Fortran programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap

Here we develop simple numerical algorithms for both stationary and non-stationary solutions of the time-dependent Gross-Pitaevskii (GP) equation describing the properties of Bose-Einstein condensates at ultra low temperatures. In particular, we consider algorithms involving real- and imaginary-time propagation based on a split-step Crank-Nicolson method. In a one-space-variable form of the GP equation we consider the one-dimensional, two-dimensional circularly-symmetric, and the three-dimensional spherically-symmetric harmonic-oscillator traps. In the two-space-variable form we consider the GP equation in two-dimensional anisotropic and three-dimensional axially-symmetric traps. The fully-anisotropic three-dimensional GP equation is also considered. Numerical results for the chemical potential and root-mean-square size of stationary states are reported using imaginary-time propagation programs for all the cases and compared with previously obtained results. Also presented are numerical results of non-stationary oscillation for different trap symmetries using real-time propagation programs. A set of convenient working codes developed in Fortran 77 are also provided for all these cases (twelve programs in all). In the case of two or three space variables, Fortran 90/95 versions provide some simplification over the Fortran 77 programs, and these programs are also included (six programs in all).

Program summary

Program title: (i) imagetime1d, (ii) imagetime2d, (iii) imagetime3d, (iv) imagetimecir, (v) imagetimesph, (vi) imagetimeaxial, (vii) realtime1d, (viii) realtime2d, (ix) realtime3d, (x) realtimecir, (xi) realtimesph, (xii) realtimeaxialCatalogue identifier: AEDU_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDU_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 122 907No. of bytes in distributed program, including test data, etc.: 609 662Distribution format: tar.gzProgramming language: FORTRAN 77 and Fortran 90/95Computer: PCOperating system: Linux, UnixRAM: 1 GByte (i, iv, v), 2 GByte (ii, vi, vii, x, xi), 4 GByte (iii, viii, xii), 8 GByte (ix)Classification: 2.9, 4.3, 4.12Nature of problem: These programs are designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in one-, two- or three-space dimensions with a harmonic, circularly-symmetric, spherically-symmetric, axially-symmetric or anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Solution method: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation, in either imaginary or real time, over small time steps. The method yields the solution of stationary and/or non-stationary problems.Additional comments: This package consists of 12 programs, see “Program title”, above. FORTRAN77 versions are provided for each of the 12 and, in addition, Fortran 90/95 versions are included for ii, iii, vi, viii, ix, xii. For the particular purpose of each program please see the below.Running time: Minutes on a medium PC (i, iv, v, vii, x, xi), a few hours on a medium PC (ii, vi, viii, xii), days on a medium PC (iii, ix).

Program summary (1)

Title of program: imagtime1d.FTitle of electronic file: imagtime1d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 1 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in one-space dimension with a harmonic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (2)

Title of program: imagtimecir.FTitle of electronic file: imagtimecir.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 1 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with a circularly-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (3)

Title of program: imagtimesph.FTitle of electronic file: imagtimesph.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 1 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with a spherically-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (4)

Title of program: realtime1d.FTitle of electronic file: realtime1d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in one-space dimension with a harmonic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (5)

Title of program: realtimecir.FTitle of electronic file: realtimecir.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with a circularly-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (6)

Title of program: realtimesph.FTitle of electronic file: realtimesph.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with a spherically-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (7)

Title of programs: imagtimeaxial.F and imagtimeaxial.f90Title of electronic file: imagtimeaxial.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Few hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an axially-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (8)

Title of program: imagtime2d.F and imagtime2d.f90Title of electronic file: imagtime2d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Few hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (9)

Title of program: realtimeaxial.F and realtimeaxial.f90Title of electronic file: realtimeaxial.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 4 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time Hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an axially-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (10)

Title of program: realtime2d.F and realtime2d.f90Title of electronic file: realtime2d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 4 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (11)

Title of program: imagtime3d.F and imagtime3d.f90Title of electronic file: imagtime3d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 4 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Few days on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (12)

Title of program: realtime3d.F and realtime3d.f90Title of electronic file: realtime3d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum Ram Memory: 8 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Days on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.     

MontePython: Implementing Quantum Monte Carlo using Python

We present a cross-language C++/Python program for simulations of quantum mechanical systems with the use of Quantum Monte Carlo (QMC) methods. We describe a system for which to apply QMC, the algorithms of variational Monte Carlo and diffusion Monte Carlo and we describe how to implement theses methods in pure C++ and C++/Python. Furthermore we check the efficiency of the implementations in serial and parallel cases to show that the overhead using Python can be negligible.

