Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn upper bound on the Hot Spots constant
108
0
0.0
(
0
)
Authors
Stefan Steinerberger
Ask ChatGPT about the research
No Arabic abstract
Let $D subset mathbb{R}^d$ be a bounded, connected domain with smooth boundary and let $-Delta u = mu_1 u$ be the first nontrivial eigenfunction of the Laplace operator with Neumann boundary conditions. We prove $$ |u|_{L^{infty}(D)} leq 60 cdot |u|_{L^{infty}(partial D)}.$$ This shows that the Hot Spots Conjecture cannot fail by an arbitrary factor. An example of Kleefeld shows that the optimal constant is at least $1 + 10^{-3}$.
rate research
Read More
We prove on the 2D sphere and on the 2D torus the Lieb-Thirring inequalities with improved constants for orthonormal families of scalar and vector functions.
Let $G$ be a finite (not necessarily abelian) group and let $p=p(G)$ be the smallest prime number dividing $|G|$. We prove that $d(G)leq frac{|G|}{p}+9p^2-10p$, where $d(G)$ denotes the small Davenport constant of $G$ which is defined as the maximal integer $ell$ such that there is a sequence over $G$ of length $ell$ contains no nonempty one-product subsequence.
Let ${bf P}_k^{(alpha, beta)} (x)$ be an orthonormal Jacobi polynomial of degree $k.$ We will establish the following inequality begin{equation*} max_{x in [delta_{-1},delta_1]}sqrt{(x- delta_{-1})(delta_1-x)} (1-x)^{alpha}(1+x)^{beta} ({bf P}_{k}^{(alpha, beta)} (x))^2 < frac{3 sqrt{5}}{5}, end{equation*} where $delta_{-1}<delta_1$ are appropriate approximations to the extreme zeros of ${bf P}_k^{(alpha, beta)} (x) .$ As a corollary we confirm, even in a stronger form, T. Erd{e}lyi, A.P. Magnus and P. Nevai conjecture [Erd{e}lyi et al., Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614], by proving that begin{equation*} max_{x in [-1,1]}(1-x)^{alpha+{1/2}}(1+x)^{beta+{1/2}}({bf P}_k^{(alpha, beta)} (x))^2 < 3 alpha^{1/3} (1+ frac{alpha}{k})^{1/6}, end{equation*} in the region $k ge 6, alpha, beta ge frac{1+ sqrt{2}}{4}.$
Hot luminous stars show a variety of phenomena in their photospheres and winds which still lack clear physical explanation. Among these phenomena are photospheric turbulence, line profile variability (LPV), non-thermal emission, non-radial pulsations, discrete absorption components (DACs) and wind clumping. Cantiello et al. (2009) argued that a convection zone close to the stellar surface could be responsible for some of these phenomena. This convective zone is caused by a peak in the opacity associated with iron-group elements and is referred to as the iron convection zone (FeCZ). Assuming dynamo action producing magnetic fields at equipartition in the FeCZ, we investigate the occurrence of subsurface magnetism in OB stars. Then we study the surface emergence of these magnetic fields and discuss possible observational signatures of magnetic spots. Simple estimates are made using the subsurface properties of massive stars, as calculated in 1D stellar evolution models. We find that magnetic fields of sufficient amplitude to affect the wind could emerge at the surface via magnetic buoyancy. While at this stage it is difficult to predict the geometry of these features, we show that magnetic spots of size comparable to the local pressure scale height can manifest themselves as hot, bright spots. Localized magnetic fields could be widespread in those early type stars that have subsurface convection. This type of surface magnetism could be responsible for photometric variability and play a role in X-ray emission and wind clumping.
The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $Game (D_n({bfSigma}^0_2))$ for some natural number $ngeq 1$, where $Game$ is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space $2^omega$ into the class ${bfDelta}^1_2$, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ${bfDelta}^1_2$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
