Infinite String Research Articles

An anyon model

We construct an infinite-dimensional dynamical Hamiltonian system that can be interpreted as a localized structure (“quasiparticle”) on the plane E2. The model is based on the theory of an infinite string in the Minkowski space E1,3formulated in terms of the second fundamental forms of the worldsheet. The model phase space H is parameterized by the coordinates, which are interpreted as “internal” (E(2)-invariant) and “external” (elements of T*E2) degrees of freedom. The construction is nontrivial because H contains a finite number of constraints entangling these two groups of coordinates. We obtain the expressions for the energy and for the effective mass of the constructed system and the formula relating the proper angular momentum and the energy. We consider a possible interpretation of the proposed construction as an anyon model.

Effects of Finite Water Depth on Natural Frequencies of Suspended Water Tanks

Linear sloshing problems (inviscid irrotational flows) in suspended tanks are revisited, with an intention to address some issues in the previous study based on shallow water wave theory. Time-periodic solutions are considered, which describe the synchronized oscillation of the water and tank, reached after the initial transient dies out. The solutions are developed for arbitrary water depths, and separate explicitly the propagating and evanescent wave components of the fluid motion, illuminating clearly the physics and converging rapidly. At the limit of infinite string length, these solutions describe the sloshing motions in tanks that are free to oscillate horizontally on a frictionless plane. Various effects on the lowest sloshing mode are discussed, emphasizing the physical interpretations and examining the limitations of the shallow water approximations. Comparisons with existing laboratory experiments are made, showing agreements with the analysis.

A comparison between LQR control for a long string of SISO systems and LQR control of the infinite spatially invariant version

In this paper we consider a long string of SISO systems which in the limit becomes a scalar infinite spatially invariant system. We compare the LQR control for long-but-finite strings with the LQR control for the corresponding infinite strings. We give analytical and numerical examples where these are quite different and we investigate the cause. In addition, we obtain sufficient conditions for the LQR solutions to be similar as the length of the string increases.

Large N reduction with the twisted Eguchi-Kawai model

We examine the breaking of Z N symmetry recently reported for the Twisted Eguchi-Kawai model (TEK). We analyse the origin of this behaviour and propose simple modifications of twist and lattice action that could avoid the problem. Our results show no sign of symmetry breaking and allow us to obtain values of t he large N infinite volume string tension in agreement with extrapolations from results based upon straightforward methods.

Gravitational waves from kinks on infinite cosmic strings

Gravitational waves emitted by kinks on infinite strings are investigated using detailed estimations of the kink distribution on infinite strings. We find that gravitational waves from kinks can be detected by future pulsar timing experiments such as SKA for an appropriate value of the string tension, if the typical size of string loops is much smaller than the horizon at their formation. Moreover, the gravitational wave spectrum depends on the thermal history of the Universe and hence it can be used as a probe into the early evolution of the Universe.

Distributional Convergence for the Number of Symbol Comparisons Used by QuickSort (Extended Abstract)

Most previous studies of the sorting algorithm $\mathtt{QuickSort}$ have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the $n$ independent and identically distributed (iid) keys are each represented as a sequence of symbols from a probabilistic source and that $\mathtt{QuickSort}$ operates on individual symbols, and we measure the execution cost as the number of symbol comparisons. Assuming only a mild "tameness'' condition on the source, we show that there is a limiting distribution for the number of symbol comparisons after normalization: first centering by the mean and then dividing by $n$. Additionally, under a condition that grows more restrictive as $p$ increases, we have convergence of moments of orders $p$ and smaller. In particular, we have convergence in distribution and convergence of moments of every order whenever the source is memoryless, i.e., whenever each key is generated as an infinite string of iid symbols. This is somewhat surprising: Even for the classical model that each key is an iid string of unbiased ("fair'') bits, the mean exhibits periodic fluctuations of order $n$.

On the expressiveness and complexity of randomization in finite state monitors

In this article, we introduce the model of finite state probabilistic monitors (FPM), which are finite state automata on infinite strings that have probabilistic transitions and an absorbing reject state. FPMs are a natural automata model that can be seen as either randomized run-time monitoring algorithms or as models of open, probabilistic reactive systems that can fail. We give a number of results that characterize, topologically as well as with respect to their computational power, the sets of languages recognized by FPMs. We also study the emptiness and universality problems for such automata and give exact complexity bounds for these problems.

