Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in 2+1 dimensions

Riemann theta functions are used to construct one-periodic and two-periodic wave solutions to a class of (2+1)-dimensional Hirota bilinear equations. The basis for the involved solution analysis is the Hirota bilinear formulation, and the particular dependence of the equations on independent variables guarantees the existence of one-periodic and two-periodic wave solutions involving an arbitrary purely imaginary Riemann matrix. The resulting theory is applied to two nonlinear equations possessing Hirota bilinear forms: $u_t+u_{xxy}-3uu_y-3u_xv=0$ and $u_t+u_{xxxxy}-(5u_{xx}v+10u_{xy}u-15u^2v)_x=0$ where $v_x=u_y$, thereby yielding their one-periodic and two-periodic wave solutions describing one dimensional propagation of waves

Optimal Uncertainty Quantification

We propose a rigorous framework for Uncertainty Quantification (UQ) in which the UQ objectives and the assumptions/information set are brought to the forefront. This framework, which we call \emph{Optimal Uncertainty Quantification} (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large, we show that under general conditions they have finite-dimensional reductions. As an application, we develop \emph{Optimal Concentration Inequalities} (OCI) of Hoeffding and McDiarmid type. Surprisingly, these results show that uncertainties in input parameters, which propagate to output uncertainties in the classical sensitivity analysis paradigm, may fail to do so if the transfer functions (or probability distributions) are imperfectly known. We show how, for hierarchical structures, this phenomenon may lead to the non-propagation of uncertainties or information across scales. In addition, a general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact and on the seismic safety assessment of truss structures, suggesting the feasibility of the framework for important complex systems. The introduction of this paper provides both an overview of the paper and a self-contained mini-tutorial about basic concepts and issues of UQ.Comment: 90 pages. Accepted for publication in SIAM Review (Expository Research Papers). See SIAM Review for higher quality figure

Scanning Tunnelling Spectroscopic Studies of Dirac Fermions in Graphene and Topological Insulators

We report novel properties derived from scanning tunnelling spectroscopic (STS) studies of Dirac fermions in graphene and the surface state (SS) of a strong topological insulator (STI), Bi_2Se_3. For mono-layer graphene grown on Cu by chemical vapour deposition (CVD), strain-induced scalar and gauge potentials are manifested by the charging effects and the tunnelling conductance peaks at quantized energies, respectively. Additionally, spontaneous time-reversal symmetry breaking is evidenced by the alternating anti-localization and localization spectra associated with the zero-mode of two sublattices while global time-reversal symmetry is preserved under the presence of pseudo-magnetic fields. For Bi_2Se_3 epitaxial films grown on Si(111) by molecular beam epitaxy (MBE), spatially localized unitary impurity resonances with sensitive dependence on the energy difference between the Fermi level and the Dirac point are observed for samples thicker than 6 quintuple layers (QL). These findings are characteristic of the SS of a STI and are direct manifestation of strong topological protection against impurities. For samples thinner than 6-QL, STS studies reveal the openup of an energy gap in the SS due to overlaps of wave functions between the surface and interface layers. Additionally, spin-preserving quasiparticle interference wave-vectors are observed, which are consistent with the Rashba-like spin-orbit splitting

A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws

In this article we consider one-dimensional random systems of hyperbolic conservation laws. We first establish existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws which involve random initial data and random flux functions. Based on these results we present an a posteriori error analysis for a numerical approximation of the random entropy admissible solution. For the stochastic discretization, we consider a non-intrusive approach, the Stochastic Collocation method. The spatio-temporal discretization relies on the Runge--Kutta Discontinuous Galerkin method. We derive the a posteriori estimator using continuous reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. We conclude with various numerical examples investigating the scaling properties of the residuals and illustrating the efficiency of the proposed adaptive algorithm

Spin alignment of vector meson in e+e- annihilation at Z0 pole

We calculate the spin density matrix of the vector meson produced in e+e- annihilation at Z^0 pole. We show that the data imply a significant polarization for the antiquark which is created in the fragmentation process of the polarized initial quark and combines with the fragmenting quark to form the vector meson. The direction of polarization is opposite to that of the fragmenting quark and the magnitude is of the order of 0.5. A qualitative explanation of this result based on the LUND string fragmentation model is given.Comment: 15 pages, 2 fgiures; submitted to Phys. Rev.

