${{ \mathcal H }}_{\infty }$ synchronization of chaotic Hopfield networks with time-varying delay: a resilient DOF control approach*

This paper focuses on the issue of resilient dynamic output-feedback (DOF) control for ${{ \mathcal H }}_{\infty }$ synchronization of chaotic Hopfield networks with time-varying delay. The aim is to determine a DOF controller with gain perturbations ensuring that the ${{ \mathcal H }}_{\infty }$ norm from the external disturbances to the synchronization error is less than or equal to a prescribed bound. A delay-dependent criterion for the ${{ \mathcal H }}_{\infty }$ synchronization is derived by employing the Lyapunov functional method together with some recent inequalities. Then, with the help of some decoupling techniques, sufficient conditions on the existence of the resilient DOF controller are developed for both the time-varying and constant time-delay cases. Lastly, an example is used to illustrate the applicability of the results obtained.     

One-dimensional $\mathcal{PT}$-symmetric acoustic heterostructure

The explorations of parity-time ($\mathcal{PT}$)-symmetric acoustics have resided at the frontier in physics, and the pre-existing accessing of exceptional points typically depends on Fabry-Perot resonances of the coupling interlayer sandwiched between balanced gain and loss components. Nevertheless, the concise $\mathcal{PT}$-symmetric acoustic heterostructure, eliminating extra interactions caused by the interlayer, has not been researched in depth. Here we derive the generalized unitary relation for one-dimensional (1D) $\mathcal{PT}$-symmetric heterostructure of arbitrary complexity, and demonstrate four disparate patterns of anisotropic transmission resonances (ATRs) accompanied by corresponding spontaneous phase transitions. As a special case of ATR, the occasional bidirectional transmission resonance reconsolidates the ATR frequencies that split when waves incident from opposite directions, whose spatial profiles distinguish from a unitary structure. The derived theoretical relation can serve as a predominant signature for the presence of $\mathcal{PT}$ symmetry and $\mathcal{PT}$-symmetry-breaking transition, which may provide substantial support for the development of prototype devices with asymmetric acoustic responses.     

${ \mathcal P }{ \mathcal T }$-symmetry berry phases,topology and ${ \mathcal P }{ \mathcal T }$-symmetry breaking

We study the complex Berry phases in non-Hermitian systems with parity- and time-reversal $\left({ \mathcal P }{ \mathcal T }\right)$ symmetry. We investigate a kind of two-level system with ${ \mathcal P }{ \mathcal T }$ symmetry. We find that the real part of the the complex Berry phases have two quantized values and they are equal to either 0 or π, which originates from the topology of the Hermitian eigenstates. We also find that if we change the relative parameters of the Hamiltonian from the unbroken-${ \mathcal P }{ \mathcal T }$-symmetry phase to the broken-${ \mathcal P }{ \mathcal T }$-symmetry phase, the imaginary part of the complex Berry phases are divergent at the exceptional points. We exhibit two concrete examples in this work, one is a two-level toys model, which has nontrivial Berry phases; the other is the generalized Su–Schrieffer–Heeger (SSH) model that has physical loss and gain in every sublattice. Our results explicitly demonstrate the relation between complex Berry phases, topology and ${ \mathcal P }{ \mathcal T }$-symmetry breaking and enrich the field of the non-Hermitian physics.     

Spherical-separability of Non-Hermitian Hamiltonians and Pseudo- PT\mathcal{PT} -symmetry

Non-Hermitian but -symmetrized spherically-separable Dirac and Schr?dinger Hamiltonians are considered. It is observed that the descendant Hamiltonians H _r , H _θ , and H _φ play essential roles and offer some “user-feriendly” options as to which one (or ones) of them is (or are) non-Hermitian. Considering a -symmetrized H _φ , we have shown that the conventional Dirac (relativistic) and Schr?dinger (non-relativistic) energy eigenvalues are recoverable. We have also witnessed an unavoidable change in the azimuthal part of the general wavefunction. Moreover, setting a possible interaction V(θ)≠0 in the descendant Hamiltonian H _θ would manifest a change in the angular θ-dependent part of the general solution too. Whilst some -symmetrized H _φ Hamiltonians are considered, a recipe to keep the regular magnetic quantum number m, as defined in the regular traditional Hermitian settings, is suggested. Hamiltonians possess properties similar to the -symmetric ones (here the non-Hermitian -symmetric Hamiltonians) are nicknamed as pseudo- -symmetric.     

