Operational Transformation

A new family of solvable potentials related to the Schroedinger-Riccati equation has been investigated. This one-dimensional potential family depends on parameters and is restricted to the real interval. It is shown that this potential class, which is a rather general class of solvable potentials related to the hypergeometric functions, can be generalized to even wider classes of solvable potentials. As a consequence the nonlinear Schroedingertype equation has been obtained.

We study the exactly solvable quantum system of two particles confined in a three-dimensional harmonic trap and interacting via finite-range soft-core interaction by means of the separation of variables and ansatz method. Supposing the solution in the form of ansatz $\Psi(r)=r^{a}e^{-\lambda r^2} \psi(r)$ we transform the time independent Schrodinger equation into Kummer's differential equation whose solution are given in the form of confluent hypergeometric function. We also discuss that in the absence of central force potential, the quantum system map into the problem of two quantum particle trapped in one-dimension harmonic oscillator and interacting through finite distance soft-core potential. In such special case the eigen value equation become the Weber's differential equation and its solution are also given in the form of confluent hypergeometric function.

A unified approach, for solving a wide class of single and many-body quantum problems, commonly encountered in literature is developed based on a recently proposed method for finding solutions of linear differential equations. Apart from dealing with exactly and quasi-exactly solvable problems, the present approach makes transparent various properties of the familiar orthogonal polynomials and also the construction of their

We make use of a recently developed method to, not only obtain the exactly known eigenstates and eigenvalues of a number of quasi-exactly solvable Hamiltonians, but also construct a convergent approximation scheme for locating those levels, not amenable to analytical treatments. The fact that, the above method yields an expansion of the wave functions in terms of corresponding energies, enables one to treat energy as a variational parameter, which can be effectively used for the identification of the eigenstates. It is particularly useful for the quasi-exactly solvable systems, where the ground state is known and a number of eigenstates are bounded, both below and above. The efficacy of the procedure is illustrated by obtaining, the low-lying excited states of a prototypical double-well potential, where the conventional techniques are not very reliable. Our approach yields the approximate eigenfunctions and eigenvalues, whose accuracy can be improved to any desired level, in a contr...

A method is developed to construct the solutions of one and many variable, linear differential equations of arbitrary order. Using this, the N -particle Sutherland model, with pair-wise inverse sine-square interactions among the particles, is shown to be equivalent to free particles on a circle. Applicability of our method to many other few and many-body problems is also illustrated.

We present here a supersymmetric (SUSY) approach for determining excitation energies within the context of a quantum Monte Carlo scheme. By using the fact that SUSY quantum mechanics gives rises to a series of isospectral Hamiltonians, we show that Monte Carlo ground-state calculations in the SUSY partners can be used to reconstruct accurately both the spectrum and states of an arbitrary Schrödinger equation. Since the ground-state of each partner potential is node-less, we avoid any "node"-problem typically associated with the Monte Carlo technique. While we provide an example of using this approach to determine the tunneling states in a double-well potential, the method is applicable to any 1D potential problem. We conclude by discussing the extension to higher dimensions.

Essentially, the Darboux proposition is based on the covariance properties of ordinary and partial differential equations with respect to a gauge transformation in the special case of second order differential equations of the Sturm-Liouville type. In this work, the one-dimensional Schrödinger equation with a position-dependent mass (SEPDM) is transformed into a Schrödinger-like equation with a position-independent mass (SLEPIM) for an effective potential which incorporates the spatially dependent mass. Therefore, taking advantage of the similarity between the SLEPIM and the Sturm-Liouville differential equation it is shown the application of the Darboux transform to the SEPDM problem.

In recent years collaborative editing systems such as wikis, GoogleDocs and version control systems became very popular. In order to improve reliability, fault-tolerance and availability shared data is replicated in these systems. User misbehaviors can make the system inconsistent or bring corrupted updates to replicated data. Solutions to secure data history of state-based replication exist, however they are hardly applied to operation-based replication. In this paper we propose an approach to secure log in operation-based ...

Following the program of Moggi, the semantics of a simple non-deterministic functional language with recursion and failure is described by a monad. We show that this monad cannot be any of the known power domain constructions, because they do not handle non-termination properly. Instead, a novel construction is proposed and investigated. It embodies both nondeterminism (choice and failure) and possible non-termination caused by recursion.

