Area-proportional Euler diagrams have many applications, for example they are often used for visualizing data in medical and biological domains. There have been a number of recent research efforts to automatically draw Euler diagrams when the areas of the regions are not considered, leading to a range of different drawing techniques. By contrast, substantially less progress has been made on the...

We reformulate the one-dimensional complex Ginzburg-Landau equation as a fourth order ordinary differential equation in order to find stationary spatiallyperiodic solutions. Using this formalism, we prove the existence and stability of stationary modulated-amplitude wave solutions. Approximate analytic expressions and a comparison with numerics are given.

This paper applies the recently developed theory of discrete nonholonomic mechanics to the study of discrete nonholonomic left-invariant dynamics on Lie groups. The theory is illustrated with the discrete versions of two classical nonholonomic systems, the Suslov top and the Chaplygin sleigh. The preservation of the reduced energy by the discrete flow is observed and the discrete momentum conse...

We use the fractional integrals to describe fractal solid. We suggest to consider the fractal solid as special (fractional) continuous medium. We replace the fractal solid with fractal mass dimension by some continuous model that is described by fractional integrals. The fractional integrals are considered as approximation of the integrals on fractals. We derive fractional generalization of the...

We prove the Euler-Lagrange delta-differential equations for problems of the calculus of variations on arbitrary time scales with delta-integral functionals depending on higher-order delta derivatives.

Let X,Y be Banach modules over a C∗-algebra and let r1, . . . , rn ∈ R be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital C∗-algebra: ∑n j 1 f −rjxj ∑ 1≤i≤n,i / j rixi 2 ∑n i 1 rif xi nf ∑n i 1 rixi . We show that if ∑n i 1 ri / 0, ri, rj / 0 for some 1 ≤ i < j ≤ n and a mapping f : X → Y satisfies the functional equation...

In 1940, Ulam [1] proposed the famous Ulam stability problem of linear mappings. In 1941, Hyers [2] considered the case of approximately additive mappings f : E→ E′, where E and E′ are Banach spaces and f satisfies Hyers inequality ‖ f (x+ y)− f (x)− f (y)‖ ≤ ε for all x, y ∈ E. It was shown that the limit L(x) = limn→∞ 2−n f (2nx) exists for all x ∈ E and that L : E→ E′ is the unique additive ...

Two infinite sequences of minimal surfaces in space are constructed using symmetry analysis. In particular, explicit formulas are obtained for the selfintersecting minimal surface that fills the trefoil knot. UDC 514.763.85, 517.972.6 Introduction. In this paper we consider the Euler–Lagrange minimal surface equation

Let Φ(u, v) = ∑ ∞ m=0 ∑ ∞ n=0 cmnu v. Bouwkamp and de Bruijn found that there exists a power series Ψ(u, v) satisfying the equation tΨ(tz, z) = log (∑ ∞ k=0 t k k! exp(kΦ(kz, z)) ) . We show that this result can be interpreted combinatorially using hypergraphs. We also explain some facts about Φ(u, 0) and Ψ(u, 0), shown by Bouwkamp and de Bruijn, by using hypertrees, and we use Lagrange inversi...