Elastic contact stiffness and contact resistance for the Weierstrass profile

The Weierstrass series comprises a system of superposed self-affine sine waves that can be used to define a simple idealization of a two-dimensional fractal rough surface profile. The load-compliance relation for the contact of this profile with a rigid plane is here estimated using Westergaard's solution for the contact of a single sine wave with a plane and various approximations concerning the interaction of the different terms in the series. These approximations are compared with a numerical solution for the contact of the profile defined by the first few terms of the series. Once the load-compliance relation is established, the electrical contact resistance can be determined, using an analogy between the conduction and incremental elastic contact problems. The results show that these simple estimates give quite good predictions of the relations between load, compliance and contact resistance. They also confirm that these relations are largely determined by the coarse scale features of the surface profile, in contrast to the predictions of classical asperity model theories.     

Generating Signals with Multiscale Time Irreversibility: The Asymmetric Weierstrass Function

Time irreversibility (asymmetry with respect to time reversal) is an important property of many time series derived from processes in nature. Some time series (e.g., healthy heart rate dynamics) demonstrate even more complex, multiscale irreversibility, such that not only the original but also coarse-grained time series are asymmetric over a wide range of scales. Several indices to quantify multiscale asymmetry have been introduced. However, there has been no simple generator of model time series with "tunable" multiscale asymmetry to test such indices. We introduce an asymmetric Weierstrass function W(A) (constructed from asymmetric sawtooth functions instead of cosine waves) that can be used to construct time series with any given value of the multiscale asymmetry. We show that multiscale asymmetry appears to be independent of other multiscale complexity indices, such as fractal dimension and multiscale entropy. We further generalize the concept of multiscale asymmetry by introducing time-dependent (local) multiscale asymmetry and provide examples of such time series. The W(A) function combines two essential features of complex fluctuations, namely fractality (self-similarity) and irreversibility (multiscale time asymmetry); moreover, each of these features can be tuned independently. The proposed family of functions can be used to compare and refine multiscale measures of time series asymmetry.     

New Exact Solutions to (2+1)-Dimensional Variable Coefficients Broer-Kaup Equations

In this paper, using the variable coefficient generalized projected Ricatti equation expansion method, we present explicit solutions of the (2 1)-dimensional variable coefficients Broer-Kaup (VCBK) equations. These solutions include Weierstrass function solution, solitary wave solutions, soliton-like solutions and trigonometric function solutions.Among these solutions, some are found for the first time. Because of the three or four arbitrary functions, rich localized excitations can be found.     

Weierstrass Representation for Surfaces in the Three-Dimensional Heisenberg Group

The authors define the Gauss map of surfaces in the three-dimensional Heisenberg group and give a representation formula for surfaces of prescribed mean curvature. Furthermore, a second order partial differential equation for the Gauss map is obtained, and it is shown that this equation is the complete integrability condition of the representation.     

The Self-conjugate Maximal Surfaces in L~(3*)

in this note we study the maximal surfaces which are congruent with theirconjugate ones in L3, and give a criterion for this kind of surfaces and some examples.     

Classification of Refinable Splines

A refinable spline is a compactly supported refinable function that is piecewise polynomial. Refinable splines, such as the well-known B-splines, play a key role in computer aided geometric design. So far all studies on refinable splines have focused on positive integer dilations and integer translations, and under this setting a rather complete classification was obtained in [12]. However, refinable splines do not have to have integer dilations and integer translations. The classification of refinable splines with noninteger dilations and arbitrary translations is studied in this paper. We classify completely all refinable splines with integer translations and arbitrary dilations. Our study involves techniques from number theory and complex analysis.     

Soap films spanning rectangular prisms

In this paper, we consider soap films spanning rectangular prisms with regular n-gon bases. As the number of edges n varies, we show that there are significant changes in the qualitative properties of the spanning soap films as well as a change in the number of spanning soap films whose existence we can prove: We can find two nontrivial soap films for n = 3, 4, 5 but only one for n ≥ 6. We also prove some results concerning the interval of aspect ratios through which the soap films exist: The interval is finite if n = 3, 4, 5 and infinite if n ≥ 6. Furthermore, for n > 6, we have that the spanning soap film converges to a soap film spanning the vertical lines through the vertices of a regular n-gon as the aspect ratio goes to infinity. We can also make sense of the case n = ∞. Here, we discover some interesting singly and triply periodic soap films spanning singly and doubly periodic sets of vertical lines or spanning singly periodic sets of vertical line segments connected by pairs of parallel, horizontal lines. Finally, for n = 3, 4, 5, 6, we can derive parameterizations for the spanning soap films, and these parameterizations are explicit up to knowing the aspect ratio.      

On three differential equations associated with Chebyshev polynomials of degrees 3, 4 and 6

We shall study the differential equation y'~2= T_n(y)-(1-2μ~2);where μ~2 is a constant, T_n(x) are the Chebyshev polynomials with n = 3, 4, 6.The solutions of the differential equations will be expressed explicitly in terms of the Weierstrass elliptic function which can be used to construct theories of elliptic functions based on _2F_1(1/4, 3/4; 1; z),_2F_1(1/3, 2/3; 1; z), _2F_1(1/6, 5/6; 1; z) and provide a unified approach to a set of identities of Ramanujan involving these hypergeometric functions.