A method for solving dual and triple Fourier-Bessel series equations is proposed. It is based on a novel operator transforming Bessel functions into the sine function and on an inversion formula analogous to one for Bessel series. As a result, the dual and triple equations are transformed into the Fredholm integral equations of the second kind or into the singular integral equations of a well-known type. The suggested approach differs from the existing and provides new possibilities for applications. This is demonstrated by the torsional problem for an annular punch in contact with an inhomogeneous elastic cylinder. The asymptotic solution is derived as a distance between the punch edge and the lateral cylinder surface is short provided that the punch hole is small. A number of novel results for series containing Bessel functions is obtained as well. (~) 2004 Elsevier Ltd. All rights reserved.

We make a nodal analysis of the processes of level crossings in a model of quantum mechanics with a PT -symmetric double well. We prove the existence of infinite crossings with their selection rules. At the crossing, before the PT-symmetry breaking and the localization, we have a total P-symmetry breaking of the states.

It is well known that matrix elements of common operators play an important role in quantum mechanics. In this paper, analytic expressions for the matrix elements of some operators are obtained in the Rosen-Morse and Manning-Rosen basis. With such matrix elements, the Rosen-Morse and Manning-Rosen oscillators become a practical building block in molecular theory and a simultaneous treatment by introduction of the Jacobi polynomials.

This paper deals with a new class of Fredholm integral equations of the first kind associated with Hankel transforms of integer order. Analysis of the equations is based on operators transforming Bessel functions of the first kind into kernels of Weber-Orr integral transforms. Their inverse operators are established by means of new inversion theorems for the Hankel and Weber-Orr integral transforms of functions belonging to L 1 and L 2 . These operators together with the proven Paley-Wiener's theorem for the Weber-Orr transform enable to regularize the equations and, in special cases, to derive explicit solutions. The integral equations analyzed in this paper can be employed instead of dual integral equations usually treated with the Cooke-Lebedev method. An example manifests that it may be preferable because of the possibility to control norms of operators in the regularized equations.

A Case Study of a Community Based Participatory Approach to Agroforestry Research and Development in Western Kenya ___________________________________________________________________________ A journey of a thousand miles begins with a single step, so the saying by Chinese Philosopher Confucious goes. This journey began in September 2003 when I came to Wageningen University to enroll for a PhD Programme under the Participatory Approaches and Upscaling Programme (PAU) coordinated by Technology and Agrarian Development (TAD) Chair Group. It has been indeed a long journey full of ups and downs, but everything that has a begginning also has an end. It has finally come to an end and I wish to acknowledge the support of people and institutions that have contributed to my being able to complete this journey. First and foremost I wish to thank the Rockefeller Foundation, for providing financial support both for my fieldwork in Kenya and my stay in the Netherlands. My gratitude also goes to Dr. Paul Konuche -Director, Kenya Forestry Research Institute for granting me study leave. I also wish to thank the Centre Director, Muguga Regional Research Centre, Mr Elly Mwanza for logistical support; he was always willing to assist. He provided a vehicle whenever I wanted to go to the field, even on short notice. Asante sana! I wish to acknowledge the support I received from my supervision team. My promoter, Prof. Paul Richards, co-promoters Dr. Paul Hebinck and Dr. Steven Franzel all put in many hours to ensure the completion of this thesis. It was indeed an honour to work with Paul Richards. He introduced me to realism and realist evaluation that have formed the basis for my thesis. There are many times I got confused and overwhelmed, but I now look back and appreciate the fact that confusion is part of the process. Paul, your philosophical and critical comments/suggestions and final editing of the thesis have made it what it is today. Thank you! And to you Paul Hebinck, your optimistic nature and enthusiasm often made me feel so at ease even when the going got tough. You always had time for me even when you had other deadlines to meet. The frequent meetings we had and your suggestions greatly improved the thesis. Asante sana! To Steve, thank you for making it possible for me to use the World Agroforestry Centre (ICRAF) facilities in Nairobi and for your suggestions and valuable comments during the initial stages of my fieldwork and also when I was writing articles for journals. Your encouraging words often gave me the motivation to write more papers. Dank u wel! To Dr. Conny Almekinders, the coordinator of the PAU Programme, no amount of words can really describe my appreciation. Your vigour, energy, and support in all aspects of my stay in the Netherlands are highly appreciated; from identification of relevant courses, field visits, finding a suitable co-promoter, financial matters and even taking time to show us (PhD students in the programme) many interesting sites in the Netherlands. Thank you for incorporating personal development and peer learning in the programme, it indeed made me understand myself better and how to relate with others in addition to creating an environment of sharing ideas with my fellow students which greatly improved the development of my research proposal during the early stages. Dank u wel! Inge Ruisch, thank you for adminstrative support (tickets, making sure I had a room in Bornsesteeg whenever I came to the Netherlands, allowances etc.) and for being a friend, your occasional loud laughter, something we hardly hear of in the Netherlands, often made me feel like I was back in Kenya. Thanks Inge for taking me to many social places; it indeed made my stay in the Netherlands enjoyable. Asante sana Inge! Na Mungu akubariki. My colleagues from the third cohort; Dr.