Understanding Partial P T $\mathcal {P}\mathcal {T}$ Symmetry as Weighted Composition Conjugation in Reproducing Kernel Hilbert Space: An application to Non-hermitian Bose-Hubbard Type Hamiltonian in Fock space

International Journal of Theoretical Physics - A new kind of symmetry behaviour introduced as partial $\mathcal {P}\mathcal {T}$ -symmetry(henceforth $\partial _{\mathcal {P}\mathcal {T}}$ ) is...     

Geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics

A new type of quantum theory known as time-dependent $\mathcal{PT}$-symmetric quantum mechanics has received much attention recently. It has a conceptually intriguing feature of equipping the Hilbert space of a $\mathcal{PT}$-symmetric system with a time-varying inner product. In this work, we explore the geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics. We find that a geometric phase can emerge naturally from the cyclic evolution of a $\mathcal{PT}$-symmetric system, and further formulate a series of related differential-geometry concepts, including connection, curvature, parallel transport, metric tensor, and quantum geometric tensor. These findings constitute a useful, perhaps indispensible, tool to investigate geometric properties of $\mathcal{PT}$-symmetric systems with time-varying system's parameters. To exemplify the application of our findings, we show that the unconventional geometric phase [Phys. Rev. Lett. 91 187902 (2003)], which is the sum of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase unveiled in this work.     

Extracting the low-energy constant ${L}_{0}^{r}$ at three flavors from pion-kaon scattering

Based on our analysis of the contributions from the connected and disconnected contraction diagrams to the pion-kaon scattering amplitude, we provide the first determination of the only free low-energy constant at ${ \mathcal O }({p}^{4})$, known as ${L}_{0}^{r}$, in SU (4∣1) Partially-Quenched Chiral Perturbation theory using the data from the Extended Twisted Mass collaboration, ${L}_{0}^{r}(\mu ={M}_{\rho })=0.77(20)(25)(7)(7)(2)\cdot {10}^{-3}$. The theory uncertainties originate from the unphysical scattering length, the physical low-energy constants, the higher-order chiral corrections, the (lattice) meson masses and the pion decay constant, respectively.     

Two-electron and one-photon transitions in highly charged nickel-like ions

This paper calculates the transition wavelengths and probabilities of the two-electron and one-photon (TEOP) transition from the $(3{\rm s}^{-1}_{1/2}4{\rm d}_{j})_{J=1,2}$ to $(3{\rm p}^{-1}_{3/2}4{\rm s}_{1/2})_{J=1}$ and the $(3{\rm p}^{-1}_{1/2}4{\rm s}_{1/2})_{J=1}$ to $(3{\rm d}^{-1}_{j}4{\rm d}_{j'})_{J=1,2}$ for highly charged Ni-like ions with atomic number $Z$ in the range $47\leq Z\leq92$. In the calculations, the multi-configuration Dirac--Fock method and corresponding program packages GRASP92 and REOS99 were used, and the relativistic effects, correlation effects and relaxation effects were considered systematically. It is found that the TEOP transitions are very sensitive to the correlation of electrons, and the probabilities will be enhanced sharply in some special $Z$ regions along the isoelectronic sequence. The present TEOP transition wavelengths are compared with the available data from some previous publications, good agreement is obtained.     

