Let f be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever J(f) belongs to a smooth curve, it also belongs to a circle. Then we discuss rational functions whose Julia sets belong to a circle. MSC classes: 37F10, 30D05. A simpl...

In this paper we study homogenization problem for strong Markov processes on Rd having infinitesimal generators Lf(x)=?Rd(f(x+z)?f(x)???f(x),z?1{|z|?1})k(x,z)?(dz)+?b(x),?f(x)?,f?Cb2(Rd) in periodic media, where ? is a nonnegative measure that does not charge the origin 0, satisfies ?Rd(1?|z|2)?(dz)<? and can be singular with respect to Lebesgue Rd. Under proper scaling show scaled converge wea...

We prove that the error term Λ(R) in the Weyl asymptotic formula #{En ≤ R } = VolM 4π R + Λ(R), for the Laplace operator on a surface of revolution M satisfying a twist hypothesis, has the form Λ(R) = RF (R) where F (R) is an almost periodic function of the Besicovitch class B, and the Fourier series of F (R) in B is ∑ γ A(γ) cos(|γ|R− φ) where the sum goes over all closed geodesics on M , and ...

1 Motivation We want to numerically approximate coefficients in a Fourier series. The first step is to see how the trapezoidal rule applies when numerically computing the integral (2π) −1 2π 0 F (t)dt, where F (t) is a continuous, 2π-periodic function. Applying the trapezoidal rule with the stepsize taken to be h = 2π/n for some integer n ≥ 1 results in (2π) −1 2π 0 F (t)dt ≈ 1 n n−1 j=0 Y j , ...

abstract background prevalence of hereditary blood diseases such as sickle cell anemia, sickle thalassemia and thalassemia major are high in khuzestan province. sickle cell anemia and beta-thalassemia are predominantly common in iranian arabs. pulmonary complications account for a large proportion of morbidity and mortality in patients with and sickle cell disease. periodic lung function assess...

An example is given of a cellular automaton map F implementing a binary counter whose non-wandering set , n(F), is st rictly larger than the closure, fI(F ), of the set of points periodic under F.

We present a multidimensional generalization of the Sharkovskĭı Theorem concerning the appearance of periodic points for the self-maps on the real line. Introduction Let f : X → X be a map. We say that x ∈ X is a periodic point of period n if f(x) = x and f(x) 6= x for k = 1, . . . , n− 1. We begin with recalling the Sharkovskĭı Theorem. Theorem 1. Let the ordering of positive integers be as fo...

We study the equation (?) ẋ(t) = −h(x(t− 1)) + f(x(t)) for t ≥ 0, x|[−1,0] = x0, where h is an odd function defined by h(y) is equal to a if 0 < y < c, equal to b if y ≥ c, a > b > 0 and c > 0 and f is an odd C function such that sup |f(x)| < b. We first consider the equation ẋ(t) = −h(x(t− 1)), corresponding to f ≡ 0. We find the admissible shapes of rapidlyoscillating symmetric periodic solut...

This paper gives experimental proof of an intriguing physical effect: periodic on–off switching of MOS transistors in a CMOS ring oscillator reduces their intrinsic 1=f noise and hence the oscillator’s close-in phase noise. More specifically, it is shown that the 1=f phase noise is dependent on the gate-source voltage of the MOS transistors in the off state. Measurement results, corrected for w...

We answer the following question: given any n ∈ N, which is the minimum number of endpoints en of a tree admitting a zero-entropy map f with a periodic orbit of period n? We prove that en = s1s2 ···sk − ∑k i=2 sisi+1 ···sk, where n = s1s2 ···sk is the decomposition of n into a product of primes such that si ≤ si+1 for 1 ≤ i < k. As a corollary, we get a criterion to decide whether a map f defin...