Intention preservation is an important aspect of consistency maintenance in real-time collaborative editing systems. The multi-version approach is able to preserve individual users' concurrent conflicting intentions in a consistent way. Various multi-versioning schemes have been proposed. In ...

In a recent paper Cari\~nena et al analyzed a non-polynomial one-dimensional quantum potential representing an oscillator which they argued was intermediate between the harmonic and isotonic oscillators. In particular they proved that it is Schr\"odinger soluble, and explicitly obtained the wavefunctions and energies of the bound states. In this paper we show that these results can be obtained much more

S. Bouveret3, S. de Givry1, F. Heras2, J. Larrosa2, E. Rollon2, M. Sanchez1, T. Schiex1, G. Verfaillie3, and M. Zytnicki1 ... 1 INRA, Toulouse, France 2 Dep. LSI, UPC, Barcelona, Spain 3 ONERA, Toulouse, France ... Abstract. This document gives a brief description of the key techniques used in four different versions of toolbar/toulbar2 solvers submitted to the Max-CSP competition 2006. ... All the solvers exploit an initial upper bound found by a local search solver : maxwalksat [13] (with 5 tries) for toolbar/MaxSAT and INCOP4 [11] for the other solvers. The solvers ...

The min-degree heuristic is used to compute a tree decomposition. The root cluster is chosen as minimizing tree height then maximizing cluster product of domain sizes. Cluster separators larger than 5 are removed, by merging clusters. The same preprocessing as toolbar for n-ary constraints is done. The value heuristic chooses the last value in the best solution found so far before sorting values by increasing unary cost.

This paper presents new, efficient, massively pipelined algorithms for several list manipulation operations. Transformational programming is used in the development of these algorithms from clear functional specifications to networks of linearly connected communicating processes in CSP. The derivation of each algorithm is achieved by transforming the specification into an instance of a generic parallel functional form called pipe pattern and then refining this into CSP. The approach is demonstrated by transforming quadratic functional algorithms for sorting, removing duplicates, and calculating the difference between two lists, into pipelined versions running in linear time with a linear number of processors. The refinement from functions to CSP processes is based on a formal treatment given in earlier work by the author. Derivation and reasoning are conducted using Bird-Meertens Formalism.

Abstract: We briefly present the supersymmetric double Darboux method and next apply it to the continuum of the quantum Taub cosmological model as a toy model in order to generate a one-parameter family of bosonic Taub potentials and the corresponding wavefunctions. Recently, extensive work has been devoted to generating quasi-isospectral potentials by the methods of supersymmetric quantum mechanics (SUSYQM) [1]. This work starts from the one dimensional (1D) Schrödinger equation with a given potential (called “bosonic ” potential V−), whose “ground-state ” (nodeless) wave function is known too. The method allows one to generate families of new potentials, which may look quite different form the original one, but have the same discrete spectrum up to the ground state. SUSYQM is in fact based on a procedure already known to Darboux [2]. We shall use an extension of SUSYQM formalism based on the general Riccati equation and on a state U0(x) of a potential V−(x), by which one can gener...

