scholarly journals Relaxation of some multi-well problems

2001 ◽  
Vol 131 (2) ◽  
pp. 279-320 ◽  
Author(s):  
Kaushik Bhattacharya ◽  
Georg Dolzmann
Keyword(s):  
Thin Films ◽  
Phase Transitions ◽  
Young Measure ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Convex Hulls ◽  
Quasiconvex Functions ◽  
Two Dimensional ◽  
Existence Of Minimizers

Mathematical models of phase transitions in solids lead to the variational problem, minimize ∫Ω W (Du) dx, where W has a multi-well structure, i.e. W = 0 on a multi-well set K and W > 0 otherwise. We study this problem in two dimensions in the case of equal determinant, i.e. for K = SO(2)U1 ∪ … ∪SO(2)Uk or K = O(2)U1 ∪ … ∪ O(2)Uk for U1, … , Uk ∈ M2×2 with det Ui = δ in three dimensions when the matrices Ui are essentially two-dimensional and also for K = SO(3)Û1 ∪ … ∪ SO(3)Ûk for U1, … , Uk ∈ M3×3 with , which arises in the study of thin films. Here, Ûi denotes the (3×2) matrix formed with the first two columns of Ui. We characterize generalized convex hulls, including the quasiconvex hull, of these sets, prove existence of minimizers and identify conditions for the uniqueness of the minimizing Young measure. Finally, we use the characterization of the quasiconvex hull to propose ‘approximate relaxed energies’, quasiconvex functions which vanish on the quasiconvex hull of K and grow quadratically away from it.

Theory of 2-D complex seismic trace analysis

Geophysics ◽  
1996 ◽  
Vol 61 (1) ◽  
pp. 264-272 ◽  
Author(s):  
Arthur E. Barnes
Keyword(s):  
Phase Velocity ◽  
Group Velocity ◽  
Hilbert Transform ◽  
Instantaneous Frequency ◽  
Trace Analysis ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Instantaneous Amplitude ◽  
Dip Angle

The ideas of 1-D complex seismic trace analysis extend readily to two dimensions. Two‐dimensional instantaneous amplitude and phase are scalars, and 2-D instantaneous frequency and bandwidth are vectors perpendicular to local wavefronts, each defined by a magnitude and a dip angle. The two independent measures of instantaneous dip correspond to instantaneous apparent phase velocity and group velocity. Instantaneous phase dips are aliased for steep reflection dips following the same rule that governs the aliasing of 2-D sinusoids in f-k space. Two‐dimensional frequency and bandwidth are appropriate for migrated data, whereas 1-D frequency and bandwidth are appropriate for unmigrated data. The 2-D Hilbert transform and 2-D complex trace attributes can be efficiently computed with little more effort than their 1-D counterparts. In three dimensions, amplitude and phase remain scalars, but frequency and bandwidth are 3-D vectors with magnitude, dip angle, and azimuth.

scholarly journals Averaging over Narain moduli space

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Alexander Maloney ◽  
Edward Witten
Keyword(s):  
Einstein Gravity ◽  
Two Dimensions ◽  
Three Dimensions ◽  
One Dimension ◽  
Simons Theory ◽  
Two Dimensional ◽  
Dual Theory ◽  
Recent Developments ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract Recent developments involving JT gravity in two dimensions indicate that under some conditions, a gravitational path integral is dual to an average over an ensemble of boundary theories, rather than to a specific boundary theory. For an example in one dimension more, one would like to compare a random ensemble of two-dimensional CFT’s to Einstein gravity in three dimensions. But this is difficult. For a simpler problem, here we average over Narain’s family of two-dimensional CFT’s obtained by toroidal compactification. These theories are believed to be the most general ones with their central charges and abelian current algebra symmetries, so averaging over them means picking a random CFT with those properties. The average can be computed using the Siegel-Weil formula of number theory and has some properties suggestive of a bulk dual theory that would be an exotic theory of gravity in three dimensions. The bulk dual theory would be more like U(1)2D Chern-Simons theory than like Einstein gravity.

Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour

1995 ◽  
Vol 305 ◽  
pp. 281-305 ◽  
Author(s):  
P. C. Matthews ◽  
M. R. E. Proctor ◽  
N. O. Weiss
Keyword(s):  
Magnetic Field ◽  
Shear Flow ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Nonlinear Behaviour ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Mean Flow ◽  
The Magnetic Field

