A rigid graph for every set

A graph G is called rigid if the identical mapping V(G)→V(G) is the only homomorphism G→G. In this note we give a simple construction of a rigid oriented graph on every set. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 108–110, 2002     

On the local structure of doubly laced crystals

Let g be a Lie algebra all of whose regular subalgebras of rank 2 are type A₁×A₁, A₂, or C₂, and let B be a crystal graph corresponding to a representation of g. We explicitly describe the local structure of B, confirming a conjecture of Stembridge.     

Pauli-Fierz Hamiltonians defined as quadratic forms

We study Pauli-Fierz Hamiltonians-self-adjoint operators describing a small quantum system interacting with a bosonic field. Using quadratic form techniques, we extend the results of Dereziński-Gérard and Gérard about the self-adjointness, the location of the essential spectrum and the existence of a ground state to a large class of Pauli-Fierz Hamiltonians.     

Energy crisis or a new soliton in the noncommutative CP(1) model?

The non-commutative (NC) CP(1) model is studied from field theory perspective. Our formalism and definition of the NC CP(1) model differs crucially from the existing one [Phys. Lett. B 498 (2001) 277, hep-th/0203125, hep-th/0303090].

Due to the U(1) gauge invariance, the Seiberg–Witten map is used to convert the NC action to an action in terms of ordinary spacetime degrees of freedom and the subsequent theory is studied. The NC effects appear as (NC parameter) θ-dependent interaction terms. The expressions for static energy, obtained from both the symmetric and canonical forms of the energy momentum tensor, are identical, when only spatial noncommutativity is present. Bogomolny analysis reveals a lower bound in the energy in an unambiguous way, suggesting the presence of a new soliton. However, the BPS equations saturating the bound are not compatible to the full variational equation of motion. This indicates that the definitions of the energy momentum tensor for this particular NC theory (the NC theory is otherwise consistent and well defined), are inadequate, thus leading to the “energy crisis”.

A collective coordinate analysis corroborates the above observations. It also shows that the above mentioned mismatch between the BPS equations and the variational equation of motion is small.     

Location of terror response facilities: A game between state and terrorist

We study a leader follower game with two players: a terrorist and a state where the later one installs facilities that provide support in case of a terrorist attack. While the Terrorist attacks one of the metropolitan areas to maximize his utility, the State, which acts as a leader, installs the facilities such that the metropolitan area attacked is the one that minimizes her disutility (i.e., minimizes ‘loss’). We solve the problem efficiently for one facility and we formulate it as a mathematical programming problem for a general number of facilities. We demonstrate the problem via a case study of the 20 largest metropolitan areas in the United States.     

Optimisation of fault-tolerant fabric-cutting schedules using genetic algorithms and fuzzy set theory

In apparel industry, manufacturers developed standard allowed minutes (SAMs) databases on various manufacturing operations in order to facilitate better scheduling, while effective production schedules ensure smoothness of downstream operations. As apparel manufacturing environment is fuzzy and dynamic, rigid production schedules based on SAMs become futile in the presence of any uncertainty. In this paper, a fuzzification scheme is proposed to fuzzify the static standard time so as to incorporate some uncertainties, in terms of both job-specific and human related factors, into the fabric-cutting scheduling problem. A genetic optimisation procedure is also proposed to search for fault-tolerant schedules using genetic algorithms, such that makespan and scheduling uncertainties are minimised. Two sets of real production data were collected to validate the proposed method. Experimental results indicate that the genetically optimised fault-tolerant schedules not only improve the operation performance but also minimise the scheduling risks.     

A Modified Szilard Engine: Measurement, Information, and Maxwell's Demon

Using an isolated measurement process, we calculate the effect measurement has on entropy for the multi-cylinder Szilard engine. We find that the system of cylinders possesses an entropy associated with cylinder total energy states, and that it records information transferred at measurement. Contrary to other's results, we find that the apparatus loses entropy due to measurement. The Second Law of Thermodynamics may be preserved if Maxwell's demon gains entropy moving the engine partition.     

Generalized 3G theorem and application to relativistic stable process on non-smooth open sets

Let G(x,y) and GD(x,y) be the Green functions of rotationally invariant symmetric α-stable process in Rd and in an open set D, respectively, where 0<α<2. The inequality GD(x,y)GD(y,z)/GD(x,z)?c(G(x,y)+G(y,z)) is a very useful tool in studying (local) Schrödinger operators. When the above inequality is true with c=c(D)∈(0,∞), then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded κ-fat open set, which includes a bounded John domain. The 3G we consider is of the form GD(x,y)GD(z,w)/GD(x,w), where y may be different from z. When y=z, we recover the usual 3G. The 3G form GD(x,y)GD(z,w)/GD(x,w) appears in non-local Schrödinger operator theory. Using our generalized 3G theorem, we give a concrete class of functions belonging to the non-local Kato class, introduced by Chen and Song, on κ-fat open sets. As an application, we discuss relativistic α-stable processes (relativistic Hamiltonian when α=1) in κ-fat open sets. We identify the Martin boundary and the minimal Martin boundary with the Euclidean boundary for relativistic α-stable processes in κ-fat open sets. Furthermore, we show that relative Fatou type theorem is true for relativistic stable processes in κ-fat open sets. The main results of this paper hold for a large class of symmetric Markov processes, as are illustrated in the last section of this paper. We also discuss the generalized 3G theorem for a large class of symmetric stable Lévy processes.     

Numerical solution of differential equations using Sinc method based on the interpolation of the highest derivatives

In general, we will use the numerical differentiation when dealing with the differential equations. Thus the differential equations can be transformed into algebraic equations and then we can get the numerical solutions. But as we all have known, the numerical differentiation process is very sensitive to even a small level of errors. In contrast it is expected that on average the numerical integration process is much less sensitive to errors. In this paper, based on the Sinc method we provide a new method using Sinc method incorporated with the double exponential transformation based on the interpolation of the highest derivatives (SIHD) for the differential equations. The error in the approximation of the solution is shown to converge at an exponential rate. The numerical results show that compared with the exiting results, our method is of high accuracy, of good convergence with little computational efforts. It is easy to treat nonhomogeneous mixed boundary condition for our method, which is unlike the traditional Sinc method.     

Some applications of thermal field theory to quark-gluon plasma

We briefly introduce the thermal field theory within imaginary time formalism, the hard thermal loop perturbation theory and some of its applications to the physics of the quark-gluon plasma, possibly created in relativistic heavy-ion collisions