In this paper we consider pro-p Poincar\'e groups of dimension 2 (so-called Demuskin groups) on which a finite (abelian) group of order prime to p acts. In particular, we are interested in free pro-p quotient which are invariant under this action.

Kay Wingberg

Let $G$ be a compact connected Lie group and $p : E\to \Sigma A$ be a principal G-bundle with a characteristic map $\alpha : A\to G$, where $A=\Sigma A_{0}$ for some $A_{0}$. Let $\{K_{i}{\to} F_{i-1}{\hookrightarrow} F_{i} \,|\, 1{\le} i {\le} n,\, F_{0}{=} \{\ast\} \; F_{1}{=} \Sigma{K_{1}} \; \text{and}\; F_{n}{\simeq} G \}$ be a cone-decomposition of $G$ of length $m$ and $F'_{1}=\Sigma{K'_{1}} \subset F_{1}$ with $K'_{1} \subset K_{1}$ which satisfy $F_{i}F'_{1} \subset F_{i+1}$ up to homotopy for any $i$. Our main result is as follows: we have $\operatorname{cat}(X) \le m{+}1$, if firstly the characteristic map $\alpha$ is compressible into $F'_{1}$, secondly the Berstein-Hilton Hopf invariant $H_{1}(\alpha)$ vanishes in $[A, \Omega F'_1{\ast}\Omega F'_1]$ and thirdly $K_{m}$ is a sphere. We apply this to the principal bundle $\mathrm{SO}(9)\hookrightarrow\mathrm{SO}(10)\to S^{9}$ to determine L-S category of $\mathrm{SO}(10)$.

Norio Iwase Kai Kikuchi Toshiyuki Miyauchi

A narrow near-threshold proton-emitting resonance (Ex = 11.4 MeV, J$^{\pi}$ = 1/2$^{+}$ and $\Gamma_{p}$ = 4.4 keV) was directly observed in $^{11}$B via proton resonance scattering. This resonance was previously inferred in the $\beta$-delayed proton emission of the neutron halo nucleus $^{11}$Be. The good agreement between both experimental results serves as a ground to confirm the existence of such exotic decay and the particular behavior of weakly bound nuclei coupled to the continuum. $R$-matrix analysis shows a sizable partial decay width for both, proton and $\alpha$ emission channels.

Y. Ayyad W. Mittig T. Tang B. Olaizola G. Potel N. Rijal N. Watwood H. Alvarez-Pol D. Bazin M. Caamaño J. Chen M. Cortesi B. Fernández-Domínguez S. Giraud P. Gueye S. Heinitz R. Jain B. P. Kay E. A. Maugeri B. Monteagudo F. Ndayisabye S. N. Paneru J. Pereira E. Rubino C. Santamaria D. Schumann J. Surbrook L. Wagner J. C. Zamora V. Zelevinsky

Absolute continuity implies uniform continuity, but generally not vice versa. In this short note, we present one sufficient condition for a uniformly continuous function to be absolutely continuous, which is the following theorem: For a uniformly continuous function f defined on an interval of the real line, if it is piecewise convex, then it is also absolutely continuous.

Kai Yang Chenhong Zhu

We obtain closed formulas, in terms of Littlewood-Richardson coefficients, for the canonical basis elements of the Fock space representation of $U_v(\hat{\mathfrak{sl}}_e)$ which are labelled by partitions having 'locally small' $e$-quotients and arbitrary $e$-cores. We further show that, upon evaluation at $v=1$, this gives the corresponding decomposition numbers of the $q$-Schur algebras in characteristic $l$ (where $q$ is a primitive $e$-th root of unity if $l \ne e$ and $q=1$ otherwise) whenever $l$ is greater than the size of each constituent of the $e$-quotient.

Kai Meng Tan

In 1938 E. T. Bell introduced "The Iterated Exponential Integers". He proved that these numbers may be expressed by polynomials with rational coefficients. However, Bell gave no formulas for any of the coefficients except the trivial one, which is always 1. Our task has been to find the coefficient of the leading term, giving asymptotic information about these numbers.