Convection in a compressible fiuid with an imposed vertical magnetic field is studied numerically in a three-dimensional Cartesian geometry with periodic lateral boundary conditions. Attention is restricted to the mildly nonlinear regime, with parameters chosen first so that convection at onset is steady, and then so that it is oscillatory.Steady convection occurs in the form of two-dimensional rolls when the magnetic field is weak. These rolls can become unstable to a mean horizontal shear flow, which in two dimensions leads to a pulsating wave in which the direction of the mean flow reverses. In three dimensions a new pattern is found in which the alignment of the rolls and the shear flow alternates.If the magnetic field is sufficiently strong, squares or hexagons are stable at the onset of convection. Both the squares and the hexagons have an asymmetrical topology, with upflow in plumes and downflow in sheets. For the squares this involves a resonance between rolls aligned with the box and rolls aligned digonally to the box. The preference for three-dimensional flow when the field is strong is a consequence of the compressibility of the layer- for Boussinesq magnetoconvection rolls are always preferred over squares at onset.In the regime where convection is oscillatory, the preferred planform for moderate fields is found to be alternating rolls - standing waves in both horizontal directions which are out of phase. For stronger fields, both alternating rolls and two-dimensional travelling rolls are stable. As the amplitude of convection is increased, either by dcereasing the magnetic field strength or by increasing the temperature contrast, the regular planform structure seen at onset is soon destroyed by secondary instabilities.

Characterization of β-transition of poly(tert-butyl methacrylate) thin films by two-dimensional infrared correlation spectral analysis

2002 ◽  
Vol 29 (1-2) ◽  
pp. 73-77 ◽  
Author(s):  
Hyeon Suk Shin ◽  
Young Mee Jung ◽  
Taihyun Chang ◽  
Yukihiro Ozaki ◽  
Seung Bin Kim
Keyword(s):  
Thin Films ◽  
Spectral Analysis ◽  
Butyl Methacrylate ◽  
Two Dimensional ◽  
Tert Butyl

[Ni(cyclam)(OCOR)2], a finite molecular complex: hydrogen-bonded supramolecular aggregation in one, two and three dimensions, and coordination polymers in one and two dimensions

2001 ◽  
Vol 58 (1) ◽  
pp. 78-93 ◽  
Author(s):  
Choudhury M. Zakaria ◽  
George Ferguson ◽  
Alan J. Lough ◽  
Christopher Glidewell
Keyword(s):  
Hydrogen Bonding ◽  
Coordination Polymer ◽  
Hydrogen Bonds ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Compound I ◽  
Supramolecular Aggregation ◽  
Linear Coordination

In the complexes [Ni(cyclam)(OCOR)2] (cyclam = 1,4,8,11-tetraazacyclotetradecane), where (RCOO)− is 2-naphtho-ate [bis-(2-naphthoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (I), monoclinic P21/c, Z′ = 0.5], 3,5-dinitrobenzoate [bis-(3,5-dinitrobenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (II), triclinic P\bar 1, Z′ = 0.5], 4-nitrobenzoate [bis-(4-nitrobenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (III), monoclinic P21/n, Z′ = 0.5], 3-hydroxybenzoate [bis-(3-hydroxybenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (IV), monoclinic P21/c, Z′ = 0.5] and 4-aminobenzo-ate [bis-(4-aminobenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (V), monoclinic C2/c, Z′ = 0.5], the Ni lies on a centre of inversion with monodentate carboxylato ligands occupying trans sites. Compound (I) consists of isolated molecules. In (II) and (III), N—H...O hydrogen bonds link the complexes into chains. Compounds (IV) and (III) form two- and three-dimensional structures generated entirely by hard hydrogen bonds. The 5-hydroxyisophthalate(2−) anion forms a hydrated complex, [Ni(cyclam)(5-hydroxyisophthalate)(H2O)]·4H2O {[aqua-(5-hydroxyisophthalato)-1,4,8,11-tetraazacyclotetradecanenickel(II)] tetrahydrate, (VI), monoclinic Cc, Z′ = 1}, in which the monodentate carboxylato ligand and a water molecule occupy trans sites at Ni: extensive hydrogen bonding links the molecular aggregates into a three-dimensional framework. The terephthalate(2−) anion forms a hydrated linear coordination polymer {catena-poly[terephthalato-1,4,8,11-tetraazacyclotetradecanenickel(II)] monohydrate, (VII), monoclinic C2/c, Z′ = 0.5}. In 1,2,4,5-benzenecarboxylate tris[1,4,8,11-tetraazacyclotetradecanenickel(II)] diperchlorate hydrate (VIII), [Ni(cyclam)]3·[1,2,4,5-benzenetetracarboxylate(4−)]·[ClO4]2·-[H2O]3, there are two distinct Ni sites: [Ni(cyclam)]2+ and centrosymmetric [C10H2O8]4− units form a two-dimensional coordination polymer, whose sheets are linked by centrosymmetric [Ni(cyclam)(H2O)2]2+ cations.