A simple supersymmetric approach recently used by Dutt, Gangopadhyaya, and Sukhatme (hep-th/9611087, Am. J. Phys., to appear) for spherical harmonics is generalized to Gegenbauer and Jacobi equations. The coordinate transformation procedure is well known to the authors writing on supersymmetric quantum mechanics (see section 5 in [1]). Essentially, one starts with a one-dimensional Schrödinger equation and tries to obtain a new one by a coordinate transformation. In a recent work, Dutt, Gangopadhyaya and Sukhatme (DGS) [2] used a coordinate transformation to recast the associated Legendre equation in the Schrödinger form and then employed the concept of shape invariance to derive properties of spherical harmonics in a simple way. In the following, after presenting the case of spherical harmonics, we generalize the DGS scheme to Gegenbauer and Jacobi equations. 1. Spherical harmonics The equation for the associated Legendre polynomials dy dθ2 + cotθ dy dθ + [l(l + 1) − m 2 sin θ ]y =...

Recently, the monomiality principle has been extended to qpolynomials, namely, the q-monomiality principle of q-Appell polynomials has been considered. Also, certain results including the monomiality properties of the q-Gould-Hopper polynomials are derived and applications of monomiality are explored for a few members of the q-Appell polynomial families. In this paper, the primary purpose of this paper is to define 2-variable q-Laguerre-Appell polynomials by applying the q-monomiality principle techniques and to study their quasi-monomial properties and applications. We provide some operational identities and quasi-monomial features. Also, we derive some q-differential equations of these polynomials. As applications, using the operational identity of 2-variable q-Laguerre-Appell polynomials we draw specific conclusions regarding several q-Laguerre-Appell polynomial families. Furthermore, we define the family of q-Laguerre-Sheffer polynomials by an operational approach and give some of its fundamental properties.

During collaborative writing, shared documents are replicated on geographically distant sites. Each user works on an individual copy. This results in divergent copies. Merging techniques such as those proposed by the Operational Transformation (OT) approach reconcile the differences among the replicas and ensure their convergence. Although the merging techniques resolve conflicting syntax, they do not help preserve coherence which is an important aspect of an effective document. Therefore, we investigate the use of ideas from narrative-based writing to improve the coherence of the document during collaborative editing. Narrativebased writing is a new technique for planning documents that enhances the implicit story conveyed by a document to the readers; thereby improving coherence. This paper presents a discussion of this investigation.

During collaborative writing, shared documents are replicated on geographically distant sites. Each user works on an individual copy. This results in divergent copies. Merging techniques such as those proposed by the Operational Transformation (OT) approach reconcile the differences among the replicas and ensure their convergence. Although the merging techniques resolve conflicting syntax, they do not help preserve coherence which is an important aspect of an effective document. Therefore, we investigate the use of ideas from narrative-based writing to improve the coherence of the document during collaborative editing. Narrativebased writing is a new technique for planning documents that enhances the implicit story conveyed by a document to the readers; thereby improving coherence. This paper presents a discussion of this investigation.

Real-time collaboration is being offered by multiple libraries and APIs (Google Drive Real-time API, Microsoft Real-Time Communications API, TogetherJS, ShareJS), rapidly becoming a mainstream option for webservices developers. However, they are offered as centralised services running in a single server, regardless if they are free/ open source or proprietary software. After re-engineering Apache Wave (former Google Wave), we can now provide the first decentralised and federated free/open source alternative. The new API allows to develop new real-time collaborative web applications in both JavaScript and Java environments.

A solution is presented for the problem of realizing a discrete-time LTI state-space model of minimal McMillan degree such that its ÿrst N expansion coe cients in terms of generalized orthonormal basis match a given sequence. The basis considered, also known as the Hambo basis, can be viewed as a generalization of the more familiar Laguerre and two-parameter Kautz constructions, allowing general dynamic information to be incorporated in the basis. For the solution of the problem use is made of the properties of the Hambo operator transform theory that underlies the basis function expansion. As corollary results compact expressions are found by which the Hambo transform and its inverse can be computed e ciently. The resulting realization algorithms can be applied in an approximative sense, for instance, for computing a low-order model from a large basis function expansion that is obtained in an identiÿcation experiment.

