Let be a locally compact Hausdorff space. We show that any local -linear map (where" local" is a weaker notion than -linearity) between Banach -modules are" nearly -linear" and" nearly bounded". As an application, a local -linear map between Hilbert -modules is automatically -linear. If, in addition, contains no isolated point, then any -linear map between Hilbert -modules is automatically bounded. Another application is that if a sequence of maps between two Banach spaces" preserve -sequences"(or" preserve ultra- -sequences"), then is bounded for large enough and they have a common bound. Moreover, we will show that if is a bijective" biseparating" linear map from a" full" essential Banach -module into a" full" Hilbert -module (where is another locally compact Hausdorff space), then is" nearly bounded"(in fact, it is automatically bounded if or contains no isolated point) and there exists a homeomorphism such that ( ).

We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to symmetric cone constraints. The algorithm is based on a distance-like function that is an extension of the Kullback-Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithms for convex programming with nonnegative orthant cone constraints, we show that, under some mild assumptions, the sequence generated by the proposed algorithm is bounded and every accumulation point is a solution of the considered problem. In addition, we also present a dual application of the proposed algorithm to the symmetric cone linear program, leading to a multiplier method which is shown to possess similar properties as the exponential multiplier method (Tseng and Bertsekas in Math. Program. 60:119, 1993) holds.

Let G={h_t|t∈ℝ} be a continuous flow on a connected n-manifold M. The flow G is said to be strongly reversible by an involution τ if h_{−t}=τh_tτ for all t∈ℝ, and it is said to be periodic if h_s= identity for some s∈ℝ^∗. A closed subset K of M is called a global section for G if every orbit G(x) intersects K in exactly one point. In this paper, we study how the two properties “strongly reversible” and “has a global section” are related. In particular, we show that if G is periodic and strongly reversible by a reflection, then G has a global section.

TTM (tame transformation method) is a type of multivariate public key cryptosystem. In 2007, the inventor of TTM proposed two new instances of TTM to resist the existing attacks, in particular, the Nie et al attack. The two instances are claimed to achieve a security of 2109 against Nie et al attack. Through computation, we found that the instance II satisfied First Order Linearization Equations. After finding all linearization equations, we can perform a ciphertext-only attack to break it.

In this paper we introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments.