On Intrinsically Random ℤ2-Actions on a Lebesgue Space

It is shown that a ${\mathbb{Z}}^2 $ -action on a Lebesgue space is intrinsically random (IR) iff it is a Kolmogorov action (K-action). As a consequence we obtain the fact that the ${\mathbb{Z}}^2 $ -action defined by the Lorentz gas is an IR-action and the ${\mathbb{Z}}^2 $ -action defined by the ideal gas is not an IR-action.     

Scaling properties of ℤ k- 1 actions on the circle

We derive universal scaling properties for ^k–1 actions on the circle whose generators have rotation numbers algebraic of degreek. As fork=2 these properties can be explained for arbitraryk in terms of a renormalization group transformation. It has at least one trivial fixed point corresponding to an action whose generators are pure rotations. The spectrum of the linearized transformation in this fixed point is analyzed completely. The fixed point is hyperbolic with a (k–1)-dimensional unstable manifold. In the casek=2 the known results are therefore recovered.     

Fermionic fields on ℤ N -curves

The line bundles of degreeg–1 on _N -curves corresponding to 1/N nonsingular characteristics are considered. The determinants of Dirac operators defined on these line bundles are evaluated in terms of branch points. The generalization of Thomae's formula for _N -curves is derived.     

On Effective Conductivity on ℤ d Lattice

We study the effective conductivity _e for a random wire problem on the d-dimensional cubic lattice ^d , d2 in the case when random conductivities on bonds are independent identically distributed random variables. We give exact expressions for the expansion of the effective conductivity in terms of the moments of the disorder parameter up to the 5th order. In the 2D case using the duality symmetry we also derive the 6th order expansion. We compare our results with the Bruggeman approximation and show that in the 2D case it coincides with the exact solution up to the terms of 4th order but deviates from it for the higher order terms.     

Kolmogorov–Sinaj Entropy on MV-Algebras

There is given a construction of the entropy of a dynamical system on arbitrary MV-algebra M. If M is the MV-algebra of characteristic functions of a σ-algebra (isomorphic to the σ-algebra), then the construction leads to the Kolmogorov–Sinaj entropy. If M is the MV-algebra (tribe) of fuzzy sets, then the construction coincides with the Maličky modification of the Kolmogorov–Sinaj entropy for fuzzy sets (Maličky and Riečan, 1986; Riečan and Mundici, 2002; Riečan and Neubrunn, 1997).     

Energy Spectrum of Helium Confined to a Two-Dimensional Space

Making use of the adiabatic hyperspherical approach, we report a calculation for the energy spectrum of the ground and low-excited states of a two-dimensional helium in a magnetic field. The results show that the ground and low-excited states of helium in low-dimensional space are more stable than those in three-dimensional space and there may exist more bound states.     

Inversion of Rayleigh Wave in a Stratified Half Space

Inversion of Rayleigh waves is studied for the first time by using the genetic algorithm based on the intensities of the fundamental and/or higher modes. The relations of all modes to the medium structure are investigated. It is emphasized that the higher modes and mode jumping should be considered in the inversion when the low-velocity     

Quantum Randomness as a Result of Random Fluctuations at the Planck Time Scale?

We show that the mathematical formalism of the quantum statistical model can be interpreted as a method for approximation of classical (measure-theoretic) averages on the infinite-dimensional phase space. The technique of approximation is based on the Taylor expansion of functionals of classical fields. To find the order of the deviation of quantum statistical predictions from the classical predictions, we use the time-scaling arguments. We show that quantum randomness might be considered as the result of random fluctuations at the Planck time-scale.     

Hydrodynamic equations for attractive particle systems on ℤ

Hydrodynamic properties for a class of nondiffusive particle systems are investigated. The method allows one to study local equilibria for a class of asymmetric zero-range processes, and applies as well to other models, such as asymmetric simple exclusion and misanthropes. Attractiveness is an essential ingredient. The hydrodynamic equations present shock wave phenomena. Preservation of local equilibrium is proven to hold away from the shocks. The problem of breakdown of local ergodicity at the shocks, which was investigated by D. Wick in a particular model, remains open in this more general setup.     

A Change of Coordinates on the Large Phase Space¶of Quantum Cohomology

The Gromov–Witten invariants of a smooth, projective variety V, when twisted by the tautological classes on the moduli space of stable maps, give rise to a family of cohomological field theories and endow the base of the family with coordinates. We prove that the potential functions associated to the tautological ψ classes (the large phase space) and the κ classes are related by a change of coordinates which generalizes a change of basis on the ring of symmetric functions. Our result is a generalization of the work of Manin–Zograf who studied the case where V is a point. We utilize this change of variables to derive the topological recursion relations associated to the κ classes from those associated to the ψ classes. Received: 2 August 1999 / Accepted: 30 September 2000     

Symplectic Structures of Moduli Space¶of Higgs Bundles over a Curve and Hilbert Scheme¶of Points on the Canonical Bundle

