Compact finite difference - Fourier spectral method for three-dimensional incompressible Navier-Stokes equations

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.

Study on the Performance of a Compact Compound Oil/Gas/Water Separator for Offshore Oil Industry

High-efficiency separation of the oil/gas/water mixtures is a significant issue in offshore oil industry. To reduce the total cost by means of reduction in weight and space compared with conventional separators, a novel compact compound oil/gas/water separator is developed. The research works on oil-gas-water separation by compound separating techniques is described in this paper. The innovative separator is a gravity settling tank with helical pipes within and T-shaped pipes outside. Both experiments and numerical simulations are presented to study the separating performance and efficiency of the helical pipes, which are the main part of the separator.

Smooth sets for borel equivalence relations and the covering property for σ-ideals of compact sets

This thesis is divided into three chapters. In the first chapter we study the smooth sets with respect to a Borel equivalence realtion E on a Polish space X. The collection of smooth sets forms σ-ideal. We think of smooth sets as analogs of countable sets and we show that an analog of the perfect set theorem for Σ¹₁ sets holds in the context of smooth sets. We also show that the collection of Σ¹₁ smooth sets is ∏¹₁ on the codes. The analogs of thin sets are called sparse sets. We prove that there is a largest ∏¹₁ sparse set and we give a characterization of it. We show that in L there is a ∏¹₁ sparse set which is not smooth. These results are analogs of the results known for the ideal of countable sets, but it remains open to determine if large cardinal axioms imply that ∏¹₁ sparse sets are smooth. Some more specific results are proved for the case of a countable Borel equivalence relation. We also study I(E), the σ-ideal of closed E-smooth sets. Among other things we prove that E is smooth iff I(E) is Borel.

In chapter 2 we study σ-ideals of compact sets. We are interested in the relationship between some descriptive set theoretic properties like thinness, strong calibration and the covering property. We also study products of σ-ideals from the same point of view. In chapter 3 we show that if a σ-ideal I has the covering property (which is an abstract version of the perfect set theorem for Σ¹₁ sets), then there is a largest ∏¹₁ set in I^int (i.e., every closed subset of it is in I). For σ-ideals on 2^ω we present a characterization of this set in a similar way as for C₁, the largest thin ∏¹₁ set. As a corollary we get that if there are only countable many reals in L, then the covering property holds for Σ¹₂ sets.

Fetntosecond pulse shaping with space-to-time conversion based on planar optics

We propose a novel structure of planar optical configuration for implementation of the space-to-time conversion for femtosecond pulse shaping. The previous apparatuses of femtosecond pulse shaping are 4f Fourier-transforming type system that is usually large, expensive, difficult to align. The planar integration of free-space optical systems on solid substrates is an optical module with the attractive advantages of compact, reliable and robust. This apparatus is analyzed in details and the design of the particular lens for femtosecond pulse shaping based on planar optics is presented. (c) 2006 Elsevier GmbH. All rights reserved.

On the Cesari fixed point method in a Banach space

In a paper published in 1961, L. Cesari [1] introduces a method which extends certain earlier existence theorems of Cesari and Hale ([2] to [6]) for perturbation problems to strictly nonlinear problems. Various authors ([1], [7] to [15]) have now applied this method to nonlinear ordinary and partial differential equations. The basic idea of the method is to use the contraction principle to reduce an infinite-dimensional fixed point problem to a finite-dimensional problem which may be attacked using the methods of fixed point indexes.

The following is my formulation of the Cesari fixed point method:

Let B be a Banach space and let S be a finite-dimensional linear subspace of B. Let P be a projection of B onto S and suppose Г≤B such that pГ is compact and such that for every x in PГ, P^-1x∩Г is closed. Let W be a continuous mapping from Г into B. The Cesari method gives sufficient conditions for the existence of a fixed point of W in Г.

Let I denote the identity mapping in B. Clearly y = Wy for some y in Г if and only if both of the following conditions hold:

(i) Py = PWy.

