Usual Lattice Research Articles

Searching for a trail of evidence in a maze

Consider a graph with a set of vertices and oriented edges connecting pairs of vertices. Each vertex is associated with a random variable and these are assumed to be independent. In this setting, suppose we wish to solve the following hypothesis testing problem: under the null, the random variables have common distribution N(0,1) while under the alternative, there is an unknown path along which random variables have distribution N(�, 1), � > 0, and distribution N(0,1) away from it. For which values of the mean shiftcan one reliably detect and for which values is this impossible? This paper develops detection thresholds for two types of common graphs which exhibit a different behavior. The first is the usual regular lattice with vertices of the form {(i, j) : 0 ≤ i, −i ≤ j ≤ i and j has the parity of i} and oriented edges (i, j) → (i+1, j+s) where s = ±1. We show that for paths of length m start- ing at the origin, the hypotheses become distinguishable (in a minimax sense) ifm ≫ √ log m, while they are not ifm ≪ log m. We derive equivalent results in a Bayesian setting where one assumes that all paths are equally likely; there the asymptotic threshold ism ≈ m 1/4 . We obtain corresponding results for trees (where the threshold is of order 1 and independent of the size of the tree), for distributions other than the Gaussian, and for other graphs. The concept of predictability profile, first introduced by Benjamini, Pemantle and Peres, plays a crucial role in our analysis.

Photonic-Crystal Heterostructure Waveguides

Photonic crystals (PC) are periodic dielectric structures that, if suitably designed, prohibit light propagation within a frequency band even though the constituent materials may be transparent in this range. One of the remarkable applications of such artificial photonic materials has been to provide guiding of light. Ultimately, to guide light in directions other than straight lines, PC waveguide (PCW) bends are needed, and thus are expected to be essential building blocks of photonic integrated circuits. While bending light through large angles is possible with conventional waveguides, the corner radii of such bends cannot typically be reduced to the electromagnetic wavelength, which hinders the realization of extremely compact devices. And though sharp two-dimensional (2-D) PCW bends have been proposed, the transmission is typically low and/or narrow band. Here we focus on PCWs obtained by introducing line defects in otherwise period 2-D PCs with the aims of enhancing the typical poor and low-bandwidth transmission through tight bends. We show how PCW bends occurring at heterojunctions between different PCs may enable unprecedented flexibility in meeting these aims. The deformation introduced to the usual PC lattice lifted off the angle constraint and resulted in the power transmission greater than 90% over in the 95-nm bandwidth

The chance that a convex body is lattice-point free: A relative of Buffon's needle problem

Given a convex body K ⊂Rd, what is the probability that a randomly chosen congruent copy, K*, of K is lattice-point free, that is, K*∩Zd = ∅︁? Here Zd is the usual lattice of integer points in Rd. Luckily, the underlying probability is well defined since integer translations of K can be factored out. The question came up in connection with integer programming. We explain what the answer is for convex bodies of large enough volume. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007

Characteristics of reaction-diffusion on scale-free networks

We examine some characteristic properties of reaction-diffusion processes of the A+A-->0 type on scale-free networks. Due to the inhomogeneity of the structure of the substrate, as compared to usual lattices, we focus on the characteristics of the nodes where the annihilations occur. We show that at early times the majority of these events take place on low-connectivity nodes, while as time advances the process moves towards the high-connectivity nodes, the so-called hubs. This pattern remarkably accelerates the annihilation of the particles, and it is in agreement with earlier predictions that the rates of reaction-diffusion processes on scale-free networks are much faster than the equivalent ones on lattice systems.

Mean-field theory of the Potts gas

We consider a gas of classical particles in R d having q distinct colours, interacting via a mean-field Potts potential, and subject to an external field; a colour-independent molecular interaction of mean-field type is also admitted. In contrast to the usual lattice Potts model, the Potts gas exhibits the specific volume v as a parameter, in addition to colour-ordering. The v-dependence of the Potts gas is studied in detail, and the complete phase diagram is derived. It turns out that, for q ≥ 3, the transition to colour-ordering implies a jump of density. For q = 2, this transition is continuous but may become discontinuous under the influence of a suitable molecular interaction.

