Table of contents

The knapsack problem is an NP-complete combinatorial optimization problem with inequality constraints. Using the replica method of statistical physics, we study the space of its solutions for a large problem size. It turns out that this problem is closely related to the theory of the binary perceptron.

We generalize the A-B₂ surface reaction model of ZGB to include the Eley-Rideal process, and introduce a 'partial simulation' method for computer simulations. Numerical results show that the second-order phase transition disappears when the ER process takes part in the kinetics.

In the final jammed state of the random sequential adsorption of dimers on a one-dimensional lattice, the average gap neighbouring a particle that originally adsorbed at time t is e^-te-e^e(-tt). Thus, the average final gap adjacent to the first particle to adsorb is e^-1, while that for the last particle is infinitesimally small. This result shows that there remains a (statistically) measurable imprint of the sequence of arrival frozen into the RSA system.

Consideration is given to the treatment of the nonlinear partial differential equation describing the diffusion of an assembly of particles which simultaneously coagulate or annihilate. It is shown how a class of solutions may be obtained in terms of the solutions of an ordinary differential equation and detailed application is made to the case of an initially localized particle distribution.

The N=4 periodic closure of the factorization chain is considered. It is shown that the nonlinear operator algebra corresponding to this closure can be transformed into the quadratic Hahn algebra. As a result, the three-term recurrence coefficients for the Hahn polynomials provide a special realization of the N=4 periodic factorization chain.

The non-local conserved currents for the restricted quantum conformal sl(2) Toda field system are obtained non-perturbatively, and the relations between these results and the perturbative BLs are discussed.

A path integral formulation in the representation of new coherent states for the Lie superalgebra osp(l/2, R) is introduced. By the use of the completeness relation of the new coherent states, a path integral expression for the transition amplitude between two osp(l/2, R) new coherent states for a Hamiltonian which is linear in the generators of the superalgebra are obtained. In the classical limit the equations of motion for the system are derived.

Quadratic Poisson brackets on a vector space equipped with a bilinear multiplication are studied. A notion of a bracket compatible with the multiplication is introduced and an effective criterion of such compatibility is given. Among compatible brackets, a subclass of coboundary brackets is described, and such brackets are enumerated in a number of examples.

A path-integral formulation in the representation of coherent states for the supergroup Osp(1 mod 2) is introduced. An expression for the transition amplitude connecting two Osp(1 mod 2) coherent states is constructed and the corresponding canonical equations of motion derived. A set of generalized Poisson brackets is introduced.

We construct two Osp(n mod 2m) solutions of the graded Yang-Baxter equation by using the algebraic braid-monoid approach. The factorizable S-matrix interpretation of these solutions is also discussed.

This letter presents exact surface of section reduction of quantum mechanics. The main theoretical result is a decomposition of the resolvent of the autonomous bound Hamiltonian H,(E-H)^-1 in terms of the four propagators which propagate from and/or to Hilbert space over the configuration space (CS) to and/or from Hilbert space over the configurational surface of section (CSOS), which has one dimension less than CS. The exact energy quantization condition can be expressed solely in terms of the CSOS-CSOS propagator. All these newly defined energy-dependent (CS/CSOS)-(CS/CSOS) propagators are expressed in terms of the resolvent of the related scattering Hamiltonian.

We present a dynamical Monte Carlo algorithm which is applicable to systems satisfying a clustering condition: during the dynamical evolution the system is mostly trapped in deep local minima (as happens in glasses, pinning problems etc). We compare the algorithm to the usual Monte Carlo algorithm using, as an example the Bernasconi model. In this model, a straightforward implementation of the algorithm gives an improvement of several orders of magnitude in computational speed with respect to a recent, already very efficient, implementation of the algorithm of Bortz et al. (1975).

The dynamics of segregation of a binary mixture AB with chemically active components are studied. For simplicity, the study is confined to chemical reactions in which the reactants and end-products consist of only A or B. A phenomenological model and detailed numerical results for the reaction AB to or from BB in the specific case when the forward and backward reactions proceed at the same rate are presented.

The author develops a novel scheme of statistical inference whereby statistical weights are assigned to folding pathways. Evidence is presented that supports the fact that this scheme accounts for the robustness and expediency of biopolymer folding processes. The essential properties of folding are captured by showing that the weight is concentrated over a very limited domain of closely related folding pathways. To make probabilistic inferences, the author constructively defines a measure eta over the space of folding pathways. Such a scheme stands in contrast to traditional methods built upon a Boltzmann measure over conformation space. In order to implement and validate this new approach the author combines analytical theory and computations that successfully reproduce pulse-chase kinetic experiments. The author first presents a rigorous analytical result by proving that an appropriate measure exists over the space of folding pathways. This existence theorem is shown to hold in two general scenarios: (i) the unbiased folding (UF) scenario, in which the complete chain starts its search in conformation space in an unbiased manner; (ii) the sequential folding (SF) scenario, in which the chain starts searching in conformation space concurrently with its own sequential assembling by progressive incorporation of monomers. A systematic coarse-graining simplification of the space of folding pathways is implemented to make the computations feasible and to validate the author's theory as a means of accounting for the expedient way of searching for the functionally competent conformation.

