Complex Hyperspherical Equations, Nodal-Partitioning, and First-Excited-State Theorems in ℝ n

Three problems related to the spherical quantum billiard in are considered. In the first, a compact form of the hyperspherical equations leads to their complex contracted representation. Employing these contracted equations, a proof is given of Courant's nodal-symmetry intersection theorem for diagonal eigenstates of spherical-like quantum billiards in . The second topic addresses the first-excited-state theorem for the spherical quantum billiard in . Wavefunctions for this system are given by the product form, ( )Z _q+()Y ⁽ⁿ⁾ , where is dimensionless displacement, is angular-momentum number, qis an integer function of dimension, Z() is either a spherical Bessel function (nodd) or a Bessel function of the first kind (neven) and represents (n– 1) independent angular components. Generalized spherical harmonics are written . It is found that the first excited state (i.e., the second eigenstate of the Laplacian) for the spherical quantum billiard in is n-fold degenerate and a first excited state for this quantum billiard exists which contains a nodal bisecting hypersurface of mirror symmetry. These findings establish the first-excited-state theorem for the spherical quantum billiard in . In a third study, an expression is derived for the dimension of the th irreducible representation (irrep) of the rotation group O(n) in by enumerating independent degenerate product eigenstates of the Laplacian.     

Critical dynamics of φ2n with random fields in the tree approximation

In the tree approximation, the critical relaxational dynamics of ²ⁿ with quenched random fields is calculated by renormalization near the upper critical dimensiond _u=4+2/(n–1). Although the relation between the characteristic frequency _c q and the wave-numberq is as usual,unusual results are obtained for the asymptotic decay of the order-parameter correlation functionS(q,t).Dedicated to Prof. Dr. W. Döring on the occasion of his 75th birthday     

Solutions with high dimensional singular set,to a conformally invariant elliptic equation in ℝ4 and in ℝ6

We construct positive weak solutions to the equation , where –₀ denotes the conformal Laplacian on then-sphere (n=4, 6), having singular sets of Hausdorff dimension greater than or equal to .     

Nonlinear sigma model,Zakharov-Shabat method,and new exact forms of the minimal surfaces in ℝ3

General formulas for the construction of exact solutions of the equation of the minimal surface in ?³, which appears in various physical problems, have been derived by the Zakharov-Shabat “dressing” method. Particular examples are considered.     

Period doubling bifurcations for families of maps on ℝ n

Infinite sequences of period doubling bifurcations in one-parameter families (1-pf) of maps enjoy very strong universality properties: This is known numerically in a multitude of cases and has been shown rigorously for certain 1-pf of maps on the interval. These bifurcations occur in 1-pf of analytic maps at values of the parameter tending to a limit with the asymptotically geometric ratio 1 /4.6692 ....In this paper we indicate the main steps of a proof that the same is true for 1-pf of analytic maps from ⁿ to ⁿ , whose restriction to ⁿ is real.Work supported by the Fonds National Suisse, and by the National Science Foundation under Grant PHY-79-16812.     

Around Polygons in ℝ3 and S3

We survey certain moduli spaces in low dimensions and some of the geometric structures that they carry, and then construct identifications among all of these spaces. In particular, we identify the moduli spaces of polygons in ℝ³ and S ³, the moduli space of restricted representations of the fundamental group of a punctured 2-sphere, the moduli space of flat connections on a punctured sphere, the moduli space of parabolic bundles on a sphere, the moduli space of weighted points on ℂℙ¹ and the symplectic quotient of SO(3) acting diagonally on (S ²) ⁿ . All of these spaces depend on parameters and some the above identifications require the parameters to be small. One consequence of this work is that these spaces are all biholomorphic with respect to the most natural complex structures they can each be given. Received: 20 September 1999 / Accepted: 28 November 2000     

The spectrum of operators elliptic along the orbits of ℝ n actions

It is shown that a periodic elliptic operator on ⁿ has no eigenvalues off of the set of discontinuities of its spectral density function. The methods involve operator algebras and are based on a spectral duality principal first introduced by J. Bellisard and D. Testard. A version of the spectral duality theorem is proved which relates the point spectrum of a certain family of operators to the continuous spectrum of an associated family.Research sponsored in part by NSF grants     

The Klein-Gordon Operator on Möbius Strip Domains and the Klein Bottle in ℝ n

In this paper we present explicit formulas for the fundamental solution to the Klein-Gordon operator on some higher dimensional generalizations of the Möbius strip and the Klein bottle with values in distinct pinor bundles. The fundamental solution is described in terms of generalizations of the Weierstraß $\wp $ -function that are adapted to the context of these geometries. The explicit formulas for the kernel then allow us to express all solutions to the homogeneous and inhomogeneous Klein-Gordon problem with given boundary data in the context of these manifolds. In the case of the Klein bottle we are able to describe all null solutions of the Klein-Gordon equation in terms of finite linear combinations of the fundamental solution and its partial derivatives.     

