scholarly journals ANALYSIS OF AN INVERSE PROBLEM ARISING IN PHOTOLITHOGRAPHY

2012 ◽  
Vol 22 (05) ◽  
pp. 1150026 ◽  
Author(s):  
LUCA RONDI ◽  
FADIL SANTOSA
Keyword(s):  
Inverse Problem ◽  
Variational Formulation ◽  
Diffractive Optics ◽  
Shape Design ◽  
Two Dimensional ◽  
Convergence Properties ◽  
Approximate Problem ◽  
The Relationship ◽  
Well Posed ◽  
Dimensional Domain

We consider the inverse problem of determining an optical mask that produces a desired circuit pattern in photolithography. We set the problem as a shape design problem in which the unknown is a two-dimensional domain. The relationship between the target shape and the unknown is modeled through diffractive optics. We develop a variational formulation that is well-posed and propose an approximation that can be shown to have convergence properties. The approximate problem can serve as a foundation to numerical methods, much like the Ambrosio–Tortorelli's approximation of the Mumford–Shah functional in image processing.

scholarly journals Fluid velocity profile reconstruction for non-Newtonian shear dispersive flow

2001 ◽  
Vol 5 (2) ◽  
pp. 87-104 ◽  
Author(s):  
Paul R. Shorten ◽  
David J. N. Wall
Keyword(s):  
Inverse Problem ◽  
Velocity Profile ◽  
Fluid Velocity ◽  
Average Concentration ◽  
Two Dimensional ◽  
Cross Sectional ◽  
Dispersive Flow ◽  
Ill Posed ◽  
Profile Reconstruction ◽  
Well Posed

An inverse problem associated with the mass transport of a material concentration down a pipe where the flowing non-Newtonian medium has a two-dimensional velocity profile is examined. The problem of determining the two-dimensional fluid velocity profile from temporally varying cross-sectional average concentration measurements at upstream and downstream locations is considered. The special case of a known input upstream concentration with a time zero step, and a strictly decreasing velocity profile is shown to be a well-posed problem. This inverse problem is in general ill-posed and mollification is used to obtain a well conditioned problem.

Relationship between the intrinsic radial distribution function for an isotropic field of particles and lower-dimensional measurements

2002 ◽  
Vol 459 ◽  
pp. 93-102 ◽  
Author(s):  
GRETCHEN L. HOLTZER ◽  
LANCE R. COLLINS
Keyword(s):  
Inverse Problem ◽  
Distribution Function ◽  
Numerical Simulations ◽  
Power Law ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Two Dimensional ◽  
Dimensional Measurements ◽  
Well Posed ◽  
Lower Dimensional

In this paper, we present relationships between the intrinsic radial distribution function (RDF) for a three-dimensional, isotropic system of particles and the lower-dimensional RDFs obtained experimentally from either two-dimensional or one-dimensional sampling of the data. The lower-dimensional RDFs are shown to be equivalent to integrals of the three-dimensional function, and as such contain less information than their three-dimensional counterpart. An important consequence is that the lower-dimensional RDFs are attenuated at separation distances below the characteristic length scale of the measurement. In addition, the inverse problem (calculating the three-dimensional RDF from the lower-dimensional measurements) is not well posed. However, recent results from direct numerical simulations (Reade & Collins 2000) showed that the three-dimensional RDF for aerosol particles in a turbulent flow field obeys a power-law dependence on r for r [Lt ] η, where η is the Kolmogorov scale of the turbulence. In this case, the inverse problem is well posed and it is possible to obtain the prefactor and exponent of the power law from one- or two-dimensional measurements. A procedure for inverting the data is given. All of the relationships derived in this paper have been validated by data derived from direct numerical simulations.

scholarly journals A Framework for Approximation of the Stokes Equations in an Axisymmetric Domain

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Niklas Ericsson
Keyword(s):  
Fourier Expansion ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Three Dimensional ◽  
Countable Family ◽  
Angular Variable ◽  
Two Dimensional ◽  
Sobolev Norms ◽  
Well Posed ◽  
Variational Formulations

