Let $E_1$ and $E_2$ be $overline{mathbb{Q}}$-nonisogenous, semistable elliptic curves over $mathbb{Q}$, having respective conductors $N_{E_1}$ and $N_{E_2}$ and both without complex multiplication. For each prime $p$, denote by $a_{E_i}(p) := p+1-#E_
i(mathbb{F}_p)$ the trace of Frobenius. Under the assumption of the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power $L$-functions $L(s, mathrm{Sym}^i E_1otimesmathrm{Sym}^j E_2)$ where $i,jin{0,1,2}$, we prove an explicit result that can be stated succinctly as follows: there exists a prime $p mid N_{E_1}N_{E_2}$ such that $a_{E_1}(p)a_{E_2}(p)<0$ and [ p < big( (32+o(1))cdot log N_{E_1} N_{E_2}big)^2. ] This improves and makes explicit a result of Bucur and Kedlaya. Now, if $Isubset[-1,1]$ is a subinterval with Sato-Tate measure $mu$ and if the symmetric power $L$-functions $L(s, mathrm{Sym}^k E_1)$ are functorial and satisfy GRH for all $k le 8/mu$, we employ similar techniques to prove an explicit result that can be stated succinctly as follows: there exists a prime $p mid N_{E_1}$ such that $a_{E_1}(p)/(2sqrt{p})in I$ and [ p < left((21+o(1)) cdot mu^{-2}log (N_{E_1}/mu)right)^2. ]
Let $mathcal{I} subset mathbb{N}$ be an infinite subset, and let ${a_i}_{i in mathcal{I}}$ be a sequence of nonzero real numbers indexed by $mathcal{I}$ such that there exist positive constants $m, C_1$ for which $|a_i| leq C_1 cdot i^m$ for all $i i
n mathcal{I}$. Furthermore, let $c_i in [-1,1]$ be defined by $c_i = frac{a_i}{C_1 cdot i^m}$ for each $i in mathcal{I}$, and suppose the $c_i$s are equidistributed in $[-1,1]$ with respect to a continuous, symmetric probability measure $mu$. In this paper, we show that if $mathcal{I} subset mathbb{N}$ is not too sparse, then the sequence ${a_i}_{i in mathcal{I}}$ fails to obey Benfords Law with respect to arithmetic density in any sufficiently large base, and in fact in any base when $mu([0,t])$ is a strictly convex function of $t in (0,1)$. Nonetheless, we also provide conditions on the density of $mathcal{I} subset mathbb{N}$ under which the sequence ${a_i}_{i in mathcal{I}}$ satisfies Benfords Law with respect to logarithmic density in every base. As an application, we apply our general result to study Benfords Law-type behavior in the leading digits of Frobenius traces of newforms of positive, even weight. Our methods of proof build on the work of Jameson, Thorner, and Ye, who studied the particular case of newforms without complex multiplication.
The first measurements of the coherence factor R_{K_S^0Kpi} and the average strong--phase difference delta^{K_S^0Kpi} in D^0 to K_S^0 K^mppi^pm decays are reported. These parameters can be used to improve the determination of the unitary triangle ang
le gamma in B^- rightarrow $widetilde{D}K^-$ decays, where $widetilde{D}$ is either a D^0 or a D^0-bar meson decaying to the same final state, and also in studies of charm mixing. The measurements of the coherence factor and strong-phase difference are made using quantum-correlated, fully-reconstructed D^0D^0-bar pairs produced in e^+e^- collisions at the psi(3770) resonance. The measured values are R_{K_S^0Kpi} = 0.70 pm 0.08 and delta^{K_S^0Kpi} = (0.1 pm 15.7)$^circ$ for an unrestricted kinematic region and R_{K*K} = 0.94 pm 0.12 and delta^{K*K} = (-16.6 pm 18.4)$^circ$ for a region where the combined K_S^0 pi^pm invariant mass is within 100 MeV/c^2 of the K^{*}(892)^pm mass. These results indicate a significant level of coherence in the decay. In addition, isobar models are presented for the two decays, which show the dominance of the K^*(892)^pm resonance. The branching ratio {B}(D^0 rightarrow K_S^0K^+pi^-)/{B}(D^0 rightarrow K_S^0K^-pi^+) is determined to be 0.592 pm 0.044 (stat.) pm 0.018 (syst.), which is more precise than previous measurements.
We report a comprehensive study of ultrafast carrier dynamics in single crystals of multiferroic BiFeO$_{3}$. Using femtosecond optical pump-probe spectroscopy, we find that the photoexcited electrons relax to the conduction band minimum through elec
tron-phonon coupling with a $sim$1 picosecond time constant that does not significantly change across the antiferromagnetic transition. Photoexcited electrons subsequently leave the conduction band and primarily decay via radiative recombination, which is supported by photoluminescence measurements. We find that despite the coexisting ferroelectric and antiferromagnetic orders in BiFeO$_{3}$, the intrinsic nature of this charge-transfer insulator results in carrier relaxation similar to that observed in bulk semiconductors.
We report the synthesis of a new oxychalcogenide HgOCuSe sample. The resistivity decreases as a function of $T^{1.75}$ with decreasing temperature from room temperature down to around 80 K. There exists a very sharp ferromagnetic-like phase transitio
n at around 60 K under a field of $H$ = 100 Oe. Contrary to the usual ferromagnetic materials, the descending and ascending branches of the magnetic hysteresis curve, at 30 K, are reversed in the whole irreversible field range and the reverse irreversibility decreases at 5 K.
Recently, two consecutive phase transitions were observed, upon cooling, in an antiferromagnetic spinel GeNi$_2$O$_4$ at $T_{N1}=12.1$ K and $T_{N2}=11.4$ K, respectively cite{matsuno, crawford}. Using unpolarized and polarized elastic neutron scatte
ring we show that the two transitions are due to the existence of frustrated minority spins in this compound. Upon cooling, at $T_{N1}$ the spins on the $<111>$ kagome planes order ferromagnetically in the plane and antiferromagnetically between the planes (phase I), leaving the spins on the $<111>$ triangular planes that separate the kagome planes frustrated and disordered. At the lower $T_{N2}$, the triangular spins also order in the $<111>$ plane (phase II). We also present a scenario involving exchange interactions that qualitatively explains the origin of the two purely magnetic phase transitions.
