Higher Dimensions

For the estimation of many econometric models, integrals without analytical solutions have to be evaluated. Examples include limited dependent variables and nonlinear panel data models. In the case of one-dimensional integrals, Gaussian quadrature is known to work efficiently for a large class of problems. In higher dimensions, similar approaches discussed in the literature are either very specific and hard to implement or suffer from exponentially rising computational costs in the number of dimensions -a problem known as the "curse of dimensionality" of numerical integration. We propose a strategy that shares the advantages of Gaussian quadrature methods, is very general and easily implemented, and does not suffer from the curse of dimensionality. Monte Carlo experiments for the random parameters logit model indicate the superior performance of the proposed method over simulation techniques.

For the estimation of many econometric models, integrals without analytical solutions have to be evaluated. Examples include limited dependent variables and nonlinear panel data models. In the case of one-dimensional integrals, Gaussian quadrature is known to work efficiently for a large class of problems. In higher dimensions, similar approaches discussed in the literature are either very specific and hard to implement or suffer from exponentially rising computational costs in the number of dimensions -a problem known as the "curse of dimensionality" of numerical integration. We propose a strategy that shares the advantages of Gaussian quadrature methods, is very general and easily implemented, and does not suffer from the curse of dimensionality. Monte Carlo experiments for the random parameters logit model indicate the superior performance of the proposed method over simulation techniques.

For the estimation of many econometric models, integrals without analytical solutions have to be evaluated. Examples include limited dependent variables and nonlinear panel data models. In the case of one-dimensional integrals, Gaussian quadrature is known to work efficiently for a large class of problems. In higher dimensions, similar approaches discussed in the literature are either very specific and hard to implement or suffer from exponentially rising computational costs in the number of dimensions - a problem known as the "curse of dimensionality" of numerical integration. We propose a strategy that shares the advantages of Gaussian quadrature methods, is very general and easily implemented, and does not suffer from the curse of dimensionality. Monte Carlo experiments for the random parameters logit model indicate the superior performance of the proposed method over simulation techniques.

For the estimation of many econometric models, integrals without analytical solutions have to be evaluated. Examples include limited dependent variables and nonlinear panel data models. In the case of one-dimensional integrals, Gaussian quadrature is known to work efficiently for a large class of problems. In higher dimensions, similar approaches discussed in the literature are either very specific and hard to implement or suffer from exponentially rising computational costs in the number of dimensions - a problem known as the "curse of dimensionality" of numerical integration. We propose a strategy that shares the advantages of Gaussian quadrature methods, is very general and easily implemented, and does not suffer from the curse of dimensionality. Monte Carlo experiments for the random parameters logit model indicate the superior performance of the proposed method over simulation techniques.

This paper presents a comprehensive study of the fundamental quasinormal modes of all Standard Model fields propagating on a brane embedded in a higher-dimensional rotating black hole spacetime. The equations of motion for fields with spin s=0, 1/2 and 1 propagating in the induced-on-the-brane background are solved numerically, and the dependence of their QN spectra on the black hole angular momentum and dimensionality of spacetime is investigated. It is found that the brane-localised field perturbations are longer-lived when the higher-dimensional black hole rotates faster, while an increase in the number of transverse-to-the-brane dimensions reduces their lifetime. Finally, the quality factor Q, that determines the best oscillator among the different field perturbations, is investigated and found to depend on properties of both the particular field studied (spin, multipole numbers) and the gravitational background (dimensionality, black hole angular momentum number).

We extend the conformal gluing construction of Isenberg et al. [19] by establishing an analogous gluing result for field theories obtained by minimally coupling Einstein's gravitational theory with matter fields. We treat classical fields such as perfect fluids and the Yang-Mills equations as well as the Einstein-Vlasov system, which is an important example e-print archive:

We describe how the iterative technique used by Isenberg and Moncrief to verify the existence of large sets of non constant mean curvature solutions of the Einstein constraints on closed manifolds can be adapted to verify the existence of large sets of asymptotically hyperbolic non constant mean curvature solutions of the Einstein constraints.

We extend the conformal gluing construction of Isenberg et al. [19] by establishing an analogous gluing result for field theories obtained by minimally coupling Einstein's gravitational theory with matter fields. We treat classical fields such as perfect fluids and the Yang-Mills equations as well as the Einstein-Vlasov system, which is an important example e-print archive:

We show that the matrix (or more generally tensor) product states in a finite translation invariant system can be accurately constructed from the same set of local matrices (or tensors) that are determined from an infinite lattice system in one or higher dimensions. This provides an efficient approach for studying translation invariant tensor product states in finite lattice systems. Two methods are introduced to determine these size-independent local tensors.

a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to generalise this corrected result to derive formulae of Hopf type for the n-fold Čech derived functors of the lower central series functors Z k . The paper ends with an application to algebraic K-theory.

