Antineutrino science in KamLAND

The primary goal of KamLAND is a search for the oscillation of \({\bar{\nu }}_\mathrm{e}\) ’s emitted from distant power reactors. The long baseline, typically 180 km, enables KamLAND to address the oscillation solution of the “solar neutrino problem” with \({\bar{\nu }}_{e} \) ’s under laboratory conditions. KamLAND found fewer reactor \({\bar{\nu }}_{e} \) events than expected from standard assumptions about \(\overline{\nu }_e\) propagation at more than 9 \(\sigma \) confidence level (C.L.). The observed energy spectrum disagrees with the expected spectral shape at more than 5 \(\sigma \) C.L., and prefers the distortion from neutrino oscillation effects. A three-flavor oscillation analysis of the data from KamLAND and KamLAND + solar neutrino experiments with CPT invariance, yields \(\Delta m_{21}^2 \) = [ \(7.54_{-0.18}^{+0.19} \times \) 10 \(^{-5}\) eV \(^{2}\) , \(7.53_{-0.18}^{+0.19} \times \) 10 \(^{-5}\) eV \(^{2}\) ], tan \(^{2}\theta _{12}\) = [ \(0.481_{-0.080}^{+0.092} \) , \(0.437_{-0.026}^{+0.029} \) ], and sin \(^{2}\theta _{13}\) = [ \(0.010_{-0.034}^{+0.033} \) , \(0.023_{-0.015}^{+0.015} \) ]. All solutions to the solar neutrino problem except for the large mixing angle region are excluded. KamLAND also demonstrated almost two cycles of the periodic feature expected from neutrino oscillation effects. KamLAND performed the first experimental study of antineutrinos from the Earth’s interior so-called geoneutrinos (geo \({\bar{\nu }}_{e} \) ’s), and succeeded in detecting geo \({\bar{\nu }}_{e} \) ’s produced by the decays of \(^{238}\) U and \(^{232}\) Th within the Earth. Assuming a chondritic Th/U mass ratio, we obtain \(116_{-27}^{+28} {\bar{\nu }}_{e}\) events from \(^{238}\) U and \(^{232}\) Th, corresponding a geo \({\bar{\nu }}_{e}\) flux of \(3.4_{-0.8}^{+0.8}\times \) 10 \(^{6}\) cm \(^{-2}\)  s \(^{-1}\) at the KamLAND location. We evaluate various bulk silicate Earth composition models using the observed geo \({\bar{\nu }}_{e} \) rate.     

Comprehensive Bayesian analysis of rare (semi)leptonic and radiative \varvec{B} decays

The available data on \(|\Delta B| = |\Delta S| = 1\) decays are in good agreement with the Standard Model when permitting subleading power corrections of about \(15\,\%\) at large hadronic recoil. Constraining new-physics effects in \(\mathcal {C}_{7}^{\mathrm {}}\) , \(\mathcal {C}_{9}^{\mathrm {}}\) , \(\mathcal {C}_{10}^{\mathrm {}}\) , the data still demand the same size of power corrections as in the Standard Model. In the presence of chirality-flipped operators, all but one of the power corrections reduce substantially. The Bayes factors are in favor of the Standard Model. Using new lattice inputs for \(B\rightarrow K^*\) form factors and under our minimal prior assumption for the power corrections, the favor shifts toward models with chirality-flipped operators. We use the data to further constrain the hadronic form factors in \(B\rightarrow K\) and \(B\rightarrow K^*\) transitions.     

