scholarly journals On the behaviour of the sup- and inf-convolutions of a function near the boundary

1996 ◽  
Vol 53 (3) ◽  
pp. 515-520
Author(s):  
Timothy R. Cranny
Keyword(s):  
Second Order ◽  
Boundary Potential ◽  
Potential Applications ◽  
Regularity Results

The study of nonclassical solutions for elliptic and parabolic PDE's often involves the use of regularisation processes such as the sup- and inf-convolutions. In this note we study the behaviour of these regularised functions near the boundary of the domain, and derive constraints on the appropriate second-order sub- and superdifferentials on and near the boundary. Potential applications to regularity results are also noted.

scholarly journals Second order elliptic boundary value problems in spaces with homogeneous norms

1980 ◽  
Vol 29 (1) ◽  
pp. 1-13 ◽  
Author(s):  
A. J. Pryde
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Partial Differential Operator ◽  
Second Order ◽  
Elliptic Boundary Value ◽  
First Order ◽  
Constant Coefficients ◽  
Regularity Results ◽  
Homogeneous Norms

AbstractWe consider the interior and Dirichiet problems and problems with first order boundary conditions, for a second order homogeneous elliptic partial differential operator with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. From there we obtain a priori estimates and regularity results for boundary value problems in Sobolev spaces.

Electrostatique de systèmes cylindriques de révolution où le potentiel sur certaines parties de la frontière est un polynôme du second degré de la cote suivant l'axe

1981 ◽  
Vol 59 (2) ◽  
pp. 202-206 ◽  
Author(s):  
Colette Berger
Keyword(s):  
Second Order ◽  
Real System ◽  
Elementary System ◽  
Boundary Potential ◽  
Rapid Calculation ◽  
Order Polynomial ◽  
Analytical Expressions ◽  
Derivatives Of ◽  
Aberration Corrected

The systems studied are equivalent to indefinite cylinders with rotationally symmetrical potentials. The analytical expressions for the derivatives of the axial potential with respect to z are given for two elementary systems called "Electrode" and "Interval." This analysis is also applied to one further elementary system in which the boundary potential is zero except in one region where its dependence on z follows a second-order polynomial. For a complex real system, the analytical expressions of the derivatives of the axial potential are obtained by superposition of elementary systems. This method renders possible a rapid calculation of the field, even for systems with large electrode separations. Furthermore, it may be used for the design of aberration-corrected systems.[Journal translation]

Kinetics studies of rapid strain-promoted [3+2] cycloadditions of nitrones with bicyclo[6.1.0]nonyne

2014 ◽  
Vol 92 (4) ◽  
pp. 337-340 ◽  
Author(s):  
Douglas A. MacKenzie ◽  
John Paul Pezacki
Keyword(s):  
Rate Constants ◽  
Substituent Effect ◽  
Reaction Rates ◽  
Reaction Vessel ◽  
Second Order ◽  
Cyclic Nitrones ◽  
Organic Azides ◽  
Kinetics Studies ◽  
Bioorthogonal Labeling ◽  
Potential Applications

Strain-promoted alkyne−nitrone cycloaddition (SPANC) reactions represent a bioorthogonal labeling strategy that is both very rapid and at the same time efficient and selective. Nitrones provide increased reaction rates as well as greater susceptibility toward stereoelectronic modification when compared with organic azides. We find that strain-promoted cycloadditions of cyclic nitrones with bicyclo[6.1.0]nonyne react with second-order rate constants as large as 1.49 L mol−1 s−1 at 25 °C. These reactions display rate constants that are up to 37-fold greater than those of the analogous reactions of benzyl azide with bicyclo[6.1.0]nonyne. We observed that reactions of nitrones with bicyclo[6.1.0]nonyne showed a stronger dependence on substituent effect for the reaction, as evidenced by a larger Hammett ρ value, than that for biaryl-aza-cyclooctanone. We demonstrate the ability to stereoelectronically tune the reactivity of nitrones towards different cyclooctynes in SPANC reactions. This ability to introduce selectivity into different SPANC reactions through substituent provides the opportunity to perform multiple SPANC reactions in one reaction vessel and opens up potential applications in multiplex labeling.

APPLICATION OF THE RENORMALIZATION GROUP TO A SECOND-ORDER QCD PHASE TRANSITION

1992 ◽  
Vol 07 (16) ◽  
pp. 3911-3925 ◽  
Author(s):  
FRANK WILCZEK
Keyword(s):  
Phase Transition ◽  
Renormalization Group ◽  
Strange Quark ◽  
Tricritical Point ◽  
Universality Class ◽  
Second Order ◽  
Chiral Phase ◽  
Quark Masses ◽  
Potential Applications ◽  
Second Order Transition

An earlier suggestion that the chiral phase transition in QCD for two flavors of massless quarks might be a second-order transition has gained credibility as a result of recent numerical simulations. One can test this hypothesis, and draw very specific quantitative consequences from it, using universality and renormalization group ideas. This hypothetical second order phase transition is in the universality class of a four component isotropic Heisenberg antiferromagnet—a model which has been investigated intensely by condensed matter physicists. Existing calculations can be adapted to yield predictions for critical exponents governing the QCD transition. The perturbation due to small equal quark masses maps onto an external staggered magnetic field; that due to unequal quark masses is effective only in second order and generates a quadratic anisotropy. Several other potential applications of the renormalization group to related questions are suggested, including a model of the tricritical point which arises with finite strange quark mass, and a model for the dynamic critical behavior.

Unveiling electrically tunable characteristics of second-order dispersion in graphene-silicon nitride waveguides

2019 ◽  
Vol 33 (05) ◽  
pp. 1950053 ◽  
Author(s):  
Peng Xie ◽  
Mantian Xue ◽  
Yu Wen ◽  
Xiaofeng Li ◽  
Xinyu Wang ◽  
...  
Keyword(s):  
Silicon Nitride ◽  
Graphene Layer ◽  
Optoelectronic Devices ◽  
Second Order ◽  
Core Material ◽  
The Core ◽  
Large Bandwidth ◽  
Waveguide Model ◽  
Potential Applications ◽  
Order Dispersion

In this paper, we proposed a graphene-silicon nitride (GSN) waveguide model, which was unveiled to unveil its second-order dispersion (SOD) characteristics. The influences of the different thicknesses of graphene layer and the different heights from graphene to the core material of silicon nitride on SOD were investigated in detail. The tunability of SOD via controlling the bias voltage applied to the graphene layer was demonstrated and a 50 nm wavelength tuning was achieved with a small perturbation in voltage while keeping the geometric structure of the waveguide unchanged. Moreover, a flat SOD curve of the GSN waveguide was obtained with a large bandwidth of 700 nm between the two zero-dispersion wavelengths (ZDWs). These results provided significant insights for potential applications of graphene-related optoelectronic devices, integrated optics, and optical communications.

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