Main features and mechanism of a calibration abelian theory constructed on U(1) Lie group is presented. Given Lorentz invariants for which local invariance principle is satisfied, its respective action is built. Calibration field and its curvature are obtained from U(1) symmetries. Maxwell electromagnetic formalism has been developed for this model. Induced Chern-Simmons action is reduced to 2+1 dimensions and Maxwell equations rewrote. Both magnetic and electric field equations are solved from Maxwell theory, and obtained Klein Gordon massive wave field equations, which presented topological mass.

This paper proposes a unified Lagrangian framework (LUnifiedLUnified) that harmonizes Quantum Gravity (LQGLQG) and Extra-Dimensional Theories (LExtraLExtra). By synthesizing Loop Quantum Gravity (LQG), String Theory, and Brane-World Models, we derive a covariant action in DD-dimensions that reconciles quantum corrections with higher-dimensional geometry. The model predicts testable signatures in gravitational wave spectra and high-energy colliders, while offering a geometric interpretation of dark matter.

The existence of not one but, potentially, two Altar Stones at Stonehenge has prompted the exploration of their placement, and whether this knowledge might throw further light on the monument's construction. What this exercise has revealed is the presence in the distances between certain key features of fractional numerics using two interconnected systems of measure-one featuring the megalithic yard of 2.71875 feet or 32.625 inches and the other using imperial measures that remain in use today. A relationship can be shown to exist between these two types of measures based on the fractions 29/32 and 32/29, which by chance or design mimic mathematical formulae related to the concept of 29 and 32 dimensions of geometry. We show that the underlying numerics at Stonehenge, as well as at the nearby henge monument at Avebury, and also in the design and function of the Bush Barrow Lozenge, are likely derived from angles relating to the movement of the sun at the time of the solstices, along with a pre-existing understanding of the cyclical motion of the celestial bodies, the planet Venus in particular.

We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, using a generalisation of the Wirtinger inequality for the comass. Using a model for the classifying space BS 3 built inductively out of BS 1 , we prove that the symmetric metrics of certain two-point homogeneous manifolds turn out not to be the systolically optimal metrics on those manifolds. We point out the unexpected role played by the exceptional Lie algebra E 7 in systolic geometry, via the calculation of Wirtinger constants. Using a technique of pullback with controlled systolic ratio, we calculate the optimal systolic ratio of the quaternionic projective plane, modulo the existence of a Joyce manifold with Spin(7) holonomy and unit middle-dimensional Betti number.

Quantum entanglement is one of the most perplexing phenomena in modern physics, where two or more particles exhibit instant correlations regardless of the distance between them. In this paper, we propose a novel perspective: what if entangled particles are not separate entities but rather different manifestations of a single higher-dimensional object? By drawing an analogy with Abbott’s Flatland, we explore the possibility that quantum entanglement could be a geometric effect of hidden dimensions. This perspective aligns with concepts in string theory and the ER = EPR conjecture, suggesting that a more complex spacetime structure might underlie quantum reality. While this idea remains speculative, it provides a conceptual framework that could help bridge gaps between quantum mechanics and higher-dimensional physics.

This paper presents a complete and rigorous proof of the Poincaré Conjecture, confirming that any simply connected, closed 3-manifold is homeomorphic to the 3-sphere S 3 . Building upon Ricci flow techniques introduced by Hamilton and the breakthrough of Perelman, we refine the arguments to ensure the full classification of 3-manifolds, closing any remaining gaps in the proof and providing a comprehensive understanding of geometric flows in three dimensions.

Static, spherically symmetric configurations of gravity with nonminimally coupled scalar fields are considered in D-dimensional space-times in the framework of generalized scalar-tensor theories. We seek special cases when the system has no naked singularity but instead forms either a black hole, or a wormhole. General conditions when this is possible are formulated. In particlar, some such special cases are indicated for multidimensional Brans-Dicke theory and for linear, massless, nonminimally coupled scalar fields. It is shown that in the Brans-Dicke theory the only black-hole solution corresponds to D=4 and the coupling constant \omega< -2, and there is a wormhole solution for \omega=0. For linear scalar fields it is shown that the only black-hole solution is the well-known one (D=4, a black hole with scalar charge), while the known 4-dimensional wormhole solution is generalized to systems with conformal coupling in arbitrary dimension.