Split supersymmetry under GUT and current dark matter constraints

There are four types of two-Higgs doublet models under a discrete \(Z_2\) symmetry imposed to avoid tree-level flavor-changing neutral current, i.e. type-I, type-II, type-X, and type-Y models. We investigate the possibility to discriminate the four models in the light of the flavor physics data, including \(B_s\) – \(\bar{B}_s\) mixing, \(B_{s,d} \rightarrow \mu ^+ \mu ^-\) , \(B\rightarrow \tau \nu \) and \(\bar{B} \rightarrow X_s \gamma \) decays, the recent LHC Higgs data, the direct search for charged Higgs at LEP, and the constraints from perturbative unitarity and vacuum stability. After deriving the combined constraints on the Yukawa interaction parameters, we have shown that the correlation between the mass eigenstate rate asymmetry \(A_{\Delta \Gamma }\) of \(B_{s} \rightarrow \mu ^+ \mu ^-\) and the ratio \(R=\mathcal{B}(B_{s} \rightarrow \mu ^+ \mu ^-)_\mathrm{exp}/ \mathcal{B}(B_{s} \rightarrow \mu ^+ \mu ^-)_\mathrm{SM}\) could be a sensitive probe to discriminate the four models with future precise measurements of the observables in the \(B_{s} \rightarrow \mu ^+ \mu ^-\) decay at LHCb.     

Optimal N-Point Configurations on the Sphere: “Magic” Numbers and Smale’s 7th Problem

This paper inquires into the concavity of the map \(N\mapsto v_s(N)\) from the integers \(N\ge 2\) into the minimal average standardized Riesz pair-energies \(v_s(N)\) of \(N\) -point configurations on the sphere \(\mathbb {S}^2\) for various \(s\in \mathbb {R}\) . The standardized Riesz pair-energy of a pair of points on \(\mathbb {S}^2\) a chordal distance \(r\) apart is \(V_s(r)= s^{-1}\left( r^{-s}-1 \right) \) , \(s \ne 0\) , which becomes \(V_0(r) = \ln \frac{1}{r}\) in the limit \(s\rightarrow 0\) . Averaging it over the \(\left( \begin{array}{c} N\\ 2\end{array}\right) \) distinct pairs in a configuration and minimizing over all possible \(N\) -point configurations defines \(v_s(N)\) . It is known that \(N\mapsto v_s(N)\) is strictly increasing for each \(s\in \mathbb {R}\) , and for \(s<2\) also bounded above, thus “overall concave.” It is (easily) proved that \(N\mapsto v_{-2}^{}(N)\) is even locally strictly concave, and that so is the map \(2n\mapsto v_s(2n)\) for \(s<-2\) . By analyzing computer-experimental data of putatively minimal average Riesz pair-energies \(v_s^x(N)\) for \(s\in \{-1,0,1,2,3\}\) and \(N\in \{2,\ldots ,200\}\) , it is found that the map \(N\mapsto {v}_{-1}^x(N)\) is locally strictly concave, while \(N\mapsto {v}_s^x(N)\) is not always locally strictly concave for \(s\in \{0,1,2,3\}\) : concavity defects occur whenever \(N\in {\mathcal {C}}^{x}_+(s)\) (an \(s\) -specific empirical set of integers). It is found that the empirical map \(s\mapsto {\mathcal {C}}^{x}_+(s),\ s\in \{-2,-1,0,1,2,3\}\) , is set-theoretically increasing; moreover, the percentage of odd numbers in \({\mathcal {C}}^{x}_+(s),\ s\in \{0,1,2,3\}\) is found to increase with \(s\) . The integers in \({\mathcal {C}}^{x}_+(0)\) are few and far between, forming a curious sequence of numbers, reminiscent of the “magic numbers” in nuclear physics. It is conjectured that these new “magic numbers” are associated with optimally symmetric optimal-log-energy \(N\) -point configurations on \(\mathbb {S}^2\) . A list of interesting open problems is extracted from the empirical findings, and some rigorous first steps toward their solutions are presented. It is emphasized how concavity can assist in the solution to Smale’s \(7\) th Problem, which asks for an efficient algorithm to find near-optimal \(N\) -point configurations on \(\mathbb {S}^2\) and higher-dimensional spheres.     