Three Essays on Contingent Claims Pricing

This dissertation consists of three research topics in contemporary financial option pricing theories and their applications. The common theme of those topics involves the pricing of financial claims whose value become path-dependent when using the usual lattice approximating schemes. The first essay explores the potential of transformation and other schemes in constructing a sequence of simple binomial processes that weakly converges to the desired diffusion limit. Convergence results are established for the valuation of both European and American contingent claims when the underlying asset prices are approximated by simple binomial processes. It is also demonstrated how to construct reflecting or absorbing binomial processes to approximate diffusions with boundaries. Numerical examples demonstrate that the proposed simple approximations not only converge, but also give more accurate results then existing methods such as Nelson and Ramaswamy (1990), especially for longer maturities. Our purpose in essay 2 is two-fold. First we extend some of the simple lattice-approximation methods for one-dimensional diffusions to higher dimensions and develop special lattices to approximate perfectly correlated diffusions. We then examine current modelling issues of the term structure of interest rates, and demonstrate how to apply the approximation techniques developed here to handle path-dependence and multi-sources of uncertainty in these models. The last essay analyzes the investment decisions of insured banks under fixed-rate deposit insurance. The model takes into account the charter value and allows banks to dynamically revise their asset portfolios. Trade-offs exists between preserving the charter and exploiting deposit insurance. The optimal bank portfolio problem is solved analytically for a constant charter value. In any audit period, banks maximize their risk exposure before some critical time and act cautiously thereafter. The corresponding deposit insurance is shown to be a put option that matures at this critical time rather than at the audit date.

On gonihedric loops and quantum gravity

We present an analysis of the gonihedric loop model, a reformulation of the two dimensional gonihedric spin model, using two different techniques. First, the usual regular lattice statistical physics problem is mapped onto a height model and studied analytically. Second, the gravitational version of this loop model is studied via matrix models techniques. Both methods lead to the conclusion that the model has $c_{matter}=0$ for all values of the parameters of the model. In this way it is possible to understand the absence of a continuous transition.

Magnetic order of the two-dimensional antiferromagnetic ¼-depleted square lattice

For a two-dimensional Heisenberg antiferromagnetic -depleted square lattice, it has been analytically proved that its ground state possessessimultaneously ferromagnetic and antiferromagnetic long-range orders, and exhibitsferrimagnetism. Numerical simulations of finite lattices using exact diagonalization showthat the depletion strengthens the short-range spin–spin correlations whereas it weakens thelong-range ones. The -depleted square lattice has a smaller susceptibility. This means that the geometricstructure of the usual two-dimensional square lattice is more advantageous for establishingmagnetic long-range order. By comparison of these two kinds of lattices, it is quite reliablyinferred that Néel order exists in the ground state of the usual two-dimensional spin- Heisenberg antiferromagnetic square lattice.

Ising model simulation in directed lattices and networks

On directed lattices, with half as many neighbours as in the usual undirected lattices, the Ising model does not seem to show a spontaneous magnetisation, at least for lower dimensions. Instead, the decay time for flipping of the magnetisation follows an Arrhenius law on the square and simple cubic lattice. On directed Barabási–Albert networks with two and seven neighbours selected by each added site, Metropolis and Glauber algorithms give similar results, while for Wolff cluster flipping the magnetisation decays exponentially with time.

Study of the 2-d [formula omitted] models at [formula omitted] and π

We present numerical results for 2-d CP(N-1) models at \theta=0 and \pi obtained in the D-theory formulation. In this formulation we construct an efficient cluster algorithm and we show numerical evidence for a first order transition for CP(N-1\geq 2) models at \theta = \pi. By a finite size scaling analysis, we also discuss the equivalence in the continuum limit of the D-theory formulation of the 2-d CP(N-1) models and the usual lattice definition.

Statistical thermodynamics of binary systems with variable valent states of one of the components

AbstractThis report contains a generalization of the usual lattice model of multicomponent systems. The generalization allows us to take into account the following factors: (i) the short‐range parts of interatomic repulsions; which are not identical for different pairs of atoms (therefore, it is impossible to take into account the repulsions by means of usual ideal lattice introduction); (ii) the long‐range interatomic potentials are taken into account by means of effective fields approximation; (iii) the existence of the vacancies as one of the components in the system; (iv) the existence of comparatively stable multi‐atomic complexes; and (v) some other factors due to specifics of constituents and their interactions, such as variable valence states and chemical reactions. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Structural transition of a Wigner crystal on a liquid substrate

The physics of an electron solid, held on a cryogenic liquid surface by a pressing electric field, is examined in a low-density regime that has not been explored before. We consider the effect of the pressing field in distorting the surface at the position of each electron and hence inducing an attractive force between the electrons. The system behavior is described in terms of an interplay between the repulsive Coulomb interaction and the attractive surface-induced interaction between individual electrons. For small densities and large enough pressing fields, we find a parameter regime where a square lattice is more favorable than the usual triangular lattice; we map out the first-order transition curve separating the two lattice geometries at zero temperature. In addition, our description allows an alternate static perspective on the charge-density wave instability of the system, corresponding to the formation of multi-electron dimples.

Wigner lattice of ripplopolarons in a multielectron bubble in helium

The properties of ripplonic polarons in a multielectron bubble in liquid helium are investigated on the basis of a path-integral variational method. We find that the two-dimensional electron gas can form deep dimples in the helium surface, or ripplopolarons, to solidify as a Wigner crystal. We derive the experimental conditions of temperature, pressure and number of electrons in the bubble for this phase to be realized. This predicted state is distinct from the usual Wigner lattice of electrons: it melts by dissociation of the ripplopolarons when the electrons shed their localizing dimple as the pressure on the multielectron bubble drops below a critical value.