In recent years collaborative editing systems such as wikis, GoogleDocs and version control systems became very popular. In order to improve reliability, fault-tolerance and availability shared data is replicated in these systems. User misbehaviors can make the system inconsistent or bring corrupted updates to replicated data. Solutions to secure data history of state-based replication exist, however they are hardly applied to operation-based replication. In this paper we propose an approach to secure log in operation-based ...

The Low Energy Ion Ring (LEIR) is a central piece for LHC ion operation at CERN, transforming long Linac3 pulses into high density bunches needed for LHC. The first phase of LEIR commissioning successfully attained its goal of providing the so-called "early ion beam" (one bunch of 2.25 108 Lead ions) needed for the first LHC ion runs with reduced

The first LHC Pb ion run took place at 1.38 A TeV/c per beam in autumn 2010. After a short period of runningin, the injector chain was able to fill the collider with up to 137 bunches per ring, with an intensity of 108 Pb82+ ions/bunch, about 50% higher than the design value. This yielded a luminosity of 3×1025 Hz/cm2, allowing the experiments to accumulate just under 10 b-1 each during the four week run. We review the performance of the individual links of the injector chain, and address the main issues limiting the LHC luminosity, in view of reaching 1026 Hz/cm2 in 2011, and substantially beyond when the LHC energy increases after the long shutdown in 2013-14.

Virtual Arc Consistency (VAC) is a recent local consistency for processing Cost Function Networks (or Weighted Constraint Networks) that exploits a simple but powerful connection with classical Constraint Networks. It has allowed to close hard frequency assignment benchmarks and is capable of directly solving networks of submodular functions. The algorithm enforcing VAC is an iterative algorithm that solves a sequence of classical Constraint Networks. In this work, we show that Dynamic Arc Consistency algorithms can be suitably injected in the virtual arc consistency iterative algorithm, providing noticeable speedups.

Nowadays, real-time editing systems are catching on. Tools such as Etherpad or Google Docs enable multiple authors at dispersed locations to collaboratively write shared documents. In such systems, a replication mechanism is required to ensure consistency when merging concurrent changes performed on the same document. Current editing systems make use of operational transformation (OT), a traditional replication mechanism for concurrent document editing. Recently, Commutative Replicated Data Types (CRDTs) were introduced as a new class of replication mechanisms whose concurrent operations are designed to be natively commutative. CRDTs, such as WOOT, Logoot, Treedoc, and RGAs, are expected to be substitutes of replication mechanisms in collaborative editing systems. This paper demonstrates the suitability of CRDTs for real-time collaborative editing. To reflect the tendency of decentralised collaboration, which can resist censorship, tolerate failures, and let users have control over documents, we collected editing logs from real-time peer-to-peer collaborations. We present our experiment results obtained by replaying those editing logs on various CRDTs and an OT algorithm implemented in the same environment.

Often collaborative graphical systems lag behind in features with well accepted single-user applications. The frequently used operations of group/ungroup offered by almost every single-user graphical editor have not been considered by the collaborative graphical editing systems. In this paper we present an algorithm for consistency maintenance in collaborative graphical editing dealing not only with simple operations such as create, delete, move, change colour or position, but also with group/ungroup operations. The system relying on this algorithm is customizable, offering three policy modes for dealing with conflict: null-effect based policy, priority based policy and multi-versioning based policy.

The handling of conflicts in collaborative editing over hierarchical documents in peer-to-peer networks is a very important issue. In this paper we show how the multi-level editing approach that recursively applies a tombstone transformational approach over the document levels offers support for flexible conflict definition and resolution.

Existing collaborative environments such as Wiki or version control systems allow users to work in isolation, but they do not prevent blind modifications. For instance, users may concurrently perform the same task or they might work on obsolete versions of shared documents. We propose a novel writing mode for avoiding blind modifications by providing real-time information about group activities. Changes performed concurrently are filtered according to user privacy preferences and depicted in their local documents.

In recent years collaborative editing systems such as wikis, GoogleDocs and version control systems became very popular. In order to improve reliability, fault-tolerance and availability shared data is replicated in these systems. User misbehaviors can make the system inconsistent or bring corrupted updates to replicated data. Solutions to secure data history of state-based replication exist, however they are hardly applied to operation-based replication. In this paper we propose an approach to secure log in operation-based ...

