Integral Mixed Cayley Graphs over Abelian Groups

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $\Gamma$ be an abelian group. The mixed Cayley graph $Cay(\Gamma,S)$ is a mixed graph on the vertex set $\Gamma$ and edge set $\left\{ (a,b): b-a\in S \right\}$, where $0\not\in S$. We characterize integral mixed Cayley graph $Cay(\Gamma,S)$ over an abelian group $\Gamma$ in terms of its connection set $S$.

First measurement of the $$C\!P$$-violating phase in $${{B} ^0_{s}} \!\rightarrow {{J /\psi }} ($$ $$\rightarrow $$ $$e ^+e ^-$$)$$\phi $$ decays

A flavour-tagged time-dependent angular analysis of $${{B} ^0_{s}} \!\rightarrow {{J /\psi }} \phi $$ B s 0 → J / ψ ϕ decays is presented where the $${J /\psi }$$ J / ψ meson is reconstructed through its decay to an $$e ^+e ^-$$ e + e - pair. The analysis uses a sample of pp collision data recorded with the LHCb experiment at centre-of-mass energies of 7 and $$8\text {\,Te V} $$ 8 \,Te V , corresponding to an integrated luminosity of $$3 \text {\,fb} ^{-1} $$ 3 \,fb - 1 . The $$C\!P$$ C P -violating phase and lifetime parameters of the $${B} ^0_{s} $$ B s 0 system are measured to be $${\phi _{{s}}} =0.00\pm 0.28\pm 0.07\text {\,rad}$$ ϕ s = 0.00 ± 0.28 ± 0.07 \,rad , $${\Delta \Gamma _{{s}}} =0.115\pm 0.045\pm 0.011\text {\,ps} ^{-1} $$ Δ Γ s = 0.115 ± 0.045 ± 0.011 \,ps - 1 and $${\Gamma _{{s}}} =0.608\pm 0.018\pm 0.012\text {\,ps} ^{-1} $$ Γ s = 0.608 ± 0.018 ± 0.012 \,ps - 1 where the first uncertainty is statistical and the second systematic. This is the first time that $$C\!P$$ C P -violating parameters are measured in the $${{B} ^0_{s}} \!\rightarrow {{J /\psi }} \phi $$ B s 0 → J / ψ ϕ decay with an $$e ^+e ^-$$ e + e - pair in the final state. The results are consistent with previous measurements in other channels and with the Standard Model predictions.

Measurement of the CP-violating phase $$\phi _s$$ in $$ B_{s}^{0} \rightarrow J/\psi \phi $$ decays in ATLAS at 13 TeV

A measurement of the $$ B_{s}^{0} \rightarrow J/\psi \phi $$ B s 0 → J / ψ ϕ decay parameters using $$ 80.5\, \mathrm {fb^{-1}} $$ 80.5 fb - 1 of integrated luminosity collected with the ATLAS detector from 13 $$\text {Te}\text {V}$$ Te proton–proton collisions at the LHC is presented. The measured parameters include the CP-violating phase $$\phi _{s} $$ ϕ s , the width difference $$ \Delta \Gamma _{s}$$ Δ Γ s between the $$B_{s}^{0}$$ B s 0 meson mass eigenstates and the average decay width $$ \Gamma _{s}$$ Γ s . The values measured for the physical parameters are combined with those from $$ 19.2\, \mathrm {fb^{-1}} $$ 19.2 fb - 1 of 7 and 8 $$\text {Te}\text {V}$$ Te data, leading to the following: $$\begin{aligned} \phi _{s}= & {} -0.087 \pm 0.036 ~\mathrm {(stat.)} \pm 0.021 ~\mathrm {(syst.)~rad} \\ \Delta \Gamma _{s}= & {} 0.0657 \pm 0.0043 ~\mathrm {(stat.)}\pm 0.0037 ~\mathrm {(syst.)~ps}^{-1} \\ \Gamma _{s}= & {} 0.6703 \pm 0.0014 ~\mathrm {(stat.)}\pm 0.0018 ~\mathrm {(syst.)~ps}^{-1} \end{aligned}$$ ϕ s = - 0.087 ± 0.036 ( stat . ) ± 0.021 ( syst . ) rad Δ Γ s = 0.0657 ± 0.0043 ( stat . ) ± 0.0037 ( syst . ) ps - 1 Γ s = 0.6703 ± 0.0014 ( stat . ) ± 0.0018 ( syst . ) ps - 1 Results for $$\phi _{s} $$ ϕ s and $$ \Delta \Gamma _{s}$$ Δ Γ s are also presented as 68% confidence level contours in the $$\phi _{s} $$ ϕ s –$$ \Delta \Gamma _{s}$$ Δ Γ s plane. Furthermore the transversity amplitudes and corresponding strong phases are measured. $$\phi _{s} $$ ϕ s and $$ \Delta \Gamma _{s}$$ Δ Γ s measurements are in agreement with the Standard Model predictions.

