Algebraic Treatment Research Articles

Karlin–McGregor-like formula in a simple time-inhomogeneous birth–death process

Algebraic discussions are developed to derive transition probabilities for a simple time-inhomogeneous birth–death process. Algebraic probability theory and Lie algebraic treatments make it easy to treat the time-inhomogeneous cases. As a result, an expression based on the Charlier polynomials is obtained, which can be considered as an extension of a famous Karlin–McGregor representation for a time-homogeneous birth–death process.

Algebraic Treatment of 𝓟𝓣-Symmetric Coupled Oscillators

The purpose of this paper is the discussion of a pair of coupled linear oscillators that has recently been proposed as a model of a system of two optical resonators. By means of an algebraic approach we show that the frequencies of the classical and quantum-mechanical interpretations of the optical phenomenon are exactly the same. Consequently, if the classical frequencies are real, then the quantum-mechanical eigenvalues are also real.

La suite exacte de Mayer–Vietoris en cohomologie de Čech

This article is about the foundation of the Mayer–Vietoris exact sequence in Čech cohomology, with respect to the augmented Čech complex, often called “the stable Koszul complex”. Our treatment is elementary and uses neither the local cohomology of Grothendieck nor the widespread noetherian local cohomology.We use explicit objects giving us an elementary algebraic treatment in a general nonnoetherian context, i.e. two finite sequences a̲, b̲ of a commutative ring and some Čech complexes built from these sequences. More precisely, our strategy consists in producing a short exact sequence of “type Čech“ complexes having the expected cohomologies. Then arises a simplicial complex on the set (a̲,b̲,ab̲) with a relatively surprising combinatory.

Algebraic treatment of alpha-cluster nuclei

I discuss an algebraic treatment of alpha-cluster nuclei in the framework of the Algebraic Cluster Model. It is suggested to treat 12C as an oblate top with symmetry (equilateral triangle) and 16O as a spherical top with tetrahedral symmetry (regular tetrahedron). The structure of the rotational bands is a consequence of the underlying symmetry, and can be used to help determine the geometrical configuration of the α-particles.

Theory of open quantum systems with bath of electrons and phonons and spins: Many-dissipaton density matrixes approach

This work establishes a strongly correlated system-and-bath dynamics theory, the many-dissipaton density operators formalism. It puts forward a quasi-particle picture for environmental influences. This picture unifies the physical descriptions and algebraic treatments on three distinct classes of quantum environments, electron bath, phonon bath, and two-level spin or exciton bath, as their participating in quantum dissipation processes. Dynamical variables for theoretical description are no longer just the reduced density matrix for system, but remarkably also those for quasi-particles of bath. The present theoretical formalism offers efficient and accurate means for the study of steady-state (nonequilibrium and equilibrium) and real-time dynamical properties of both systems and hybridizing environments. It further provides universal evaluations, exact in principle, on various correlation functions, including even those of environmental degrees of freedom in coupling with systems. Induced environmental dynamics could be reflected directly in experimentally measurable quantities, such as Fano resonances and quantum transport current shot noise statistics.

Lattice structures for attractors I

We describe the basic lattice structures of attractors and repellers in dynamical systems.The structure of distributive lattices allows for an algebraic treatment of gradient-like dynamics in general dynamical systems, both invertible and noninvertible. We separate those properties which rely solely on algebraic structures from those that require some topological arguments, in order to lay a foundation forthe development of algorithms to manipulate these structures computationally.

Handling Algebraic Effects

Algebraic effects are computational effects that can be represented by an equational theory whose operations produce the effects at hand. The free model of this theory induces the expected computational monad for the corresponding effect. Algebraic effects include exceptions, state, nondeterminism, interactive input/output, and time, and their combinations. Exception handling, however, has so far received no algebraic treatment. We present such a treatment, in which each handler yields a model of the theory for exceptions, and each handling construct yields the homomorphism induced by the universal property of the free model. We further generalise exception handlers to arbitrary algebraic effects. The resulting programming construct includes many previously unrelated examples from both theory and practice, including relabelling and restriction in Milner's CCS, timeout, rollback, and stream redirection.

Airports and airlines economics and policy: An interpretive review of recent research

This paper provides an interpretive review of recent research, concentrating on three areas where new and important insights have been derived during the last years. We present the main ideas in a general unifying framework—there is somewhat more emphasis on a unifying analytical structure, as well as a sufficiently detailed algebraic treatment, of each topic area than in most other surveys. The part on “airport congestion and pricing” is a natural sequel of the survey paper by Basso and Zhang (2007), who concentrated on theoretical airport pricing studies. We have the obvious opportunity to include more recent material, leading to a more complete guide to the literature. Similarly, the part on “airline alliances” contains up-to-date research on alliance modeling and policy. The part on “airport privatization and regulation” has not been covered by other surveys before. We also elaborate important avenues for future research.

Algebraic treatment of the time-dependent Jaynes–Cummings Hamiltonian including nonlinear terms

A generalization of the time-dependent Jaynes–Cummings Hamiltonian is considered including nonlinear terms in the field and/or an intensity-dependent coupling term. Using algebraic methods we find the corresponding time evolution operator of the problem as a product of exponential operators. Also, we study the effect of the atomic motion in the cavity on its quantum state. We found that, when the atom is vibrating classically with a small amplitude A0 (in comparison with the effective length of the cavity L), i.e. A0 ≪ L, the effects of the atom’s motion are negligible but when A0 ∼ L/4, the state of the atom changes drastically. Our study is devoted to a Jaynes–Cummings Hamiltonian with an additional Kerr medium.

