On the Ill-posed Hyperbolic Systems with a Multiplicity Change Point of Not Less Than the Third Order

2010 ◽  
pp. 147-174
Author(s):  
Valeri V. Kucherenko ◽  
Andriy Kryvko
Keyword(s):  
Change Point ◽  
Hyperbolic Systems ◽  
Third Order ◽  
The Third ◽  
Ill Posed

Existence theorem for hyperbolic systems with a multiplicity change point of at most third order

2009 ◽  
Vol 85 (1-2) ◽  
pp. 128-132
Author(s):  
V. V. Kucherenko ◽  
A. V. Krivko
Keyword(s):  
Existence Theorem ◽  
Change Point ◽  
Hyperbolic Systems ◽  
Third Order

scholarly journals The Third-Order Approximation Godunov Method for the Equations of Gas Dynamics

2019 ◽  
pp. 71-83
Author(s):  
Eugene Vasilev ◽  
Tatiana Vasileva ◽  
Dmitriy Kolybelkin ◽  
Boris Krasovitov
Keyword(s):  
Riemann Problem ◽  
Order Approximation ◽  
Hyperbolic Systems ◽  
Godunov Method ◽  
Systems Of Nonlinear Equations ◽  
Numerical Verification ◽  
Time Step ◽  
Third Order ◽  
The Third ◽  
Space And Time

The paper suggests a new modification of Godunov difference method with the 3rd order approximation in space and time for hyperbolic systems of conservation laws. The diёerence scheme uses the simultaneous discretization of the equations in space and time without of Runge — Kutta stages. An exact or approximate solution of Riemann problem is applied to calculate numerical fluxes between cells. Before the time step, corrections to the arguments of the Riemann problem providing third-order approximations for linear systems are calculated. After the time step, the numerical solution correction procedure is applied to eliminate the second-order error arising from the nonlinearity of the equations. The paper presents the results of experimental numerical verification of the method approximation order on the exact smooth solution inside the fan of the expansion wave. The test results completely confirm the third order of the presented method. The proposed approach of constructing third-order difference schemes can be used for inhomogeneous and two-dimensional hyperbolic systems of nonlinear equations.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

scholarly journals Ultrafast Third-Order Nonlinear Optical Response Excited by fs Laser Pulses at 1550 nm in GaN Crystals

Materials ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 3194
Author(s):  
Adrian Petris ◽  
Petronela Gheorghe ◽  
Tudor Braniste ◽  
Ion Tiginyanu
Keyword(s):  
Harmonic Generation ◽  
Nonlinear Optical ◽  
Laser Pulses ◽  
Third Harmonic Generation ◽  
High Repetition Rate ◽  
Third Order ◽  
Third Harmonic ◽  
The Third ◽  
High Repetition ◽  
Gan Crystal

The ultrafast third-order optical nonlinearity of c-plane GaN crystal, excited by ultrashort (fs) high-repetition-rate laser pulses at 1550 nm, wavelength important for optical communications, is investigated for the first time by optical third-harmonic generation in non-phase-matching conditions. As the thermo-optic effect that can arise in the sample by cumulative thermal effects induced by high-repetition-rate laser pulses cannot be responsible for the third-harmonic generation, the ultrafast nonlinear optical effect of solely electronic origin is the only one involved in this process. The third-order nonlinear optical susceptibility of GaN crystal responsible for the third-harmonic generation process, an important indicative parameter for the potential use of this material in ultrafast photonic functionalities, is determined.

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

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