In 1907, M. Petrovitch initiated the study of a class of entire functions all whose finite sections (i.e., truncations) are real-rooted polynomials. He was motivated by previous studies of E. Laguerre on uniform limits of sequences of real-rooted polynomials and by an interesting result of G. H. Hardy. An explicit description of this class in terms of the coefficients of a series is impossible since it is determined by an infinite number of discriminant inequalities, one for each degree. However, interesting necessary or sufficient conditions can be formulated. In particular, J. I. Hutchinson has shown that an entire function $p\left(x\right)=a_0+a_{1}x+\cdots +a_n{x}^n+\cdots$ with strictly positive coefficients has the property that all of its finite segments $a_i{x}^i+a_{i+1}{x}^{i+1}+\cdots +a_j{x}^j$ have only real roots if and only if $a_i^2/a_{i-1}{a}_{i+1}\ge 4$ for $i=1,2,\dots$ . In the present paper, we give sharp lower bounds on the ratios $a_i^2/a_{i-1}{a}_{i+1}$ ($i=1,2,\dots$) for the class considered by M. Petrovitch. In particular, we show that the limit of these minima when $i\to \infty$ equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the Laguerre–Pólya class $\mathcal{L-P}I$. We also explain the relation between Newton’s and Hutchinson’s inequalities and the logarithmic image of the set of all real-rooted polynomials with positive coefficients.

In this paper we consider the cohomology of a closed arithmetic hyperbolic $3$-manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of $SL(2,\mathbb{C})$. The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomology grows exponentially as the local system grows. We also consider the twisted Ruelle zeta function of a closed arithmetic hyperbolic $3$-manifold, and we express the leading coefficient of its Laurent expansion at the origin in terms of the orders of the torsion subgroups of the cohomology.

Given martingales $W$ and $Z$ such that $W$ is differentially subordinate to $Z$, Burkholder obtained the sharp inequality $E|W{|}^p\le (p^{\ast }-1{)}^{p}E|Z{|}^p$, where $p^{\ast }=max\hspace{0.17em}\{p,p/(p-1\left)\right\}$. What happens if one of the martingales is also a conformal martingale? Bañuelos and Janakiraman proved that if $p\ge 2$ and $W$ is a conformal martingale differentially subordinate to any martingale $Z$, then $E|W{|}^p\le [(p^2-p)/2{]}^{p/2}E|Z{|}^p$. In this paper, we establish that if $p\ge 2$, $Z$ is conformal, and $W$ is any martingale subordinate to $Z$, then $\mathbb{E}|W{|}^p\le [\sqrt{2}(1-z_p)/z_p{]}^p\mathbb{E}|Z{|}^p$, where $z_p$ is the smallest positive zero of a certain solution of the Laguerre ordinary differential equation. We also prove the sharpness of this estimate and an analogous one in the dual case for $1<p<2$. Finally, we give an application of our results. Previous estimates on the $L^p$-norm of the Beurling–Ahlfors transform give at best $\Vert B{\Vert }_p\lesssim \sqrt{2}p$ as $p\to \infty$. We improve this to $\Vert B{\Vert }_p\lesssim 1.3922p$ as $p\to \infty$.

For a locally compact group $G$, let $A\left(G\right)$ denote its Fourier algebra, and let $M_{0}A\left(G\right)$ denote the space of completely bounded Fourier multipliers on $G$. The group $G$ is said to have the Approximation Property (AP) if the constant function $1$ can be approximated by a net in $A\left(G\right)$ in the weak-∗ topology on the space $M_{0}A\left(G\right)$. Recently, Lafforgue and de la Salle proved that $SL(3,\mathbb{R})$ does not have the AP, implying the first example of an exact discrete group without it, namely, $SL(3,\mathbb{Z})$. In this paper we prove that $Sp(2,\mathbb{R})$ does not have the AP. It follows that all connected simple Lie groups with finite center and real rank greater than or equal to two do not have the AP. This naturally gives rise to many examples of exact discrete groups without the AP.

We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra, and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and its ideal consisting of derivations with positive degrees. As an application of the last case, and by making use of a theorem of Kontsevich, we obtain a new proof of the vanishing theorem of Harer concerning the top rational cohomology group of the mapping class group with respect to its virtual cohomological dimension.