A general product of tensors with applications

We define a general product of two $n$-dimensional tensors $\mathbb {A}$ and $\mathbb {B}$ with orders $m\ge 2$ and $k\ge 1$, respectively. This product is a generalization of the usual matrix product, and satisfies the associative law. Using this product, many concepts and known results of tensors can be simply expressed and/or proved, and a number of applications of this product will be given. Using this tensor product and some properties on the resultant of a system of homogeneous equations on $n$ variables, we define the similarity and congruence of tensors (which are also the generalizations of the corresponding relations for matrices), and prove that similar tensors have the same characteristic polynomials. We study two special kinds of similarity: permutational similarity and diagonal similarity, and their applications in the study of the spectra of hypergraphs and nonnegative irreducible tensors. We define the direct product of tensors (in matrix case it is also called the Kronecker product), and give its applications in the study of the spectra of two kinds of the products of hypergraphs. We also give applications of this tensor product in the study of nonnegative tensors, including a characterization of primitive tensors, the upper bounds of primitive degrees and the cyclic indices of some nonnegative irreducible tensors.Comment: 17 page

Equidistribution theorems on strongly pseudoconvex domains

This work consists of two parts. In the first part, we consider a compact connected strongly pseudoconvex CR manifold $X$ with a transversal CR $S^{1}$ action. We establish an equidistribution theorem on zeros of CR functions. The main techniques involve a uniform estimate of Szeg\H{o} kernel on $X$. In the second part, we consider a general complex manifold $M$ with a strongly pseudoconvex boundary $X$. By using classical result of Boutet de Monvel-Sj\"ostrand about Bergman kernel asymptotics, we establish an equidistribution theorem on zeros of holomorphic functions on $\overline M$.Comment: 26 page

A possible relation between flare activity in superluminous supernovae and gamma-ray bursts

Significant undulations appear in the light curve of a recently discovered super-luminous supernova (SLSN) SN 2015bn after the first peak, while the underlying profile of the light curve can be well explained by a continuous energy supply from a central engine, possibly the spin-down of a millisecond magnetar. We propose that these undulations are caused by an intermittent pulsed energy supply, indicating an energetic flare activity of the central engine of the SLSN. Many post-burst flares were discovered during X-ray afterglow observations of Gamma-Ray Bursts (GRBs). We find that the SLSN flares described here approximately obey the empirical correlation between the luminosity and time scale of GRB flares, extrapolated to the relevant longer time scales of SLSN flares. This confirms the possible connection between these two different phenomena as previously suggested.Comment: 6 pages, 4 figures, accepted for publication in MNRA

Shock breakout driven by the remnant of a neutron star binary merger: An X-ray precursor of mergernova emission

A supra-massive neutron star (NS) spinning extremely rapidly could survive from a merger of NS-NS binary. The spin-down of this remnant NS that is highly magnetized would power the isotropic merger ejecta to produce a bright mergernova emission in ultraviolet/optical bands. Before the mergernova, the early interaction between the NS wind and the ejecta can drive a forward shock propagating outwards into the ejecta. As a result, a remarkable amount of heat can be accumulated behind the shock front and the final escaping of this heat can produce a shock breakout emission. We describe the dynamics and thermal emission of this shock with a semi-analytical model. It is found that sharp and luminous breakout emission appears mainly in soft X-rays with a luminosity of $\sim10^{45}~\rm erg~s^{-1}$ at a few hours after the merger, by leading the mergernova emission as a precursor. Therefore, detection of such an X-ray precursor would provide a smoking-gun evidence for identifying NS-powered mergernovae and distinguishing them from the radioactive-powered ones (i.e., kilonovae or macronovae). The discovery of NS-powered mergernovae would finally help to confirm the gravitational wave signals due to the mergers and the existence of supra-massive NSs.Comment: 7 pages, 4 figures, accepted by Ap

Extended thermodynamics and microstructures of four-dimensional charged Gauss-Bonnet black hole in AdS space