Spectrum generating algebras for few-body systems

I discuss an algebraic treatment of few-body systems based on spectrum generating algebras for the relative motion of the clusters. Particular attention is paid to the formalism for identical clusters including a discussion of the permutation symmetry and a study of special solutions which are shown to correspond to the harmonic oscillator, the deformed oscillator, the oblate symmetric top and the spherical top with tetrahedral symmetry.

Extension of HPL to complex arguments

In this paper we describe the extension of the Mathematica package HPL to treat harmonic polylogarithms of complex arguments. The harmonic polylogarithms have been introduced by Remiddi and Vermaseren [E. Remiddi, J.A.M. Vermaseren, Int. J. Modern Phys. A 15 (2000) 725, hep-ph/9905237] and have many applications in high energy particle physics. New version program summary Program title: HPL Catalogue identifier: ADWX_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADWX_v2_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 13 610 No. of bytes in distributed program, including test data, etc.: 1 055 706 Distribution format: tar.gz Programming language: Mathematica 7/8. Computer: All computers running Mathematica. Operating system: Operating systems running Mathematica. Supplementary material: Additional “high weight” MinimalSet files available. Classification: 4.7. Catalogue identifier of previous version: ADWX_v1_0 Journal reference of previous version: Comput. Phys. Comm. 174 (2006) 222 Does the new version supersede the previous version?: Yes Nature of problem: Computer algebraic treatment of the harmonic polylogarithms which appear in the evaluation of Feynman diagrams. Solution method: Mathematica implementation. Reasons for new version: Added treatment of complex arguments. Details in arXiv:hep-ph/0703052. Summary of revisions: Added treatment of complex arguments. Details in arXiv:hep-ph/0703052. Running time: A few seconds for each function.

An algebraic treatment of imprecise probabilities

Abstract This is a survey paper about an algebraic approach to imprecise probabilities. In the first part of it, we outline the work by Walley on imprecise probabilities and the more algebraic approach of Fedel et al.. Then, in the second part we will present some work in progress about a general treatment of upper and lower probabilities over many-valued events and of upper and lower previsions of gambles, by means of Universal Algebra.

Lie algebraic approach and quantum treatment of an anisotropic charged particle via the quadratic invariant

We consider the problem of a charged harmonic oscillator under the influence of a constant magnetic field. The system is assumed to be anisotropic and the magnetic field is applied along z-axis. A canonical transformation is invoked to remove the interaction term and the system is reduced to a model contains two uncoupled harmonic oscillators. Two classes of real and complex quadratic invariants (constants of motion) are obtained. We employ the Lie algebraic technique to find the most general solution for the wave-function for both real and complex invariants. The quadratic invariant is also used to derive two classes of creation and annihilation operators from which the wave-functions in the coherent states and number states are obtained. Some discussion related to the advantage of using the quadratic invariants to solve the Cauchy problem instead of the direct use of the Hamiltonian itself is also given.

A Noncommutative Enumeration Problem

We tackle the combinatorics of coloured hard-dimer objects. This is achieved by identifying coloured hard-dimer configurations with a certain class of rooted trees that allow for an algebraic treatment in terms of noncommutative formal power series. A representation in terms of matrices then allows to find the asymptotic behaviour of these objects.

In Memoriam: James K. Knowles 1931–2009

In Memoriam: James K. Knowles 1931–2009

Split quaternion nonlinear adaptive filtering

A split quaternion learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of three- and four-dimensional signals is proposed. The derivation takes into account the non-commutativity of the quaternion product, an aspect neglected in the derivation of the existing learning algorithms. It is shown that the additional information taken into account by a rigorous treatment of quaternion algebra provides improved performance on hypercomplex processes. A rigorous analysis of the convergence of the proposed algorithms is also provided. Simulations on both benchmark and real-world signals support the approach.

A Nominal Axiomatization of the Lambda Calculus

The lambda calculus is fundamental in computer science. It resists an algebraic treatment because of capture-avoidance sideconditions. Nominal algebra is a logic of equality designed for specifications involving binding. We axiomatize the lambda calculus using nominal algebra, demonstrate how proofs with these axioms reflect the informal arguments on syntax and we prove the axioms to be sound and complete. We consider both non-extensional and extensional versions (alpha-beta and alpha-beta-eta equivalence). This connects the nominal approach to names and binding with the view of variables as a syntactic convenience for describing functions. The axiomatization is finite, close to informal practice and it fits into a context of other research such as nominal rewriting and nominal sets.

A Markov model for analyzing the evolution of bladder carcinoma

A Markovian approach to analyze different states of the superficial vesical carcinoma is considered, taking into account up to two recurrences and the possibility of progression. So, three transient states are considered: free of disease, first, and second recurrence; and an absorbent state, the progression. A methodology based in phase-type distributions is also used, that allows the usual quantities of interest in survival studies to be expressed in a well-structured form. This type of distribution has shown its utility in queue theory, and has the advantage that mathematical expressions can be presented in a closed form that allows algebraic treatment.

Lie algebraic treatment of the quadratic invariants for a quantum system

We consider the problem of the time-dependent degenerate parametric amplifier. We obtain the quadratic invariant and use it to derive the wave function via its su(1, 1) algebraic basis and a unitary transformation to the time-dependent Schrodinger equation for the parametric amplifier. We obtain the real and the complex invariants, which we use to solve the time-dependent Cauchy problem. Using different integrability conditions, we find the most general solution, which we analyze extensively, providing details of the calculations.

ON MODULAR SYMBOLS AND THE COHOMOLOGY OF HECKE TRIANGLE SURFACES

The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalized from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the standard algorithms for the computation of holomorphic modular forms. Precise and explicit connections are established to the cohomology of Hecke triangle surfaces and group cohomology. A general commutative ring is used as coefficient ring in view of applications to the computation of modular forms over rings different from the complex numbers.