The repetition threshold is the smallest real number α such that there exists an infinite word over a k-letter alphabet that avoids repetition of exponent strictly greater than α. This notion can be generalized to graph classes. In this paper, we completely determine the repetition thresholds for caterpillars and caterpillars of maximum degree 3. Additionally, we present bounds for the repetiti...

We find the threshold between avoidable and unavoidable repetitions in circular words over k letters for any k > 6. Namely, we show that the number CRT(k) = ⌈k/2⌉+1 ⌈k/2⌉ satisfies the following properties. For any n there exists a k-ary circular word of length n containing no repetition of exponent greater than CRT(k). On the other hand, k-ary circular words of some lengths must have a repetit...

Critical exponent γ ⪰ 1.1 characterizes the behavior of mechanical compressibility a real fluid when temperature approaches critical one. It results in zero Gaussian curvature local shape point on thermodynamic equation state surface, which imposes new constraint upon construction potential from empirical data. All known equations suffer weakness that is negative definite instead zero.

We study the balance between the effect of spatial inhomogeneity of the potential in the dissipative term and the focusing nonlinearity. Sharp critical exponent results will be presented in the case of slow decaying potential.