Ivar Henning Skau Kai Forsberg Kristensen

The neutron-rich nucleus $^{144}$Ba ($t_{1/2}$=11.5 s) is expected to exhibit some of the strongest octupole correlations among nuclei with mass numbers $A$ less than 200. Until now, indirect evidence for such strong correlations has been inferred from observations such as enhanced $E1$ transitions and interleaving positive- and negative-parity levels in the ground-state band. In this experiment, the octupole strength was measured directly by sub-barrier, multi-step Coulomb excitation of a post-accelerated 650-MeV $^{144}$Ba beam on a 1.0-mg/cm$^2$ $^{208}$Pb target. The measured value of the matrix element, $\langle 3_1^- \| \mathcal{M}(E3) \| 0_1^+ \rangle=0.65(^{+17}_{-23})$ $e$b$^{3/2}$, corresponds to a reduced $B(E3)$ transition probability of 48($^{+25}_{-34}$) W.u. This result represents an unambiguous determination of the octupole collectivity, is larger than any available theoretical prediction, and is consistent with octupole deformation.

B. Bucher S. Zhu C. Y. Wu R. V. F. Janssens D. Cline A. B. Hayes M. Albers A. D. Ayangeakaa P. A. Butler C. M. Campbell M. P. Carpenter C. J. Chiara J. A. Clark H. L. Crawford M. Cromaz H. M. David C. Dickerson E. T. Gregor J. Harker C. R. Hoffman B. P. Kay F. G. Kondev A. Korichi T. Lauritsen A. O. Macchiavelli R. C. Pardo A. Richard M. A. Riley G. Savard M. Scheck D. Seweryniak M. K. Smith R. Vondrasek A. Wiens

The highly forbidden $^2$S$_{1/2} \rightarrow ^2$F$_{7/2}$ electric octupole transition in $^{171}$Yb$^+$ is a potential candidate for a redefinition of the SI second. We present a measurement of the absolute frequency of this optical transition, performed using a frequency link to International Atomic Time to provide traceability to the SI second. The $^{171}$Yb$^+$ optical frequency standard was operated for 76% of a 25-day period, with the absolute frequency measured to be 642 121 496 772 645.14(26) Hz. The fractional uncertainty of $4.0 \times 10 ^{-16}$ is comparable to that of the best previously reported measurement, which was made by a direct comparison to local caesium primary frequency standards.

Charles F. A. Baynham Rachel M. Godun Jonathan M. Jones Steven A. King Peter B. R. Nisbet-Jones Fred Baynes Antoine Rolland Patrick E. G. Baird Kai Bongs Patrick Gill Helen S. Margolis

Mean-field theory is an approximation replacing an extended system by a few variables. For depinning of elastic manifolds, these are the position of its center of mass $u$, and the statistics of the forces $F(u)$. There are two proposals to model the latter: as a random walk (ABBM model), or as uncorrelated forces at integer $u$ (discretized particle model, DPM). While for many experiments ABBM (in the literature misleadingly equated with mean-field theory) makes quantitatively correct predictions, the microscopic disorder force-force correlations cannot grow linearly, and thus unboundedly as a random walk. Even the effective (renormalized) disorder forces which do so at small distances are bounded at large distances. We propose to model forces as an Ornstein Uhlenbeck process. The latter behaves as a random walk at small scales, and is uncorrelated at large ones. By connecting to results in both limits, we solve the model largely analytically, allowing us to describe in all regimes the distributions of velocity, avalanche size and duration. To establish experimental signatures of this transition, we study the response function, and the correlation function of position $u$, velocity $\dot u$ and forces $F$ under slow driving with velocity $v>0$. While at $v=0$ force or position correlations have a cusp at the origin, this cusp is rounded at a finite driving velocity. We give a detailed analytic analysis for this rounding by velocity, which allows us, given experimental data, to extract the time-scale of the response function, and to reconstruct the force-force correlator at $v=0$. The latter is the central object of the field theory, and as such contains detailed information about the universality class in question. We test our predictions by careful numerical simulations extending over up to ten orders in magnitude.

Cathelijne ter Burg Kay Joerg Wiese

The entropic uncertainty relation proven by Maassen and Uffink for arbitrary pairs of two observables is known to be non-optimal. Here, we call an uncertainty relation optimal, if the lower bound can be attained for any value of either of the corresponding uncertainties. In this work we establish optimal uncertainty relations by characterising the optimal lower bound in scenarios similar to the Maassen-Uffink type. We disprove a conjecture by Englert et al. and generalise various previous results. However, we are still far from a complete understanding and, based on numerical investigation and analytical results in small dimension, we present a number of conjectures.

Kais Abdelkhalek René Schwonnek Hans Maassen Fabian Furrer Jörg Duhme Philippe Raynal Berthold-Georg Englert Reinhard F. Werner