Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA remark on a paper by Hundertmark and Simon
158
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove a sharp Lieb-Thirring type inequality for Jacobi matrices, thereby settling a conjecture of Hundertmark and Simon. An interesting feature of the proof is that it employs a technique originally used by Hundertmark-Laptev-Weidl concerning sums of singular values for compact operators.
rate research
Read More
In an interesting recent paper [1] (A. Acharya, Stress of a spatially uniform dislocation density field, J. Elasticity 137 (2019), 151--155), Acharya proved that the stress produced by a spatially uniform dislocation density field in a body comprising a nonlinear elastic material may fail to vanish under no loads. The class of counterexamples constructed in [1] is essentially $2$-dimensional: it works with the subgroup $mathcal{O}(2) oplus langle{bf Id}rangle subset mathcal{O}(3)$. The objective of this note is to extend Acharyas result in [1] to $mathcal{O}(3)$, subject to an additional structural assumption and less regularity requirements.
The conjecture by Steklov was solved negatively by Rakhmanov in 1979. His original proof was based on the formula for orthogonal polynomial obtained by adding point masses to the measure of orthogonality. In this note, we show how this polynomial can be obtained by applying the method developed recently for proving the sharp lower bounds for the problem by Steklov.
In their recent paper titled Large Associative Memory Problem in Neurobiology and Machine Learning [arXiv:2008.06996] the authors gave a biologically plausible microscopic theory from which one can recover many dense associative memory models discussed in the literature. We show that the layers of the recent MLP-mixer [arXiv:2105.01601] as well as the essentially equivalent model in [arXiv:2105.02723] are amongst them.
We apply a recent result of Borichev-Golinskii-Kupin on the Blaschke-type conditions for zeros of analytic functions on the complex plane with a cut along the positive semi-axis to the problem of the eigenvalues distribution of the Fredholm-type analytic operator-valued functions.
We prove some refinements of concentration compactness principle for Sobolev space $W^{1,n}$ on a smooth compact Riemannian manifold of dimension $n$. As an application, we extend Aubins theorem for functions on $mathbb{S}^{n}$ with zero first order moments of the area element to higher order moments case. Our arguments are very flexible and can be easily modified for functions satisfying various boundary conditions or belonging to higher order Sobolev spaces.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