Singular $$ \mathcal{R}$$ -Matrices and Drinfeld's Comultiplication

We compute the $\mathcal{R}$ -matrix which intertwines two dimensional evaluation representations with Drinfeld comultiplication for ${\text{U}}_q \left( {\widehat{{\text{sl}}}_{\text{2}} } \right)$ . This $\mathcal{R}$ -matrix contains terms proportional to the δ-function. We construct the algebra $A\left( \mathcal{R} \right)$ generated by the elements of the matrices L^±(z) with relations determined by $\mathcal{R}$ . In the category of highest-weight representations, there is a Hopf algebra isomorphism between $A\left( \mathcal{R} \right)$ and an extension $\overline {\text{U}} _q \left( {\widehat{{\text{sl}}}_{\text{2}} } \right)$ of Drinfeld's algebra.     

Gap solitons of spin-orbit-coupled Bose-Einstein condensates in $\mathcal{PT}$ periodic potential

We numerically investigate the gap solitons in Bose-Einstein condensates (BECs) with spin-orbit coupling (SOC) in the parity-time ($\mathcal{PT}$)-symmetric periodic potential. We find that the depths and periods of the imaginary lattice have an important influence on the shape and stability of these single-peak gap solitons and double-peak gap solitons in the first band gap. The dynamics of these gap solitons are checked by the split-time-step Crank-Nicolson method. It is proved that the depths of the imaginary part of the $\mathcal{PT}$-symmetric periodic potential gradually increase, and the gap solitons become unstable. But the different periods of imaginary part hardly affect the stability of the gap solitons in the corresponding parameter interval.     

Study of semileptonic decay of ${\bar{B}}_{s}^{0} \rightarrow \phi {l}^{+}{l}^{-}$ in QCD sum rule

In this work we study the semileptonic decay of ${\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-}$ (l=e, μ, τ) with the QCD sum rule method. We calculate the ${\bar{B}}_{s}^{0}\to \phi $ translation form factors relevant to this semileptonic decay, then the branching ratios of ${\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-}$ (l=e, μ, τ) decays are calculated with the form factors obtained here. Our result for the branching ratio of ${\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-}$ agree very well with the recent experimental data. For the unmeasured decay modes such as ${\bar{B}}_{s}^{0}\to \phi {e}^{+}{e}^{-}$ and ${\bar{B}}_{s}^{0}\to \phi {\tau }^{+}{\tau }^{-}$, we give theoretical predictions.     

Some Properties of Generalized Quantum Operations

Let $\mathcal{B}(\mathcal{H})$ be the set of all bounded linear operators on the separable Hilbert space  $\mathcal{H}$ . A (generalized) quantum operation is a bounded linear operator defined on  $\mathcal{B}(\mathcal{H})$ , which has the form $\varPhi_{\mathcal{A}}(X)=\sum_{i=1}^{\infty}A_{i}XA_{i}^{*}$ , where $A_{i}\in\mathcal{B}(\mathcal{H})$ (i=1,2,…) satisfy $\sum_{i=1}^{\infty}A_{i}A_{i}^{*}\leq \nobreak I$ in the strong operator topology. In this paper, we establish the relationship between the (generalized) quantum operation $\varPhi_{\mathcal{A}}$ and its dual $\varPhi_{\mathcal {A}}^{\dag}$ with respect to the set of fixed points and the noiseless subspace. In particular, we also partially characterize the extreme points of the set of all (generalized) quantum operations and give some equivalent conditions for the correctable quantum channel.     