The 𝒥-Radical and the Pierce Radicals of a Lattice-Ordered Ring

The 𝒥-radical of a lattice-ordered ring Ris the ℓ-ring analogue of the Jacobson radical of a ring. It is shown that if the matrix ring R n has the usual lattice order, then 𝒥(R n ) = 𝒥(R) n . The connection between an element abeing right ℓ-quasi-regular and the inequality a ○ x ≤ 0 is also investigated. For squares in an f-ring the connection is an equivalence. In general it is still an equivalence provided xis the sum of elements from a larger f-ring whose absolute values lie in R. It is also shown that the vanishing of annihilators in an f-ring is inherited by enough totally ordered homomorphic images to give a subdirect product decomposition.

Nonreciprocal microwave band-gap structures.

An electrically controlled nonreciprocal electromagnetic band-gap material is proposed and studied. The new material is a periodic three-dimensional regular lattice of small magnetized ferrite spheres. In this paper, we consider plane electromagnetic waves in this medium and design an analytical model for the material parameters. An analytical solution for plane-wave reflection from a planar interface is also presented. In the proposed material, a new electrically controlled stop band appears for one of the two circularly polarized eigenwaves in a frequency band around the ferrimagnetic resonance frequency. This frequency can be well below the usual lattice band gap, which allows the realization of rather compact structures. The main properties of the material are outlined.

Electromagnetism and gauge theory on the permutation group S3

Using noncommutative geometry we do U(1) gauge theory on the permutation group S 3. Unlike usual lattice gauge theories the use of a non–Abelian group here as spacetime corresponds to a background Riemannian curvature. In this background we solve spin 0, 1 2 and spin 1 equations of motion, including the spin 1 or ‘photon’ case in the presence of sources, i.e. a theory of classical electromagnetism. Moreover, we solve the U(1) Yang–Mills theory (this differs from the U(1) Maxwell theory in noncommutative geometry), including the moduli space of flat connections. We show that the Yang–Mills action has a simple form in terms of Wilson loops in the permutation group, and we discuss aspects of the quantum theory.

A proof of Weinberg’s conjecture on lattice-ordered matrix algebras

Let F \mathbf {F} be a subfield of the field of real numbers and let F n \mathbf {F}_{n} ( n ≥ 2 n \geq 2 ) be the n × n n \times n matrix algebra over F \mathbf {F} . It is shown that if F n \mathbf {F}_{n} is a lattice-ordered algebra over F \mathbf {F} in which the identity matrix 1 is positive, then F n \mathbf {F}_{n} is isomorphic to the lattice-ordered algebra F n \mathbf {F}_{n} with the usual lattice order. In particular, Weinberg’s conjecture is true.

Co-crystalline hydrogen bonded solids formed by helical tubuland diols

The helical tubulands are a group of alicyclic dialcohols whose inclusion compounds crystallise in the chiral space group P3121 (or enantiomorph P3221) with a hydrogen bonded network structure containing parallel guest-filled tubes. Protic organic guest molecules are also included within these host tubes, with the exception of phenols which form co-crystalline adducts. The diol 3 includes non-protic guests within the usual tubular lattice, but forms a 1∶1 adduct with hydroquinone in space group C2/c whose crystal structure is entirely different from typical diol–phenol compounds like (2)2·(hydroquinone) in space group P21/c. Unexpectedly, diol 3 also forms the unprecedented co-crystalline diol–alcohol compound (3)·(methanol). This substance does have close structural similarities to the earlier co-crystalline diol–phenol compounds in space group P21/c.

Disorder-activated infrared modes and surface depletion layer in highly Si-doped hexagonal GaN

Three infrared-active low-polar modes are reported for highly Si-doped hexagonal (α-) GaN. The 0.8–1.6 μm thick films, grown by metal organic vapor phase epitaxy or molecular beam epitaxy on (0001) sapphire substrates, were studied by infrared spectroscopic ellipsometry. For GaN epilayers with free-electron concentration N⩾8×1018 cm−3 we observe, besides the usual GaN transverse-optical lattice modes and coupled longitudinal-optical phonon-plasmon modes, a band of additional modes at 567.4±2.5, 752.5±0.9, and 855.0±0.9 cm−1. We tentatively assign the first one to the disorder-activated high E2 GaN mode and the third mode to an acoustic-optical combination band, whereas the origin of the second mode remains unclear. Furthermore, the ellipsometric spectra of highly n-conductive Si-doped GaN reveal thin carrier-depleted regions at the sample surface.

Rigidity of the Critical Phases on a Cayley Tree

We discuss statistical mechanics on nonamenable graphs, and we study the features of the phase transition, which are due to nonamenability. For the Ising model on the usual lattice it is known that below the critical temperature fluctuations of magnetization are much less likely in the states with nonzero magnetic field than in the pure states with zero field. We show that on the Cayley tree the corresponding fluctuations have the same order. 2000 Math. Subj. Class. 60F10, 82B20.