The Hardy uncertainty principle says that no function is better localized together with its Fourier transform than the Gaussian. The textbook proof of the result, as well as one of the original proofs by Hardy, refers to the Phragm\'en-Lindel\"of theorem. In this note we first describe the connection of the Hardy uncertainty to the Schr\"odinger equation, and give a new proof of Hardy's result which is based on this connection and the Liouville theorem. The proof is related to the second proof of Hardy, which has been underservedly forgotten. Then we survey the recent results on dynamical versions of Hardy's theorem.

Aingeru Fernández-Bertolin Eugenia Malinnikova

Some problems involving the classical Hardy function $$ Z(t) := \zeta(1/2+it)\bigl(\chi(1/2+it)\bigr)^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s) $$ are discussed. In particular we discuss the odd moments of $Z(t)$, the distribution of its positive and negative values and the primitive of $Z(t)$. Some analogous problems for the mean square of $|\zeta(1/2+it)|$ are also discussed.

Aleksandar Ivić

A recent experimental proposal by Ahnert and Payne [S.E. Ahnert and M.C. Payne, Phys. Rev. A 70, 042102 (2004)] outlines a method to measure the weak value predictions of Aharonov in Hardy's paradox. This proposal contains flaws such as the state preparation method and the procedure for carrying out the requisite weak measurements. We identify previously published solutions to some of the flaws.

J. S. Lundeen K. J. Resch A. M. Steinberg

The half-life of 10C has been measured to be 19.310(4)s, a result with 0.02% precision, which is a factor of three improvement over the best previous result. Since 10C is the lightest superallowed 0+ --> 0+ beta emitter, its ft value has the greatest weight in setting an upper limit on the possible presence of scalar currents.

V. E. Iacob J. C. Hardy V. Golovko J. Goodwin N. Nica H. I. Park L. Trache R. E. Tribble

In this paper we study the extension problem for the sublaplacian on a $H$-type group and use the solutions to prove trace Hardy and Hardy inequalities for fractional powers of the sublaplacian.

L. Roncal S. Thangavelu

It has been proposed that the ability to perform joint weak measurements on post-selected systems would allow us to study quantum paradoxes. These measurements can investigate the history of those particles that contribute to the paradoxical outcome. Here, we experimentally perform weak measurements of joint (i.e. nonlocal) observables. In an implementation of Hardy's Paradox, we weakly measure the locations of two photons, the subject of the conflicting statements behind the Paradox. Remarkably, the resulting weak probabilities verify all these statements but, at the same time, resolve the Paradox.

J. S. Lundeen A. M. Steinberg

We present a test with which to evaluate the calculated isospin-symmetry-breaking corrections to superallowed 0+-to-0+ nuclear beta decay. The test is based on the corrected experimental Ft values being required to satisfy conservation of the vector current (CVC). When applied to six sets of published calculations, the test demonstrates quantitatively that only one set -- the one based on the shell model with Saxon-Woods radial wave functions -- provides satisfactory agreement with CVC. This test can easily be applied to any sets of calculated correction terms that are produced in future.

I. S. Towner J. C. Hardy

We have measured the K-shell internal conversion coefficient, alpha-K, for the 65.7-keV M4 transition in 119Sn to be 1621(25). This result agrees well with Dirac-Fock calculations in which the effect of the K-shell atomic vacancy is accounted for, and disagrees with calculations in which the vacancy is ignored. This extends our precision tests of theory to Z = 50, the lowest Z yet measured.

N. Nica J. C. Hardy V. E. Iacob M. Bencomo V. Horvat H. I. Park M. Maguire S. Miller M. B. Trzhaskovskaya

We present a parameterization of the statistical rate function, f, for 20 superallowed 0+-to-0+ nuclear beta transitions between T=1 analog states, and for 18 superallowed "mirror" transitions between analog T=1/2 states. All these transitions are of interest in the determination of V_{ud}. Although most of the transition Q_{EC} values have been measured, their precision will undoubtedly be improved in future. Our parameterization allows a user to easily calculate the corresponding new f value to high precision (+/-0.01%) without complicated computing.

I. S. Towner J. C. Hardy

In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy-Sobolev fractional equation with critical growth \begin{equation*}\label{0.1} \left\{% \begin{array}{ll} (-\Delta)^{s} u-\ds\frac{\mu u}{|x|^{2s}}=|u|^{2^*_s-2}u+au, & \hbox{$\text{in}~ \Omega$},\vspace{0.1cm} u=0,\,\, &\hbox{$\text{on}~\partial \Omega$}, \\ \end{array}% \right. \end{equation*} provided $N>6s$, $\mu\geq0$, $0< s<1$, $2^*_s=\frac{2N}{N-2s}$, $a>0$ is a constant and $\Omega$ is an open bounded domain in $\R^N$ which contains the origin.

Chunhua Wang Jing Yang Jing Zhou