Numerical modeling of collapse in ideal incompressible hydrodynamics

The appearance of a singularity in the velocity-field vorticity ω at an isolated point irrespective of the symmetry of initial distribution is demonstrated numerically. The behavior of maximal vorticity |ω| near the collapse point is well approximated by the dependence (t ₀?t)^?1, where t ₀ is the collapse time. This is consistent with the interpretation of collapse as the breaking of vortex lines.     

Rise of proton proton total cross section and inclusive spectra

The origin of the rise of p-p total cross section (σ_t(pp)) is investigated by making use of the inclusive sum rule. The contribution to the rise of σ_t(pp) from the development of sharp peak near the kinematical boundary in the process p+p→p+ “anything” is almost completely canceled out by the decrease of the cross section in other x-region between s=45 (GeV/c)² and s=2820 (GeV/c)². Then the net contribution to the rise of σ_t(pp) from the process p+p→p+ “anything” is very likely to be zero. The true origin of the rise of σ_t(pp) is found to be the increase of inclusive cross sections at x ? 0 between PS and ISR energies. The contribution from the process p+p→π+ “anything” at x ? 0 to the rise of σ_t(pp) is estimated to be 3.2 mb, that from the process p+p→K + “anything”, 0.71 mb and that from the process p+p→$\text{p}$ “anything”, 0.62 mb. According to our conclusion we expect that all total cross sections will rise with energy at high energy.     

Measurement of high momentum transfer π−p → π0n at 5.9 GeV/c

The differential cross section for π^?p → π⁰n has been measured in the t range 1.8 ? |t| ? 8.2 (GeV/c)² by a counter-spark chamber experiment detecting the neutron and both π⁰ decay photons. A broad minimum was found, centered at |t| = 5.2 (GeV/c)².     

Instability of potential jet streams

A Hamiltonian version has been formulated for the model of a potential jet stream of a homogeneous incompressible fluid with a free boundary. In the framework of this model, instability regimes have been analyzed. It has been shown that self-similar solutions with a compact support can be dominant structures. Analysis of the instability mechanism shows that two collapse scenarios are possible. The first scenario occurs without the deformation of the shape and leads to an intensification of the vortex sheet according to the law (t ₀ ? t)^?1, where t ₀ is the collapse time. The second scenario leads to the formation of a singularity for the surface shape and to a decrease in the intensity of the vortex sheet according to the laws (t ₀ ? t)^?1/5 and (t ₀ ? t)^1/5, respectively. The integral collapse criterion has been found.     

Expanding Domain Limit for Incompressible Fluids in the Plane

The general class of problems we consider is the following: Let Ω ₁ be a bounded domain in \({\mathbb{R}^d}\) for d ≥ 2 and let u ⁰ be a velocity field on all of \({\mathbb{R}^d}\) . Suppose that for all R ≥ 1 we have an operator \({\mathcal{T}_R}\) that projects u ⁰ restricted to RΩ ₁ (Ω ₁ scaled by R) into a function space on RΩ ₁ for which the solution to some initial value problem is well-posed with \({\mathcal{T}_{R}u^0}\) as the initial velocity. Can we show that as R → ∞ the solution to the initial value problem on RΩ ₁ converges to a solution in the whole space? We answer this question when d  =  2 for weak solutions to the Navier-Stokes and Euler equations. For the Navier-Stokes equations we assume the lowest regularity of u ⁰ for which one can obtain adequate control on the pressure. For the Euler equations we assume the lowest feasible regularity of u ⁰ for which uniqueness of solutions to the Euler equations is known (thus, we allow “slightly unbounded” vorticity). In both cases, we obtain strong convergence of the velocity and the vorticity as R → ∞ and, for the Euler equations, the flow. Our approach yields, in principle, a bound on the rates of convergence.     

A nonlinear scenario for development of vortex layer instability in gravity field

A Hamiltonian version of contour dynamics is formulated for models of constant-vorticity plane flows with interfaces. The proposed approach is used as a framework for a nonlinear scenario for instability development. Localized vortex blobs are analyzed as structural elements of a strongly perturbed wall layer of a vorticity-carrying fluid with free boundary in gravity field. Gravity and vorticity effects on the geometry and velocity of vortex structures are examined. It is shown that compactly supported nonlinear solutions (compactons) are candidates for the role of particle-like vortex structures in models of flow breakdown. An analysis of the instability mechanism demonstrates the possibility of a self-similar collapse. It is found that the vortex shape stabilizes at the final stage of the collapse, while the vortex sheet strength on its boundary increases as (t ₀ ? t)^?1, where t ₀ is the collapse time.     

