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A parallelotope is a polytope whose translation copies fill space without gaps and intersections by interior points. Voronoi conjectured that each parallelotope is an affine image of the Dirichlet domain of a lattice, which is a Voronoi polytope. We give several properties of a parallelotope and prove that each of them is equivalent to it is an affine image of a Voronoi polytope.

Mutation testing is a software testing technique that is considered to be very powerful, but is very expensive to apply. One open problem in mutation testing is how to automatically detect equivalent mutant programs. Currently, equivalent mutants are detected by hand, which makes it a very expensive and time-consuming process, and restricts the use of mutation testing. This paper presents a tec...

We prove that the set of properties describable by a uniform sequence of rst-order sentences using at most k + 1 distinct variables is exactly equal to the set of properties checkable by a Turing machine in DSPACEn k ] (where n is the size of the universe). This set is also equal to the set of properties describable using an iterative deenition for a nite set of relations of arity k. This is a ...

We show that a quantum system possessing an exact antilinear symmetry, in particular PT -symmetry, is equivalent to a quantum system having a Hermitian Hamiltonian. We construct the unitary operator relating an arbitrary non-Hermitian Hamiltonian with exact PT -symmetry to a Hermitian Hamiltonian. We apply our general results to PT symmetry in finite-dimensions and give the explicit form of the...

Neutrinos are present in the natural environment and are also produced by particle accelerators. A recent hypothesis has also been proposed that asserts that ionizing radiation due to neutrinos from certain astronomical events may have led to the extinction of some biological species. Thus, it is of interest to be able to estimate the dose equivalent due to these weakly interacting particles. P...

The Borsuk-Ulam theorem is a fundamental result in algebraic topology, with applications to various areas of Mathematics. A classical application of the Borsuk-Ulam theorem is the Ham Sandwich theorem: The volumes of any n compact sets in R can always be simultaneously bisected by an (n− 1)-dimensional hyperplane. In this paper, we demonstrate the equivalence between the Borsuk-Ulam theorem and...

In this paper, we propose an efficient method to break H2-MAC, by using a generalized birthday attack to recover the equivalent key, under the assumption that the underlying hash function is secure (collision resistance). We can successfully recover the equivalent key of H2-MAC in about 2n/2 on-line MAC queries and 2n/2 off-line hash computations with great probability. This attack shows that t...

Let 8 denote the real line. If TQ& and r £ 8 , we set {t+r: / £ J H } = r [ r ] . In [ l ] we have proved these two theorems: (BK) Let S C S , T C 8 , S be at most enumerable and T be of first category. Then 8 contains a residual subset R such that SC\T\r\ is empty for every rÇzR. (BM) Let S C S , 2"C8, S be at most enumerable and T be of measure zero. Then 8 contains a subset R such that 8— Ri...

We find evidence that a certain class of reaction-diffusion (RD) systems can exhibit chemical turbulence equivalent to Nikolaevskii turbulence. We study an extended complex Ginzburg-Landau (CGL) equation derived from this class of RD systems. First, we show numerically that the power spectrum of this CGL equation, in the neighborhood of a codimension-two Turing-Benjamin-Feir point, is qualitati...