Manifolds with non-negative Ricci curvature and Nash inequalities

We prove that for any complete n-dimensional Riemannian manifold with nonnegative Ricci curvature, if the Nash inequality is satisfied, then it is diffeomorphic to $R^{n}$l.Comment: five pages, latex fil

General Sobolev Inequality on Riemannian Manifold

Let M be a complete n-dimensional Riemannian manifold, if the sobolev inqualities hold on M, then the geodesic ball has maximal volume growth; if the Ricci curvature of M is nonnegative, and one of the general Sobolev inequalities holds on M, then M is diffeomorphic to $R^{n}$.Comment: 4 page

Quantum steerability based on joint measurability

Occupying a position between entanglement and Bell nonlocality, Einstein-Podolsky-Rosen (EPR) steering has attracted increasing attention in recent years. Many criteria have been proposed and experimentally implemented to characterize EPR-steering. Nevertheless, only a few results are available to quantify steerability using analytical results. In this work, we propose a method for quantifying the steerability in two-qubit quantum states in the two-setting EPR-steering scenario, using the connection between joint measurability and steerability. We derive an analytical formula for the steerability of a class of X-states. The sufficient and necessary conditions for two-setting EPRsteering are presented. Based on these results, a class of asymmetric states, namely, one-way steerable states, are obtained.Comment: 8 pages, 5 figure

On the kk-error linear complexity of binary sequences derived from polynomial quotients

We investigate the $k$-error linear complexity of $p^2$-periodic binary sequences defined from the polynomial quotients (including the well-studied Fermat quotients), which is defined by $$q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p ~ \mathrm{with} 0 \le q_{p,w}(u) \le p-1, ~u\ge 0,$$ where $p$ is an odd prime and $1\le w<p$. Indeed, first for all integers $k$, we determine exact values of the $k$-error linear complexity over the finite field \F_2 for these binary sequences under the assumption of f2 being a primitive root modulo $p^2$, and then we determine their $k$-error linear complexity over the finite field \F_p for either $0\le k<p$ when $w=1$ or $0\le k<p-1$ when $2\le w<p$. Theoretical results obtained indicate that such sequences possess `good' error linear complexity.Comment: 2 figure

Linear complexity problems of level sequences of Euler quotients and their related binary sequences

The Euler quotient modulo an odd-prime power $p^r~(r>1)$ can be uniquely decomposed as a $p$-adic number of the form $$\frac{u^{(p-1)p^{r-1}} -1}{p^r}\equiv a_0(u)+a_1(u)p+\ldots+a_{r-1}(u)p^{r-1} \pmod {p^r},~ \gcd(u,p)=1,$$ where $0\le a_j(u)<p$ for $0\le j\le r-1$ and we set all $a_j(u)=0$ if $\gcd(u,p)>1$. We firstly study certain arithmetic properties of the level sequences $(a_j(u))_{u\ge 0}$ over $\mathbb{F}_p$ via introducing a new quotient. Then we determine the exact values of linear complexity of $(a_j(u))_{u\ge 0}$ and values of $k$-error linear complexity for binary sequences defined by $(a_j(u))_{u\ge 0}$.Comment: 16 page

High-order Green Operators on the Disk and the Polydisc

In this paper, we give the explicit expressions of high-order Green operators on the disk and the polydisc, and hence the kernel functions of high-order Green operators are also presented. As applications, we present the explicit integral expressions of all the solutions for linear high-order partial differential equations in the disk

Regret vs. Communication: Distributed Stochastic Multi-Armed Bandits and Beyond

In this paper, we consider the distributed stochastic multi-armed bandit problem, where a global arm set can be accessed by multiple players independently. The players are allowed to exchange their history of observations with each other at specific points in time. We study the relationship between regret and communication. When the time horizon is known, we propose the Over-Exploration strategy, which only requires one-round communication and whose regret does not scale with the number of players. When the time horizon is unknown, we measure the frequency of communication through a new notion called the density of the communication set, and give an exact characterization of the interplay between regret and communication. Specifically, a lower bound is established and stable strategies that match the lower bound are developed. The results and analyses in this paper are specific but can be translated into more general settings

Conservation law for Uncertainty relations and quantum correlations

Uncertainty principle, a fundamental principle in quantum physics, has been studied intensively via various uncertainty inequalities. Here we derive an uncertainty equality in terms of linear entropy, and show that the sum of uncertainty in complementary local bases is equal to a fixed quantity. We also introduce a measure of correlation in a bipartite state, and show that the sum of correlations revealed in a full set of complementary bases is equal to the total correlation in the bipartite state. The surprising simple equality relations we obtain imply that the study on uncertainty principle and correlations can rely on the use of linear entropy, a simple quantity that is very convenient for calculation

Spectral density of mixtures of random density matrices for qubits

We derive the spectral density of the equiprobable mixture of two random density matrices of a two-level quantum system. We also work out the spectral density of mixture under the so-called quantum addition rule. We use the spectral densities to calculate the average entropy of mixtures of random density matrices, and show that the average entropy of the arithmetic-mean-state of $n$ qubit density matrices randomly chosen from the Hilbert-Schmidt ensemble is never decreasing with the number $n$. We also get the exact value of the average squared fidelity. Some conjectures and open problems related to von Neumann entropy are also proposed.Comment: 21 pages, LaTex, 6 figure

Genuine Multipartite Entanglement of Superpositions

We investigate how the genuine multipartite entanglement is distributed among the components of superposed states. Analytical lower and upper bounds for the usual multipartite negativity and the genuine multipartite entanglement negativity are derived. These bounds are shown to be tight by detailed examples.Comment: 5 page