Program summary

Program title: MontePythonCatalogue identifier: ADZP_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZP_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 49 519No. of bytes in distributed program, including test data, etc.: 114 484Distribution format: tar.gzProgramming language: C++, PythonComputer: PC, IBM RS6000/320, HP, ALPHAOperating system: LINUXHas the code been vectorised or parallelized?: Yes, parallelized with MPINumber of processors used: 1-96RAM: Depends on physical system to be simulatedClassification: 7.6; 16.1Nature of problem: Investigating ab initio quantum mechanical systems, specifically Bose-Einstein condensation in dilute gases of ⁸⁷RbSolution method: Quantum Monte CarloRunning time: 225 min with 20 particles (with 4800 walkers moved in 1750 time steps) on 1 AMD Opteron^TM Processor 2218 processor; Production run for, e.g., 200 particles takes around 24 hours on 32 such processors.     

Ground state of the time-independent Gross-Pitaevskii equation

We present a suite of programs to determine the ground state of the time-independent Gross-Pitaevskii equation, used in the simulation of Bose-Einstein condensates. The calculation is based on the Optimal Damping Algorithm, ensuring a fast convergence to the true ground state. Versions are given for the one-, two-, and three-dimensional equation, using either a spectral method, well suited for harmonic trapping potentials, or a spatial grid.

Program summary

Program title: GPODACatalogue identifier: ADZN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZN_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 5339No. of bytes in distributed program, including test data, etc.: 19 426Distribution format: tar.gzProgramming language: Fortran 90Computer: ANY (Compilers under which the program has been tested: Absoft Pro Fortran, The Portland Group Fortran 90/95 compiler, Intel Fortran Compiler)RAM: From <1 MB in 1D to ∼¹⁰² MB for a large 3D gridClassification: 2.7, 4.9External routines: LAPACK, BLAS, DFFTPACKNature of problem: The order parameter (or wave function) of a Bose-Einstein condensate (BEC) is obtained, in a mean field approximation, by the Gross-Pitaevskii equation (GPE) [F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phys. 71 (1999) 463]. The GPE is a nonlinear Schrödinger-like equation, including here a confining potential. The stationary state of a BEC is obtained by finding the ground state of the time-independent GPE, i.e., the order parameter that minimizes the energy. In addition to the standard three-dimensional GPE, tight traps can lead to effective two- or even one-dimensional BECs, so the 2D and 1D GPEs are also considered.Solution method: The ground state of the time-independent of the GPE is calculated using the Optimal Damping Algorithm [E. Cancès, C. Le Bris, Int. J. Quantum Chem. 79 (2000) 82]. Two sets of programs are given, using either a spectral representation of the order parameter [C.M. Dion, E. Cancès, Phys. Rev. E 67 (2003) 046706], suitable for a (quasi) harmonic trapping potential, or by discretizing the order parameter on a spatial grid.Running time: From seconds in 1D to a few hours for large 3D grids     

A basis-set based Fortran program to solve the Gross-Pitaevskii equation for dilute Bose gases in harmonic and anharmonic traps

Inhomogeneous boson systems, such as the dilute gases of integral spin atoms in low-temperature magnetic traps, are believed to be well described by the Gross-Pitaevskii equation (GPE). GPE is a nonlinear Schrödinger equation which describes the order parameter of such systems at the mean field level. In the present work, we describe a Fortran 90 computer program developed by us, which solves the GPE using a basis set expansion technique. In this technique, the condensate wave function (order parameter) is expanded in terms of the solutions of the simple-harmonic oscillator (SHO) characterizing the atomic trap. Additionally, the same approach is also used to solve the problems in which the trap is weakly anharmonic, and the anharmonic potential can be expressed as a polynomial in the position operators x, y, and z. The resulting eigenvalue problem is solved iteratively using either the self-consistent-field (SCF) approach, or the imaginary time steepest-descent (SD) approach. Iterations can be initiated using either the simple-harmonic-oscillator ground state solution, or the Thomas-Fermi (TF) solution. It is found that for condensates containing up to a few hundred atoms, both approaches lead to rapid convergence. However, in the strong interaction limit of condensates containing thousands of atoms, it is the SD approach coupled with the TF starting orbitals, which leads to quick convergence. Our results for harmonic traps are also compared with those published by other authors using different numerical approaches, and excellent agreement is obtained. GPE is also solved for a few anharmonic potentials, and the influence of anharmonicity on the condensate is discussed. Additionally, the notion of Shannon entropy for the condensate wave function is defined and studied as a function of the number of particles in the trap. It is demonstrated numerically that the entropy increases with the particle number in a monotonic way.

