We study the existence of sign-changing solutions for following supercritical problem with variable exponent and logarithmic nonlinearity$ \begin{equation*} \left\{ \begin{aligned} - \Delta u& = |u|^{2^{*}-2} u\big( \ln( \tau +|u|) \big)^{ |x|^{\beta} } & \ \mbox{in} B_{1}, \\ 0 \mbox{on} \partial \end{aligned} \right. \end{equation*} $where $ B_{1} is unit ball in \mathbb{R}^{N} $, N\g...

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...

This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The scaling exponent μ of polar codes for a memoryless channel qY |X with capacity I(qY |X) characterizes the closest gap between the capacity and non-asymptotic achievable rates in the following way: For a fixed ε ∈ (0, 1), the gap between the capacity I(qY |X) and the maximum non-asymptotic rate R n achi...

We investigate the protein expression pattern of the lamB gene in Escherichia coli LE392. The gene product LamB is an important membrane protein for maltose transport into cells but it is also exploited by bacteriophage λ for infection. Although our bacterial population is clonal, stochastic gene expression leads to a majority population with a large receptor number and a minority population wi...

Overview Last class we stated a similar reciprocity theorem for the Jacobi symbol. In this class we shall do the proof of it, discuss the algorithm, and also do the Solovay-Strassen primality testing. 1 Proof of the Reciprocity of m n The proof will just be induction on m. Recall the statement of the theorem 2 n = (−1) n 2 −1 8 m n n m = (−1) m−1 2 n−1 2 We shall just prove the second part here...

Given a smooth compact Riemannian n–manifold (M, g), we consider the equation ∆gu+ hu = |u| ∗−2−ε u, where h is a C–function on M , the exponent 2∗ := 2n/ (n− 2) is the critical Sobolev exponent, and ε is a small positive real parameter such that ε→ 0. We prove the existence of blowing-up families of sign-changing solutions which develop bubble towers at some point where the function h is great...

We investigate analytically and numerically the dynamical properties of critical Boolean networks with power-law in-degree distributions and for two choices of update functions. When the exponent of the in-degree distribution is larger than 3, we obtain results equivalent to those obtained for networks with fixed in-degree, e.g., the number of the nonfrozen nodes scales as N(2/3) with the syste...

A significant concern about Metabolic Scaling Theory (MST) in real forests relates to consistent differences between the values of power law scaling exponents of tree primary size measures used to estimate mass and those predicted by MST. Here we consider why observed scaling exponents for diameter and height relationships deviate from MST predictions across three semi-arid conifer forests in r...