scholarly journals Two-dimensional dilaton gravity theory and lattice Schwarzian theory

2019 ◽  
Vol 34 (29) ◽  
pp. 1950176
Author(s):  
Su-Kuan Chu ◽  
Chen-Te Ma ◽  
Chih-Hung Wu
Keyword(s):  
Boundary Condition ◽  
Cosmological Constant ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Gravity Theory ◽  
Boundary Theory ◽  
Two Dimensional ◽  
Dilaton Field ◽  
Dilaton Gravity ◽  
Cosmological Constants

We report a holographic study of a two-dimensional dilaton gravity theory with the Dirichlet boundary condition for the cases of nonvanishing and vanishing cosmological constants. Our result shows that the boundary theory of the two-dimensional dilaton gravity theory with the Dirichlet boundary condition for the case of nonvanishing cosmological constants is the Schwarzian term coupled to a dilaton field, while for the case of vanishing cosmological constant, a theory does not have a kinetic term. We also include the higher derivative term [Formula: see text], where [Formula: see text] is the scalar curvature that is coupled to a dilaton field. We find that the form of the boundary theory is not modified perturbatively. Finally, we show that a lattice holographic picture is realized up to the second-order perturbation of boundary cutoff [Formula: see text] under a constant boundary dilaton field and the nonvanishing cosmological constant by identifying the lattice spacing [Formula: see text] of a lattice Schwarzian theory with the boundary cutoff [Formula: see text] of the two-dimensional dilaton gravity theory.

COMPLETE STABILITY FOR STANDARD CELLULAR NEURAL NETWORKS

1999 ◽  
Vol 09 (05) ◽  
pp. 909-918 ◽  
Author(s):  
SONG-SUN LIN ◽  
CHIH-WEN SHIH
Keyword(s):  
Neural Networks ◽  
Boundary Condition ◽  
Symmetric Space ◽  
Dirichlet Boundary Condition ◽  
Periodic Boundary ◽  
Dirichlet Boundary ◽  
Periodic Boundary Conditions ◽  
Cellular Neural Networks ◽  
Two Dimensional ◽  
Space Variant

We consider cellular neural networks with symmetric space-variant feedback template. The complete stability is proved via detailed analysis on the energy function. The proof is presented for the two-dimensional case with Dirichlet boundary condition. It can be extended to other dimensions with minor adjustments. Modifications to the cases of Neumann and periodic boundary conditions are also mentioned.

Singularities of two-dimensional exterior solutions of the Helmholtz equation

1971 ◽  
Vol 69 (1) ◽  
pp. 175-188 ◽  
Author(s):  
R. F. Millar
Keyword(s):  
Boundary Condition ◽  
Convex Hull ◽  
Helmholtz Equation ◽  
Dirichlet Boundary Condition ◽  
Harmonic Functions ◽  
Dirichlet Boundary ◽  
Two Dimensional ◽  
Helmholtz Equations

AbstractA technique for locating possible singularities of two-dimensional ex-terior harmonic functions was discussed in a previous paper. In the present work, the method is generalized to exterior solutions of the Helmholtz equation. Although the procedure deviates in some of its details from the earlier exposition, the conclusions are similar. In particular, it is verified that solutions of the Laplace and Helmholtz equations that satisfy the same Dirichlet boundary condition on the same boundary, possess the same convex hull of singularities. The possibility of extending the method to more general equations is raised.

scholarly journals Solving Poisson Equation by Distributional HK-Integral: Prospects and Limitations

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Amila J. Maldeniya ◽  
Naleen C. Ganegoda ◽  
Kaushika De Silva ◽  
Sanath K. Boralugoda
Keyword(s):  
Boundary Condition ◽  
Half Space ◽  
Poisson Equation ◽  
Dirichlet Boundary Condition ◽  
Linear Functional ◽  
Dirichlet Boundary ◽  
Two Dimensional ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Test Function

In this paper, we present some properties of integrable distributions which are continuous linear functional on the space of test function D ℝ 2 . Here, it uses two-dimensional Henstock–Kurzweil integral. We discuss integrable distributional solution for Poisson’s equation in the upper half space ℝ + 3 with Dirichlet boundary condition.

scholarly journals On some classes of generalized Schrödinger equations

2020 ◽  
Vol 10 (1) ◽  
pp. 522-533
Author(s):  
Amanda S. S. Correa Leão ◽  
Joelma Morbach ◽  
Andrelino V. Santos ◽  
João R. Santos Júnior
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

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