Black holes, quantum chaos, and the Riemann hypothesis

Quantum gravity is expected to gauge all global symmetries of effective theories, in the ultraviolet. Inspired by this expectation, we explore the consequences of gauging CPT as a quantum boundary condition in phase space. We find that it provides for a natural semiclassical regularisation and discretisation of the continuous spectrum of a quantum Hamiltonian related to the Dilation operator. We observe that the said spectrum is in correspondence with the zeros of the Riemann zeta and Dirichlet beta functions. Following ideas of Berry and Keating, this may help the pursuit of the Riemann hypothesis. It strengthens the proposal that this quantum Hamiltonian captures the near horizon dynamics of the scattering matrix of the Schwarzschild black hole, given the rich chaotic spectrum upon discretisation. It also explains why the spectrum appears to be erratic despite the unitarity of the scattering matrix.

SEGMENTATION AND ANALYSIS METHOD FOR TWO-PHASE CERAMIC (HfB2-B4C) BASED ON THE DETECTION OF VIRTUAL BOUNDARIES

Microstructure of a material stores the genesis of the material and shows various properties of the material. To efficiently analyse the microstructure of a material, the segmentation of different phases or constituents is an important step. However, in general, due to the microstructure’s complexity, most of segmentation is manually done by human experts. It is challenging to automatically segment the material phases and the microstructure. In this work, we propose a method which combines the the dilation operator, GLCM (gray-level co-occurrence matrix), Hough transform and DBSCAN (density-based spatial clustering of applications with noise) for phases segmentation in the examples of certain material of eutectic HfB2-B4C ceramics. In the segmented regions, the further analysis for the microstructural elements is done with DBSCAN. The experimental results show that the proposed method achieves 95.75% segmentation accuracy for segmenting phases and 86.64% correct classification rate for the microstructure in the segmented phases. These experimental results show that our method is effective for the difficult task of the both segmentation and classification of the microstructural characteristics.

CONSTRUCTION OF THE MULTI-WAVELETS ON SOME SMOOTH PLANE CURVES VIA LENGTH-PRESERVING PROJECTION

Based on the theory of the discrete multi-wavelets in the space L2(R), the theory of the discrete multi-wavelets in the space L2(C) is presented properly in this paper, where C denotes a smooth plane curve. Firstly, the length-preserving projection is constructed, and by the length-preserving projection, the multiplicity multi-resolution analysis in the space L2(C) is defined properly and we define the dilation operator and translation operator in the space L2(C). Then, the two-scale refinement equations of multi-scaling function and multi-wavelet in the space L2(C) is deduced by using length-preserving mapping, the orthogonality is discussed, and the decomposition and reconstruction algorithm is computed. Finally, the example is given.

SCATTERING THEORY FOR LATTICE OPERATORS IN DIMENSION d ≥ 3

This paper analyzes the scattering theory for periodic tight-binding Hamiltonians perturbed by a finite range impurity. The classical energy gradient flow is used to construct a conjugate (or dilation) operator to the unperturbed Hamiltonian. For dimension d ≥ 3, the wave operator is given by an explicit formula in terms of this dilation operator, the free resolvent and the perturbation. From this formula, the scattering and time delay operators can be read off. Using the index theorem approach, a Levinson theorem is proved which also holds in the presence of embedded eigenvalues and threshold singularities.