scholarly journals Canonical Coordinates and Natural Equation for Lorentz Surfaces in R13

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3121
Author(s):  
Krasimir Kanchev ◽  
Ognian Kassabov ◽  
Velichka Milousheva
Keyword(s):  
Mean Curvature ◽  
General Type ◽  
Canonical Coordinates ◽  
Surfaces Of General Type ◽  
Isotropic Coordinates ◽  
Natural Equation ◽  
Special Cases ◽  
The Mean ◽  
First Fundamental Form ◽  
Zero Mean Curvature

We consider Lorentz surfaces in R13 satisfying the condition H2−K≠0, where K and H are the Gaussian curvature and the mean curvature, respectively, and call them Lorentz surfaces of general type. For this class of surfaces, we introduce special isotropic coordinates, which we call canonical, and show that the coefficient F of the first fundamental form and the mean curvature H, expressed in terms of the canonical coordinates, satisfy a special integro-differential equation which we call a natural equation of the Lorentz surfaces of a general type. Using this natural equation, we prove a fundamental theorem of Bonnet type for Lorentz surfaces of a general type. We consider the special cases of Lorentz surfaces of constant non-zero mean curvature and minimal Lorentz surfaces. Finally, we give examples of Lorentz surfaces illustrating the developed theory.

scholarly journals Regularity of locally convex surfaces

1990 ◽  
Vol 42 (3) ◽  
pp. 487-497 ◽  
Author(s):  
Friedmar Schulz
Keyword(s):  
Boundary Condition ◽  
Fundamental Form ◽  
Mean Curvature ◽  
Convex Surface ◽  
Gauss Curvature ◽  
Radius Vector ◽  
Convex Surfaces ◽  
The Mean ◽  
First Fundamental Form ◽  
Locally Convex

Interior estimates are derived for the C2, µ-Hölder norm of the radius vector X ∈ C1, 1 (Ω) of a locally convex surface Σ in terms of the first fundamental form IΣ, the Gauss curvature K and the integral ∫ |H| dσ. Here H is the mean curvature of Σ. The coefficients gij of IΣ are assumed to belong to the Hölder class C2, µ (Ω) for some μ, 0 < μ < 1. A boundary condition is discussed which ensures an estimate for ∫ | H | dσ.

Decoration technique and surface physics

1978 ◽  
Vol 36 (1) ◽  
pp. 436-437 ◽  
Author(s):  
H. Bethge
Keyword(s):  
Diffusion Path ◽  
Special Importance ◽  
Surface Physics ◽  
Step Structure ◽  
Initial Stage ◽  
Special Cases ◽  
Atomic Surface ◽  
The Mean ◽  
Bulk Structure ◽  
Decoration Technique

Besides the atomic surface structure, diverging in special cases with respect to the bulk structure, the real structure of a surface Is determined by the step structure. Using the decoration technique /1/ it is possible to image step structures having step heights down to a single lattice plane distance electron-microscopically. For a number of problems the knowledge of the monatomic step structures is important, because numerous problems of surface physics are directly connected with processes taking place at these steps, e.g. crystal growth or evaporation, sorption and nucleatlon as initial stage of overgrowth of thin films.To demonstrate the decoration technique by means of evaporation of heavy metals Fig. 1 from our former investigations shows the monatomic step structure of an evaporated NaCI crystal. of special Importance Is the detection of the movement of steps during the growth or evaporation of a crystal. From the velocity of a step fundamental quantities for the molecular processes can be determined, e.g. the mean free diffusion path of molecules.

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

The focusing of radiation by a random surface when the source is at a finite distance

1960 ◽  
Vol 56 (1) ◽  
pp. 27-40 ◽  
Author(s):  
M. S. Longuet-Higgins
Keyword(s):  
Mean Curvature ◽  
Joint Distribution ◽  
Total Curvature ◽  
Statistical Distribution ◽  
Constant Parameter ◽  
Random Surface ◽  
Second Derivatives ◽  
Finite Distance ◽  
The Mean ◽  
Image Position

Imagine a nearly horizontal, statistically uniform, random surface ζ(x, y), Gaussian in the sense that the second derivatives , , have a normal joint distribution. The problem considered is the statistical distribution of the quantitywhere J and Ω denote the mean curvature and total curvature of the surface, respectively, and ν is a constant parameter.

Functionals of Bounded Frechet Variation

1949 ◽  
Vol 1 (2) ◽  
pp. 153-165 ◽  
Author(s):  
Marston Morse ◽  
William Transue
Keyword(s):  
Representation Theory ◽  
Spectral Theory ◽  
General Type ◽  
Integrodifferential Equations ◽  
Vector Spaces ◽  
Variational Theory ◽  
Characteristic Equations ◽  
Special Cases ◽  
Jacobi Equations ◽  
Normed Vector

In a series of papers which will follow this paper the authors will present a theory of functionals which are bilinear over a product A × B of two normed vector spaces A and B. This theory will include a representation theory, a variational theory, and a spectral theory. The associated characteristic equations will include as special cases the Jacobi equations of the classical variational theory when n = 1, and self-adjoint integrodifferential equations of very general type. The bilinear theory is oriented by the needs of non-linear and non-bilinear analysis in the large.

scholarly journals Thermodynamic and Elastic Properties of Interstitial Alloy FeC with BCC Structure at Zero Pressure

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Bui Duc Tinh ◽  
Nguyen Quang Hoc ◽  
Dinh Quang Vinh ◽  
Tran Dinh Cuong ◽  
Nguyen Duc Hien
Keyword(s):  
Nearest Neighbor ◽  
Young Modulus ◽  
Thermodynamic Quantities ◽  
Atom Concentration ◽  
Main Metal ◽  
Special Cases ◽  
Analytic Expressions ◽  
Rigidity Modulus ◽  
Bcc Structure ◽  
The Mean

The analytic expressions for the thermodynamic and elastic quantities such as the mean nearest neighbor distance, the free energy, the isothermal compressibility, the thermal expansion coefficient, the heat capacities at constant volume and at constant pressure, the Young modulus, the bulk modulus, the rigidity modulus, and the elastic constants of binary interstitial alloy with body-centered cubic (BCC) structure, and the small concentration of interstitial atoms (below 5%) are derived by the statistical moment method. The theoretical results are applied to interstitial alloy FeC in the interval of temperature from 100 to 1000 K and in the interval of interstitial atom concentration from 0 to 5%. In special cases, we obtain the thermodynamic quantities of main metal Fe with BCC structure. Our calculated results for some thermodynamic and elastic quantities of main metal Fe and alloy FeC are compared with experiments.

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