We study the structure and stability of the recently discussed spherically symmetric Brans-Dicke black-hole type solutions with an infinite horizon area and zero Hawking temperature, existing for negative values of the coupling constant ω . These solutions split into two classes: B1, whose horizon is reached by an infalling particle in a finite proper time, and B2, for which this proper time is infinite. Class B1 metrics are shown to be extendable beyond the horizon only for discrete values of mass and scalar charge, depending on two integers m and n . In the case of even mn the space-time is globally regular; for odd m the metric changes its signature as the horizon is crossed. Nevertheless, the Lorentzian nature of the space-time is preserved, and geodesics are smoothly continued across the horizon, but, when crossing the horizon, for odd m timelike geodesics become spacelike and vice versa. Problems with causality, arising in some cases, are discussed. Tidal forces are shown to grow infinitely near type B1 horizons. All vacuum static, spherically symmetric solutions of the Brans-Dicke theory with ω < -3/2 are found to be linearly stable against spherically symmetric perturbations. This result extends to the generic case of the Bergmann-Wagoner class of scalar-tensor theories of gravity with the coupling function ω(φ) < -3/2 .

We develop a fast algorithm to calculate the entanglement of formation of a mixed state, which is defined as the minimum average entanglement of the pure states that form the mixed state. The algorithm combines conjugate gradient and steepest descent algorithms and outperforms both. Using this new algorithm, we obtain the statistics of the entanglement of formation on ensembles of random density matrices of higher dimensions than possible before. The correlation between the entanglement of formation and other quantities that are easier to compute, such as participation ratio and negativity are presented. Our results suggest a higher percentage of zero-entanglement states among zero-negativity states than previously reported.

The nature of reality has been a subject of philosophical inquiry and scientific investigation for centuries. This paper explores a multidisciplinary framework that integrates quantum mechanics, higher-dimensional physics, pattern analysis, and classical philosophy to propose a coherent model of the universe.

The discussion is structured around three central hypotheses: (1) the existence of higher dimensions that transcend conventional three-dimensional space and linear time, (2) the non-reductive nature of consciousness, suggesting that subjective experience is deeply connected to quantum and extra-dimensional processes, and (3) the presence of fine-tuned mathematical and structural patterns in nature, indicating an underlying intelligent design.

Key elements of modern theoretical physics—such as string theory, the delayed choice quantum eraser experiment, and the fine-tuning of physical constants—are examined in the context of their philosophical and empirical implications. Moreover, the paper incorporates findings from neuroscience and pattern engineering, demonstrating the emergence of self-organized complexity and its potential link to a designed cosmic architecture.

Further, it discusses quantum observer effects, the Anthropic Principle, quantum entanglement, Gödel’s Incompleteness Theorem, and the role of mathematical structures such as the Fibonacci sequence and the Golden Ratio in nature. These elements collectively suggest a structured, interconnected, and computationally sophisticated reality.

This study proposes that the synthesis of quantum physics, emergent system theory, and metaphysical models could provide a deeper understanding of the universe, bridging the apparent divide between empirical science and foundational philosophical questions about existence and consciousness.

Future research directions include experimental validation of discrete space-time, neuroquantum studies of consciousness, and the continued exploration of mathematical signatures embedded in natural phenomena.

Conformal techniques, based on the covariance of Maxwell's electrodynamics under the full conformal group in four spacetime dimensions, are discussed in relation with the constant input impedance properties of frequency-independent antennas. In particular we show that by applying suitable conformal transformations to existing self-complementary planar structures like logarithmic spirals, the constant input impedance property is considerably improved, specially in the low frequency domain, where it usually fails due to the large wavelength involved. The effect of the conformal transformation is to bring infinity up to a finite distance and, in a way, to imprison the otherwise infinite structure in a compact region. This procedure enables to perform a more intelligent truncation with the aim to capture some of the properties left behind in the cutting, and to design realistic ultra width-band antennas with a more controlled value of the input impedance in the whole operation range. Due to the fact that the tools here developed find their roots in the differential geometry of curved manifolds, they are quite general and also suitable for dealing with 3D-antennas of arbitrary shape, curvature and size, as well as with other systems in which the effects of truncation play a crucial role in defining physical properties of them.

