scholarly journals Comparison Theorems for Small Deviations of Random Series

2003 ◽  
Vol 8 (0) ◽  
Author(s):  
Fuchang Gao ◽  
Jan Hannig ◽  
Fred Torcaso
Keyword(s):  
Comparison Theorems ◽  
Random Series ◽  
Small Deviations

Microstructural Evolution in Reduced TiO2

1981 ◽  
Vol 39 ◽  
pp. 74-75
Author(s):  
W. T. Donlon ◽  
S. Shinozaki ◽  
E. M. Logothetis ◽  
W. Kaizer
Keyword(s):  
Microstructural Evolution ◽  
Point Defects ◽  
Extended Defects ◽  
Small Deviations ◽  
Controlled Atmospheres ◽  
Limited Solubility ◽  
Thin Foils ◽  
Heat Treated ◽  
Porous Polycrystalline ◽  
Crystallographic Shear

Since point defects have a limited solubility in the rutile (TiO2) lattice, small deviations from stoichiometry are known to produce crystallographic shear (CS) planes which accomodate local variations in composition. The material used in this study was porous polycrystalline TiO2 (60% dense), in the form of 3mm. diameter disks, 1mm thick. Samples were mechanically polished, ion-milled by conventional techniques, and initially examined with the use of a Siemens EM102. The electron transparent thin foils were then heat-treated under controlled atmospheres of CO/CO2 and H2 and reexamined in the same manner.The “as-received” material contained mostly TiO2 grains (∼5μm diameter) which had no extended defects. Several grains however, aid exhibit a structure similar to micro-twinned grains observed in reduced rutile. Lattice fringe images (Fig. 1) of these grains reveal that the adjoining layers are not simply twin related variants of a single TinO2n-1 compound. Rather these layers (100 - 250 Å wide) are alternately comprised of stoichiometric TiO2 (rutile) and reduced TiO2 in the form of Ti8O15, with the Ti8O15 layers on either side of the TiO2 being twin related.

Solution of heat and mass transfer problems with non-linear coefficients

2019 ◽  
Vol 5 (4) ◽  
pp. 10-20
Author(s):  
Boris G. Aksenov ◽  
Yuri E. Karyakin ◽  
Svetlana V. Karyakina
Keyword(s):  
Mass Transfer ◽  
Exact Solution ◽  
Numerical Methods ◽  
Heat And Mass Transfer ◽  
Unknown Function ◽  
Comparison Theorems ◽  
Upper And Lower Bounds ◽  
Maximum Deviation ◽  
Approximate Methods ◽  
Nonmonotonic Dependence

Equations, which have nonlinear nonmonotonic dependence of one of the coefficients on an unknown function, can describe processes of heat and mass transfer. As a rule, existing approximate methods do not provide solutions with acceptable accuracy. Numerical methods do not involve obtaining an analytical expression for the unknown function and require studying the convergence of the algorithm used. The value of absolute error is uncertain. The authors propose an approximate method for solving such problems based on Westphal comparison theorems. The comparison theorems allow finding upper and lower bounds of the unknown exact solution. A special procedure developed for the stepwise improvement of these bounds provide solutions with a given accuracy. There are only a few problems for equations with nonlinear nonmonotonic coefficients for which the exact solution has been obtained. One of such problems, presented in this article, shows the efficiency of the proposed method. The results prove that the proposed method for obtaining bounds of the solution of a nonlinear nonmonotonic equation of parabolic type can be considered as a new method of the approximate analytical solution having guaranteed accuracy. In addition, the proposed here method allows calculating the maximum deviation from the unknown exact solution of the results of other approximate and numerical methods.

scholarly journals Comparison of Direct and Adjoint k and α-Eigenfunctions

2021 ◽  
Vol 2 (2) ◽  
pp. 132-151
Author(s):  
Vito Vitali ◽  
Florent Chevallier ◽  
Alexis Jinaphanh ◽  
Andrea Zoia ◽  
Patrick Blaise
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Energy Transport ◽  
Distinct Point ◽  
Point Of View ◽  
Critical Systems ◽  
Small Deviations ◽  
Power Iteration ◽  
Continuous Energy ◽  
Fission Probability