Finite-dimensional integrable systems associated with Davey-Stewartson I equation

For the Davey-Stewartson I equation, which is an integrable equation in 1+2 dimensions, we have already found its Lax pair in 1+1 dimensional form by nonlinear constraints. This paper deals with the second nonlinearization of this 1+1 dimensional system to get three 1+0 dimensional Hamiltonian systems with a constraint of Neumann type. The full set of involutive conserved integrals is obtained and their functional independence is proved. Therefore, the Hamiltonian systems are completely integrable in Liouville sense. A periodic solution of the Davey-Stewartson I equation is obtained by solving these classical Hamiltonian systems as an example.Comment: 18 pages, LaTe

WRF-CMAQ two-way coupled system with aerosol feedback: software development and preliminary results

Air quality models such as the EPA Community Multiscale Air Quality (CMAQ) require meteorological data as part of the input to drive the chemistry and transport simulation. The Meteorology-Chemistry Interface Processor (MCIP) is used to convert meteorological data into CMAQ-ready input. Key shortcoming of such one-way coupling include: excessive temporal interpolation of coarsely saved meteorological input and lack of feedback of atmospheric pollutant loading on simulated dynamics. We have developed a two-way coupled system to address these issues. A single source code principle was used to construct this two-way coupling system so that CMAQ can be consistently executed as a stand-alone model or part of the coupled system without any code changes; this approach eliminates maintenance of separate code versions for the coupled and uncoupled systems. The design also provides the flexibility to permit users: (1) to adjust the call frequency of WRF and CMAQ to balance the accuracy of the simulation versus computational intensity of the system, and (2) to execute the two-way coupling system with feedbacks to study the effect of gases and aerosols on short wave radiation and subsequent simulated dynamics. Details on the development and implementation of this two-way coupled system are provided. When the coupled system is executed without radiative feedback, computational time is virtually identical when using the Community Atmospheric Model (CAM) radiation option and a slightly increased (~8.5 %) when using the Rapid Radiative Transfer Model for GCMs (RRTMG) radiation option in the coupled system compared to the offline WRF-CMAQ system. Once the feedback mechanism is turned on, the execution time increases only slightly with CAM but increases about 60 % with RRTMG due to the use of a more detailed Mie calculation in this implementation of feedback mechanism. This two-way model with radiative feedback shows noticeably reduced bias in simulated surface shortwave radiation and 2 m temperatures as well improved correlation of simulated ambient ozone and PM<sub>2.5</sub> relative to observed values for a test case with significant tropospheric aerosol loading from California wildfires

A refined invariant subspace method and applications to evolution equations

The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. A two-component nonlinear system of dissipative equations was analyzed to shed light on the resulting theory, and two concrete examples are given to find invariant subspaces associated with 2nd-order and 3rd-order linear ordinary differential equations and their corresponding exact solutions with generalized separated variables.Comment: 16 page

Heavy fermions and two loop electroweak corrections to b→s+γb\rightarrow s+\gamma

Applying effective Lagrangian method and on-shell scheme, we analyze the electroweak corrections to the rare decay $b\rightarrow s+\gamma$ from some special two loop diagrams in which a closed heavy fermion loop is attached to the virtual charged gauge bosons or Higgs. At the decoupling limit where the virtual fermions in inner loop are much heavier than the electroweak scale, we verify the final results satisfying the decoupling theorem explicitly when the interactions among Higgs and heavy fermions do not contain the nondecoupling couplings. Adopting the universal assumptions on the relevant couplings and mass spectrum of new physics, we find that the relative corrections from those two loop diagrams to the SM theoretical prediction on the branching ratio of $B\rightarrow X_{_s}\gamma$ can reach 5% as the energy scale of new physics $\Lambda_{_{\rm NP}}=200$ GeV.Comment: 30 pages,4 figure