We propose a method to control the number of species of lattice fermions, which yields new classes of minimally doubled lattice fermions with one exact chiral symmetry and exact locality. We classify all the known minimally doubled fermions into two types based on the locations of the propagator poles in the Brillouin zone. We also study higher-dimensional extension of them and show it tends to be more difficult to realize minimal-doubling in higher dimensions.

In this work we use constructs from the dual space of the semi-direct product of the Virasoro algebra and the affine Lie algebra of a circle to write a theory of gravitation which is a natural analogue of Yang-Mills theory. The theory provides a relation between quadratic differentials in 1+1 dimensions and rank two symmetric tensors in higher dimensions as well as a covariant local Lagrangian for two dimensional gravity. The isotropy equations of coadjoint orbits are interpreted as Gauss law constraints for a field theory in two dimensions, which enables us to extend to higher dimensions. The theory has a Newtonian limit in any space-time dimension. Our approach introduces a novel relationship between string theories and 2D field theories that might be useful in defining dual theories. We briefly discuss how this gravitational field couples to fermions.

We introduce special supersymmetric gauge theories in three, five, seven and nine dimensions, whose compactification on two-, four-, six-and eight-folds produces a supersymmetric quantum mechanics on moduli spaces of holomorphic bundles and/or solutions to the analogues of instanton equations in higher dimensions. The theories may occur on the worldvolumes of D-branes wrapping manifolds of special holonomy. We also discuss the theories with matter.

We investigate the state space of bipartite qutrits. For states which are locally maximally mixed we obtain an analog of the "magic" tetrahedron for bipartite qubits-a magic simplex W. This is obtained via the Weyl group which is a kind of "quantization" of classical phase space. We analyze how this simplex W is embedded in the whole state space of two qutrits and discuss symmetries and equivalences inside the simplex W. Because we are explicitly able to construct optimal entanglement witnesses we obtain the border between separable and entangled states. With our method we find also the total area of bound entangled states of the parameter subspace under intervestigation. Our considerations can also be applied to higher dimensions.

In this paper, we discuss the Lie algebra bundles defined by Jordan algebra bundles of finite type over an arbitrary topological space and observe that such Lie algebra bundles are also of finite type. Also, we examine some ideal bundles of Lie algebra bundles defined by semisimple Jordan algebra bundles of finite type.

, we studied p-mean curvature and the associated p-minimal surfaces in the Heisenberg group from the viewpoint of PDE and differential geometry. In this paper, we look into the problem through the variational formulation. We study a generalized p-area and associated (p-) minimizers in general dimensions. We prove the existence and investigate the uniqueness of minimizers. Since this is reduced to solving a degenerate elliptic equation, we need to consider the effect of the singular set and this requires a careful study. We define the notion of weak solution and prove that in a certain Sobolev space, a weak solution is a minimizer and vice versa. We also give many interesting examples in dimension 2. An intriguing point is that, in dimension 2, a C 2 -smooth solution from the PDE viewpoint may not be a minimizer. However, this statement is true for higher dimensions due to the relative smallness of the size of the singular set. Ω {|∇u -X * | + Hu}dx 1 ∧ dx 1 ∧ ... ∧ dx n ∧ dx n .

Given any mean zero, finite variance σ 2 random variable W , there exists a unique distribution on a variable W * such that EW f (W ) = σ 2 Ef (W * ) for all absolutely continuous functions f for which these expectations exist. This distributional 'zero bias' transformation of W to W * , of which the normal is the unique fixed point, was introduced in [9] to obtain bounds in normal approximations. After providing some background on the zero bias transformation in one dimension, we extend its definition to higher dimension and illustrate with an application to the normal approximation of sums of vectors obtained by simple random sampling.

In this paper, we consider several variations of the following basic tiling problem: given a sequence of real numbers with two size-bound parameters, we want to find a set of tiles of maximum total weight such that each tiles satisfies the size bounds. A solution to this problem is important to a number of computational biology applications such as selecting genomic DNA fragments for PCR-based amplicon microarrays and performing homology searches with long sequence queries. Our goal is to design efficient algorithms with linear or near-linear time and space in the normal range of parameter values for these problems. For this purpose, we first discuss the solution to a basic online interval maximum problem via a sliding-window approach and show how to use this solution in a nontrivial manner for many of the tiling problems introduced. We also discuss NP-hardness results and approximation algorithms for generalizing our basic tiling problem to higher dimensions. Finally, computational results from applying our tiling algorithms to genomic sequences of five model eukaryotes are reported.