The moduli space of triples of the form (E,θ,s) are considered, where (E,θ) is a Higgs bundle on a fixed Riemann surface X, and s is a nonzero holomorphic section of E. Such a moduli space admits a natural map to the moduli space of Higgs bundles simply by forgetting s. If (Y,L) is the spectral data for the Higgs bundle (E,θ), then s defines a section of the line bundle L over Y. The divisor of this section gives a point of a Hilbert scheme, parametrizing 0-dimensional subschemes of the total space of the canonical bundle K _X , since Y is a curve on K _X . The main result says that the pullback of the symplectic form on the moduli space of Higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the Hilbert scheme using the map that sends any triple (E,θ,s) to the divisor of the corresponding section of the line bundle on the spectral curve. Received: 15 January 2000 / Accepted: 25 March 2001     

Methods of KAM-theory for long-range quasi-periodic operators on ℤ v . Pure point spectrum

We consider the class of quasi-periodic self-adjoint operators?(x)) = \(\hat D(x) + \hat V(x)\) ,x∈S ¹=?¹/?¹, on a multi-dimensional lattice ? ^v , with the matrix elements $$\hat D_{mn} (x) = \delta _{mn} D(x + n\omega ), \hat V_{mn} (x) = V(m - n, x + n\omega )$$ , whereD(x+1) =D(x), V(n, s+1) =V(n, x), ω ∈ ? ^v and |V(n, x)| ≤ εe ^?r|n|,r > 0. We prove that, if ε is small enough,V(n,·) andD(·) satisfy some conditions of smoothness, andD(·) is non-degenerate, then for a.e. ω and for a.e.x∈S ¹ the operator?(x) has pure point spectrum. All its eigenfunctions belong tol ¹(? ^v ).     

Global Solutions of Einstein—Dirac Equation on the Conformal Space

The difference between the Riemann and Lorentz spinor manifolds of four dimensions is that the Dirac operator of the former is elliptic and that of the latter is hyperbolic.Moreover the spinor group of the former is a compact group and that of the latter is a noncompact group,which is isomorphic to SL(2,C).Hence the results and their interpretation coming from the two theories would be different.In this short note we study only the Lorentz spinor manifold and,especially,the solutions of Einstein-Dirac equations on the conformal space,which is closely related to the AdS/CFT correspondence.     

The Navier–Stokes Equations in a Space of Bounded Functions

We establish a blow-up rate of the Navier–Stokes equations subject to the non-slip boundary condition for a certain class of domains including bounded and exterior domains.     

The Â-genus as a Projective Volume form on the Derived Loop Space

In the present work, we extend our previous work with Gwilliam by realizing \(\hat {A}(X)\) as the projective volume form associated to the BV operator in our quantization of a one-dimensional sigma model. We also discuss the associated integration/expectation map. We work in the formalism of L _∞ spaces, objects of which are computationally convenient presentations for derived stacks. Both smooth and complex geometry embed into L _∞ spaces and we specialize our results in both of these cases.     

Simulation of Space Charge Effect in Field Emission from a Nanowire

At present, field emission from nanowires (NWs) is under intense investigation. In experiments, the dependence of field emission current from an NW on applied voltage (I-V behavior) often deviates from the Fowler-Nordheim (FN) theory. We simulate the role of the space charge effect in the I-V behaviour of an NW. We determine the threshold electric field, at which the space charge effect starts to manifest itself, under different NW work functions. Our simulation result has confirmed that space charge effect is one of the possible factors responsible for the nonlinearity of the FN plots of an NW at high fields. Furthermore, our calculation reveals that the threshold field is related to the NW work function in an almost linear manner.     

Self-Deflection of Dark Screening Spatial Solitons Based on Higher-Order Space Charge Field

The effects of higher-order space charge field on the self-deflection of dark screening spatial solitons in biased photorefractive crystals are numerically investigated under steady-state conditions. The expression for an induced space-charge electric field including higher-order space-charge field terms is obtained. Numerical results indicate that dark solitons possess a self-deflection process during propagation, and the solitons always bend in the direction of the c axis of the crystal. The self-deflection of dark solitons can experience considerable increase especially in the regime of high bias field strengths.     

A Note on the Problem of Proper Time in Weyl Space–Time

We discuss the question of whether or not a general Weyl structure is a suitable mathematical model of space–time. This is an issue that has been in debate since Weyl formulated his unified field theory for the first time. We do not present the discussion from the point of view of a particular unification theory, but instead from a more general standpoint, in which the viability of such a structure as a model of space–time is investigated. Our starting point is the well known axiomatic approach to space–time given by Elhers, Pirani and Schild (EPS). In this framework, we carry out an exhaustive analysis of what is required for a consistent definition for proper time and show that such a definition leads to the prediction of the so-called “second clock effect”. We take the view that if, based on experience, we were to reject space–time models predicting this effect, this could be incorporated as the last axiom in the EPS approach. Finally, we provide a proof that, in this case, we are led to a Weyl integrable space–time as the most general structure that would be suitable to model space–time.     

Trace Construction of a Basis for the Solution Space¶of sl N qKZ Equation

The trace of intertwining operators over the level one irreducible highest weight modules of the quantum affine algebra of type A_Nу⁽¹⁾ is studied. It is proved that the trace function gives a basis of the solution space of the qKZ equation at a generic level. The highest-highest matrix elements of the composition of intertwining operators are explicitly determined as rational functions up to an overall scalar function. The integral formula for the trace is presented.