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In this paper, we will systematically study who indeed are the hard lattice cases in LLL reduction. The “hard” cases here mean for their special geometric structures, with a comparatively high “failure probability” that LLL can not solve SVP even by using a powerful relaxation factor. We define the perfect lattice as the “Beauty”, which is given by basis of vectors of the same length with the mutual angles of any two vectors to be exactly 60°. Simultaneously the “Beasts” lattice is defined as the lattice close to the Beauty lattice. There is a relatively high probability (e.g. 15.0% in 3 dimensions) that our “Beasts” bases can withstand the exact-arithmetic LLL reduction (relaxation factors δ close to 1), comparing to the probability (corresponding <0.01%) when apply same LLL on random bases from TU Darmstadt SVP Challenge. Our theoretical proof gives us a direct explanation of this phenomenon. Moreover, we give rational Beauty bases of 3 and 8 dimensions, an irrational Beauty bases of general high dimensions. We also give a general way to construct Beasts lattice bases from the Beauty ones. Experimental results show the Beasts bases derived from Beauty can withstand LLL reduction by a stable probability even for high dimensions. Our work in a way gives a simple and direct way to explain how to build a hard lattice in LLL reduction.

Let <i>A</i> and <i>B</i> be non-negative self-adjoint operators in a separable Hilbert space such that their form sum <i>C</i> is densely defined. It is shown that the Trotter product formula holds for imaginary parameter values in the <i>L</i> ²-norm, that is, one has <div class="gsh_dspfr"> \lim_{no+\infty} \int\limits^T_{-T} \left\|\left(e^{-itA/n}e^{-itB/n} ight)^nh - e^{-itC}hight\|^2dt = 0

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Nonlinear dynamics of wave packets in parity-time-symmetric optical lattices near the phase-transition point is analytically studied. A nonlinear Klein–Gordon equation is derived for the envelope of these wave packets. A variety of phenomena known to exist in this envelope equation are shown to also exist in the full equation, including wave blowup, periodic bound states, and solitary wave solutions.