(ii) y = (P + (I - P)W)y.

Definition. The Cesari fixed paint method applies to (Г, W, P) if and only if the following three conditions are satisfied:

(1) For each x in PГ, P + (I - P)W is a contraction from P^-1x∩Г into itself. Let y(x) be that element (uniqueness follows from the contraction principle) of P^-1x∩Г which satisfies the equation y(x) = Py(x) + (I-P)Wy(x).

(2) The function y just defined is continuous from PГ into B.

(3) There are no fixed points of PWy on the boundary of PГ, so that the (finite- dimensional) fixed point index i(PWy, int PГ) is defined.

Definition. If the Cesari fixed point method applies to (Г, W, P) then define i(Г, W, P) to be the index i(PWy, int PГ).

The three theorems of this thesis can now be easily stated.

Theorem 1 (Cesari). If i(Г, W, P) is defined and i(Г, W, P) ≠0, then there is a fixed point of W in Г.

Theorem 2. Let the Cesari fixed point method apply to both (Г, W, P₁) and (Г, W, P₂). Assume that P₂P₁=P₁P₂=P₁ and assume that either of the following two conditions holds:

(1) For every b in B and every z in the range of P₂, we have that ‖b=P₂b‖ ≤ ‖b-z‖

(2)P₂Г is convex.

Then i(Г, W, P₁) = i(Г, W, P₂).

Theorem 3. If Ω is a bounded open set and W is a compact operator defined on Ω so that the (infinite-dimensional) Leray-Schauder index i_LS(W, Ω) is defined, and if the Cesari fixed point method applies to (Ω, W, P), then i(Ω, W, P) = i_LS(W, Ω).

Theorems 2 and 3 are proved using mainly a homotopy theorem and a reduction theorem for the finite-dimensional and the Leray-Schauder indexes. These and other properties of indexes will be listed before the theorem in which they are used.

All-optical switching in a vertical coupler space switch employing photocarrier-induced nonlinearity

A novel compact integrated nonlinear optical switch is demonstrated. Using a high-power picosecond pulse of 5-ps pulsewidth and 250-MHz repetition rate, all-optical switching with a contrast ratio of 23 dB has been achieved using an in-fiber input power < 14 dBm (100 pJ/pulse). The switch speed depends on the carrier sweep-out time, which can be reduced to the 10 ps range by either applying a reverse bias or by introduction of carrier recombination centers in the active layer.

On the use of combined compact scheme for numerical solution of the two-layer oceanic shallow water equations

Many types of oceanic physical phenomena have a wide range in both space and time. In general, simplified models, such as shallow water model, are used to describe these oceanic motions. The shallow water equations are widely applied in various oceanic and atmospheric extents. By using the two-layer shallow water equations, the stratification effects can be considered too. In this research, the sixth-order combined compact method is investigated and numerically implemented as a high-order method to solve the two-layer shallow water equations. The second-order centered, fourth-order compact and sixth-order super compact finite difference methods are also used to spatial differencing of the equations. The first part of the present work is devoted to accuracy assessment of the sixth-order super compact finite difference method (SCFDM) and the sixth-order combined compact finite difference method (CCFDM) for spatial differencing of the linearized two-layer shallow water equations on the Arakawa's A-E and Randall's Z numerical grids. Two general discrete dispersion relations on different numerical grids, for inertia-gravity and Rossby waves, are derived. These general relations can be used for evaluation of the performance of any desired numerical scheme. For both inertia-gravity and Rossby waves, minimum error generally occurs on Z grid using either the sixth-order SCFDM or CCFDM methods. For the Randall's Z grid, the sixth-order CCFDM exhibits a substantial improvement , for the frequency of the barotropic and baroclinic modes of the linear inertia-gravity waves of the two layer shallow water model, over the sixth-order SCFDM. For the Rossby waves, the sixth-order SCFDM shows improvement, for the barotropic and baroclinic modes, over the sixth-order CCFDM method except on Arakawa's C grid. In the second part of the present work, the sixth-order CCFDM method is used to solve the one-layer and two-layer shallow water equations in their nonlinear form. In one-layer model with periodic boundaries, the performance of the methods for mass conservation is compared. The results show high accuracy of the sixth-order CCFDM method to simulate a complex flow field. Furthermore, to evaluate the performance of the method in a non-periodic domain the sixth-order CCFDM is applied to spatial differencing of vorticity-divergence-mass representation of one-layer shallow water equations to solve a wind-driven current problem with no-slip boundary conditions. The results show good agreement with published works. Finally, the performance of different schemes for spatial differencing of two-layer shallow water equations on Z grid with periodic boundaries is investigated. Results illustrate the high accuracy of combined compact method.