Generalized Coherent States of a Particle in a Time-Dependent Linear Potential

We derive, with an invariant operator method and unitary transformation approach, that the Schrödinger equation with a time-dependent linear potential possesses an infinite string of shape-preseving wave-packet states |ϕα,λ(t)⟩ having classical motion. The qualitative properties of the invariant eigenvalue spectrum (discrete or continuous) are described separately for the different values of the frequency ω of a harmonic oscillator. It is also shown that, for a discrete eigenvalue spectrum, the states |ϕα,n(t)⟩ could be obtained from the coherent state |ϕα,0(t)⟩.

Electrodynamic Casimir effect in a medium-filled wedge

We re-examine the electrodynamic Casimir effect in a wedge defined by two perfect conductors making dihedral angle alpha=pi/p. This system is analogous to the system defined by a cosmic string. We consider the wedge region as filled with an azimuthally symmetric material, with permittivity and permeability epsilon1, micro1 for distance from the axis r<a, and epsilon2, micro2 for r>a. The results are closely related to those for a circular-cylindrical geometry, but with noninteger azimuthal quantum number mp. Apart from a zero-mode divergence, which may be removed by choosing periodic boundary conditions on the wedge, and may be made finite if dispersion is included, we obtain finite results for the free energy corresponding to changes in a for the case when the speed of light is the same inside and outside the radius a , and for weak coupling, |epsilon1-epsilon2|<<1, for purely dielectric media. We also consider the radiation produced by the sudden appearance of an infinite cosmic string, situated along the cusp line of the pre-existing wedge.

Description of braids in terms of first-order spectral problems

We consider the representation of colored braids in terms of world lines of return points of an infinite string in the space E1,2. We propose a description of braids by a pair of first-order spectral problems for 2 × 2 matrices.

Provably correct runtime monitoring

Runtime monitoring is an established technique to enforce a wide range of program safety and security properties. We present a formalization of monitoring and monitor inlining, for the Java Virtual Machine. Monitors are security automata given in a special-purpose monitor specification language, ConSpec. The automata operate on finite or infinite strings of calls to a fixed API, allowing local dependencies on parameter values and heap content. We use a two-level class file annotation scheme to characterize two key properties: (i) that the program is correct with respect to the monitor as a constraint on allowed program behavior, and (ii) that the program has a copy of the given monitor embedded into it. As the main application of these results we sketch a simple inlining algorithm and show how the two-level annotations can be completed to produce a fully annotated program which is valid in the standard sense of Floyd/Hoare logic. This establishes the mediation property that inlined programs are guaranteed to adhere to the intended policy. Furthermore, validity can be checked efficiently using a weakest precondition based annotation checker, thus preparing the ground for on-device checking of policy adherence in a proof-carrying code setting.

Depth as Randomness Deficiency

Depth of an object concerns a tradeoff between computation time and excess of program length over the shortest program length required to obtain the object. It gives an unconditional lower bound on the computation time from a given program in absence of auxiliary information. Variants known as logical depth and computational depth are expressed in Kolmogorov complexity theory. We derive quantitative relation between logical depth and computational depth and unify the different depth notions by relating them to A. Kolmogorov and L. Levin’s fruitful notion of randomness deficiency. Subsequently, we revisit the computational depth of infinite strings, study the notion of super deep sequences and relate it with other approaches.

Vorton construction and dynamics

Vortons are closed loops of superconducting cosmic strings carrying current and charge. In this paper we present the first numerical construction of vortons in the global version of Witten's U ( 1 ) × U ( 1 ) theory. An energy minimization procedure is used to compute stationary vortons for a range of charges and currents, and the associated vorton radius is calculated. It is found that the standard analysis based on infinite straight cosmic strings does not provide a good description of the vorton cross-section. The computed solutions are used as initial conditions in an axially symmetric time evolution code, which verifies that the solutions are indeed stationary, and are stable to axially symmetric perturbations. Perturbations which preserve the axial symmetry excite oscillatory modes and produce an evolution which eventually returns to a stationary vorton. Finally, the constructed vorton solutions are used as initial conditions in a full ( 3 + 1 )-dimensional simulation and an instability to non-axial perturbations is found. The instability produces a pinching and bending of the vorton which results in its destruction.