Local geometric properties of the lightlike Killing magnetic curves in de Sitter 3-space

<abstract><p>In this article, we mainly discuss the local differential geometrical properties of the lightlike Killing magnetic curve $ \mathit{\boldsymbol{\gamma }}(s) $ in $ \mathbb{S}^{3}_{1} $ with a magnetic field $ \boldsymbol{ V} $. Here, a new Frenet frame $ \{\mathit{\boldsymbol{\gamma }}, \boldsymbol{ T}, \boldsymbol{ N}, \boldsymbol{ B}\} $ is established, and we obtain the local structure of $ \mathit{\boldsymbol{\gamma }}(s) $. Moreover, the singular properties of the binormal lightlike surface of the $ \mathit{\boldsymbol{\gamma }}(s) $ are given. Finally, an example is used to understand the main results of the paper.</p></abstract>

New approach to determine the surface and interface thermodynamic properties of H-β-zeolite/rhodium catalysts by inverse gas chromatography at infinite dilution

The thermodynamic surface properties and Lewis acid–base constants of H-β-zeolite supported rhodium catalysts were determined by using the inverse gas chromatography technique at infinite dilution. The effect of the temperature and the rhodium percentage supported by zeolite on the acid base properties in Lewis terms of the various catalysts were studied. The dispersive component of the surface energy of Rh/H-β-zeolite was calculated by using both the Dorris and Gray method and the straight-line method. We highlighted the role of the surface areas of n-alkanes on the determination of the surface energy of catalysts. To this aim various molecular models of n-alkanes were tested, namely Kiselev, cylindrical, Van der Waals, Redlich–Kwong, geometric and spherical models. An important deviation in the values of the dispersive component of the surface energy $${\gamma }_{s}^{d}$$ γ s d determined by the classical and new methods was emphasized. A non-linear dependency of $${\gamma }_{s}^{d}$$ γ s d with the specific surface area of catalysts was highlighted showing a local maximum at 1%Rh. The study of RTlnVn and the specific free energy ∆Gsp(T) of n-alkanes and polar solvents adsorbed on the various catalysts revealed the important change in the acid properties of catalysts with both the temperature and the rhodium percentage. The results proved strong amphoteric behavior of all catalysts of the rhodium supported by H-β-zeolite that actively react with the amphoteric solvents (methanol, acetone, tri-CE and tetra-CE), acid (chloroform) and base (ether) molecules. It was shown that the Guttmann method generally used to determine the acid base constants KA and KD revealed some irregularities with a linear regression coefficient not very satisfactory. The accurate determination of the acid–base constants KA, KD and K of the various catalysts was obtained by applying Hamieh’s model (linear regression coefficients approaching r2 ≈ 1.000). It was proved that all acid base constants determined by this model strongly depends on the rhodium percentage and the specific surface area of the catalysts.

Статистические свойства трехмерных полиэдров Клейна

Пусть $\Gamma$ - $s$-мерная решетка из $\mathbb R^s$. Выпуклые оболочки ненулевых узлов из $\Gamma$, содержащихся в каждом ортанте, называются полиэдрами Клейна решетки $\Gamma$. Эта конструкция была введена Ф. Клейном (1895 г.) в связи с обобщением классического алгоритма непрерывных дробей на многомерный случай. В. И. Арнольд сформулировал ряд задач о статистических и геометрических свойствах полиэдров Клейна. В двумерном случае соответствующие результаты вытекают из теории непрерывных дробей. В работе выводится асимптотическая формула для среднего значения $f$-вектора (количество граней, ребер и вершин) трехмерных полиэдров Клейна. Усреднение проводится по полиэдрам Клейна трехмерных целочисленных решеток с определителем из отрезка $[1,R]$, где $R$ - растущий параметр. Библиография: 27 названий.

New criteria for oscillation of nonlinear neutral differential equations

The aim of this work is to offer sufficient conditions for the oscillation of neutral differential equation second order $$ \bigl( r ( t ) \bigl[ \bigl( y ( t ) +p ( t ) y \bigl( \tau ( t ) \bigr) \bigr) ^{\prime } \bigr] ^{\gamma } \bigr) ^{\prime }+f \bigl( t,y \bigl( \sigma ( t ) \bigr) \bigr) =0, $$(r(t)[(y(t)+p(t)y(τ(t)))′]γ)′+f(t,y(σ(t)))=0, where $\int ^{\infty }r^{-1/\gamma } ( s ) \,\mathrm{d}s= \infty $∫∞r−1/γ(s)ds=∞. Based on the comparison with first order delay equations and by employ the Riccati substitution technique, we improve and complement a number of well-known results. Some examples are provided to show the importance of these results.

On certain sums over ordinates of zeta-zeros II

International audience Let γ denote the imaginary parts of complex zeros ρ = β + iγ of ζ(s). The problem of analytic continuation of the function $G(s) :=\sum_{\gamma >0} {\gamma}^{-s}$ to the left of the line $\Re{s} = −1 $ is investigated, and its Laurent expansion at the pole s = 1 is obtained. Estimates for the second moment on the critical line $\int_{1}^{T} {| G (\frac{1}{2} + it) |}^2 dt $ are revisited. This paper is a continuation of work begun by the second author in [Iv01].

Hadronic contribution of light by light scattering in the energy spectrum of muonic hydrogen

The contribution from scalar meson (S) exchange to the potential of the muon-proton interaction in muonic hydrogen induced by the scalar meson coupling to the two photon state is calculated. An estimate of transition form factor S → γγ is given on the basis of the quark model and experimental data on the decay widths $${\Gamma _{S\gamma \gamma }}$$. It is shown that scalar meson exchange contribution to the Lamb shift in muonic hydrogen ∆ELs(2P – 2S) is large and important for a comparison with precise experimental data.