Instability of pressure-driven gas–liquid two-layer channel flows in two and three dimensions

2018 ◽  
Vol 849 ◽  
pp. 1-34 ◽  
Author(s):  
Lennon Ó Náraigh ◽  
Peter D. M. Spelt
Keyword(s):  
Flow Pattern ◽  
Linear Theory ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Channel Flows ◽  
Three Dimensions ◽  
Nonlinear Behaviour ◽  
Two Dimensional ◽  
Vorticity Field ◽  
Flow Pattern Map

We study unstable waves in gas–liquid two-layer channel flows driven by a pressure gradient, under stable stratification, not assumed to be set in motion impulsively. The basis of the study is direct numerical simulation (DNS) of the two-phase Navier–Stokes equations in two and three dimensions for moderately large Reynolds numbers, accompanied by a theoretical description of the dynamics in the linear regime (Orr–Sommerfeld–Squire equations). The results are compared and contrasted across a range of density ratios $r=\unicode[STIX]{x1D70C}_{liquid}/\unicode[STIX]{x1D70C}_{gas}$. Linear theory indicates that the growth rate of small-amplitude interfacial disturbances generally decreases with increasing $r$; at the same time, the cutoff wavenumbers in both streamwise and spanwise directions increase, leading to an ever-increasing range of unstable wavenumbers, albeit with diminished growth rates. The analysis also demonstrates that the most dangerous mode is two-dimensional in all cases considered. The results of a comparison between the DNS and linear theory demonstrate a consistency between the two approaches: as such, the route to a three-dimensional flow pattern is direct in these cases, i.e. through the strong influence of the linear instability. We also characterize the nonlinear behaviour of the system, and we establish that the disturbance vorticity field in two-dimensional systems is consistent with a mechanism proposed previously by Hinch (J. Fluid Mech., vol. 144, 1984, p. 463) for weakly inertial flows. A flow-pattern map constructed from two-dimensional numerical simulations is used to describe the various flow regimes observed as a function of density ratio, Reynolds number and Weber number. Corresponding simulations in three dimensions confirm that the flow-pattern map can be used to infer the fate of the interface there also, and show strong three-dimensionality in cases that exhibit violent behaviour in two dimensions, or otherwise the development of behaviour that is nearly two-dimensional behaviour possibly with the formation of a capillary ridge. The three-dimensional vorticity field is also analysed, thereby demonstrating how streamwise vorticity arises from the growth of otherwise two-dimensional modes.

Cathodoluminescence characterization of two-dimensional interface structure of quantum wells

1990 ◽  
Vol 48 (4) ◽  
pp. 754-755
Author(s):  
Kazumi Wada
Keyword(s):  
Quantum Wells ◽  
Optical Transition ◽  
Two Dimensions ◽  
Interface Roughness ◽  
Luminescence Spectroscopy ◽  
Two Dimensional ◽  
Quantum Well Structures ◽  
Interface Structures ◽  
Nondestructive Characterization

Exotic properties shown by quantum well structures, typical structures of future electron devices, are sensitive to interface roughness. Extensive studies are, thus, focused on characterization of interface structures. Recent improvement in quantum wire fabrication technology demands for characterizing not only perpendicular-interfaces to the growth direction but also parallel-ones (sidewall-interfaces). Such sophistication needs innovation in two-dimensional and nondestructive characterization technology.In device structures, interfaces are generally located deep in bulk. STM which visualize surface atoms can not monitor such interface. It is, thus, difficult to two- dimensionally characterize the interfaces.Interface steps induce well width fluctuation, which modulates optical transition energy between ground subbands in conduction and valence bands. Thus, interface step structures can be characterized by luminescence spectroscopy. Cathodoluminescence basically meets demand for nondestructive characterization of interface structures in two dimensions.

scholarly journals THE “DUAL” VARIABLES OF YANG-MILLS THEORY AND LOCAL GAUGE-INVARIANT VARIABLES

1996 ◽  
Vol 11 (32) ◽  
pp. 5701-5728 ◽  
Author(s):  
ORI GANOR ◽  
J. SONNENSCHEIN
Keyword(s):  
Degrees Of Freedom ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Six Degrees Of Freedom ◽  
Local Gauge ◽  
Dual Variables ◽  
Topological Part

After adding auxiliary fields and integrating out the original variables, the Yang-Mills action can be expressed in terms of local gauge-invariant variables. This method reproduces the known solution of the two-dimensional SU (N) theory. In more than two dimensions the action splits into a topological part and a part proportional to αs. We demonstrate the procedure for SU (2) in three dimensions where we reproduce a gravitylike theory. We discuss the four-dimensional case as well. We use a cubic expression in the fields as a space-time metric to obtain a covariant Lagrangian. We also show how the four-dimensional SU (2) theory can be expressed in terms of a local action with six degrees of freedom only.

DONOR BINDING ENERGY IN TWO DIMENSIONS

1994 ◽  
Vol 08 (17) ◽  
pp. 1041-1043 ◽  
Author(s):  
E. S. ACKLEH ◽  
G. D. MAHAN ◽  
JI-WEI WU
Keyword(s):  
Binding Energy ◽  
Electron Density ◽  
Electron Gas ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Dimensional Electron ◽  
Two Dimensional Electron Gas ◽  
Donor Binding Energy ◽  
Dimensional Electron Gas

We calculate numerically the binding energy of an electron bound to a proton in two-dimensional electron gas. Unlike the case in three dimensions, in 2D the binding energy is nonzero for all values of electron density.

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