Aberration correction for free space optical communications using rectangular zernike modal wavefront sensing

A time multiplexed rectangular Zernike modal wavefront sensor based on a nematic phase-only liquid crystal spatial light modulator and specially designed for a high power two-electrode tapered laser diode which is a compact and novel free space optical communication source is used in an adaptive beam steering free space optical communication system, enabling the system to have 1.25 GHz modulation bandwidth, 4.6° angular coverage and the capability of sensing aberrations within the system and caused by atmosphere turbulence up to absolute value of 0.15 waves amplitude and correcting them in one correction cycle. Closed-loop aberration correction algorithm can be implemented to provide convergence for larger and time varying aberrations. Improvement of the system signal-to-noise-ratio performance is achieved by aberration correction. To our knowledge, it is first time to use rectangular orthonormal Zernike polynomials to represent balanced aberrations for high power rectangular laser beam in practice. © 2014 IEEE.

Australia Telescope Compact Array 1.2cm Observations of the Massive Star Forming Region G305.2+0.2

We report on Australia Telescope Compact Array observations of the massive star-forming region G305.2+0.2 at 1.2 cm. We detected emission in five molecules towards G305A, confirming its hot core nature. We determined a rotational temperature of 26 K for methanol. A non-local thermodynamic equilibrium excitation calculation suggests a kinematic temperature of the order of 200 K. A time-dependent chemical model is also used to model the gas-phase chemistry of the hot core associated with G305A. A comparison with the observations suggest an age of between 2 × 104 and 1.5 × 105 yr. We also report on a feature to the south-east of G305A which may show weak Class I methanol maser emission in the line at 24.933 GHz. The more evolved source G305B does not show emission in any of the line tracers, but strong Class I methanol maser emission at 24.933 GHz is found 3 arcsec to the east. Radio continuum emission at 18.496 GHz is detected towards two H ii regions. The implications of the non-detection of radio continuum emission towards G305A and G305B are also discussed.

Compact operators without extended eigenvalues

A complex number lambda is called an extended eigenvalue of a bounded linear operator T on a Banach space B if there exists a non-zero bounded linear operator X acting on B such that XT = lambda TX. We show that there are compact quasinilpotent operators on a separable Hilbert space, for which the set of extended eigenvalues is the one-point set {1}.

Decay measures on locally compact abelian topological groups

Source: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS Volume: 131 Pages: 1257-1273 Part: Part 6 Published: 2001 Times Cited: 5 References: 23 Citation MapCitation Map beta Abstract: We show that the Banach space M of regular sigma-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces M-D and M-ND, where M-D is the set of measures mu is an element of M whose Fourier transform vanishes at infinity and M-ND is the set of measures mu is an element of M such that nu is not an element of MD for any nu is an element of M \ {0} absolutely continuous with respect to the variation \mu\. For any corresponding decomposition mu = mu(D) + mu(ND) (mu(D) is an element of M-D and mu(ND) is an element of M-ND) there exist a Borel set A = A(mu) such that mu(D) is the restriction of mu to A, therefore the measures mu(D) and mu(ND) are singular with respect to each other. The measures mu(D) and mu(ND) are real if mu is real and positive if mu is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from M-D and M-ND.

