An asymmetric Kadison's inequality

Some inequalities for positive linear maps on matrix algebras are given, especially asymmetric extensions of Kadison's inequality and several operator versions of Chebyshev's inequality. We also discuss well-known results around the matrix geometric mean and connect it with complex interpolation.Comment: To appear in LA

Some New Inequalities of Dirichlet Eigenvalues for Laplace Operator with any Order

In this paper, we establish several inequalities of Dirichlet eigenvalues for Laplace operator $\Delta$ with any order on \emph{n}-dimensional Euclidean space. These inequalities are more general than known Yang's inequalities and contain new consequences. To obtain them, we borrow the approach of Illias and Makhoul, and use a generalized Chebyshev's inequality

Improved Chebyshev inequality: new probability bounds with known supremum of PDF

In this paper, we derive new probability bounds for Chebyshev's inequality if the supremum of the probability density function is known. This result holds for one-dimensional or multivariate continuous probability distributions with finite mean and variance (covariance matrix). We also show that the similar result holds for specific discrete probability distributions.Comment: 7 pages, 2 figure

On certain generalizations of the Smarandache function

The famous Smarandache function is an arithmetical function, connected to the number of divisors of n, and other important number theoretic function