Program summary

Title of program:bose.xCatalogue identifier:ADWZ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWZ_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format:tar.gzComputers:PC's/Linux, Sun Ultra 10/Solaris, HP Alpha/Tru64, IBM/AIXProgramming language used:mostly Fortran 90Number of bytes in distributed program, including test data, etc.:27 313Number of lines in distributed program, including test data, etc.:28 015Card punching code:ASCIINature of physical problem:It is widely believed that the static properties of dilute Bose condensates, as obtained in atomic traps, can be described to a fairly good accuracy by the time-independent Gross-Pitaevskii equation. This program presents an efficient approach of solving this equation.Method of solution:The solutions of the Gross-Pitaevskii equation corresponding to the condensates in atomic traps are expanded as linear combinations of simple-harmonic oscillator eigenfunctions. Thus, the Gross-Pitaevskii equation which is a second-order nonlinear differential equation, is transformed into a matrix eigenvalue problem. Thereby, its solutions are obtained in a self-consistent manner, using methods of computational linear algebra.Unusual features of the program:None     

Efficient parallel implementation of Bose Hubbard model: Exact numerical ground states and dynamics of gaseous Bose-Einstein condensates

We present a parallel implementation of the Bose Hubbard model, using imaginary time propagation to find the lowest quantum eigenstate and real time propagation for simulation of quantum dynamics. Scaling issues, performance of sparse matrix-vector multiplication, and a parallel algorithm for determining nonzero matrix elements are described. Implementation of imaginary time propagation yields an O(N) linear convergence on a single processor and slightly better than ideal performance on up to 160 processors for a particular problem size. The determination of the nonzero matrix elements is intractable using sequential non-optimized techniques for large problem sizes. Thus, we discuss a parallel algorithm that takes advantage of the intrinsic structural characteristics of the Fock-space matrix representation of the Bose Hubbard Hamiltonian and utilizes a parallel implementation of a Fock state look up table to make this task solvable within reasonable timeframes. Our parallel algorithm demonstrates near ideal scaling on thousand of processors. We include results for a matrix 22.6 million square, with 202 million nonzero elements, utilizing 2048 processors.     

Dependence of grip strength on shoulder position and its implications for ergonomics practice

Grip strength (GS) variability due to positional changes in the upper extremity joints is of importance while designing workstations and work methods. This study was conducted to analyze the GS variations due to positional changes at shoulder joint when some important variables were under control. The GSs of dominant and nondominant hands were measured in eight shoulder (0°, 45°, 90°, and 135° of flexion and abduction) and standard test positions (STP). One hundred and thirteen subjects 20–30 years old completed the study. At the dominant side, no significant difference was observed in the pairwise comparisons between STP and the others. Maximum and minimum GSs were obtained in 0° abduction and 45° flexion and abduction, respectively. At the nondominant side, GSs were significantly lower (p < 0.001) in the corresponding test positions and demonstrated more variability. The findings of this study can contribute to the available knowledge to guide occupational ergonomists in their practices.     

High‐speed Marching Cubes using HistoPyramids

We present an implementation approach for Marching Cubes (MC) on graphics hardware for OpenGL 2.0 or comparable graphics APIs. It currently outperforms all other known graphics processing units (GPU)‐based iso‐surface extraction algorithms in direct rendering for sparse or large volumes, even those using the recently introduced geometry shader (GS) capabilites. To achieve this, we outfit the Histogram Pyramid (HP) algorithm, previously only used in GPU data compaction, with the capability for arbitrary data expansion. After reformulation of MC as a data compaction and expansion process, the HP algorithm becomes the core of a highly efficient and interactive MC implementation. For graphics hardware lacking GSs, such as mobile GPUs, the concept of HP data expansion is easily generalized, opening new application domains in mobile visual computing. Further, to serve recent developments, we present how the HP can be implemented in the parallel programming language CUDA (compute unified device architecture), by using a novel 1D chunk/layer construction.     

Excited Bose–Einstein condensates: Quadrupole oscillations and dark solitons

We study the dynamics of atomic Bose–Einstein condensates (BECs), when the quadrupole mode is excited. Within the Thomas–Fermi approximation, we derive an exact first-order system of differential equations that describes the parameters of the BEC wave function. Using perturbation theory arguments, we derive explicit analytical expressions for the phase, density and width of the condensate. Furthermore, it is found that the observed oscillatory dynamics of the BEC density can even reach a quasi-resonance state when the trap strength varies according to a time-periodic driving term. Finally, the dynamics of a dark soliton on top of a breathing BEC are also briefly discussed.     