Dual-mode compact structure comprising of side-fed bifilar and quadrifilar helix antenna

A side-fed bifilar helix antenna can be integrated with a quadrifilar helix antenna in a piggy back configuration in order to achieve a dual-mode radiating structure. The overall length of the structure is 0.44 lambda at the resonant frequency (1.54 GHz) of the space mode antenna and 0.39 lambda at the resonant frequency (1.34 GHz) of the terrestrial mode antenna. The computed results are validated by experimental data.

Compact quadrifilar helix antenna

Introduction: The quadrifilar helix antenna (QHA) is used widely for terrestrial [1] and space communication systems [2], where it is necessary to generate a circularly polarised cardioid-shaped radiation pattern with a high front-to-back ratio and low cross-polarisation. The radiating structure comprises four helical conductors which are excited in phase quadrature at the feed point, which is usually located at the centre of the top radials. The physical size of the quadrifilar antenna can be reduced by dielectric loading [3] or by meandering the printed linear elements [4]. However, in the former arrangement dielectric absorption reduces the radiation efficiency of the antenna, and the latter technique is not suitable for constructing free standing wire structures, which are normally used for spacecraft payloads in the VHF and UHF bands [2]. This Letter shows that a significant reduction in the axial length of a 1/2 turn half-wavelength QHA can be achieved by modifying the geometry of the helices in the region around the midpoint where a current null exists. Simulated and experimental results at L band are used to show that a size reduction of up to 15% is possible without significantly degrading the pattern shape and the bandwidth.

The rotational broadening of V395 Carinae. Implications on the compact object's mass

Context: The masses previously obtained for the X-ray binary 2S 0921-630 inferred a compact object that was either a high-mass neutron star or low-mass black-hole, but used a previously published value for the rotational broadening (v sin i) with large uncertainties. Aims: We aim to determine an accurate mass for the compact object through an improved measurement of the secondary star's projected equatorial rotational velocity. Methods: We have used UVES echelle spectroscopy to determine the v sin i of the secondary star (V395 Car) in the low-mass X-ray binary 2S 0921-630 by comparison to an artificially broadened spectral-type template star. In addition, we have also measured v sin i from a single high signal-to-noise ratio absorption line profile calculated using the method of Least-Squares Deconvolution (LSD). Results: We determine v sin i to lie between 31.3±0.5 km s-1 to 34.7±0.5 km s-1 (assuming zero and continuum limb darkening, respectively) in disagreement with previous results based on intermediate resolution spectroscopy obtained with the 3.6 m NTT. Using our revised v sin i value in combination with the secondary star's radial velocity gives a binary mass ratio of 0.281±0.034. Furthermore, assuming a binary inclination angle of 75° gives a compact object mass of 1.37±0.13 M_?. Conclusions: We find that using relatively low-resolution spectroscopy can result in systemic uncertainties in the measured v sin i values obtained using standard methods. We suggest the use of LSD as a secondary, reliable check of the results as LSD allows one to directly discern the shape of the absorption line profile. In the light of the new v sin i measurement, we have revised down the compact object's mass, such that it is now compatible with a canonical neutron star mass.

Generation of a train of ultrashort pulses from a compact birefringent crystal array

A linear array of n calcite crystals is shown to allow the generation of a high contrast (> 10: 1) train of 2(n) high energy (> 100 mu J) pulses from a single ultrafast laser pulse. Advantage is taken of the pulse-splitting properties of a single birefringent crystal, where an incident laser pulse can be split into two pulses with orthogonal polarizations and equal intensity, separated temporally in proportion to the thickness of the crystal traversed and the difference in refractive indices of the two optic axes. In the work presented here an array of seven calcite crystals of sequentially doubled thickness is used to produce a train of 128 pulses, each of femtosecond duration. Readily versatile properties such as the number of pulses in the train and variable mark-space ratio are realized from such a setup. (c) 2007 Optical Society of America