Investigation of conductivity of adhesive layer including indium tin oxide particles for multi-junction solar cells

We report connection conductivity ( \(C_{\rm c}\) ) of adhesive which including \(\hbox {In}_2\hbox {O}_3\) – \(\hbox {SnO}_2\) (ITO) particles developed for fabrication of stacked-type-multi-junction solar cells. The commercial 20- \(\upmu \) m sized ITO particles were heated in vacuum at temperature ranging from 800 to 1,300  \(^{\circ }{\rm C}\) for 10 min to increase \(C_{\rm c}\) . 6.2 wt% ITO particles were dispersed in commercial Cemedine adhesive gel to form 100 samples structured with n-type Si/adhesive/n-type Si (n-Si sample) and p-type Si/adhesive/p-type Si (p-Si sample). Current density as a function of voltage (J–V) characteristics gave \(C_{\rm c}\) . It ranged from 4.3 to 1.0 S/cm \(^2\) for the n-Si sample with 800 \(^{\circ }{\rm C}\) heat-treated ITO particles. Its standard deviation was 0.59 S/cm \(^2\) . On the other hand, it ranged from 2.0 to 0.6 S/cm \(^2\) for the p-Si sample with 800  \(^{\circ }{\rm C}\) heat-treated ITO particles. Its standard deviation was 0.22 S/cm \(^2\) . The distribution of \(C_{\rm c}\) mainly resulted from contact efficiency of ITO particles to substrate. We theoretically estimated that present \(C_{\rm c}\) achieved a low loss of the power conversion efficiency ( \(E_{\rm ff}\) ) lower than 0.3 % in the application of fabrication of multi-junction solar cell with an intrinsic \(E_{\rm ff}\) of 30 % and an open circuit voltage above 1.9 V.     

Femtosecond relaxation kinetics of highly excited \mathrm{M}_{\mathrm{Na}}^{**} states in CaF2 at 3.2 eV and 4.7 eV

Femtosecond (fs) laser pulses at variable delay times allowed us to track the fast non-radiative transitions between the manifold of highly excited $\mathrm{M}_{\mathrm{Na}}^{**}$ states to the lower lying fluorescent $\mathrm{M}_{\mathrm{Na}}^{*}$ state in CaF₂. Two distinct $\mathrm{M}_{\mathrm{Na}}^{**}$ states of the manifold at 3.16?eV ( $\mathrm{M}_{\mathrm{Na}2}^{**}$ ) and 4.73?eV ( $\mathrm{M}_{\mathrm{Na}3}^{**}$ ) were populated using the second (SH) and third harmonics (TH) of fs laser light at 785?nm. The population kinetics of the fluorescent $\mathrm{M}_{\mathrm{Na}}^{*}$ state in the 2?eV excitation energy range was revealed by depleting its fluorescence centered at 740?nm using fundamental near infrared (NIR) fs laser pulses. The related time constants for $\mathrm{M}_{\mathrm{Na}2,3}^{**}{\sim}{>} \mathrm{M}_{\mathrm{Na}}^{*}$ relaxation amounted to 1.0±0.14?ps and 3.0±0.3?ps upon SH and TH excitation, respectively.     

Is the CNMSSM more credible than the CMSSM?

“Post-sphaleron baryogenesis”, a fresh and profound mechanism of baryogenesis accounts for the matter–antimatter asymmetry of our present universe in a framework of Pati–Salam symmetry. We attempt here to embed this mechanism in a non-SUSY SO(10) grand unified theory by reviving a novel symmetry breaking chain with Pati–Salam symmetry as an intermediate symmetry breaking step and as well to address post-sphaleron baryogenesis and neutron–antineutron oscillation in a rational manner. The Pati–Salam symmetry based on the gauge group \(\mathrm{SU}(2)_L \times \mathrm{SU}(2)_{R} \times \mathrm{SU}(4)_C\) is realized in our model at \(10^{5}\) – \(10^{6}\)  GeV and the mixing time for the neutron–antineutron oscillation process having \(\Delta B=2\) is found to be \(\tau _{n-\bar{n}} \simeq 10^{8}\) – \(10^{10}\)  s with the model parameters, which is within the reach of forthcoming experiments. Other novel features of the model include low scale right-handed \(W^{\pm }_R\) , \(Z_R\) gauge bosons, explanation for neutrino oscillation data via the gauged inverse (or extended) seesaw mechanism and most importantly TeV scale color sextet scalar particles responsible for an observable \(n\) – \(\bar{n}\) oscillation which may be accessible to LHC. We also look after gauge coupling unification and an estimation of the proton lifetime with and without the addition of color sextet scalars.     