One‐dimensional wave animation using Mathematica

AbstractThe work presents how one‐dimensional wave phenomenon is animated. Several methods including the D'Alembert solution, the diamond rule, the Laplace transform and the convolution integral, are employed in the Mathematica animation. All the analytical derivations were carried out by using the symbolic software. Several examples, including an infinite string with a spring, mass and damper as well as a semi‐infinite string, two‐media string, string and beam subject to support motions, were demonstrated to show the validity of the present formulation. Parameter study of impedance ratio and mass, spring, and dashpot was also examined to see the transmission and reflection coefficient. © 2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 17: 323–339, 2009; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/cae.20224

Effective Control Synthesis for Partially Observed Discrete-Event Systems

Control under partial observations is studied within the infinite-string supervisory control framework. It is shown that, provided that there are no pure liveness assumptions embodied in the plant model (technically, the $\omega$-language generated by the plant is the limit of the finite-string language generated), the problem of synthesizing a centralized supervisor reduces to the case of complete observations; the result involves constructions similar to those used to “determinize” nondeterministic automata on infinite strings.

Generalisations of long wave theories for pre-stressed compressible elastic plates

Generalisations of classical bending and extension are established for pre-stressed compressible elastic plates. In respect of the analogue of extension, the associated quasi-front is shown to be either advancing or receding, contrasting with the classical case. For the generalisation of bending, the long wave limit of the fundamental mode is non-zero; thus, unlike its classical counterpart, an associated quasi-front can, therefore, exist and is again noted to be either advancing or receding. In both cases appropriate leading order and higher order corrected governing equations are obtained. The ideas are illustrated through investigation of a model problem involving impact edge loading. For the generalised theory of bending, the leading order governing equation for the mid-surface deflection is used to establish the classical equation for wave propagation along an infinite string, with its second order refinement providing a second order correction. Motion within the vicinity of the thickness shear and thickness stretch resonance frequencies is also investigated. Special cases, in which either a stretch resonance and shear resonance frequency are very close, or the speeds of longitudinal and shear waves are very close, are also discussed.

Characteristics of discrete transfinite time Turing machine models: Halting times, stabilization times, and Normal Form theorems

We give an acount of the basic determinants of the courses of computation of the Infinite Time Turing Machine model of Hamkins and Kidder, a model of computation which allows for transfinitely many steps of computation, and therefore may accept and output infinite strings of bits. We provide, inter alia, a Normal form Theorem, and a characterisation of which ordinals start gaps in halting times of such machines.

Statistical mechanics of strings with Y-junctions

We investigate the Hagedorn transitions of string networks with Y-junctions as may occur, for example, with $(p,q)$ cosmic superstrings. In a simplified model with three different types of string, the partition function reduces to three generalized coupled XY models. We calculate the phase diagram and show that, as the system is heated, the lightest strings first undergo the Hagedorn transition despite the junctions. There is then a second, higher, critical temperature above which infinite strings of all tensions, and junctions, exist. Conversely, on cooling to low temperatures, only the lightest strings remain, but they collapse into small loops.

Rise and fall of hidden string order of lattice bosons

We investigate the ground-state properties of a newly discovered phase of one-dimensional lattice bosons with extended interactions [E. G. Dalla Torre et al., Phys. Rev. Lett. 97, 260401 (2006)]. The new phase, termed the Haldane insulator in analogy with the gapped phase of spin-1 chains, is characterized by a nonlocal order parameter, which can only be written as an infinite string in terms of the bosonic densities. We show that the string order can nevertheless be probed with physical fields that couple locally, via the effect those fields have on the quantum phase transitions separating the exotic phase from the conventional Mott and density wave phases. Using a field theoretical analysis, we show that a perturbation that breaks lattice inversion symmetry gaps the critical point separating the Mott and Haldane phases and eliminates the sharp distinction between them. This is remarkable given that neither of these phases involves broken inversion symmetry. We also investigate the evolution of the phase diagram with the tunable coupling between parallel chains in an optical lattice setup. We find that interchain tunneling destroys the direct phase transition between the Mott and Haldane insulators by establishing an intermediate superfluid phase. On the other hand, coupling the chains only by weak repulsive interactions does not modify the structure of the phase diagram. The theoretical predictions are confirmed with numerical calculations using the density matrix renormalization group.

Dynamics of a generic Brownian motion: Recursive aspects

We study the local fluctuations of Brownian motions which are represented by infinite binary strings which are random in the sense of Kolmogorov–Chaitin. We show how the dynamical properties of such a Brownian motion at a point depend on its recursive properties.