A journey of a thousand miles begins with a single step, so the saying by Chinese Philosopher Confucious goes. This journey began in September 2003 when I came to Wageningen University to enroll for a PhD Programme under the Participatory Approaches and Upscaling Programme (PAU) coordinated by Technology and Agrarian Development (TAD) Chair Group. It has been indeed a long journey full of ups and downs, but everything that has a begginning also has an end. It has finally come to an end and I wish to acknowledge the support of people and institutions that have contributed to my being able to complete this journey. First and foremost I wish to thank the Rockefeller Foundation, for providing financial support both for my fieldwork in Kenya and my stay in the Netherlands. My gratitude also goes to Dr. Paul Konuche-Director, Kenya Forestry Research Institute for granting me study leave. I also wish to thank the Centre Director, Muguga Regional Research Centre, Mr Elly Mwanza for logistical support; he was always willing to assist. He provided a vehicle whenever I wanted to go to the field, even on short notice. Asante sana! I wish to acknowledge the support I received from my supervision team. My promoter, Prof. Paul Richards, co-promoters Dr. Paul Hebinck and Dr. Steven Franzel all put in many hours to ensure the completion of this thesis. It was indeed an honour to work with Paul Richards. He introduced me to realism and realist evaluation that have formed the basis for my thesis. There are many times I got confused and overwhelmed, but I now look back and appreciate the fact that confusion is part of the process. Paul, your philosophical and critical comments/suggestions and final editing of the thesis have made it what it is today. Thank you! And to you Paul Hebinck, your optimistic nature and enthusiasm often made me feel so at ease even when the going got tough. You always had time for me even when you had other deadlines to meet. The frequent meetings we had and your suggestions greatly improved the thesis. Asante sana! To Steve, thank you for making it possible for me to use the World Agroforestry Centre (ICRAF) facilities in Nairobi and for your suggestions and valuable comments during the initial stages of my fieldwork and also when I was writing articles for journals. Your encouraging words often gave me the motivation to write more papers. Dank u wel! To Dr. Conny Almekinders, the coordinator of the PAU Programme, no amount of words can really describe my appreciation. Your vigour, energy, and support in all aspects of my stay in the Netherlands are highly appreciated; from identification of relevant courses, field visits, finding a suitable co-promoter, financial matters and even taking time to show us (PhD students in the programme) many interesting sites in the Netherlands. Thank you for incorporating personal development and peer learning in the programme, it indeed made me understand myself better and how to relate with others in addition to creating an environment of sharing ideas with my fellow students which greatly improved the development of my research proposal during the early stages. Dank u wel! Inge Ruisch, thank you for adminstrative support (tickets, making sure I had a room in Bornsesteeg whenever I came to the Netherlands, allowances etc.) and for being a friend, your occasional loud laughter, something we hardly hear of in the Netherlands, often made me feel like I was back in Kenya. Thanks Inge for taking me to many social places; it indeed made my stay in the Netherlands enjoyable. Asante sana Inge! Na Mungu akubariki. My colleagues from the third cohort; Dr.

Conflict-free Replicated Data Types (CRDTs) allow collaborative access to an app's data. We describe a novel CRDT operation, for-each on the list of CRDTs, and demonstrate its use in collaborative apps. Our for-each operation applies a given mutation to each element of a list, including elements inserted concurrently. This often preserves user intention in a way that would otherwise require custom CRDT algorithms. We give example applications of our for-each operation to collaborative rich-text, recipe, and slideshow editors. CCS Concepts: • Theory of computation → Distributed algorithms; • Human-centered computing → Collaborative and social computing systems and tools.

The harmonic oscillator is a model which has several important applications in both classical and quantum mechanics. It serves as a prototype in the mathematical treatment of such diverse phenomena as elasticity, acoustics, AC circuits, molecular and crystal vibrations, electromagnetic fields and optical properties of matter.