Reunion Probabilities of N One-Dimensional Random Walkers with Mixed Boundary Conditions

In this work we extend the results of the reunion probability of \(N\) one-dimensional random walkers to include mixed boundary conditions between their trajectories. The level of the mixture is controlled by a parameter \(c\) , which can be varied from \(c=0\) (independent walkers) to \(c\rightarrow \infty \) (vicious walkers). The expressions are derived by using Quantum Mechanics formalism (QMf) which allows us to map this problem into a Lieb-Liniger gas (LLg) of \(N\) one-dimensional particles. We use Bethe ansatz and Gaudin’s conjecture to obtain the normalized wave-functions and use this information to construct the propagator. As it is well-known, depending on the boundary conditions imposed at the endpoints of a line segment, the statistics of the maximum heights of the reunited trajectories have some connections with different ensembles in Random Matrix Theory. Here we seek to extend those results and consider four models: absorbing, periodic, reflecting, and mixed. In all four cases, the probability that the maximum height is less or equal than \(L\) takes the form \(F_N(L)=A_N\sum _{\varvec{k}\in \Omega _{\text {B}}} \mathrm{e}^{-\sum _{j=1}^Nk_j^2}\mathcal {V}_N(\varvec{k})\) , where \(A_N\) is a normalization constant, \(\mathcal {V}_N(\varvec{k})\) contains a deformed and weighted Vandermonde determinant, and \(\Omega _{\text {B}}\) is the solution set of quasi-momenta \(\varvec{k}\) obeying the Bethe equations for that particular boundary condition.     

Maximal Distance Travelled by N Vicious Walkers Till Their Survival

We consider N Brownian particles moving on a line starting from initial positions \(\mathbf{{u}}\equiv \{u_1,u_2,\ldots u_N\}\) such that \(0 . Their motion gets stopped at time \(t_s\) when either two of them collide or when the particle closest to the origin hits the origin for the first time. For \(N=2\) , we study the probability distribution function \(p_1(m|\mathbf{{u}})\) and \(p_2(m|\mathbf{{u}})\) of the maximal distance travelled by the \(1^{\text {st}}\) and \(2^{\text {nd}}\) walker till \(t_s\) . For general N particles with identical diffusion constants \(D\) , we show that the probability distribution \(p_N(m|\mathbf{u})\) of the global maximum \(m_N\) , has a power law tail \(p_i(m|\mathbf{{u}}) \sim {N^2B_N\mathcal {F}_{N}(\mathbf{u})}/{m^{\nu _N}}\) with exponent \(\nu _N =N^2+1\) . We obtain explicit expressions of the function \(\mathcal {F}_{N}(\mathbf{u})\) and of the N dependent amplitude \(B_N\) which we also analyze for large N using techniques from random matrix theory. We verify our analytical results through direct numerical simulations.     

Dispersion dependence of two-photon absorption transition on frequency in Si PIN photodetector

The variation of two-photon absorption (TPA) coefficient \(\beta _{\mathrm{TPA}} (\omega )\) of Si excited at difference photon energy was investigated. The TPA coefficient was measured by using a picosecond pulsed laser with the wavelength could be tuned in a wide photon-energy range. An equivalent RC circuit model was adapted to derive the TPA coefficient \(\beta _{\mathrm{TPA}} (\omega )\) . The results showed that \(\beta _{\mathrm{TPA}} (\omega )\) varied from \(4.2 \times 10^{-4}\) to \(1.17 \times 10^{-3 }\)  cm/GW in the transparent wavelength region \(1.80<\lambda <1.36\,\upmu \) m of Si. The increasing tendency of \(\beta _{\mathrm{TPA}} (\omega )\) with the incident photon energy can be qualitatively interpreted as the photon energy increases from \(E_{\mathrm{ig}}/2\) to nearly \(E_{\mathrm{ig}}\) , the electrons excited from the valance band find an increasing availability of conduction band states. Comparing with the high-energy side transitions, the TPA coefficient in low-energy side is about 10 times too small. This can be attributed that the TPA transition in low-energy side is the process of photon-assisted electron transitions from valence to conduction band occurring between different points in k-space, while is direct transition in high-energy side.     

