We show that all $q$-semimultiplicative sequences are asymptotically orthogonal to the M\"obius function, thus proving the Sarnak conjecture for this class of sequences. This generalises analogous results for the sum-of-digits function and other digital sequences which follow from previous work of Mauduit and Rivat.

Jakub Konieczny

We prove that strongly $b$-multiplicative functions of modulus $1$ along squares are asymptotically orthogonal to the M\"obius function. This provides examples of sequences having maximal entropy and satisfying this property.

Michael Drmota Christian Mauduit Joël Rivat Lukas Spiegelhofer

In this paper, we prove analogues of the Dirichlet theorem on arithmetic progressions and the Siegel--Walfisz theorem for the digital reverses of primes for arbitrary bases, which the authors obtained in the previous paper but only for large bases. The proof is based on a generalization of the result of Martin--Mauduit--Rivat (2014) on the exponential sums over primes with the so-called ``digital'' functions.

Gautami Bhowmik Yuta Suzuki

There exist operators $A$ such that : for any sequence of contractions $\{A_n\}$, there is a total sequence of mutually orthogonal projections $\{E_n\}$ such that $\Sigma E_nAE_n=\bigoplus A_n$.

Jean-Christophe Bourin

We review some convexity inequalities for Hermitian matrices an add one more to the list.

Jean-christophe Bourin

Subaddivity type matrix inequalities for concave funcions and symetric norms are given.

Jean-Christophe Bourin Eun-Young Lee

Several subadditivity results and conjectures are given for matrices (or operators), block-matrices, concave functions and norms.

Jean-Christophe Bourin

We study various convex functions on $R^n$ associated with positive definite matrices. This yiels some exotic Holder matrix inequalities.

Jean-Christophe Bourin Jingjing Shao

A refinement of a trace inequality of McCarthy establishing the uniform convexity of the Schatten $p$-classes for p>2 is proved

Jean-Christophe Bourin Eun-Young Lee

Three measures of pseudorandomness of finite binary sequences were introduced by Mauduit and S\'ark\"ozy in 1997 and have been studied extensively since then: the normality measure, the well-distribution measure, and the correlation measure of order r. Our main result is that the correlation measure of order r for random binary sequences converges strongly, and so has a limiting distribution. This solves a problem due to Alon, Kohayakawa, Mauduit, Moreira, and R\"odl. We also show that the best known lower bounds for the minimum values of the correlation measures are simple consequences of a celebrated result due to Welch, concerning the maximum nontrivial scalar products over a set of vectors.

Kai-Uwe Schmidt