On Some Formulas for Families of Curves and Surfaces and Aminov’s Divergent Representations

2018 ◽  
Vol 39 (1) ◽  
pp. 114-120 ◽  
Author(s):  
A. G. Megrabov
Keyword(s):  
Curves And Surfaces ◽  
Families Of Curves

scholarly journals On Tangency in Equisingular Families of Curves and Surfaces

2020 ◽  
Vol 71 (2) ◽  
pp. 485-505
Author(s):  
Arturo Giles Flores ◽  
O N Silva ◽  
J Snoussi
Keyword(s):  
Parameter Space ◽  
Tangent Cones ◽  
Curves And Surfaces ◽  
Families Of Curves ◽  
The Family ◽  
Surface Singularities ◽  
Whitney Equisingularity

Abstract We study the behavior of limits of tangents in topologically equivalent spaces. In the context of families of generically reduced curves, we introduce the $s$-invariant of a curve and we show that in a Whitney equisingular family with the property that the $s$-invariant is constant along the parameter space, the number of tangents of each curve of the family is constant. In the context of families of isolated surface singularities, we show through examples that Whitney equisingularity is not sufficient to ensure that the tangent cones of the family are homeomorphic. We explain how the existence of exceptional tangents is preserved by Whitney equisingularity but their number can change.

scholarly journals Motion of curves and surfaces and nonlinear evolution equations in (2+1) dimensions

1998 ◽  
Vol 39 (7) ◽  
pp. 3765-3771 ◽  
Author(s):  
M. Lakshmanan ◽  
R. Myrzakulov ◽  
S. Vijayalakshmi ◽  
A. K. Danlybaeva
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Curves And Surfaces

Digital Jordan Curves and Surfaces with Respect to a Closure Operator

2021 ◽  
Vol 179 (1) ◽  
pp. 59-74
Author(s):  
Josef Šlapal
Keyword(s):  
Direct Product ◽  
Jordan Curve ◽  
Closure Operator ◽  
Jordan Curve Theorem ◽  
Closure Operators ◽  
Ternary Relation ◽  
Digital Space ◽  
Jordan Curves ◽  
Curves And Surfaces ◽  
Digital Plane

In this paper, we propose new definitions of digital Jordan curves and digital Jordan surfaces. We start with introducing and studying closure operators on a given set that are associated with n-ary relations (n > 1 an integer) on this set. Discussed are in particular the closure operators associated with certain n-ary relations on the digital line ℤ. Of these relations, we focus on a ternary one equipping the digital plane ℤ2 and the digital space ℤ3 with the closure operator associated with the direct product of two and three, respectively, copies of this ternary relation. The connectedness provided by the closure operator is shown to be suitable for defining digital curves satisfying a digital Jordan curve theorem and digital surfaces satisfying a digital Jordan surface theorem.

MEASUREMENT OF THE COMPLEX DIELECTRIC CONSTANT OF LIQUIDS AT CENTIMETER AND MILLIMETER WAVELENGTHS

1957 ◽  
Vol 35 (9) ◽  
pp. 995-1003 ◽  
Author(s):  
A. G. Mungall ◽  
John Hart
Keyword(s):  
Dielectric Constant ◽  
Methyl Alcohol ◽  
Free Space ◽  
Normal Incidence ◽  
Complex Dielectric Constant ◽  
Families Of Curves ◽  
Millimeter Wavelengths ◽  
Centimeter Wave ◽  
Space Technique ◽  
Wave Region

The measurement of the complex dielectric constant of lossy liquids in the millimeter and centimeter wave region by a free-space technique is described. The method involves the measurement of absorption per wavelength and of reflectance at normal incidence. Families of curves are given for the relations between these two quantities and the real and imaginary parts of the complex dielectric constant. Results for ethyl and methyl alcohol at 9 and 13 mm. wavelength are compared with those obtained by waveguide techniques.

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