Local structure of Fe-doped In 2 O 3 films investigated by X-ray absorption fine structure spectroscopy

$(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ $(x=0.07, 0.09, 0.16, 0.22, 0.31)$ films were deposited on Si (100) substrates by RF-magnetron sputtering technique. The influence of Fe doping on the local structure of films was investigated by X-ray absorption spectroscopy (XAS) at Fe K-edge and L-edge. For the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ films with $x=0.07, 0.09 \mbox{ and } 0.16$ , Fe ions dissolve into $\mathrm{In}_{2}\mathrm{O}_{3}$ and substitute for $\mathrm{In}^{3+}$ sites with a mixed-valence state ( $\mathrm{Fe}^{2+}/\mathrm{Fe}^{3+}$ ) of Fe ions. However, a secondary phase of Fe metal clusters is formed in the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ films with $x=0.22 \mbox{ and } 0.31$ . The qualitative analyses of Fe-K edge extended X-ray absorption fine structure (EXAFS) reveal that the Fe–O bond length shortens and the corresponding Debye–Waller factor ( $\sigma^{2}$ ) increases with the increase of Fe concentration, indicating the relaxation of oxygen environment of Fe ions upon substitution. The anomalously large structural disorder and very short Fe–O distance are also observed in the films with high Fe concentration. Linear combination fittings at Fe L-edge further confirm the coexistence of $\mathrm{Fe}^{2+}$ and $\mathrm{Fe}^{3+}$ with a ratio of ${\sim}3:2$ ( $\mathrm{Fe}^{2+}: \mathrm{Fe}^{3+}$ ) for the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ film with $x=0.16$ . However, a significant fraction ( ${\sim}40~\mbox{at\%}$ ) of the Fe metal clusters is found in the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ film with $x=0.31$ .     

Spectroscopic properties of Ho 3 + -doped K–Sr–Al phosphate glasses

Trivalent holmium-doped K–Sr–Al phosphate glasses ( $\mathrm{P}_{2}\mathrm{O}_{5}$ – $\mathrm{K}_{2}\mathrm{O}$ –SrO– $\mathrm{Al}_{2}\mathrm{O}_{3}$ – $\mathrm{Ho}_{2}\mathrm{O}_{3}$ ) were prepared, and their spectroscopic properties have been evaluated using absorption, emission, and excitation measurements. The Judd–Ofelt theory has been used to derive spectral intensities of various absorption bands from measured absorption spectrum of 1.0 mol% $\mathrm{Ho}_{2}\mathrm{O}_{3}$ -doped K–Sr–Al phosphate glass. The Judd–Ofelt intensity parameters ( $\varOmega_{\lambda}$ , $\times10^{-20}~\mathrm{cm}^{2}$ ) have been determined of the order of $\varOmega_{2} = 11.39$ , $\varOmega_{4} = 3.59$ , and $\varOmega_{6} = 2.92$ , which in turn used to derive radiative properties such as radiative transition probability, radiative lifetime, branching ratios, etc. for excited states of $\mathrm{Ho}^{3+}$ ions. The radiative lifetimes for the ${}^{5}F_{4}$ , ${}^{5}S_{2}$ , and ${}^{5}F_{5}$ levels of $\mathrm{Ho}^{3+}$ ions are found to be 169, 296, and 317 μs, respectively. The stimulated emission cross-section for 2.05-μm emission was calculated by the McCumber theory and found to be $9.3\times10^{-2 1}~\mathrm{cm}^{2}$ . The wavelength-dependent gain coefficient with population inversion rate has been evaluated. The results obtained in the titled glasses are discussed systematically and compared with other $\mathrm{Ho}^{3+}$ -doped systems to assess the possibility for visible and infrared device applications.     