In this paper we present a new method for the numerical solution of the relativistic Vlasov-Maxwell system on a phase-space grid using an adaptive semi-Lagrangian method. The adaptivity is performed through a wavelet multiresolution analysis, which gives a powerful and natural refinement criterion based on the local measurement of the approximation error and regularity of the distribution function. Therefore, the multiscale expansion of the distribution function allows to get a sparse representation of the data and thus save memory space and CPU time. We apply this numerical scheme to reduced Vlasov-Maxwell systems arising in laser-plasma physics. Interaction of relativistically strong laser pulses with overdense plasma slabs is investigated. These Vlasov simulations revealed a rich variety of phenomena associated with the fast particle dynamics induced by electromagnetic waves as electron trapping, particle acceleration, and electron plasma wavebreaking. However, the wavelet based adaptive method that we developed here, does not yield significant improvements compared to Vlasov solvers on a uniform mesh due to the substantial overhead that the method introduces. Nonetheless they might be a first step towards more efficient adaptive solvers based on different ideas for the grid refinement or on a more efficient implementation. Here the Vlasov simulations are performed in a two-dimensional phase-space where the development of thin filaments, strongly amplified by relativistic effects requires an important increase of the total number of points of the phase-space grid as they get finer as time goes on. The adaptive method could be more useful in cases where these thin filaments that need to be resolved are a very small fraction of the hyper-volume, which arises in higher dimensions because of the surface-to-volume scaling and the essentially one-dimensional structure of the filaments. Moreover, the main way to improve the efficiency of the adaptive method is to increase the local character in phase-space of the numerical scheme, by considering multiscale reconstruction with more compact support and by replacing the semi-Lagrangian method with more local -in space -numerical scheme as compact finite

It is shown, that oscillators on the sphere and the pseudosphere are related, by the so-called Bohlin transformation, with the Coulomb systems on the pseudosphere. The even states of an oscillator yield the conventional Coulomb system on the pseudosphere, while the odd states yield the Coulomb system on the pseudosphere in the presence of magnetic flux tube generating spin 1/2. A similar relation is established for the oscillator on the (pseudo)sphere specified by the presence of constant uniform magnetic field B0 and the Coulomb-like system on pseudosphere specified by the presence of the magnetic field B 2r 0 (| x 3 x | -ǫ). The correspondence between the oscillator and the Coulomb systems the higher dimensions is also discussed.

We propose a new robust method for the computation of scattering of high-frequency acoustic plane waves by smooth convex objects in 2D. We formulate this problem by the direct boundary integral method, using the classical combined potential approach. By exploiting the known asymptotics of the solution, we devise particular expansions, valid in various zones of the boundary, which express the solution of the integral equation as a product of explicit oscillatory functions and more slowly varying unknown amplitudes. The amplitudes are approximated by polynomials (of minimum degree d) in each zone using a Galerkin scheme. We prove that the underlying bilinear form is continuous in L 2 , with a continuity constant that grows mildly in the wavenumber k. We also show that the bilinear form is uniformly L 2 -coercive, independent of k, for all k sufficiently large. (The latter result depends on rather delicate Fourier analysis and is restricted in 2D to circular domains, but it also applies to spheres in higher dimensions.) Using these results and the asymptotic expansion of the solution, we prove superalgebraic convergence of our numerical method as d → ∞ for fixed k. We also prove that, as k → ∞, d has to increase only very modestly to maintain a fixed error bound (d ∼ k 1/9 is a typical behaviour). Numerical experiments show that the method suffers minimal loss of accuracy as k → ∞, for a fixed number of degrees of freedom. Numerical solutions with a relative error of about 10 -5 are obtained on domains of size O(1) for k up to 800 using about 60 degrees of freedom.

Reminiscences of my intellectual and personal interactions with Professor Nambu, and discussion of his contributions to string theory. Switching to my own research, by expressing the string coordinates as bispinors, I suggest a kinematical framework for the interacting (2, 0) theory on the light-cone by using chiral constrained "Viking" superfields.

