scholarly journals Turbulence of generalised flows in two dimensions

2019 ◽  
Vol 883 ◽  
Author(s):  
Simon Thalabard ◽  
Jérémie Bec
Keyword(s):  
Two Dimensions ◽  
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Stresses in Multilayered Thin Films

MRS Bulletin ◽  
1999 ◽  
Vol 24 (2) ◽  
pp. 34-38 ◽  
Author(s):  
Robert C. Cammarata ◽  
John C. Bilello ◽  
A. Lindsay Greer ◽  
Karl Sieradzki ◽  
Steven M. Yalisove
Keyword(s):  
Thin Films ◽  
Angular Displacement ◽  
Incident Beam ◽  
Two Dimensions ◽  
Bragg Angle ◽  
Radius Of Curvature ◽  
Wave Source ◽  
Contour Maps ◽  
Film Substrate ◽  
Image Position

Almost all thin films deposited on a substrate are in a state of stress. Fifty years ago pioneering work concerning the measurement of thin-film stresses was conducted by Brenner and Senderoff. They electroplated a metal film onto a thin metal substrate strip fixed at one end and measured the deflection of the free end of the substrate with a micrometer. Using a beam-bending analysis, they were able to calculate a residual stress from the measured deflection of the bimetallic film-substrate system. A variety of other, more sensitive methods of measuring the curvature of the surface of a film-substrate system have since been developed using, for example, capacitance measurements and interferometry techniques.When a monochromatic x-ray beam is incident onto a curved single crystal, the diffraction condition is satisfied only for regions of the crystal where the inclination angle with respect to the incident beam exactly matches the Bragg angle. When a parallel beam plane-wave source is used, the diffracted beam from a particular set of (hkl) planes gives rise to a single narrow-contour band. If the crystal is rocked by an angle ω, the contour band will move by a certain distance D. The radius of curvature R of the crystal lattice planes is given bywhere θ is the Bragg angle. Equal rocking angles produce equivalent D values for uniform curvature, or varied D values for nonuniform curvature. Using this procedure, detailed contour maps of the angular displacement field of the crystal can be mapped in two dimensions.

scholarly journals Improved Bounds for Incidences Between Points and Circles

2014 ◽  
Vol 24 (3) ◽  
pp. 490-520 ◽  
Author(s):  
MICHA SHARIR ◽  
ADAM SHEFFER ◽  
JOSHUA ZAHL
Keyword(s):  
Three Dimensional ◽  
Combinatorial Geometry ◽  
Two Dimensions ◽  
Three Dimensions ◽  
List Type ◽  
Planar Case ◽  
Constant Degree ◽  
Common Plane ◽  
Formula Group ◽  
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We establish an improved upper bound for the number of incidences betweenmpoints andncircles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension ≥ 2, isO*(m2/3n2/3+m6/11n9/11+m+n), where theO*(⋅) notation hides polylogarithmic factors. Since all the points and circles may lie on a common plane (or sphere), it is impossible to improve the bound in ℝ3without first improving it in the plane.Nevertheless, we show that if the set of circles is required to be ‘truly three-dimensional’ in the sense that no sphere or plane contains more thanqof the circles, for someq≪n, then for any ϵ > 0 the bound can be improved to\[ O\bigl(m^{3/7+\eps}n^{6/7} + m^{2/3+\eps}n^{1/2}q^{1/6} + m^{6/11+\eps}n^{15/22}q^{3/22} + m + n\bigr). \]For various ranges of parameters (e.g., whenm= Θ(n) andq=o(n7/9)), this bound is smaller than the lower bound Ω*(m2/3n2/3+m+n), which holds in two dimensions.We present several extensions and applications of the new bound.(i)For the special case where all the circles have the same radius, we obtain the improved boundO(m5/11+ϵn9/11+m2/3+ϵn1/2q1/6+m+n).(ii)We present an improved analysis that removes the subpolynomial factors from the bound whenm=O(n3/2−ϵ) for any fixed ϵ < 0.(iii)We use our results to obtain the improved boundO(m15/7) for the number of mutually similar triangles determined by any set ofmpoints in ℝ3.Our result is obtained by applying the polynomial partitioning technique of Guth and Katz using a constant-degree partitioning polynomial (as was also recently used by Solymosi and Tao). We also rely on various additional tools from analytic, algebraic, and combinatorial geometry.