With the help of scaling methods, a general relation is established between the thermodynamic pressure and the mechanical pressure tensor of an equilibrium one-component plasma in a magnetic field. The mechanical pressure tenser is shown to be anisotropic. A general proof of the compressibility sum rule for a magnetized quantum plasma is presented. Finally, fourth-order wavenumber inequalities for the static charge correlation function are derived.

We consider one type of gauge-invariant anyon operators from the complex scalar field theory minimally coupled to the Chern-Simons term in 2+1 dimensions. In the Coulomb gauge condition, the anyonicity of operators, the multi-valued states, and the spin-statistics relation are derived by considering the proper distribution function, which appears in the definition of anyon operators. It is shown that the anyonicity is not a gauge artifact by also obtaining it in the covariant gauge.

A new critical-amplitude relation which holds for one-dimensional quantum ground-state transitions is presented. The relation yields an estimate of the sound velocity which appears in the conformal field theory as an a priori unknown parameter. With the use of this relation, the exponent eta can be explicitly determined from only the scaling fit of the off-critical energy gap. The relation is confirmed in the transverse Ising model, the S= 1/2 anisotropic XY model, and the three-state Potts model.

We compute the semiclassical magnetization and susceptibility of non-interacting electrons, confined by a smooth two-dimensional potential and subjected to a uniform perpendicular magnetic field, in the general case when their classical motion is chaotic. It is demonstrated that the magnetization per particle m(B) is directly related to the staircase function N(E), which counts the single-particle levels up to energy E. Using Gutzwiller's trace formula for N, we derive a semiclassical expression for m. Our results show that the magnetization has a non-zero average, which arises from quantum corrections to the leading-order Weyl approximation to the mean staircase and which is independent of whether the classical motion is chaotic or not. Fluctuations about the average are due to classical periodic orbits and do represent a signature of chaos. This behaviour is confirmed by numerical computations for a specific system.

A random-tiling model with octagonal symmetry is presented. The tiles are the square and various hexagons. When each vertex of the tiling is decorated by a disc, a valid disc packing is formed, which is an octagonal disc packing of maximum density under certain constraints. A simple inflation rule is given that produces one member of the random-tiling ensemble. The space group of this tiling is non-symmorphic. The projection description of this tiling involves a fractal acceptance domain with fourfold symmetry for the even nodes of a four-dimensional cubic lattice and the same acceptance domain rotated by pi /4 in perpendicular space for the odd nodes. Other tilings can be generated by an inflation rule with constrained randomness. Additional members of the random-tiling ensemble can be created from inflation-generated tilings by a set of update moves that rearrange the positions of the discs along closed loops. An update move does not always conserve the number of each kind of tile.

Spin models on quenched random graphs are related to many important optimization problems. We give a new derivation of their mean-field equations that elucidates the role of the natural order parameter in these models.

The general solutions for the factorization equations of the reflection matrices K^+or-( theta ) for the eight-vertex and six-vertex models (XYZ, XXZ and XXX chains) are found. The associated integrable magnetic Hamiltonians are derived explicitly, finding families depending on several continuous as well as discrete parameters.

Non-equilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and nearest-neighbour spin exchanges at T= infinity are investigated numerically from the point of view of a phase transition. The branching annihilating random walk of the ferromagnetic domain boundaries determines the steady state of the system for a range of parameters of the model. Critical exponents obtained by simulation are found to agree, within error, with those in Grassberger's cellular automata.

Using nonequilibrium transport theory, the thermal denaturation of DNA molecules is investigated as a preliminary step to clarify the dynamical process of DNA transcription. The distribution functions of the displacement of base pairs at different temperatures are calculated. A modified model is also proposed which can reproduce the essential features of the denaturation process. Calculations show that the Langevin equation is an effective tool for describing the dynamical process of DNA molecules, and it seems to be more advantageous than the Nose method.

We consider the scattering of a free quantum particle on a singular potential with rather arbitrary support boundary geometry. In the classical limit h(cross)=0, this problem reduces to the well known problem of chaotic scattering. The universal estimates for the stability of the scattering amplitudes are derived. The application of the obtained results to the mesoscopic systems and quantum chaos are discussed. We also discuss a possibility of experimental verification of the obtained results.

We consider classically chaotic systems with the topology of a ring threaded by quantum flux phi . Using semiclassical asymptotics, we calculate the flux-averaged autocorrelation function C( phi ) of slopes of the energy levels (persistent currents), normalized by the mean level spacing, for flux values differing by phi . Our result furnishes the uniform approximation C( phi ) approximately=-(sin²( pi phi )-1/w*²)/(sin²( pi phi )+1/w*²)². Here w*, the RMS winding number of the classical periodic orbits whose period is connected by Heisenberg's relation to the mean level spacing, is a (large) semiclassical parameter, of order 1/h(cross)^(D-12/) for a system with D freedoms.