Modal expansions based on k-eigenvalues and α-eigenvalues are commonly used in order to investigate the reactor behaviour, each with a distinct point of view: the former is related to fission generations, whereas the latter is related to time. Well-known Monte Carlo methods exist to compute the direct k or α fundamental eigenmodes, based on variants of the power iteration. The possibility of computing adjoint eigenfunctions in continuous-energy transport has been recently implemented and tested in the development version of TRIPOLI-4®, using a modified version of the Iterated Fission Probability (IFP) method for the adjoint α calculation. In this work we present a preliminary comparison of direct and adjoint k and α eigenmodes by Monte Carlo methods, for small deviations from criticality. When the reactor is exactly critical, i.e., for k0 = 1 or equivalently α0 = 0, the fundamental modes of both eigenfunction bases coincide, as expected on physical grounds. However, for non-critical systems the fundamental k and α eigenmodes show significant discrepancies.

scholarly journals Comparison theorems for fourth order differential equations

1986 ◽  
Vol 9 (1) ◽  
pp. 105-109
Author(s):  
Garret J. Etgen ◽  
Willie E. Taylor
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Fourth Order ◽  
Linear Equations ◽  
Oscillation Criterion ◽  
Comparison Theorems ◽  
Positive Continuous Function ◽  
Order Linear ◽  
All Solutions

This paper establishes an apparently overlooked relationship between the pair of fourth order linear equationsyiv−p(x)y=0andyiv+p(x)y=0, wherepis a positive, continuous function defined on[0,∞). It is shown that if all solutions of the first equation are nonoscillatory, then all solutions of the second equation must be nonoscillatory as well. An oscillation criterion for these equations is also given.

scholarly journals Comparison theorems for eigenvalues

1970 ◽  
Vol 28 (2) ◽  
pp. 289-292 ◽  
Author(s):  
A. M. Fink
Keyword(s):  
Comparison Theorems

scholarly journals NMR solution conformation of heparin-derived tetrasaccharide

1996 ◽  
Vol 318 (1) ◽  
pp. 93-102 ◽  
Author(s):  
Dmitri MIKHAILOV ◽  
Kevin H. MAYO ◽  
Ioncho R. VLAHOV ◽  
Toshihiko TOIDA ◽  
Azra PERVIN ◽  
...  
Keyword(s):  
Chair Conformation ◽  
Coupling Constants ◽  
Solution Conformation ◽  
1H And 13C Nmr ◽  
Small Deviations ◽  
J Coupling Constants ◽  
Nuclear Overhauser ◽  
Nuclear Overhauser Effects ◽  
Dynamics Simulations ◽  
Ring Conformations

The solution conformation of the homogeneous, heparin-derived tetrasaccharide ΔUA2S(1 → 4)-α-d-GlcNpS6S(1 → 4)-α-l-IdoAp2S(1 → 4)-α-d-GlcNpS6S (residues A, B, C and D respectively, where IdoA is iduronic acid) has been investigated by using 1H- and 13C-NMR. Ring conformations have been defined by J-coupling constants and inter-proton nuclear Overhauser effects (NOEs), and the orientation of one ring with respect to the other has been defined by inter-ring NOEs. NOE-based conformational modelling has been done by using the iterative relaxation matrix approach (IRMA), restrained molecular dynamics simulations and energy minimization to refine structures and to distinguish between minor structural differences and equilibria between various ring forms. Both glucosamine residues B and D are in the 4C1 chair conformation. The 6-O-sulphate group is oriented in the gauche–trans configuration in the D ring, whereas in the B ring the gauche–gauche rotomer predominates. Uronate (A) and iduronate (C) residues are mostly represented by 1H2 and 2S0 twisted boat forms, respectively, with small deviations in expected coupling constants and NOEs suggesting minor contributions from other A and C ring conformations.

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