Blow-up instability in shallow water flows with horizontally-nonuniform density

The mechanisms of instability, whose development leads to the occurrence of the collapse (blow up), have been studied in the scope of the rotating shallow water flows with horizontal density gradient. Analysis shows that collapses in such models are initiated by the Rayleigh-Taylor instability and two scenarios are possible. Both the scenarios evolve according to a power law (t ₀ ? t)^??, where t ₀ is the collapse time, with ?? = ?1, ?2, and ?? = ?2/3, ?1 for the isotropic and anisotropic collapses, respectively. The rigorous criterion of collapse is found on the base of integrals of motion.     

Quantum deformation of compactified Minkowski space

The quantum GrassmanianG(2|0; ? _q ^4|0 ) of “quantum 2-planes ? _q ^2|0 in the quantum 4-plane ? _q ^4|0 ”, which provides aq-deformation of compactified complexified Minkowski space, is proposed. A quantum analogue of classical Plücker embedding of the usual GrassmanianG(2; ?²) into a non-degenerate quadric in ??⁵ is found.     

Computable upper bounds for the adiabatic approximation errors

For a given Hermitian Hamiltonian H(s)(s∈[0,1])with eigenvalues Ek(s)and the corresponding eigenstates|Ek(s)(1 k N),adiabatic evolution described by the dilated Hamiltonian HT(t):=H(t/T)(t∈[0,T])starting from any fixed eigenstate|En(0)is discussed in this paper.Under the gap-condition that|Ek(s)-En(s)|λ0 for all s∈[0,1]and all k n,computable upper bounds for the adiabatic approximation errors between the exact solution|ψT(t)and the adiabatic approximation solution|ψadi T(t)to the Schr¨odinger equation i|˙ψT(t)=HT(t)|ψT(t)with the initial condition|ψT(0)=|En(0)are given in terms of fidelity and distance,respectively.As an application,it is proved that when the total evolving time T goes to infinity,|ψT(t)-|ψadi T(t)converges uniformly to zero,which implies that|ψT(t)≈|ψadi T(t)for all t∈[0,T]provided that T is large enough.     

On the nonrelativistic limit of the Dirac theory

The relation between a “nonrelativistic” Hamiltonian of the formH ^∞=(A+B)²+C and a corresponding family of “Dirac-Hamiltonians”H(c) in the limitc → ∞ is investigated. It is shown that the resolvent (z ?H(c))^?1 and the relativistic perturbation of isolated eigenvalues ofH ^∞ are analytic in 1/c for sufficiently large |c|.     

Continuous sample paths in quantum field theory

Nelson's free Markoff field on ? ^l+1 is a natural generalization of the Ornstein-Uhlenbeck process on ?¹, mapping a class of distributions φ(x,t) on ? ^l ×?¹ to mean zero Gaussian random variables φ with covariance given by the inner product \(\left( {\left( {m^2 - \Delta - \frac{{\partial ^2 }}{{\partial t^2 }}} \right)^{ - 1} \cdot , \cdot } \right)_2 \) . The random variables φ can be considered functions φ〈q〉=∝ φ(x,t)q(x,t)d x dt on a space of functionsq(x,t). In the O.U. case,l=0, the classical Wiener theorem asserts that the underlying measure space can be taken as the space of continuous pathst →q(t). We find analogues of this, in the casesl>0, which assert that the underlying measure space of the random variables φ which have support in a bounded region of ? ^l+1 can be taken as a space of continuous pathst →q(·,t) taking values in certain Soboleff spaces.     

Determination of the one-pion exchange potential from pn and pp charge exchange scattering

The πNN vertex function is determined from $\text{d}\text{σ}\text{d}\text{t}$ for pn → np and $\text{p}\text{p →}\text{n}\text{n}$ at 8 GeV/c in the interval 0 < ? t < 0.1 GeV². A “regularor mass” of 3.5m_π=488 MeV is found, corresponding to an “extension” of 0.40 fm of the πNN vertex. The resulting OPE potential is discussed.     

A multi-channel scattering theory for some time dependent Hamiltonians,charge transfer problem

Scattering theory for time dependent HamiltonianH(t)=?(1/2) Δ+ΣV _j (x?q _j (t)) is discussed. The existence, asymptotic orthogonality and the asymptotic completeness of the multi-channel wave operators are obtained under the conditions that the potentials are short range: |V _j (x)|≦C _j (1+|x|)^?2?ε, roughly spoken; and the trajectoriesq _j (t) are straight lines at remote past and far future, and |q _j (t)?q _k (t)| → ∞ ast → ± ∞ (j≠k).     

