We associate to a finite digraph D a lattice polytope PD whose vertices are the rows of the Laplacian matrix of D. This generalizes a construction introduced by Braun and the third author. As a consequence of the Matrix-Tree Theorem, we show that the normalized volume of PD equals the complexity of D, and PD contains the origin in its relative interior if and only if D is strongly connected. Interesting connections with other families of simplices are established and then used to describe reflexivity, the h∗-polynomial, and the integer decomposition property of PD in these cases. We extend Braun and Meyer’s study of cycles by considering cycle digraphs. In this setting, we characterize reflexivity and show there are only four non-trivial reflexive Laplacian simplices having the integer decomposition property.

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Wave dynamics in topological materials has been widely studied recently. A striking feature is the existence of robust and chiral wave propagations that have potential applications in many fields. A common way to realize such wave patterns is to utilize Dirac points which carry topological indices and is supported by the symmetries of the media. In this work, we investigate these phenomena in photonic media. Starting with Maxwell's equations with a honeycomb material weight as well as the nonlinear Kerr effect, we first prove the existence of Dirac points in the dispersion surfaces of transverse electric and magnetic Maxwell operators under very general assumptions of the material weight. Our assumptions on the material weight are almost the minimal requirements to ensure the existence of Dirac points in a general hexagonal photonic crystal. We then derive the associated wave packet dynamics in the scenario where the honeycomb structure is weakly modulated. It turns out the reduced envelope equation is generally a two-dimensional nonlinear Dirac equation with a spatially varying mass. By studying the reduced envelope equation with a domain-wall-like mass term, we realize the subtle wave motions which are chiral and immune to local defects. The underlying mechanism is the existence of topologically protected linear line modes, also referred to as edge states. However, we show that these robust linear modes do not survive with nonlinearity. We demonstrate the existence of nonlinear line modes, which can propagate in the nonlinear media based on high-accuracy numerical computations. Moreover, we also report a new type of nonlinear modes which are localized in both directions.

In 2006, Nie et al proposed an attack to break an instance of TTM cryptosystems. However, the inventor of TTM disputed this attack and he proposed two new instances of TTM to support his viewpoint. At this time, he did not give the detail of key construction --- the construction of the lock polynomials in these instances which would be used in decryption. The two instances are claimed to achieve a security of 2^{109} against Nie et al attack. In this paper, we show that these instances are both still insecure, and in fact, they do not achieve a better design in the sense that we can find a ciphertext-only attack utilizing the First Order Linearization Equations while for the previous version of TTM, only Second Order Linearization Equations can be used in the beginning stage of the previous attack. Different from previous attacks, we use an iterated linearization method to break these two instances. For any given valid ciphertext, we can find its corresponding plaintext within 2^{31} F_{2^8}-computations after performing once for any public key a computation of complexity less than 2^{44} . Our experiment result shows we have unlocked the lock polynomials after several iterations, though we do not know the detailed construction of lock polynomials.

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