Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSharp lower bounds for Coulomb energy
154
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove $L^p$ lower bounds for Coulomb energy for radially symmetric functions in $dot H^s(R^3)$ with $frac 12 <s<frac{3}{2}$. In case $frac 12 <s leq 1$ we show that the lower bounds are sharp.
rate research
Read More
An algebraic lower bound on the energy decay for solutions of the advection-diffusion equation in $mathbb{R}^d$ with $d=2,3$ is derived using the Fourier splitting method. Motivated by a conjecture on mixing of passive scalars in fluids, a lower bound on the $L^2-$ norm of the inverse gradient of the solution is obtained via gradient estimates and interpolation.
The Dirac equation in $mathbb{R}^{1,3}$ with potential Z/r is a relativistic field equation modeling the hydrogen atom. We analyze the singularity structure of the propagator for this equation, showing that the singularities of the Schwartz kernel of the propagator are along an expanding spherical wave away from rays that miss the potential singularity at the origin, but also may include an additional spherical wave of diffracted singularities emanating from the origin. This diffracted wavefront is 1-0 derivatives smoother than the main singularities and is a conormal singularity.
The dependence of the fractal dimension of global attractors for the damped 3D Euler--Bardina equations on the regularization parameter $alpha>0$ and Ekman damping coefficient $gamma>0$ is studied. We present explicit upper bounds for this dimension for the case of the whole space, periodic boundary conditions, and the case of bounded domain with Dirichlet boundary conditions. The sharpness of these estimates when $alphato0$ and $gammato0$ (which corresponds in the limit to the classical Euler equations) is demonstrated on the 3D Kolmogorov flows on a torus.
We study the $2k$-th discrete moment of the derivative of the Riemann zeta-function at nontrivial zeros to establish sharp lower bounds for all real $k geq 0$ under the Riemann hypothesis (RH).
By a result of Schur [J. Reine Angew. Math. 1911], the entrywise product $M circ N$ of two positive semidefinite matrices $M,N$ is again positive. Vybiral [Adv. Math. 2020] improved on this by showing the uniform lower bound $M circ overline{M} geq E_n / n$ for all $n times n$ real or complex correlation matrices $M$, where $E_n$ is the all-ones matrix. This was applied to settle a conjecture of Novak [J. Complexity 1999] and to positive definite functions on groups. Vybiral (in his original preprint) asked if one can obtain similar uniform lower bounds for higher entrywise powers of $M$, or for $M circ N$ when $N eq M, overline{M}$. A natural third question is to obtain a tighter lower bound that need not vanish as $n to infty$, i.e. over infinite-dimensional Hilbert spaces. In this note, we affirmatively answer all three questions by extending and refining Vybirals result to lower-bound $M circ N$, for arbitrary complex positive semidefinite matrices $M, N$. Specifically: we provide tight lower bounds, improving on Vybirals bounds. Second, our proof is conceptual (and self-contained), providing a natural interpretation of these improved bounds via tracial Cauchy-Schwarz inequalities. Third, we extend our tight lower bounds to Hilbert-Schmidt operators. As an application, we settle Open Problem 1 of Hinrichs-Krieg-Novak-Vybiral [J. Complexity, in press], which yields improvements in the error bounds in certain tensor product (integration) problems.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
