Extension of Dasgupta’s Technique for Higher Degree Approximation

In the present paper, rational wedge functions for degree two approximation have been computed over a pentagonal discretization of the domain, by using an analytic approach which is an extension of Dasgupta’s approach for linear approximation. This technique allows to avoid the computation of the exterior intersection points of the elements, which was a key component of the technique initiated by Wachspress. The necessary condition for the existence of the denominator function was established by Wachspress whereas our assertion, induced by the technique of Dasgupta, assures the sufficiency of the existence. Considering the adjoint (denominator) functions for linear approximation obtained by Dasgupta, invariance of the adjoint for degree two approximation is established. In other words, the method proposed by Dasgupta for the construction ofWachspress coordinates for linear approximation is extended to obtain the coordinates for quadratic approximation. The assertions have been supported by considering some illustrative examples.

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Correlation of Internal Torsional Motion with Overall Molecular Motion in Crystals

Acta Crystallographica Section B Structural Science ◽

10.1107/s0108768198003243 ◽

1998 ◽

Vol 54 (5) ◽

pp. 507-514 ◽

Cited By ~ 69

Author(s):

V. Schomaker ◽

K. N. Trueblood

Keyword(s):

Linear Approximation ◽

Molecular Motion ◽

Quadratic Approximation ◽

Symmetry Element ◽

Torsional Motion ◽

Mean Square ◽

Molecular Symmetry ◽

Quadratic Mean ◽

The Mean ◽

Rigid Group

The simple one-parameter Dunitz–White model for internal torsional motion accompanying overall molecular motion in crystals ignores the correlations between this torsion and the overall translation and libration. These correlations are explicitly considered here, in quadratic approximation (based on the linear approximation for displacements). For each attached rigid group (ARG) undergoing a torsional libration, there are, in addition to the mean-square libration amplitude, six correlations to be considered, three with the overall molecular libration and three with the overall translation. Because it is impossible from the observed quadratic mean displacements to distinguish the torsional motion from the overall molecular libration parallel to the torsional axis, the present analysis emphasizes the overall libration parallel to the torsional axis, which we term \Lambda. In the general case there are only six determinable parameters for each ARG. If the torsional axis of the ARG lies on a molecular symmetry element, the number of parameters is reduced to three or fewer. Examples of analyses with these correlations included, and without them, are compared.

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IMPLEMENTATION OF QUADRATIC AXIAL TRIAL FUNCTIONS IN THE HIGH-FIDELITY TRANSPORT CODE PROTEUS-MOC

EPJ Web of Conferences ◽

10.1051/epjconf/202124703015 ◽

2021 ◽

Vol 247 ◽

pp. 03015

Author(s):

Guangchun Zhang ◽

Won Sik Yang

Keyword(s):

Linear Approximation ◽

Quadratic Approximation ◽

Linear Functions ◽

Computational Time ◽

High Fidelity ◽

Test Results ◽

Memory Requirement ◽

Transport Code ◽

Axial Variation ◽

Memory Reduction

PROTEUS-MOC is a pin-resolved high-fidelity transport code, in which the axial variation of angular flux is represented in terms of orthogonal polynomials. Currently, PROTEUS-MOC employs linear functions and requires relatively fine axial meshes to achieve high accuracy, which increases the number of axial meshes and hence the memory requirement. In this study, aiming to reduce the memory requirement and potentially the computational time by allowing larger axial meshes, we have extended the PROTEUS-MOC transport solution method to quadratic trial functions. Preliminary tests for the performance of quadratic trial functions have been performed using the 3-D C5G7 benchmark problem. Test results showed that for the same axial mesh configuration with relatively large sizes, the quadratic approximation yields about 2 to 5 times more accurate pin powers than the linear approximation, depending on the degree of axial variation of angular fluxes. The quadratic approximation also allows the use of about 3 times coarser axial meshes than the linear approximation for comparable pin power accuracy, which consequently reduces the memory requirement by about 2 times. The memory reduction is not proportional because of the increased number of coefficients in each element from 2 to 3. However, the quadratic approximation did not reduce the computational time as expected because of the deteriorated performance of the pCMFD acceleration scheme due to large axial mesh sizes.

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On the self-intersections of an immersed sphere

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270003937x ◽

2007 ◽

Vol 75 (3) ◽

pp. 453-458 ◽

Cited By ~ 2

Author(s):

Albert Borbély

Keyword(s):

General Position ◽

Closed Curve ◽

The Self ◽

Necessary Condition ◽

Analogous Question ◽

Long Time ◽

Intersection Points ◽

Triple Points ◽

Gauss Code ◽

Special Case

A closed curve f: S1 → ℝ2 in general position gives rise to a word whose letters are the self-intersection points, each of them appearing exactly twice. Such a word is called a Gauss code. The problem of determining whether a given Gauss code is realisable or not was first proposed by Gauss and it has been settled a long time ago. The analogous question for immersions f: S2 → ℝ3 in general position is settled only in a special case when the immersion has no triple points. We give a necessary condition for a system of curves to be realisable by a general immersion f: S2 → ℝ3.

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The Infancy of Solar-Type Stars and its Relevance for the Origin of Life

International Astronomical Union Colloquium ◽

10.1017/s0252921100014780 ◽

1997 ◽

Vol 161 ◽

pp. 267-282 ◽

Cited By ~ 1

Author(s):

Thierry Montmerle

Keyword(s):

Main Sequence ◽

Solar Nebula ◽

Circumstellar Disks ◽

T Tauri Stars ◽

Necessary Condition ◽

Sequence Evolution ◽

T Tauri ◽

Solar Type ◽

Optical Domain ◽

The Origin Of Life

AbstractFor life to develop, planets are a necessary condition. Likewise, for planets to form, stars must be surrounded by circumstellar disks, at least some time during their pre-main sequence evolution. Much progress has been made recently in the study of young solar-like stars. In the optical domain, these stars are known as «T Tauri stars». A significant number show IR excess, and other phenomena indirectly suggesting the presence of circumstellar disks. The current wisdom is that there is an evolutionary sequence from protostars to T Tauri stars. This sequence is characterized by the initial presence of disks, with lifetimes ~ 1-10 Myr after the intial collapse of a dense envelope having given birth to a star. While they are present, about 30% of the disks have masses larger than the minimum solar nebula. Their disappearance may correspond to the growth of dust grains, followed by planetesimal and planet formation, but this is not yet demonstrated.

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Universal ESEM

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100149763 ◽

1993 ◽

Vol 51 ◽

pp. 786-787

Author(s):

G.D. Danilatos

Keyword(s):

Electron Microscope ◽

Scanning Electron Microscope ◽

High Vacuum ◽

Natural Extension ◽

Necessary Condition ◽

Environmental Scanning ◽

Detection Systems ◽

Small Gain ◽

Detection Medium ◽

Scanning Electron

The environmental scanning electron microscope (ESEM) has evolved as the natural extension of the scanning electron microscope (SEM), both historically and technologically. ESEM allows the introduction of a gaseous environment in the specimen chamber, whereas SEM operates in vacuum. One of the detection systems in ESEM, namely, the gaseous detection device (GDD) is based on the presence of gas as a detection medium. This might be interpreted as a necessary condition for the ESEM to remain operational and, hence, one might have to change instruments for operation at low or high vacuum. Initially, we may maintain the presence of a conventional secondary electron (E-T) detector in a "stand-by" position to switch on when the vacuum becomes satisfactory for its operation. However, the "rough" or "low vacuum" range of pressure may still be considered as inaccessible by both the GDD and the E-T detector, because the former has presumably very small gain and the latter still breaks down.

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