Predicting leptonic CP violation in the light of the Daya Bay result on θ 13

In the light of the recent Daya Bay result $\theta_{13}^{\mathrm{DB}}=8.8^{\circ}\pm0.8^{\circ}$ , we reconsider the model presented in Meloni et?al. (J. Phys.?G 38:015003, 2011), showing that, when all neutrino oscillation parameters are taken at their best fit values of Schwetz et?al. (New J. Phys. 10:113011,?2008) and where $\theta_{13}=\theta_{13}^{\mathrm{DB}}$ , the predicted values of the CP phase are ????±??/4.     

Antiproton low-energy collisions with Ps-atoms and true muonium atoms (μ + μ −)

Three-charge-particle collisions with participation of ultra-slow antiprotons ( \(\overline {\rm {p}}\) ) is the subject of this work. Specifically we compute the total cross sections and corresponding thermal rates of the following three-body reactions: \(\overline {\rm p}+(e^+e^-) \rightarrow \overline {\rm {H}} + e^-\) and \(\overline {\rm p}+(\mu ^+\mu ^-) \rightarrow \overline {\rm {H}}_{\mu } + \mu ^-\) , where \(e^-(\mu ^-)\) is an electron (muon) and \(e^+(\mu ^+)\) is a positron (antimuon) respectively, \(\overline {\rm {H}}=(\overline {\rm p}e^+)\) is an antihydrogen atom and \(\overline {\rm {H}}_{\mu }=(\overline {\rm p}\mu ^+)\) is a muonic antihydrogen atom, i.e. a bound state of \(\overline {\rm {p}}\) and μ ⁺. A set of two-coupled few-body Faddeev-Hahn-type (FH-type) equations is numerically solved in the framework of a modified close-coupling expansion approach.     

A flexible scintillation light apparatus for rare event searches

Compelling experimental evidences of neutrino oscillations and their implication that neutrinos are massive particles have given neutrinoless double beta decay ( \(\beta \beta 0\nu \) ) a central role in astroparticle physics. In fact, the discovery of this elusive decay would be a major breakthrough, unveiling that neutrino and antineutrino are the same particle and that the lepton number is not conserved. It would also impact our efforts to establish the absolute neutrino mass scale and, ultimately, understand elementary particle interaction unification. All current experimental programs to search for \(\beta \beta 0\nu \) are facing with the technical and financial challenge of increasing the experimental mass while maintaining incredibly low levels of spurious background. The new concept described in this paper could be the answer which combines all the features of an ideal experiment: energy resolution, low cost mass scalability, isotope choice flexibility and many powerful handles to make the background negligible. The proposed technology is based on the use of arrays of silicon detectors cooled to 120 K to optimize the collection of the scintillation light emitted by ultra-pure crystals. It is shown that with a 54 kg array of natural CaMoO \(_4\) scintillation detectors of this type it is possible to yield a competitive sensitivity on the half-life of the \(\beta \beta 0\nu \) of \(^{100}\) Mo as high as \(\sim \) \(10^{24}\)  years in only 1 year of data taking. The same array made of \(^{40}\) Ca \(^{\mathrm {nat}}\) MoO \(_4\) scintillation detectors (to get rid of the continuous background coming from the two neutrino double beta decay of \(^{48}\) Ca) will instead be capable of achieving the remarkable sensitivity of \(\sim \) \(10^{25}\)  years on the half-life of \(^{100}\) Mo \(\beta \beta 0\nu \) in only 1 year of measurement.     