Measurement of(n,α)and(n,2n)reaction cross sections at a neutron energy of 14.92±0.02 MeV for potassium and copper with uncertainty propagation

Experimentally measured neutron activation cross sections are presented for the ⁶⁵Cu(n,0)^62mCu,⁴¹ K(n,a)^38Cl,and ⁶⁵Cu(n.2n)⁶⁴Cu reactions with detailed uncertainty propagation.The neutron cross secions were measured at an incident energy of 14.92±0.02 MeV,and the neutrons were based on the(d,n)a fusion reaction.The ²⁷ Al(n,a)²⁴Na reaction was used as a reference reaction for the normalization of the neutron flux.The pre-calib-rated lead-shielded HPGe detector was used to detect the residues'γ-ray spetra.The data from the measured cross sections are compared to the previously measured cross sections from the EXFOR database,theoretically calculated cross sections using the TALYS and EMPIRE codes,and evaluated nuclear data.     

A study of (n,nα) reaction cross sections at 14.7 MeV

Cross sections for (n, nα) reactions induced by 14.7 ± 0.3 MeV neutrons on ⁵¹V, ⁹²Mo, ¹¹⁵In, ¹⁶⁵Ho and ¹⁷⁶Yb have been measured by the activation technique using radiochemical separations and high resolution γ-ray spectroscopy or low-level β⁻counting. A brief summary of the activation data reported in the literature is given and a tentative trend in the data is suggested. The contribution of the (n, nα) process generally amounts to between 10 and 15% of the (n, α) cross section; in some cases, however, it is as low as 0.5%. The summed (n, α) and (n, nα) reaction cross sections obtained by the activation technique are generally in agreement with those deduced from emitted α-particle characterisation as well as mass spectrometry. Detailed Hauser-Feshbach calculations show that the statistical model can describe the (n, nα) cross section only with partial success.     

Cubic anisotropy created by defects of “random local anisotropy” type,and phase diagram of the O(n) Model

Physics of the Solid State - The expression for the cubic-type-anisotropy constant created by defects of “random local anisotropy” type is derived. It is shown that the Imry–Ma...     

Existence,Non-existence,and Uniqueness for a Heat Equation with Exponential Nonlinearity in ℝ2

We consider a semilinear heat equation with exponential nonlinearity in ?². We prove that local solutions do not exist for certain data in the Orlicz space exp L ²(?²), even though a small data global existence result holds in the same space exp L ²(?²). Moreover, some suitable subclass of exp L ²(?²) for local existence and uniqueness is proposed.

     

Large scale structures in Rayleigh-Bénard convection at high Rayleigh numbers

Direct numerical simulations of Rayleigh-Bénard convection in a plane layer with periodic boundary conditions at Rayleigh numbers up to 10(7) show that flow structures can be objectively classified as large or small scale structures because of a gap in spatial spectra. The typical size of the large scale structures does not always vary monotonically as a function of the Rayleigh number but broadly increases with increasing Rayleigh number. A mean flow (whose average over horizontal planes differs from zero) is also excited but is weak in comparison with the large scale structures. The large scale circulation observed in experiments should therefore be a manifestation of the large scale structures identified here.     

Measurement of 232Th(n, γ) and 232Th({n}, {2n}) cross-sections at neutron energies of 13.5, 15.5 and 17.28?MeV using neutron activation techniques

The ²³²Th(n, ??) reaction cross-section at average neutron energies of 13.5, 15.5 and 17.28?MeV from the ⁷Li(p, n) reaction has been determined for the first time using activation and off-line ??-ray spectrometric technique. The ²³²Th(n, 2n) cross-section at 17.28?MeV neutron energy has also been determined using the same technique. The experimentally determined ²³²Th(n, ??) and ²³²Th(n, 2n) reaction cross-sections from the present work were compared with the evaluated data of ENDF/BVII and JENDL-4.0 and were found to be in good agreement. The present data, along with literature data in a wide range of neutron energies, were interpreted in terms of competition between ²³²Th(n, ??), (n, f), (n, nf) and (n, xn) reaction channels. The ²³²Th(n, ??) and ²³²Th(n, 2n) reaction cross-sections were also calculated theoretically using the TALYS 1.2 computer code and were found to be in good agreement with the experimental data from the present work but were slightly higher than the literature data at lower neutron energies.     

The autocorrelation of speckles in deep Fresnel diffraction region and characterizations of random self-affine fractal surfaces

This paper studies the correlation properties of the speckles in the deep Fresnel diffraction region produced by the scattering of rough self-affine fractal surfaces. The autocorrelation function of the speckle intensities is formulated by the combination of the light scattering theory of Kirchhoff approximation and the principles of speckle statistics. We propose a method for extracting the three surface parameters, i.e. the roughness w, the lateral correlation length ξ and the roughness exponent α, from the autocorrelation functions of speckles. This method is verified by simulating the speckle intensities and calculating the speckle autocorrelation function. We also find the phenomenon that for rough surfaces with α= 1, the structure of the speckles resembles that of the surface heights, which results from the effect of the peak and the valley parts of the surface, acting as micro-lenses converging and diverging the light waves.     

Decay Estimates of Solutions for the Equations of Motion of Compressible Viscous and Heat-Conductive Gases in an Exterior Domain in ℝ3

We consider the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in ³. We give the L_^qmL_^p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem.     

Excitation of isomeric states in the (γ,n), (n, 2n), and (γ, 2n) reactions on the 113In nucleus

The induced-activity method was used to measure the isomeric ratio of the yields of the (γ, n), (n, 2n), and (γ, 2n) reactions on the ¹¹³In nucleus. The energy dependence of the isomeric ratios of the yields of the photonuclear reactions ¹¹³In(γ, n)^112m,g In and ¹¹³In(γ, 2n)^111m,g In is studied in the energy range of 12–35 MeV.