A compact model for the I–V characteristics of an undoped double-gate MOSFET

An analytic model for the I–V characteristics of a symmetric, undoped, double gate MOSFET is presented. The model is two-dimensional and extends recent work by Chen and Taur. The formulae involve the LambertW function recently used by Ortiz-Conde to obtain threshold voltage approximations of an undoped single gate MOSFET. The drift diffusion equations are also solved numerically and our approximate solution for the Fermi potential is shown to be in close agreement with the exact numeric solution. We present a compact model for the complete I–V characteristics of an undoped double gate MOSFET.     

Tera-leptons’ shadows over Sinister Universe

The role of Sinister Heavy Fermions in Glashow’s SU(3) × SU(2) × SU(2)′ × U(1) model is to offer in a unique frame relic helium-like products (an ingenious candidate to the dark matter puzzle), a solution to the See-Saw mechanism for light neutrino masses as well as to the strong CP violation problem in QCD. The Sinister model requires three additional families of leptons and quarks, but only the lightest of them, the heavy U-quark and E-electron, are stable. Apparently the final neutral heliumlike (UUUEE) state is an ideal evanescent dark-matter candidate. However, it is reached by multi-body interactions in the early Universe along a tail of more manifest secondary frozen blocks. They should be now here polluting the surrounding matter. Moreover, in opposition to effective $U\bar U$ pair annihilation, there is no such an early or late tera-lepton pairs suppression because: (a) electromagnetic interactions are weaker than nuclear ones and (b) the primordial helium nucleus (⁴He)⁺⁺ is able to attract and capture (in the first three minutes) E ^? fixing it into a hybrid tera-helium ion trap. This leads to a pile up of (⁴HeE ^?)⁺ traces, a lethal compound for any Sinister Universe. This capture leaves no tera-lepton frozen in (Ep) relic, otherwise an ideal catalyzer to achieve effective late E ⁺ E ^? annihilations, possibly saving the model. The (⁴HeE ^?)⁺ Coulomb screening is also avoiding the synthesis of the desired (UUUEE) hidden dark matter gas. The (⁴HeE ^?)⁺ E ^? behave chemically like an anomalous hydrogen isotope. Also terapositronium relics (e ^? E ⁺) are over-abundant, and they behave like an anomalous hydrogen atom: these gases do not fit by many orders of magnitude the known severe bounds on hydrogen anomalous isotope, making shadows hanging over a Sinister Universe. However a surprising and resolver role for Tera-Pions in UHECR astrophysics has been revealed.     

Reactive path integral quantum simulations of molecules solvated in superfluid helium

A hybrid ab initio   path integral molecular dynamics/bosonic path integral Monte Carlo simulation method has been developed, implemented and tested, which allows for the reactive simulations of molecules, clusters or complexes solvated by superfluid ⁴${}^4$He. The simulation takes into account “on-the-fly” the electronic structure and thus the chemical reactivity of the solutes, in conjunction with the Bose–Einstein statistics, and thus the superfluid character of this peculiar solvent. This enables investigations into cryochemical reactions taking place in helium nanodroplets, such as those used in helium nanodroplet isolation (HENDI) spectroscopy.     

On some families of languages related to developmental systems

OL systems and TOL systems are the simplest mathematical models for the study of the development of biological organisms with or without a variable environment, respectively. This paper contributes to the study of the properties of the languages generated by these systems and by their generalizations. Macro OL (TOL) languages are languages obtained by substituting languages of a given type in OL (TOL) languages. We study properties of certain families of macro OL (TOL) languages in particular we show that they are full AFL's.

We observe that OL, TOL systems and many of their generalizations can be viewed as special classes of index grammars.     

On stable and explicit numerical methods for the advection–diffusion equation

In this paper two stable and explicit numerical methods to integrate the one-dimensional (1D) advection–diffusion equation are presented. These schemes are stable by design and follow the main general concept behind the semi-Lagrangian method by constructing a virtual grid where the explicit method becomes stable. It is shown that the new schemes compare well with analytic solutions and are often more accurate than implicit schemes. In particular, the diffusion-only case is explored in some detail. The error produced by the stable and explicit method is a function of the ratio between the standard deviation σ₀ of the initial Gaussian state and the characteristic virtual grid distance ΔS. Larger values of this ratio lead to very accurate results when compared to implicit methods, while lower values lead to less accuracy. It is shown that the σ₀/ΔS ratio is also significant in the advection–diffusion problem: it determines the maximum error generated by new methods, obtained with a certain combination of the advection and diffusion values. In addition, the error becomes smaller when the problem becomes more advective or more diffusive.     