On the boundary conditions in estimating ∇ω by div ω and curl ω

2018 ◽  
Vol 149 (03) ◽  
pp. 739-760
Author(s):  
Gyula Csató ◽  
Olivier Kneuss ◽  
Dhanya Rajendran
Keyword(s):  
Boundary Conditions ◽  
Normal Component ◽  
Tangential Component ◽  
Two Dimensions ◽  
Classical Boundary ◽  
Image Position ◽  
Special Case ◽  
Interpolation Result

AbstractIn this paper, we study under what boundary conditions the inequality$${\rm \Vert }\nabla \omega {\rm \Vert }_{L^2(\Omega )}^2 \les C({\rm \Vert }{\rm curl}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }{\rm div}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }\omega {\rm \Vert }_{L^2(\Omega )}^2 )$$holds true. It is known that such an estimate holds if either the tangential or normal component ofωvanishes on the boundary ∂Ω. We show that the vanishing tangential component condition is a special case of a more general one. In two dimensions, we give an interpolation result between these two classical boundary conditions.

scholarly journals Wall-to-wall optimal transport in two dimensions

2020 ◽  
Vol 889 ◽  
Author(s):  
Andre N. Souza ◽  
Ian Tobasco ◽  
Charles R. Doering
Keyword(s):  
Optimal Transport ◽  
Two Dimensions ◽  
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On the remainder in the two-dimensional central limit theorem

1970 ◽  
Vol 68 (2) ◽  
pp. 455-458
Author(s):  
J. E. A. Dunnage
Keyword(s):  
Distribution Function ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Two Dimensions ◽  
Independent Random Variables ◽  
Normal Distribution Function ◽  
Two Dimensional ◽  
Third Order ◽  
First Order ◽  
Image Position

Our object here is to refine the theorem proved in (3), and we use the notation of that paper. Let Z1, Z2, …, Zn, where Zr = (Xr, Yr), be independent random variables in two dimensions with zero first-order moments and finite third-order moments; and et the covariance matrix of Zr beWe writeLet (x, y) be the distribution function of the sum and let (x, y) be the normal distribution function having the same first- and second-order moments as (x, y).

scholarly journals Two Dimensional Analytical Considerations of Large Magnetic and Electric Fields in Laser Produced Plasmas

1986 ◽  
Vol 4 (2) ◽  
pp. 249-259 ◽  
Author(s):  
S. Eliezer ◽  
A. Loeb Loeb
Keyword(s):  
Differential Equation ◽  
Electric Fields ◽  
Electric Potential ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Electric And Magnetic Fields ◽  
Electrostatic Double Layer ◽  
Linear Differential ◽  
Image Position ◽  
Physical Branch

A simple model in two dimensions is developed and solved analytically taking into account the electric and magnetic fields in laser produced plasmas. The electric potential in this model is described by the non-linear differential equationψ = eφ/T, where eφ is the electric potential energy and T is the temperature in energy units. The physical branch ψ < 1, defined by the electron density n = no exp ψ, boundary conditions n (x = 0) = const and n (x = +∞) = 0, introduces a typical electrostatic double layer. The stationary solution of this model is consistent for − 3 ≲ ψ < 1, with electron temperatures in the KeV region and a ratio of the electric (E) to magnetic (B) fields of [E/106 v/cm]/[B/MGauss] ∼ 1.

7. Preliminary Note on a New Method of Investigating the Properties of Knots

1878 ◽  
Vol 9 ◽  
pp. 403-403 ◽  
Author(s):  
Tait
Keyword(s):  
Periodic Function ◽  
Two Dimensions ◽  
New Method ◽  
Plane Curves ◽  
Preliminary Note ◽  
Four Dimensions ◽  
Complete Representation ◽  
Image Position ◽  
Closed Plane

As we cannot have knots in two dimensions, and as Prof. Klein has proved that they cannot exist in space of four dimensions, it would appear that the investigation of their properties belongs to that class of problems for which the methods of quaternions were specially devised. The equationwhere ϕ is a periodic function, of course represents any endless curve whatever. Now the only condition to which variations of this function (looked on as corresponding to deformations of the knot) is subject, is that no two values of ρ shall ever be equal even at a stage of the deformation. Subject to this proviso, ϕ may suffer any changes whatever—retaining of course its periodicity. Some of the simpler results of a study of this novel problem in the theory of equations were given,—among others the complete representation of any knot whatever by three closed plane curves, non-autotomic and (if required) non-intersecting.

scholarly journals Texture mediated grain boundary network design in two dimensions

2016 ◽  
Vol 31 (9) ◽  
pp. 1171-1184 ◽  
Author(s):  
Oliver K. Johnson ◽  
Christopher A. Schuh
Keyword(s):  
Grain Boundary ◽  
Network Design ◽  
Two Dimensions ◽  
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Abstract

The Lavrentieff phenomenon for quadratic functionals

1995 ◽  
Vol 125 (2) ◽  
pp. 329-339 ◽  
Author(s):  
Riccardo De Arcangelis
Keyword(s):  
Sobolev Space ◽  
Weighted Sobolev Space ◽  
Integral Functional ◽  
Two Dimensions ◽  
Linear Boundary ◽  
Smooth Functions ◽  
Quadratic Functionals ◽  
Quadratic Integral ◽  
Image Position ◽  
Lavrentieff Phenomenon

The paper provides an example of an integral functional in more than two dimensions, with a symmetric and positively defined quadratic integrand q, exhibiting the Lavrentieff phenomenon on a ball B and on a linear boundary datum u0, i.e. for whichThe example is also utilised to discuss nonidentity between some relaxation procedures for a quadratic integral functional and to provide a weighted Sobolev space in the Hilbert case in which smooth functions are not dense.

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