Analysis of the \frac{1} {2}^ - and \frac{3} {2}^ - heavy and doubly heavy baryon states with QCD sum rules

In this article, we study the $\frac{1} {2}^ -$ and $\frac{3} {2}^ -$ heavy and doubly heavy baryon states $\Sigma _Q \left( {\frac{1} {2}^ - } \right)$ , $\Xi '_Q \left( {\frac{1} {2}^ - } \right)$ , $\Omega _Q \left( {\frac{1} {2}^ - } \right)$ , $\Xi _{QQ} \left( {\frac{1} {2}^ - } \right)$ , $\Omega _{QQ} \left( {\frac{1} {2}^ - } \right)$ , $\Sigma _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Xi _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Omega _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Xi _{QQ}^* \left( {\frac{3} {2}^ - } \right)$ and $\Omega _{QQ}^* \left( {\frac{3} {2}^ - } \right)$ by subtracting the contributions from the corresponding $\frac{1} {2}^ +$ and $\frac{3} {2}^ +$ heavy and doubly heavy baryon states with the QCD sum rules in a systematic way, and make reasonable predictions for their masses.     

Pesin’s Entropy Formula for Systems Between C^1 and C^{1+\alpha }

In this article we give a new observation of Pesin’s entropy formula, motivated from Mañé’s proof of (Ergod Theory Dyn Sys 1:95–102, 1981). Let \(M\) be a compact Riemann manifold and \(f:\,M\rightarrow M\) be a \(C^1\) diffeomorphism on \(M\) . If \(\mu \) is an \(f\) -invariant probability measure which is absolutely continuous relative to Lebesgue measure and nonuniformly-H \(\ddot{\text {o}}\) lder-continuous(see Definition 1.1), then we have Pesin’s entropy formula, i.e., the metric entropy \(h_\mu (f)\) satisfies $$\begin{aligned} h_{\mu }(f)=\int \sum _{\lambda _i(x)> 0}\lambda _i(x)d\mu , \end{aligned}$$ where \(\lambda _1(x)\ge \lambda _2(x)\ge \cdots \ge \lambda _{dim\,M}(x)\) are the Lyapunov exponents at \(x\) with respect to \(\mu .\) Nonuniformly-H \(\ddot{\text {o}}\) lder-continuous is a new notion from probabilistic perspective weaker than \(C^{1+\alpha }.\)      

Dilaton gravity, (quasi-) black holes,and scalar charge

Electrically charged dust is considered in the framework of Einstein–Maxwell–dilaton gravity with a Lagrangian containing the interaction term \(P(\chi )F_{\mu \nu }F^{\mu \nu }\) , where \(P(\chi )\) is an arbitrary function of the dilaton scalar field \(\chi \) , which can be normal or phantom. Without assumption of spatial symmetry, we show that static configurations exist for arbitrary functions \(g_{00} = \exp (2\gamma (x^{i}))\) ( \(i=1,2,3\) ) and \(\chi =\chi (\gamma )\) . If \(\chi = \mathrm{const}\) , the classical Majumdar–Papapetrou (MP) system is restored. We discuss solutions that represent black holes (BHs) and quasi-black holes (QBHs), deduce some general results and confirm them by examples. In particular, we analyze configurations with spherical and cylindrical symmetries. It turns out that cylindrical BHs and QBHs cannot exist without negative energy density somewhere in space. However, in general, BHs and QBHs can be phantom-free, that is, can exist with everywhere nonnegative energy densities of matter, scalar and electromagnetic fields.     