Spin polarization effect for Cr2 molecule

Density functional theory （DFT） （B3P86） of Gaussian 03 has been used to optimize the structure of the Cr2 molecule, a transition metal element molecule. The result shows that the ground state for the Cr2 molecule is a 13- multiple state, indicating that there exists a spin polarization effect in the Cr2 molecule. Meanwhile, we have not found any spin pollution because the wave function of the ground state does not mingle with wave functions of higher-energy states. So the ground state for Cr2 molecule being a 13-multiple state is indicative of spin polarization effect of the Cr2 molecule among transition metal elements, that is, there are 12 parallel spin electrons in the Cr2 molecule. The number of non-conjugated electrons is greatest. These electrons occupy different spatial orbitals so that the energy of the Cr2 molecule is minimized. It can be concluded that the effect of parallel spin in the Cr2 molecule is larger than the effect of the conjugated molecule, which is obviously related to the effect of electron d delocalization. In addition, the Murrell Sorbie potential functions with the parameters for the ground state and other states of the Cr2 molecule are derived. The dissociation energy De for the ground state of the Cr2 molecule is 0.1034eV, equilibrium bond length Re is 0.3396 nm, and vibration frequency we is 73.81cm^-1. Its force constants f2, f3 and f4 are 0.0835, -0.2831 and 0.3535 aJ. nm^-4 respectively. The other spectroscopic data for the ground state of the Cr2 molecule ωeχe, Be and αe are 1.2105, 0.0562 and 7.2938 x 10^-4cm^-1 respectively.     

Ground state for CH2 and symmetry for methane decomposition

Using the different level of methods B3P86, BLYP, B3PW91, HF, QCISD、 CASSCF （4,4） and MP2 with the various basis functions 6-311G^＊＊, D95, cc-pVTZ and DGDZVP, the calculations of this paper confirm that the ground state is X^-3B1 with C2v group for CH2. Furthermore, the three kinds of theoretical methods, i.e. B3P86、 CCSD（T, MP4） and G2 with the same basis set cc-pVTZ only are used to recalculate the zero-point energy revision which are modified by scaling factor 0.989 for the high level based on the virial theorem, and also with the correction for basis set superposition error. These results are also contrary to X^-3∑^-g for the ground state of CH2 in reference. Based on the atomic and molecular reaction statics, this paper proves that the decomposition type （1） i.e. CH4 →CH2＋H2, is forbidden and the decomposition type （2） i.e. CH4→CHa＋H is allowed for CH4. This is similar to the decomposition of SiH4.     

Effects of Quantum Fluctuations on $\mathcal{PT}$-Symmetric Solitons of a Trapped Bose Gas *

Considering the quantum fluctuation effects, the existence and stabilityof solitons in a Bose-Einstein condensate subjected in a $\mathcal{PT}$-symmetric potentialare discussed. Using the variational approach, we investigate how the quantum fluctuationaffects the self-localization and stability of the condensate with attractivetwo-body interactions. The results show that the quantum fluctuation dramaticallyinfluences the shape, width, and chemical potential of the condensate.Analytical variational computation also predicts there exists a positive critical quantumfluctuation strength $q_{c}$ with each fixed attractive two-body interaction $g_{0}$, if thequantum fluctuation strength $q_{0}$ is bigger than $q_{c}$, there is no bright solitonsolution existence. We also study the effects of the quantum fluctuations on the stabilityof solitons using the Vakhitov-Kolokolov (VK) stability criterion. A robust stable brightsoliton will always exist when the quantum fluctuation strength $q_{0}$ belongs tothe parameter regimes $q_{c}\geq q_{0}＞0$.     

Analysis of the Possible $D\bar{D}_{s0}^*(2317)$ and $D^*\bar{D}_{s1}^*(2460)$ Molecules with QCD Sum Rules *

In this article, we assume that there exist the pseudoscalar $D\bar{D}_{s0}^*(2317)$ and $D^*\bar{D}_{s1}^*(2460)$ molecular states $Z_{1,2}$ and construct the color singlet-singlet molecule-type interpolating currents to study their masses with the QCD sum rules. In calculations, we consider the contributions of the vacuum condensates up to dimension-10 and use the formula $\mu=\sqrt{M_{X/Y/Z}^{2}-(2{\mathbb{M}}_{c})^{2}}$ to determine the energy scales of the QCD spectral densities. The numerical results, $M_{Z_1}=4.61_{-0.08}^{+0.11}\,\text{GeV}$ and $M_{Z_2}=4.60_{-0.06}^{+0.07}\,\text{GeV}$, which lie above the $D\bar{D}_{s0}^*(2317)$ and $D^*\bar{D}_{s1}^*(2460)$ thresholds respectively, indicate that the $D\bar{D}_{s0}^*(2317)$ and $D^*\bar{D}_{s1}^*(2460)$ are difficult to form bound state molecular states, the $Z_{1,2}$ are probably resonance states.     

