Binomial coefficients, Catalan numbers and Lucas quotients

Let $p$ be an odd prime and let $a,m$ be integers with $a>0$ and $m \not\equiv0\pmod p$. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ mod $p^2$ for $d=0,1$; for example, $$\sum_{k=0}^{p^a-1}\frac{\binom{2k}k}{m^k}\equiv\left(\frac{m^2-4m}{p^a}\right)+\left(\frac{m^2-4m}{p^{a-1}}\right)u_{p-(\frac{m^2-4m}{p})}\pmod{p^2},$$ where $(-)$ is the Jacobi symbol, and $\{u_n\}_{n\geqslant0}$ is the Lucas sequence given by $u_0=0$, $u_1=1$ and $u_{n+1}=(m-2)u_n-u_{n-1}$ for $n=1,2,3,\ldots$. As an application, we determine $\sum_{0<k<p^a,\, k\equiv r\pmod{p-1}}C_k$ modulo $p^2$ for any integer $r$, where $C_k$ denotes the Catalan number $\binom{2k}k/(k+1)$. We also pose some related conjectures.Comment: 24 pages. Correct few typo

Super congruences and Euler numbers

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $\pi^2$, $\pi^{-2}$ and the constant $K:=\sum_{k>0}(k/3)/k^2$ (with (-) the Jacobi symbol), two of which are $$\sum_{k=1}^\infty(10k-3)8^k/(k^3\binom{2k}{k}^2\binom{3k}{k})=\pi^2/2$$ and $$\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$

Decoherence-Based Quantum Zeno Effect in a Cavity-QED System

We present a decoherence-based interpretation for the quantum Zeno effect (QZE) where measurements are dynamically treated as dispersive couplings of the measured system to the apparatus, rather than the von Neumann's projections. It is found that the explicit dependence of the survival probability on the decoherence time quantitatively distinguishes this dynamic QZE from the usual one based on projection measurements. By revisiting the cavity-QED experiment of the QZE [J. Bernu, et al., Phys. Rev. Lett, 101, 180402 (2008)], we suggest an alternative scheme to verify our theoretical consideration that frequent measurements slow down the increase of photon number inside a microcavity due to the nondemolition couplings with the atoms in large detuning.Comment: 4 pages, 3 figure

Hydrostatic pressure effects on the static magnetism in Eu(Fe0.925_{0.925}Co0.075_{0.075})2_{2}As2_{2}

The effects of hydrostatic pressure on the static magnetism in Eu(Fe$_{0.925}$Co$_{0.075}$)$_{2}$As$_{2}$ are investigated by complementary electrical resistivity, ac magnetic susceptibility and single-crystal neutron diffraction measurements. A specific pressure-temperature phase diagram of Eu(Fe$_{0.925}$Co$_{0.075}$)$_{2}$As$_{2}$ is established. The structural phase transition, as well as the spin-density-wave order of Fe sublattice, is suppressed gradually with increasing pressure and disappears completely above 2.0 GPa. In contrast, the magnetic order of Eu sublattice persists over the whole investigated pressure range up to 14 GPa, yet displaying a non-monotonic variation with pressure. With the increase of the hydrostatic pressure, the magnetic state of Eu evolves from the canted antiferromagnetic structure in the ground state, via a pure ferromagnetic structure under the intermediate pressure, finally to a possible "novel" antiferromagnetic structure under the high pressure. The strong ferromagnetism of Eu coexists with the pressure-induced superconductivity around 2 GPa. The change of the magnetic state of Eu in Eu(Fe$_{0.925}$Co$_{0.075}$)$_{2}$As$_{2}$ upon the application of hydrostatic pressure probably arises from the modification of the indirect Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the Eu$^{2+}$ moments tuned by external pressure.Comment: 9 pages, 6 figure

Time-Resolved Magnetic Relaxation of a Nanomagnet on Subnanosecond Time Scales

We present a two-current-pulse temporal correlation experiment to study the intrinsic subnanosecond nonequilibrium magnetic dynamics of a nanomagnet during and following a pulse excitation. This method is applied to a model spin-transfer system, a spin valve nanopillar with perpendicular magnetic anisotropy. Two-pulses separated by a short delay (< 500 ps) are shown to lead to the same switching probability as a single pulse with a duration that depends on the delay. This demonstrates a remarkable symmetry between magnetic excitation and relaxation and provides a direct measurement of the magnetic relaxation time. The results are consistent with a simple finite temperature Fokker-Planck macrospin model of the dynamics, suggesting more coherent magnetization dynamics in this short time nonequilibrium limit than near equilibrium

Study on the effect of metallurgical waste on the water resistance of magnesium oxysulfate cement (MOS) coatings

In this paper, in order to achieve the resource utilization of metallurgical waste, this article studied the water resistance of magnesium oxysulfate cement coating using blast furnace slag powder and iron tailings powder as fillers, and characterized its hydration products using X-ray diffraction (XRD) and scanning electron microscopy (SEM). The results indicate that an appropriate amount of slag powder and iron tailings powder can make the internal structure of the coating more dense, effectively improve the softening coefficient, and enhance water resistance

Study on the effect of metallurgical waste on the cracking resistance of magnesium oxysulfate cement coatings

In this paper, in order to achieve the resource utilization of metallurgical industry waste, the cracking resistance of magnesium oxysulfate cement coatings using granulated blast furnace slag powder and iron tailings powder as fillers was studied, x-ray Diffraction (XRD) and scanning electron microscope (SEM) were used to characterize the hydration products. The results show that an appropriate amount of slag powder and iron tailings powder can make the internal structure of the coating more compact, the surface smooth and effectively reduce the generation of cracks

Determination of the magnetic anisotropy axes of single-molecule magnets

Simple methods are presented allowing the determination of the magnetic anisotropy axes of a crystal of a single-molecule magnet (SMM). These methods are used to determine an upper bound of the easy axis tilts in a standard Mn12-Ac crystal. The values obtained in the present study are significately smaller than those reported in recent high frequency electron paramagnetic resonance (HF-EPR) studies which suggest distributions of hard-axes tilts.Comment: 10 pages, 6 figure