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In this paper, we will develop a new staggered hybridization technique for discontinuous Galerkin methods to discretize linear elastodynamic equations. The idea of hybridization is used extensively in many discontinuous Galerkin methods, but the idea of staggered hybridization is new. Our new approach offers several advantages, namely energy conservation, high-order optimal convergence, preservation of symmetry for the stress tensor, block diagonal mass matrices as well as low dispersion error. The key idea is to use two staggered hybrid variables to enforce the continuity of the velocity and the continuity of the normal component of the stress tensor on a staggered mesh. We prove the stability and the convergence of the proposed scheme in both the semi-discrete and the fully-discrete settings. Numerical results confirm the optimal rate of convergence and show that the method has a superconvergent property for dispersion. Furthermore, an application of this method to Rayleigh wave propagation is presented.

Among approaches to defining the Feynman integral, various sequential methods are probably the most popular ones. Here we consider that one based on product formulae, the original idea of which belongs to Nelson71'. This method is powerful in the sense that it enables to prove the strict version of Feynman's basic dynamical formula for a wide class of poten tials^^/including time-dependent ones/4/in comparison with other known methods (see, eg, refs./5, 6/). 0n the other, hand, the method does not correspond exactly to Feynman's heuristic prescription because it replaces the exact action along poly gonal paths by its Riemannian approximation. Moreover, one can regard as an unsatisfactory feature that it employs the" equidistant" time-interval partitions only. In the present paper we show that the last mentioned sour ce of arbitrariness can be removed: the product formula for perturbations of propagators by

We propose and analyze a multivariate encryption scheme that uses odd characteristic and an embedding in its construction. This system has a very simple core map F(X) = X^2, allowing for efficient decryption. We also discuss ways to make this decryption faster with specific parameter choices. We give heuristic arguments along with experimental data to show that this scheme resists all known attacks.

We consider the Laplacian in a strip R(0,d) with the boundary condition which is Dirichlet except at the segment of a length 2a of one of the boundaries where it is switched to Neumann. This operator is known to have a non-empty and simple discrete spectrum for any a>0. There is a sequence 0<a1<a2< of critical values at which new eigenvalues emerge from the continuum when the Neumann window expands. We find the asymptotic behavior of these eigenvalues around the thresholds showing that the gap is in the leading order proportional to (aan)2 with an explicit coefficient expressed in terms of the corresponding threshold-energy resonance eigenfunction.