Degenerate scale problem in antiplane elasticity or Laplace equation for contour shapes of triangles or quadrilaterals

This paper provides several solutions to the degenerate scale for the shapes of triangles or quadrilaterals in an exterior boundary value problem （BVP） of the antiplane elasticity or the Laplace equation. The Schwarz-Christoffel mapping is used throughout. It is found that a complex potential with a simple form in the mapping plane satisfies the vanishing displacement condition （or w ---- 0） along the boundary of the unit circle when the dimension R reaches its critical value 1. This means that the degenerate size in the physical plane is also achieved. The degenerate scales can be evaluated from the partic- ular integrals depending on certain parameters in the mapping function. The numerical results of degenerate sizes for the shapes of triangles or quadrilaterals are provided.

EVALUATION OF DEGENERATE SCALE FOR N-GON CONFIGURATION IN ANTIPLANE ELASTICITY

This paper provides a solution for the degenerate scale for N-gon configuration in antiplane elasticity using the conformal mapping function, and the lower and upper bounds for the degenerate scale for N = 3, 4, 5, 6, 7, 8, 10, 20, 30, 40, 50, 100, 150 and 200, are evaluated.

Interspecific differences and ecological correlations between scale number and skin structure in freshwater fishes

Fish skin is mainly composed of the epidermis,dermis,and its derivative scales.There is a wide diversity in scale number in fishes,but the diversity of skin structure lacks systematic histological comparison.This research aimed to improve our understanding of the functional relationship between the scale number and the skin structure in freshwater fishes and to determine which ecological factors affect the scale number and skin structure.First,we presented a method to quantify skin structure in fish and histologically quantified the skin structure of 54 freshwater fishes.Second,we collected the scale number and habitat information of 509 Cyprinidae fishes in China and explored which ecological factors were related to their scale number.Third,common carp and scaleless carp were used as models to study the effects of scale loss on swimming.We found a strong negative correlation between scale thickness and scale number.The main factor affecting the skin structure of fishes was the species’water column position,and the skin of benthic fishes was the most well-developed(thicker skin layers(dermis,epidermis)or more/larger goblet cells and club cells).The scale number was related to two factors,namely,temperature and water column position,and cold,benthic and pelagic adaptation may have contributed to increased scale numbers.Only in benthic fishes,the more well-developed their skin,the more scales.In common carp,scale loss did not affect its swimming performance.In summary,we suggest that there is a rich diversity of skin structure in freshwater fishes,and the scales of fish with well-developed skin tend to degenerate(greater number/smaller size/thinner,or even disappear),but the skin of fish with degenerated scales is not necessarily well developed.