On Gravitational Effects in the Schrödinger Equation

The Schrödinger  equation for a particle of rest mass $m$ and electrical charge $ne$ interacting with a four-vector potential $A_i$ can be derived as the non-relativistic limit of the Klein–Gordon  equation $\left( \Box '+m^2\right) \varPsi =0$ for the wave function $\varPsi $ , where $\Box '=\eta ^{jk}\partial '_j\partial '_k$ and $\partial '_j=\partial _j -\mathrm {i}n e A_j$ , or equivalently from the one-dimensional  action $S_1=-\int m ds +\int neA_i dx^i$ for the corresponding point particle in the semi-classical approximation $\varPsi \sim \exp {(\mathrm {i}S_1)}$ , both methods yielding the equation $\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2m}\eta ^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + m + n e\phi \right) \varPsi $ in Minkowski  space–time  , where $\alpha ,\beta =1,2,3$ and $\phi =-A_0$ . We show that these two methods generally yield equations  that differ in a curved background  space–time   $g_{ij}$ , although they coincide when $g_{0\alpha }=0$ if $m$ is replaced by the effective mass $\mathcal{M}\equiv \sqrt{m^2-\xi R}$ in both the Klein–Gordon  action $S$ and $S_1$ , allowing for non-minimal coupling to the gravitational  field, where $R$ is the Ricci scalar and $\xi $ is a constant. In this case $\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2\mathcal{M}'} g^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + \mathcal{M}\phi ^{(\mathrm g)} + n e\phi \right) \varPsi $ , where $\phi ^{(\mathrm g)} =\sqrt{g_{00}}$ and $\mathcal{M}'=\mathcal{M}/\phi ^{(\mathrm g)} $ , the correctness of the gravitational  contribution to the potential having been verified to linear order $m\phi ^{(\mathrm g)} $ in the thermal-neutron beam interferometry experiment due to Colella et al. Setting $n=2$ and regarding $\varPsi $ as the quasi-particle wave function, or order parameter, we obtain the generalization of the fundamental macroscopic Ginzburg-Landau equation of superconductivity to curved space–time. Conservation of probability and electrical current requires both electromagnetic gauge and space–time  coordinate conditions to be imposed, which exemplifies the gravito-electromagnetic analogy, particularly in the stationary case, when div ${{\varvec{A}}}=\hbox {div}{{\varvec{A}}}^{(\mathrm g)}=0$ , where ${{\varvec{A}}}^{\alpha }=-A^{\alpha }$ and ${{\varvec{A}}}^{(\mathrm g)\alpha }=-\phi ^{(\mathrm g)}g^{0\alpha }$ . The quantum-cosmological Schrödinger  (Wheeler–DeWitt) equation is also discussed in the $\mathcal{D}$ -dimensional  mini-superspace idealization, with particular regard to the vacuum potential $\mathcal V$ and the characteristics of the ground state, assuming a gravitational  Lagrangian   $L_\mathcal{D}$ which contains higher-derivative  terms up to order $\mathcal{R}^4$ . For the heterotic superstring theory  , $L_\mathcal{D}$ consists of an infinite series in $\alpha '\mathcal{R}$ , where $\alpha '$ is the Regge slope parameter, and in the perturbative approximation $\alpha '|\mathcal{R}| \ll 1$ , $\mathcal V$ is positive semi-definite for $\mathcal{D} \ge 4$ . The maximally symmetric ground state satisfying the field equations is Minkowski  space for $3\le {\mathcal {D}}\le 7$ and anti-de Sitter  space for $8 \le \mathcal {D} \le 10$ .     

The Colored Hofstadter Butterfly for the Honeycomb Lattice

We rely on a recent method for determining edge spectra and we use it to compute the Chern numbers for Hofstadter models on the honeycomb lattice having rational magnetic flux per unit cell. Based on the bulk-edge correspondence, the Chern number \(\sigma _\mathrm{H}\) is given as the winding number of an eigenvector of a \(2 \times 2\) transfer matrix, as a function of the quasi-momentum \(k\in (0,2\pi )\) . This method is computationally efficient (of order \(\mathcal {O}(n^4)\) in the resolution of the desired image). It also shows that for the honeycomb lattice the solution for \(\sigma _\mathrm{H}\) for flux \(p/q\) in the \(r\) -th gap conforms with the Diophantine equation \(r=\sigma _\mathrm{H}\cdot p+ s\cdot q\) , which determines \(\sigma _\mathrm{H}\mod q\) . A window such as \(\sigma _\mathrm{H}\in (-q/2,q/2)\) , or possibly shifted, provides a natural further condition for \(\sigma _\mathrm{H}\) , which however turns out not to be met. Based on extensive numerical calculations, we conjecture that the solution conforms with the relaxed condition \(\sigma _\mathrm{H}\in (-q,q)\) .