In this paper, the relation between algebraic shifting and join which was conjectured by Eran Nevo will be proved. Let σ and τ be simplicial complexes and σ*τ be their join. Let$J$_{σ}be the exterior face ideal of σ and Δ(σ) the exterior algebraic shifted complex of σ. Assume that σ*τ is a simplicial complex on [$n$]={1,2,...,$n$}. For any$d$-subset$S$⊂[$n$], let $m_{\preceq_{\textrm{rev}}S}(\sigma)$ denote the number of$d$-subsets$R$∈σ which are equal to or smaller than$S$with respect to the reverse lexicographic order. We will prove that $m_{\preceq_{\textrm{rev}}S}(\Delta(\sigma*\tau))\geq m_{\preceq_{\textup{rev}}S}(\Delta(\Delta(\sigma) *\Delta(\tau)))$ for all$S$⊂[$n$]. To prove this fact, we also prove that $m_{\preceq_{\textrm{rev}}S}(\Delta(\sigma))\geq m_{\preceq_{\textup{rev}}S}(\Delta(\Delta_{\varphi}(\sigma)))$ for all$S$⊂[$n$] and for all nonsingular matrices ϕ, where Δ_{ϕ}(σ) is the simplicial complex defined by $J_{\Delta_{\varphi}(\sigma)}=\textup{in}(\varphi(J_{\sigma}))$ .

We present an ansatz which makes the equations of motion more tractable for the simplest of Vasiliev’s four-dimensional higher spin theories. The ansatz is similar to axial gauge in electromagnetism. We present a broad class of solutions in the gauge where the spatial connection vanishes, and we discuss the lift of one of these solutions to a full spacetime solution via a gauge transformation.

A new residual-based stabilized finite element method is analyzed for solving the Stokes equations in terms of velocity and pressure, where the $H^{−1}$ norm is introduced in the measurement of the residuals to obtain a symmetric positive definite (SPD) method. The $H^{−1}$ norm is computable and can be always easily realized offline by the continuous linear finite element solution or the preconditioned counterpart of the Poisson Dirichlet problem. Although the $H^{−1}$ norm is computed in the linear element space, no matter what the finite element spaces for the velocity and the pressure are, optimal error bounds can be established when using continuous finite element pairs $R_l$−$R_m$ for velocity and pressure for any $l, m\ge 1$. Numerical experiments are performed to confirm the theoretical results obtained.

In this article, we consider the Lorentz cone complementarity problems in infinite-dimensional real Hilbert space. We establish several results that are standard and important when dealing with complementarity problems. These include proving the same growth of the FishcherBurmeister merit function and the natural residual merit function, investigating property of bounded level sets under mild conditions via different merit functions, and providing global error bounds through the proposed merit functions. Such results are helpful for further designing solution methods for the Lorentz cone complementarity problems in Hilbert space.

In this paper, we study a class of analytical solutions to the 3-D compressible Navier-Stokes equations with density-dependent viscosity coefficients, where the shear viscosity h ( ) = 0 , and the bulk viscosity g ( ) = ( > 0 ). By constructing a class of radial symmetric and self-similar analytical solutions in R N ( N 2 ) with both the continuous density condition and the stress free condition across the free boundaries separating the fluid from vacuum, we derive that the free boundary expands outward in the radial direction at an algebraic rate in time and we also have shown that such solutions exhibit interesting new information such as the formation of a vacuum at the center of the symmetry as time tends to infinity and explicit regularities, and we have large time decay estimates of the velocity