Uniqueness and perturbation bounds for sparse non-negative tensor equations

We discuss the uniqueness and the perturbation analysis for sparse non-negative tensor equations arriving from data sciences. By two different techniques, we may get better ranges of parameters to guarantee the uniqueness of the solution of the tensor equation. On the other hand, we present some perturbation bounds for the tensor equation. Numerical examples are given to show the efficiency of the theoretical results.     

Jump Conditions of a Shock with Current in Cylindrical Non-Neutral Plasma

jump conditions of the parameters （mass flow, momentum flow and energy flow） of a shock with current （thereby, electric and magnetic field） in cylindrical non-neutral plasma are presented and derived from Maxwell＇s equations and two fluid equations for electron and ion fluid. The critical Mach number for the shock existence is calculated, which depends on the shock carried current, the ion charge, and the composition of the magnetic and thermal pressure. The numerical results show that both the strength and profiles of the downstream shock parameters will be affected obviously by the shock carried current, electric and magnetic field in the two-dimensional shock.     

Tetrahydropyrido[3,4-b]pyrazine scaffolds from pentafluoropyridine

[reaction: see text] Representative polyfunctional tetrahydropyrido[3,4-b]pyrazine scaffolds have been synthesized very readily by a one-pot annelation reaction of pentafluoropyridine with appropriate diamines. The trifluorinated pyridopyrazine products react sequentially with various nucleophiles to give poly-substituted tetrahydropyridopyrazines, demonstrating the potential of the polyfluorinated ring fused pyridine system as a scaffold for the synthesis of previously inaccessible poly-substituted pyridopyrazine derivatives. This general approach has special relevance to the development of new chemical entities for the life science industries and particularly in the drug discovery arena, in which low molecular weight, polyfunctional heterocyclic derivatives are playing an increasingly important role.     

Photodegradable transient bilayered poly(phthalaldehyde) with improved shelf life

Metastable poly(phthalaldehyde) (PPHA) can be triggered to depolymerize under visible light by incorporation of photosensitive compounds, such as a photoacid generator (PAG), which can generate a strong acid in situ. However, photosensitive compounds can be thermally unstable and have limited shelf life, causing inadvertent device triggering. It can also be difficult to fabricate components that are photosensitive because special lighting conditions are needed. In this paper, nonphotosensitive PPHA films were formed and made photosensitive at the point of use. This improved the material shelf life and manufacturability by adding a second, PAG‐containing layer to the original nonphotosensitive layer at an optimal point before use. The catalytic photoacid was generated rapidly by exposure of the PAG‐containing layer to radiation. Depolymerization of PPHA via the acid catalyst was followed by diffusion of the acid into the nonphotosensitive layer causing it to depolymerize. Diffusion of the photoacid into the nonphotosensitive medium was quantified at various temperatures. Photoacid diffusion in a liquid, moving‐front caused depolymerization of the nonphotosensitive PPHA layer. The fabricated bilayer structure allowed for better stability of the structural material using PPHA while still achieving transience.     

A high‐order compact scheme for the nonlinear fractional Klein–Gordon equation

In this article, a high‐order finite difference scheme for a kind of nonlinear fractional Klein–Gordon equation is derived. The time fractional derivative is described in the Caputo sense. The solvability of the difference system is discussed by the Leray–Schauder fixed point theorem, while the stability and L_∞ convergence of the finite difference scheme are proved by the energy method. Numerical examples are provided to demonstrate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 706–722, 2015     

A compact ADI scheme for the two dimensional time fractional diffusion‐wave equation in polar coordinates

In this article, we consider finite difference schemes for two dimensional time fractional diffusion‐wave equations on an annular domain. The problem is formulated in polar coordinates and, therefore, has variable coefficients. A compact alternating direction implicit scheme with accuracy order is derived, where τ, h₁, h₂ are the temporal and spatial step sizes, respectively. The stability and convergence of the proposed scheme are studied using its matrix form by the energy method. Numerical experiments are presented to support the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1692–1712, 2015     

A nonuniform L2 formula of Caputo derivative and its application to a fractional Benjamin–Bona–Mahony-type equation with nonsmooth solutions

To recover the full accuracy of discretized fractional derivatives, nonuniform mesh technique is a natural and simple approach to efficiently resolve the initial singularities that always appear in the solutions of time-fractional linear and nonlinear differential equations. We first construct a nonuniform L2 approximation for the fractional Caputo's derivative of order 1 < α < 2 and present a global consistency analysis under some reasonable regularity assumptions. The temporal nonuniform L2 formula is then utilized to develop a linearized difference scheme for a time-fractional Benjamin–Bona–Mahony-type equation. The unconditional convergence of our scheme on both uniform and nonuniform (graded) time meshes are proven with respect to the discrete H¹-norm. Numerical examples are provided to justify the accuracy.     

High accuracy error estimates of a Galerkin finite element method for nonlinear time fractional diffusion equation

In this work, an effective and fast finite element numerical method with high-order accuracy is discussed for solving a nonlinear time fractional diffusion equation. A two-level linearized finite element scheme is constructed and a temporal–spatial error splitting argument is established to split the error into two parts, that is, the temporal error and the spatial error. Based on the regularity of the time discrete system, the temporal error estimate is derived. Using the property of the Ritz projection operator, the spatial error is deduced. Unconditional superclose result in H¹-norm is obtained, with no additional regularity assumption about the exact solution of the problem considered. Then the global superconvergence error estimate is obtained through the interpolated postprocessing technique. In order to reduce storage and computation time, a fast finite element method evaluation scheme for solving the nonlinear time fractional diffusion equation is developed. To confirm the theoretical error analysis, some numerical results are provided.     

A linearized second-order scheme for nonlinear time fractional Klein-Gordon type equations

We consider difference schemes for nonlinear time fractional Klein-Gordon type equations in this paper. A linearized scheme is proposed to solve the problem. As a result, iterative method need not be employed. One of the main difficulties for the analysis is that certain weight averages of the approximated solutions are considered in the discretization and standard energy estimates cannot be applied directly. By introducing a new grid function, which further approximates the solution, and using ideas in some recent studies, we show that the method converges with second-order accuracy in time.     

Total synthesis of seco-lateriflorone

A convergent strategy toward the synthesis of lateriflorone (5) is described. Our approach is based on biosynthetic considerations and draws on a sequence of prenylation, oxygenation and Claisen reactions for the construction of chromenequinone 6, and a tandem Claisen/Diels-Alder reaction cascade for the synthesis of caged tricycle 7. Union of fragments 6 and 7 led to the synthesis of seco-lateriflorone (49).