In this paper, we consider the problem of the existence of positive weak solutions of { (−∆)su = up in Ω u = 0 on Rn\Ω having prescribed isolated interior singularities. We prove that if n n−2s < p < p1 for some critical exponent p1 defined in the introduction which is related to the stability of the singular solution us, and if S is a closed subset of Ω, then there are infinitely many positive...

We investigate the finite repetition threshold for k-letter alphabets, k ≥ 4, that is the smallest number r for which there exists an infinite r+-free word containing a finite number of r-powers. We show that there exists an infinite Dejean word on a 4-letter alphabet (i.e. a word without factors of exponent more than 7 5 ) containing only two 7 5 -powers. For a 5-letter alphabet, we show that ...

The Scaling Law for the Discrete Kinetic Growth Percolation Model The critical exponent of the total number of finite clusters α is calculated directly without using scaling hypothesis both below and above the percolation threshold pc based on a kinetic growth percolation model in two and three dimensions. Simultaneously, we can calculate other critical exponents β and γ, and show that the scal...

Brylawski [2] has shown that loopless principal transversal matroids have critical exponent at most 2. Welsh [4] asks if a similar result holds for all transversal matroids. We answer in the affirmative by proving that all loopless transversal matroids have critical exponent at most 2. The terminology used here for matroids will in general follow Welsh 131. A rank r transversal matroid M is rep...

Abstract. We study a class of Schrödinger operators on Z with a random potential decaying as |x|, 0 < σ ≤ 1 2 , in the limit of small disorder strength λ. For the critical exponent σ = 1 2 , we prove that the localization length of eigenfunctions is bounded below by 2 − 1 4 +η , while for 0 < σ < 1 2 , the lower bound is λ− 2−η 1−2σ , for any η > 0. These estimates ”interpolate” between the low...

Let F(z) = ∑ n>1 fnz n be the generating series of the regular paperfolding sequence. For a real number α the irrationality exponent μ(α), of α, is defined as the supremum of the set of real numbers μ such that the inequality |α − p/q| < q−μ has infinitely many solutions (p, q) ∈ Z × N. In this paper, using a method introduced by Bugeaud, we prove that μ(F(1/b)) 6 275331112987 137522851840 = 2....

Brandenburg and (implicitly) Dejean introduced the concept of repetition threshold: the smallest real number α such that there exists an infinite word over a k-letter alphabet that avoids β-powers for all β > α. We generalize this concept to include the lengths of the avoided words. We give some conjectures supported by numerical evidence and prove some of these conjectures. As a consequence of...

The interior of a cell is very crowded (up to 35% protein by weight), with significant consequences for diffusive transport. The possible trajectories of a given tracer particle are constrained by the labyrinth of obstacles. For a static obstacle configuration, a percolation transition takes place at a certain critical obstacle density, so that above this density no long-range transport is poss...

We study the problem of the existence and nonexistence of positive solutions to a superlinear second–order divergence type elliptic equation with measurable coefficients −∇ · a · ∇u = u (∗), p > 1, in an unbounded cone–like domain G ⊂ R (N ≥ 3). We prove that the critical exponent p∗(a,G) = inf{p > 1 : (∗) has a positive supersolution at infinity in G } for a nontrivial cone– like domain is alw...

We consider a basic fractional differential inequality with a fractional derivative named after Hilfer and a polynomial source. A non-existence of global solutions result is proved in an appropriate space and the critical exponent is shown to be optimal.