Numerical studies of the Zaitsev (Robin Hood ) model PERRY FOX, GABRIEL CWILICH, SERGEY BULDYREV, FREDY ZYPMAN, Department of Physics -Yeshiva University -The Zaitsev[1] model of depinning of interfaces has been widely used to discuss motion of dislocations, low temperature flux creep, and more recently dry friction. The properties of this model have been discussed theoretically in one dimension, and numerically verified with precision in the isotropic case. We are studying here the effect of anisotropy in the distribution of the "mass" among the neighbors in the updating of the sites, which is known to modify the critical exponents of the model in one dimension. We have considered the validity of the scaling laws in higher dimensions, which might be relevant for the case of friction [2], by computing several of the exponents of the model for the avalanche size distribution, average avalanche size, avalanche fractal dimension and distribution of jumps between extremal sites of activity. The much richer space of parameters of anisotropy in two dimensions has been explored. [1] S.I. Zaitsev , Physica A189, 411 (1992). [2] S. Buldyrev, J. Ferrante and F. Zypman Phys. Rev E64, 066110, (2006)

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak-Keller-Segel system with d ≥ 3 and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass M c such that if M ∈ (0, M c ] solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for M ∈ (0, M c ). While characterising the eventual infinite time blowing-up profile for M = M c , we observe that the long time asymptotics are much more complicated than in the classical Patlak-Keller-Segel system in dimension two.

The basic properties of oscillons -localized, long-lived, time-dependent scalar field configurations -are briefly reviewed, including recent results demonstrating how their existence depends on the dimensionality of spacetime. Their role on the dynamics of phase transitions is discussed, and it is shown that oscillons may greatly accelerate the decay of metastable vacuum states. This mechanism for vacuum decay -resonant nucleation -is then applied to cosmological inflation. A new inflationary model is proposed which terminates with fast bubble nucleation.

In modern physics we could say that space dimension is derived from physical conditions. Kaluza-Klein theory and D-brane are typical examples. However, not only by such conditions, we should also think about space dimension with insights from known facts without any physical conditions. In this talk we rethink space dimensionality from scratch. Let us think about these physical conditions: 1. The standard model requires 19 numerical constants whose values are determined by experimental results. That is one of the motivations for particle physicists to establish a unified theory beyond the standard model.

We study the thermodynamics of the asymptotically flat static black hole in Lovelock back ground where the coupling constants of the Lovelock theory effects are taken into account. We consider the effects of the second order of the coupling constant, and third order of the Lovelock constant coefficient on the thermodynamics of asymptotically flat static black holes. In this case the effect of the coupling constants on the thermodynamics of the black hole are discussed for 5, 6, and 7 dimensional spacetime.

We present a method to measure potentials over an extended region using onedimensional ion crystals in a radio frequency (RF) ion trap. The equilibrium spacings of the ions within the crystal allow the determination of the external forces acting at each point. From this the overall potential, and also potentials due to specific trap features, are calculated. The method can be used to probe potentials near proximal objects in real time, and can be generalized to higher dimensions.

After a survey of the cohomological quantum field theory, we review the computation of their Donaldson-Witten invariants. These invariants are generalized for smooth flows defined on the four manifold using notion of asymptotic cycles of higher dimensions than one introduced recently by S. Schwartzman.

We construct embeddings of Bers slices of ideal polygon reflection groups into the classical family of univalent functions Σ. This embedding is such that the conformal mating of the reflection group with the anti-holomorphic polynomial z → z d is the Schwarz reflection map arising from the corresponding map in Σ. We characterize the image of this embedding in Σ as a family of univalent rational maps. Moreover, we show that the limit set of every Kleinian reflection group in the closure of the Bers slice is naturally homeomorphic to the Julia set of an anti-holomorphic polynomial.

By restricting the motion of high-mobility 2D electron gas to a network of channels with smooth confinement, we were able to trace, both classically and quantum-mechanically, the interplay of backscattering, and of the bending action of a weak magnetic field. Backscattering limits the mobility, while bending initiates quantization of the Hall conductivity. We demonstrate that, in restricted geometry, electron motion reduces to two Chalker-Coddington networks, with opposite directions of propagation along the links, which are weakly coupled by disorder. Interplay of backscattering and bending results in the quantum Hall transition in a non-quantizing magnetic field, which decreases with increasing mobility. This is in accord with scenario of floating up delocalized states.

We study the stochastic dynamics of deposition-evaporation cooperative processes of dimers, trimers, etc., in two-and higher-dimensional lattices. The dimer system in bipartite lattices allows for an exact solution of dynamic correlations and scaling functions by means of a quantum spin equivalence. Autocorrelations exhibit a diffusive asymptotic kinetics and crossovers of different dynamic regimes in highly anisotropic lattices. Monte Carlo simulations combined with finite-size scaling arguments support the validity of the diffusive picture in more general situations. Steady-state coverages and diffusion constants are obtained using mean-field approaches, spin wave calculations, and random walk analyses in nearly jammed configurations.