Quantum billiards in multidimensional models with fields of forms

A Bianchi type I cosmological model in (n + 1)-dimensional gravity with several forms is considered. When the electric non-composite brane ansatz is adopted, the Wheeler-DeWitt (WDW) equation for the model, written in a conformally covariant form, is analyzed. Under certain restrictions, asymptotic solutions to the WDW equation near the singularity are found, which reduce the problem to the so-called quantum billiard on the (n ? 1)-dimensional Lobachevsky space ? ^n?1. Two examples of quantum billiards are considered: a 2-dimensional quantum billiard for a 4D model with three 2-forms and a 9D quantum billiard for an 11D model with 120 4-forms, whichmimics the SM2-brane sector of D = 11 supergravity. For certain solutions, vanishing of the wave function at the singularity is proved.     

A fuzzy framework for coordinating pricing and inventory policies for deteriorating items under retailer partial trade credit financing

In this paper, we proposed a generalized economic order quantity (EOQ) – based inventory model using a trade credit policy in a fuzzy sense. The trade credit policy adopted here is a two-level trade credit policy in which the supplier offers the retailer a permissible delay period M, and the retailer, in turn, partially provides customers a permissible delay period N. This study considers fuzzy EOQ model to allow for: (1) selling price dependent demand rate which is imprecise in nature, (2) a profit maximization objective and (3) an imprecise holding cost, ordering cost, purchasing cost, interest earned and interest charged rate. Besides, the cases N ? M and N ? M are explored thoroughly. The objective function for the retailer in fuzzy sense is defuzzified using Modified Graded Mean Integration Representation Method. For the defuzzified objective function sufficient conditions for the existence and uniqueness of the optimal solution are provided. An efficient algorithm is designed to determine the optimal pricing and inventory policies for the retailer. Finally, numerical examples are presented to illustrate the proposed model and the effect of key parameters on optimal solution is examined.     

Real-time vision-based tracking and reconstruction

Many of the recent real-time markerless camera tracking systems assume the existence of a complete 3D model of the target scene. Also the system developed in the MATRIS project assumes that a scene model is available. This can be a freeform surface model generated automatically from an image sequence using structure from motion techniques or a textured CAD model built manually using a commercial software. The offline model provides 3D anchors to the tracking. These are stable natural landmarks, which are not updated and thus prevent an accumulating error (drift) in the camera registration by giving an absolute reference. However, sometimes it is not feasible to model the entire target scene in advance, e.g. parts, which are not static, or one would like to employ existing CAD models, which are not complete. In order to allow camera movements beyond the parts of the environment modelled in advance it is desired to derive additional 3D information online. Therefore, a markerless camera tracking system for calibrated perspective cameras has been developed, which employs 3D information about the target scene and complements this knowledge online by reconstruction of 3D points. The proposed algorithm is robust and reduces drift, the most dominant problem of simultaneous localisation and mapping (SLAM), in real-time by a combination of the following crucial points: (1) stable tracking of longterm features on the 2D level; (2) use of robust methods like the well-known Random Sampling Consensus (RANSAC) for all 3D estimation processes; (3) consequent propagation of errors and uncertainties; (4) careful feature selection and map management; (5) incorporation of epipolar constraints into the pose estimation. Validation results on the operation of the system on synthetic and real data are presented.     

On ruin probability minimization under excess reinsurance

The problem of ruin probability minimization in the Cramer-Lundberg risk model under excess reinsurance is studied. Together with traditional maximization of the Lundberg characteristic coefficient R is considered the problem of direct calculation of insurer’s ruin probability ? _r (x) as an initial-capital function x under the prescribed level of net-retention r. To solve this problem, we propose the excess variant of the Cramer integral equation which is an equivalent to the Hamilton-Jacobi-Bellman equation. The continuation method is used for solving this equation; by means of it is found the analytical solution to the Markov risk model. We demonstrated on a series of standard examples that with any admissible value of x the ruin probability ? _x (r): = ? _r (x) is usually a unimodal function r. A comparison of the analytic representation of ruin probability ? _r(x) with its asymptotic approximation with x → ∞ was conducted.