We used the Riemann problem method with a 3*3 matrix system to find the femtosecond single soliton solution for a perturbed nonlinear Schrodinger equation which describes bright ultrashort pulse propagation in properly tailored monomode optical fibres. Compared with the Gel'fand-Levitan-Marchenko approach, the major advantage of the Riemann problem method is that it provides the general single soliton solution in a simple and compact form. Unlike the standard nonlinear Schrodinger equation, here the single soliton solution exhibits periodic evolution patterns.

New asymptotic formulae for the two-dimensional Hermite polynomials with large values of indices are found. The applications to the photon distribution functions in squeezed one-mode mixed quantum states are considered.

Given a braid on N strings, find an algorithm which generates an Artin braid word B of minimal length. This is an important unsolved problem-a solution would give us the most economical way of notating and drawing braids. The length of an Artin word equals the number of crossings seen in a braid diagram. Minimum crossing numbers provide a measure of complexity for braids. This paper presents an algorithm for N=3. A three-dimensional configuration space for 3-braids will also be defined and analysed.

The analysis of the singularity structure of some series arising in the conjugation theory of holomorphic maps is performed using the Pade approximants. The validity of PA is checked for functions whose analytic structure is known: one observes that the numerical results are extremely sensitive to errors due to the use of a finite precision; normal or double precision used in FORTRAN codes is, in most cases, not sufficient to perform a numerical analysis of the singularity structure. In order to have significant results at high orders all the computations were carried out using codes which allow us to operate with a sufficiently large number of digits. The case of linearizable diffeomorphisms and of mappings tangent to the identity of (C,O) was considered: PA of both the direct and the inverse conjugating functions of a quadratic map to its normal form were computed and a comparison was performed with analytical results when available. When the singularity pattern was unknown, the poles and the zeros of the PA provided a coherent picture, which in some cases allowed rigorous results to be established.

A massively parallel supercomputer was used to exhaustively enumerate all of the Hamiltonian walks for simple cubic sublattices of four different sizes (up to 3*4*4). The behaviour of the logarithm of the number of walks was found to be linear in the number of vertices in the lattice. The linear fit is shown to agree also with the asymptotic limit of the Flory mean field theoretical estimate. Thus, we suggest that the fit obtained yields the number of walks for any size fragment of the cubic lattice to logarithmic accuracy. The significance of this result to the validity of polymer models is also discussed.

It is shown that when a hydrogen-like atom is treated as a two-dimensional system whose configuration space is multiply connected, then, in order to obtain the same energy spectrum as in the Bohr model, the angular momentum must be half-integral.

The solutions of the inhomogeneous wave equation are obtained. The source is distributed on a circle at rest or moving with a constant velocity. The wavefunction is found as a series of transient modes by means of incomplete separation of variables, the Riemann formula, and the integrals containing three Bessel functions.

Quantum states which evolve cyclically in their projective Hilbert space give rise to a geometric (or Aharonov-Anandan) phase. An aspect of primary interest is stable cyclic behaviour as realized under a periodic Hamiltonian. The problem has been handled by use of time-dependent transformations treated along the lines of Floquet's theory as well as in terms of exponential operators with a goal to examine the variety of initial states exhibiting cyclic behaviour. A particular case of special cyclic initial states is described which is shown to be important for nuclear magnetic resonance experiments aimed at the study of the effects of the geometric phase. An example of arbitrary spin j in a precessing magnetic field and spin j=1 subject to both axially symmetric quadrupolar interaction and a precessing magnetic field are presented. The invariant (Kobe's) geometric phase is calculated for special cyclic states.

In the adiabatic approximation the connection of the Berry phase with the quasi-classical trajectory-coherent states of the Schrodinger-type equation (with the arbitrary scalar h(cross)-(pseudo) differential operator) and the Dirac equation in the external periodic electromagnetic field is studied.

The Berezin quantization on a simply connected homogeneous Kahler manifold, which is considered as a phase space for a dynamical system, enables a description of the quantal system in a (finite-dimensional) Hilbert space of holomorphic functions corresponding to generalized coherent states. The Lie algebra associated with the manifold symmetry group is given in terms of first-order differential operators. In the classical theory, the Lie algebra is represented by the momentum maps which are functions on the manifold, and the Lie product is the Poisson bracket given by the Kahler structure. The Kahler potentials are constructed for the manifolds related to all compact semi-simple Lie groups. The complex coordinates are introduced by means of the Borel method. The Kahler structure is obtained explicitly for any unitary group representation. The cocycle functions for the Lie algebra and the Killing vector fields on the manifold are also obtained.

A solvable nonlinear 2D field model is proposed which is shown to be closely connected with the nonlinear Schrodinger chain. The conservation laws and soliton solutions for the corresponding field equations are obtained.