The Bose-Einstein condensation in a finite one-dimensional homogeneous system of noninteracting bosons

Condensation of the ideal Bose gas in a closed volume having the shape of a rectangular parallel-epiped of length L with a square base of side length l (L ? l) is theoretically studied within the framework of the Bose-Einstein statistics (grand canonical ensemble) and within the statistics of a canonical ensemble of bosons. Under the condition N(l/L)⁴ ? l, where N is the total number of gas particles, dependence of the average number of particles in the condensate on the temperature T in both statistics is expressed as a function of the ratio t=T/T ₁, where T ₁ is a certain characteristic temperature depending only on the longitudinal size L. Therefore, the condensation process exhibits a one-dimensional (1D) character. In the 1D regime, the average numbers of particles in condensates of the grand canonical and canonical ensembles coincide only in the limiting cases of t → 0 and t → ∞. The distribution function of the number of particles in the condensate of a canonical ensemble of bosons at t ≤1 has a resonance shape and qualitatively differs from the Bose-Einstein distribution. The former distribution begins to change in the region of t ～ 1 and acquires the shape of the Bose-Einstein distribution for t ? 1. This transformation proceeds gradually that is, the 1D condensation process exhibits no features characteristic of the phase transition in a 3D system. For N(l/L)⁴ ? 1, the process acquires a 3D character with respect to the average number of particles in the condensate, but the 1D character of the distribution function of the number of particles in the condensate of a canonical ensemble of bosons is retained at all N values.     

A measurement of the KL0p → KS0p reaction between 4 and 14 GeV/c in the range 0.1 ⩽|t|⩽ 2 GeV2

We present experimental data on the K_L⁰p → K_S⁰p reaction between 4 and 14 GeV/c in the range 0.1 ? |t| ? 2 GeV². This experiment has been performed at the CERN PS, using spark chambers and a large aperture magnet. The results show a break of slope at t = ?0.3 GeV². The ω trajectory deduced from the data has an intercept α(0) = 0.5 and a slope α′ = 0.88. A comparison with various models shows that the non-flip amplitude is dominant.     

An experimental test of exchange degeneracy in the hyperchange exchange reactions π+p→K+Σ+and K−→π−Σ+

For the first time, the line reversed reactions π⁺p→K⁺Σ⁺and K^?p→π^?Σ⁺ have been studied in the same apparatus. We present the differential cross sections and polarizations over a large t range and at two momenta, 7.0 and 10.1 GeV/c. The differential cross sections as a function of t are shown for the first time to cross over. Going from the lower to the higher momentum, the differences in cross section between the two reactions diminish at low |t| by about a factor 2. We find large polarizations of opposite sign for the two reactions. The momentum dependence, presented in the form of α_eff(t) for the t range 0 to ?2 (GeV/c)², is compared with the expectations from the K¹?K⁷ trajectory.     

Phases of resonant amplitudes: πN → ϱN

Using SU(6)_W symmetry in its l-broken form we find that the newly reported relative signs of πN → ?N resonant amplitudes seem to indicate a universal “SU (6)_W-like” preference for all observed multiplets. Since the corresponding πN → πΔ phases favour an “anti-SU(6)_W” solution for the 70, L^P = 1^? multiplet the l-broken SU (6)_W model now faces a serious contradiction.     

Electroproduction of neutral pions at energies above the resonance region

The reaction e p→e'pπ⁰ has been measured at W=2.55 GeV a fixed electron scattering angle of 10.3°. Two magnetic spectrometers and a lead glass hodoscope were used to detect all four final state particles. Electroproduction cross sections in the t range ?0.15 to ?1.4 (GeV/c)² at q² = ?0.22, ?0.55 and ?0.85 (GeV/c)² are presented. Above |t|=0.6 (GeV/c)² the cross sections are considerably smaller than those for photoproduction.     

Classical and quantum regimes of superfluid turbulence

We argue that turbulence in superfluids is governed by two dimensionless parameters. One of them is the intrinsic parameter q which characterizes the friction forces acting on a vortex moving with respect to the heat bath, with q^?1 playing the same role as the Reynolds number Re=UR/ν in classical hydrodynamics. It marks the transition between the “laminar” and turbulent regimes of vortex dynamics. The developed turbulence described by Kolmogorov cascade occurs when Re?1 in classical hydrodynamics, and q?1 in superfluid hydrodynamics. Another parameter of superfluid turbulence is the superfluid Reynolds number Re_s=UR/κ, which contains the circulation quantum κ characterizing quantized vorticity in superfluids. This parameter may regulate the crossover or transition between two classes of superfluid turbulence: (i) the classical regime of Kolmogorov cascade where vortices are locally polarized and the quantization of vorticity is not important; (ii) the quantum Vinen turbulence whose properties are determined by the quantization of vorticity. A phase diagram of the dynamical vortex states is suggested.