Abstract We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems.

scholarly journals Template-Free Self-Assembly of Two-Dimensional Polymers into Nano/Microstructured Materials

Molecules ◽  
2021 ◽  
Vol 26 (11) ◽  
pp. 3310
Author(s):  
Shengda Liu ◽  
Jiayun Xu ◽  
Xiumei Li ◽  
Tengfei Yan ◽  
Shuangjiang Yu ◽  
...  
Keyword(s):  
Self Assembly ◽  
Recent Progress ◽  
Two Dimensional ◽  
Subsequent Removal ◽  
The Past ◽  
2D Polymers ◽  
Template Free ◽  
Self Assembled ◽  
The Relationship ◽  
Polymer Tubes

In the past few decades, enormous efforts have been made to synthesize covalent polymer nano/microstructured materials with specific morphologies, due to the relationship between their structures and functions. Up to now, the formation of most of these structures often requires either templates or preorganization in order to construct a specific structure before, and then the subsequent removal of previous templates to form a desired structure, on account of the lack of “self-error-correcting” properties of reversible interactions in polymers. The above processes are time-consuming and tedious. A template-free, self-assembled strategy as a “bottom-up” route to fabricate well-defined nano/microstructures remains a challenge. Herein, we introduce the recent progress in template-free, self-assembled nano/microstructures formed by covalent two-dimensional (2D) polymers, such as polymer capsules, polymer films, polymer tubes and polymer rings.

EQUIVALENCE OF TWO-DIMENSIONAL GRAVITIES

1990 ◽  
Vol 05 (16) ◽  
pp. 1251-1258 ◽  
Author(s):  
NOUREDDINE MOHAMMEDI
Keyword(s):  
Gauge Theory ◽  
Explicit Solution ◽  
Light Cone ◽  
Equations Of Motion ◽  
Two Dimensional ◽  
Induced Gravity ◽  
Light Cone Gauge ◽  
Dimensional Gravity ◽  
The Relationship ◽  
Chern Simons

We find the relationship between the Jackiw-Teitelboim model of two-dimensional gravity and the SL (2, R) induced gravity. These are shown to be related to a two-dimensional gauge theory obtained by dimensionally reducing the Chern-Simons action of the 2+1 dimensional gravity. We present an explicit solution to the equations of motion of the auxiliary field of the Jackiw-Teitelboim model in the light-cone gauge. A renormalization of the cosmological constant is also given.

“Beholding the Man”: Viewing (or is it Marking?) John’s Trial Scene alongside Kitsch Art

2016 ◽  
Vol 24 (2) ◽  
pp. 245-264
Author(s):  
Andrew P. Wilson
Keyword(s):  
Two Dimensional ◽  
Devotional Art ◽  
The People ◽  
Subject And Object ◽  
Ecce Homo ◽  
Small Stories ◽  
John's Gospel ◽  
Relationship Of ◽  
The Relationship ◽  
Point Of Departure

One of the grand scenes of the Passion narratives can be found in John’s Gospel where Pilate, presenting Jesus to the people, proclaims “Behold the man”: “Ecce Homo.” But what exactly does Pilate mean when he asks the reader to “Behold”? This paper takes as its point of departure a roughly drawn picture of Jesus in the “Ecce Homo” tradition and explores the relationship of this picture to its referent in John’s Gospel, via its capacity as kitsch devotional art. Contemporary scholarship on kitsch focuses on what kitsch does, or how it functions, rather than assessing what it is. From this perspective, when “beholding” is understood not for what it reveals but for what it does, John’s scene takes on a very different significance. It becomes a scene that breaks down traditional divisions between big and small stories, subject and object as well as text and context. A kitsch perspective opens up possibilities for locating John’s narrative in unexpected places and experiences. Rather than being a two-dimensional departure from the grandeur of John’s trial scene, kitsch “art” actually provides a lens through which the themes and dynamics of the narrative can be re-viewed with an expansiveness somewhat lacking from more traditional commentary.


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