We apply the continuous variable approach to study entangled dynamics of coupled harmonic oscillators interacting with a thermal reservoir and to a deterministic creation of entanglement in an atomic ensemble located inside a high-Q ring cavity. In the case of harmonic oscillators, we show that a suitable unitary transformation of the position and momentum operators transforms the system to a set of independent harmonic oscillators with only one of them coupled to the reservoir. Working in the Wigner representation of the density operator, we find that the covariance matrix has a block diagonal form of smaller size matrices. This simple property allows to treat the problem to some extend analytically. We analyze the time evolution of an initial entanglement and find that the entanglement can persists for long times due to presence of constants of motion for the covariance matrix elements. In the case of an atomic ensemble located inside the cavity, the attention is focused on creation of one and two-mode continuous variable entangled states from the vacuum by applying laser pulses of a suitably adjusted amplitudes and phases. The pulses together with the cavity dissipation prepare the collective modes of the atomic ensemble in a desired entangled state.

Association rule learning is a popular and well-researched set of methods for discovering interesting relations between entities in large databases in real-world problems. In this regard, an intelligent offshore oil industry environment is a very complex scenario and Occupational Health and Security (OHS) is a priority issue as it is an important factor to reduce the number of accidents and incidents records. In the oil industry, there exist standards to identify and record workplace accidents and incidents in order to provide guiding means on prevention efforts, indicating specific failures or reference, means of correction of conditions or circumstances that culminated in accident. OHS's employees are in charge of analyzing the mined rules to extract knowledge. In most of cases these users has two main challenges during this process: i) to explore the measures of interestingness (confidence, lift, support, etc.) and ii) to understand and analyze the large number of association rules. In this sense, an intuitive visualization of mined rules becomes a key component in a decision-making process. In this paper, we propose a novel visualization of spatio-temporal rules that provides the big picture about risk analysis in a real world environment. Our main contribution lies in an interactive visualization of accident interpretations by means of welldefined spatio-temporal constraints, in the oil industry domain.

The problem of the characterization of all analytic potentials which give rise to isochronous oscillatory motions still open. However, there are several approaches to highlight motions with period T (E) ≡ T 0 independent on the energy. It is proposed in this paper to give necessary and sufficient conditions for a center to be isochronous. The proofs produced here are self contained. As corollaries we find again lot of known conditions previously established by different authors. This allows us to produce several class of isochronous analytic motions. We are also interested in the quantum spectrum. We then use the perturbation method WKB to derive an expression for the corrections to the equally spaced valid for analytic isochronous potentials.

We give a construction of the Lie algebras of the non-compact groups appearing in four dimensional supergravity theories in terms of boson operators. Our construction parallels very closely their emergence in supergravity and is an extension of the well-known construction of the Lie algebras of the non-compact groups SP (2n, R) and SO (2n)* from boson operators transforming like a fundamental representation of their maximal compact subgroup U (n). However this extension is nontrivial only for n/> 4 and stops at n = 8 leading to the Lie algebras of SU(4) X SU(1, 1), SU(5, 1), SO(12)* and E7(7). We then give a general construction of an infinite class of unitary irreducible representations of the respective non-compact groups (except for E7(7) and SO(12)* obtained from the extended construction). We illustrate our construction with the examples of SU(5, 1) and SO(12)*.

An elementary introduction is given to the subject of Supersymmetry in Quantum Mechanics which can be understood and appreciated by any one who has taken a first course in quantum mechanics. We demonstrate with explicit examples that given a solvable problem in quantum mechanics with n bound states, one can construct new exactly solvable n Hamiltonians having n-1,n-2,...,0 bound states. The relationship between the eigenvalues, eigenfunctions and scattering matrix of the supersymmetric partner potentials is derived and a class of reflectionless potentials are explicitly constructed. We extend the operator method of solving the one-dimensional harmonic oscillator problem to a class of potentials called shape invariant potentials. It is worth emphasizing that this class includes almost all the solvable problems that are found in the standard text books on quantum mechanics. Further, we show that given any potential with at least one bound state, one can very easily construct one continuous parameter family of potentials having same eigenvalues and s-matrix. The supersymmetry inspired WKB approximation (SWKB) is also discussed and it is shown that unlike the usual WKB, the lowest order SWKB approximation is exact for the shape invariant potentials and further, this approximation is not only exact for large quantum numbers but by construction, it is also exact for the ground state. Finally, we also construct new exactly solvable periodic potentials by using the machinery of supersymmetric quantum mechanics.