁺]+[K⁺]), the β′-Al₂O₃ phase decomposes to Kβ-Al₂O₃ 0997 0815 V 3 and a solid solution of Na β′-Al₂O₃ and K β′-Al₂O₃. For f(β)=[K⁺]/([Na⁺]+[K⁺], the ceramic consists of K β-Al₂O₃ and Naβ′-Al₂O₃ and for f(β)>[K⁺]/([Na⁺]+[K⁺]), the excess Na⁺ after Na β′-Al₂O₃ dissolves in the β phase, giving Na β-Al₂O₃/K β-Al₂O₃ solid solution and Na β′-Al₂O₃. These sequences were confirmed by measuring the dependence of the c-axis lattice parameters of β- and β′-Al₂O₃ phases on the f(β), and the change of these parameters during the ion-exchange of Na⁺ and K⁺ ions.     

Excited lepton production at LEP and HERA

General expressions for single and pair production cross sections of excited leptons (e ^*,µ^*,v ^*) are presented. Specific results are shown for a realisticSU(2)×U(1) invariant model. Pair production ine ⁺ e ^? annihilation can measure anomalous magnetic moments of excited leptons. Single production ofe ^* is dominated by thet-channel γ exchange contribution which makes its detection feasible up to masses just below thee ⁺ e ^? c.m. energy. Due to this small |t| enhancement effect, contributions from elastic and resonance scattering inep production ofe ^* are substantial. Realistic estimates of the excited lepton production cross section at HERA are given     

Weak-coupling expansion of the low-lying energy values in the SU(2) gauge theory on a torus

Relying on analytic results obtained previously, we complete the one-loop computation of the low-lying energy values of the SU(2) gauge theory in an L × L × L periodic box. The expansions are then rewritten in terms of the universal parameter z = M(0⁺) · L (M(0⁺): energy gap in the J^p = 0⁺ sector). We find that the crossover from small-volume to large-volume behaviour is likely to take place at z ? 2. Furthermore, near the crossover, the lowest energy levels (above the ground state) in the 0⁺ sector and the 2⁺ sector are practically degenerate. In each of these sectors there is one more state at about 1.5 · M(0⁺). In the 0^? sector, on the other hand, the lowest energy value is greater than 3 · M(0⁺).