Measurements of charged kaon semileptonic decay branching fractions K±→π0μ±ν and K±→π0e±ν and their ratio

Measured ratios of decay rates for , and are presented. These measurements are based on K^± decays collected in a dedicated run in 2003 by the NA48/2 experiment at CERN. The results obtained are and . Using the PDG average for the K^±→π^±π⁰ normalisation mode, both values are found to be larger than the current values given by the particle data book and lead to a larger magnitude of the |V_us| CKM element than previously accepted. When combined with the latest particle data book value of |V_ud|, the result is in agreement with unitarity of the CKM matrix. In addition, a new measured value of is compared to the semi-empirical predictions based on the latest form factor measurements. An erratum to this article can be found at     

An Infinite Genus Mapping Class Group and Stable Cohomology

We exhibit a finitely generated group whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus g with n boundary components, for any g ≥ 0 and n > 0. We construct a representation of into the restricted symplectic group of the real Hilbert space generated by the homology classes of non-separating circles on , which generalizes the classical symplectic representation of the mapping class groups. Moreover, we show that the first universal Chern class in is the pull-back of the Pressley-Segal class on the restricted linear group via the inclusion . L. F. was partially supported by the ANR Repsurf:ANR-06-BLAN-0311.     

On the Product of Real Spectral Triples

The product of two real spectral triples and , the first of which is necessarily even, was defined by A.Connes as given by and, in the even-even case, by . Generically it is assumed that the real structure obeys the relations , , , where the -sign table depends on the dimension n modulo 8 of the spectral triple. If both spectral triples obey Connes' >-sign table, it is seen that their product, defined in the straightforward way above, does not necessarily obey this -sign table. In this Letter, we propose an alternative definition of the product real structure such that the -sign table is also satisfied by the product.     

Pure annihilation decays of B_s~0→a_0~+a_0~- and B_d~0→K_0~(*+)K_0~(*-) in the PQCD approach

We study the CP-averaged branching fractions and the CP-violating asymmetries in the pure annihilation decays ofB _s~0→a_0~+a _0~-andB _d~0→K_0~(*+)K_0~(*-),where a_0[K_0~*]denotes the scalar a_0(980) and a_0(1450)[K_0~*(800)(ork)and K_0~*(1430)],with the perturbative QCD factorization approach under the assumption of two-quark structure for the a_0and K_0~*states.The numerical results show that the branching ratios of theB _d~0→K_0~(*+)K_0~(*-)decays are in the order of 10~(-6),while the decay rates of theB_s~0→a_0~+a _0~-modes are in the order of 10~(-5).In light of the measured modes with the same quark components in the pseudoscalar sector,namely,B _d~0→K~+K~-and B_s~0→p p~(+-),the predictions for the considered decay modes in this work are expected to be measured at the Large Hadron Collider beauty and/or Belle-Ⅱ experiments in the (near) future.Meanwhile,it is of great interest to find that the twist-3 distribution amplitudes φ~S and φ~T with inclusion of the Gegenbauer polynomials for the scalar a_0(1450) and K_0~*(1430)states in scenario2 contribute slightly to the branching ratios while significantly to the CP violations in the B_d~0→K_0~*(14 30)~+K_0*(14 30)~-and B_s~0→a_0(1450)~+a_0(1450)~-decays,which indicates that,compared to the asymptotic φ~Sand φ~T,these Gegenbauer polynomials could change the strong phases evidently in these pure annihilation decay channels.These predictions await for the future confirmation experimentally,which could further provide useful information to help explore the inner structure of the scalars and shed light on the annihilation decay mechanism.