The discovery of new four-dimensional black hole solutions presents a new approach to understand the Gauss-Bonnet gravity in low dimensions. In this paper, we test the Gauss-Bonnet gravity by studying the phase transition and microstructures for the four-dimensional charged AdS black hole. In the extended phase space, where the cosmological constant and the Gauss-Bonnet coupling parameter are treated as thermodynamic variables, we find that the thermodynamic first law and the corresponding Smarr formula are satisfied. Both in the canonical ensemble and grand canonical ensemble, we observe the small-large black hole phase transition, which is similar to the case of the van der Walls fluid. This phase transition can also appear in the neutral black hole system. Furthermore, we construct the Ruppeiner geometry, and find that besides the attractive interaction, the repulsive interaction can also dominate among the microstructures for the small black hole with high temperature in a charged or neutral black hole system. This is quite different from the five-dimensional neutral black hole, for which only dominant attractive interaction can be found. The critical behaviors of the normalized scalar curvature are also examined. These results will shed new light into the characteristic property of four-dimensional Gauss-Bonnet gravity.Comment: 17 pages, 9 figures, and 2 tables, references added. Accepted for publication in PR

Noncommutative Theory in Light of Neutrino Oscillation

Solar neutrino problem and atmospheric neutrino anomaly which are both long-standing issues studied intensively by physicists in the past several decades, are reckoned to be able to be solved simultaneously in the framework of the assumption of the neutrino oscillation. For the presence of the Lorentz invariance in the Standard Model, the massless neutrino can't have flavor mixing and oscillation. However, we exploit the q-deformed noncommutative theory to derive a general modified dispersion relation, which implies some violation of the Lorentz invariance. Then it is found that the application of the q-deformed dispersion relation to the neutrino oscillation can provide a sound explanation for the current data from the reactor and long baseline experiments.Comment: 8 pages,1 figure,Latex Fil

Computational Soundness Results for Stateful Applied pi Calculus

In recent years, many researches have been done to establish symbolic models of stateful protocols. Two works among them, the SAPIC tool and StatVerif tool, provide a high-level specification language and an automated analysis. Their language, the stateful applied \pi-calculus, is extended from the applied \pi-calculus by defining explicit state constructs. Symbolic abstractions of cryptography used in it make the analysis amenable to automation. However, this might overlook the attacks based on the algebraic properties of the cryptographic algorithms. In our paper, we establish the computational soundness results for stateful applied \pi-calculus used in SAPIC tool and StatVerif tool. In our approach, we build our results on the CoSP framework. For SAPIC, we embed the non-monotonic protocol states into the CoSP protocols, and prove that the resulting CoSP protocols are efficient. Through the embedding, we provide the computational soundness result for SAPIC (by Theorem 1). For StatVerif, we encode the StatVerif process into a subset of SAPIC process, and obtain the computational soundness result for StatVerif (by Theorem 2). Our encoding shows the differences between the semantics of the two languages. Our work inherits the modularity of CoSP, which allows for easily extending the proofs to specific cryptographic primitives. Thus we establish a computationally sound automated verification result for the input languages of SAPIC and StatVerif that use public-key encryption and signatures (by Theorem 3).Comment: to appear in POST 201

Area Spectrum of the Large AdS Black Hole from Quasinormal Modes

Using the new physical interpretation of quasinormal modes proposed by Maggiore, we calculate the area and entropy spectra for the 3-dimensioal and 5-dimensional large AdS black holes. The spectra are obtained by imposing the Bohr-Sommerfeld quantization condition to the adiabatic invariant quantity. With this semiclassical method, we find that the spacings of the area and entropy spectra are equidistant and independent of the AdS radius of the black hole for both the cases. However, the spacings of the spectra are not the same for different dimension of space-time. The equidistant area spectra will be broken when the black hole has other parameters (i.e., charge and angular momentum) or in a non-Einstein's gravity theory.Comment: 10 pages, no figure

Establishing a universal relation between gravitational waves and black hole lensing

Black hole lensing and gravitational waves are, respectively, closely dependent of the property of the lens and radiation source. In this letter, a universal relation between them is established for a rotating black hole acting simultaneously as a lens and a gravitational wave source, in an asymptotically flat spacetime. The relation only relies on the lens geometry and observable, while is independent of the specific nature of the black hole. Therefore, the possible gravitational wave sources could be located with modern astronomical instrument from the side of the lensing without knowing the specific nature of the black hole lens. Moreover, the low bound of the frequency of the gravitational waves can also be well determined.Comment: 6 pages, 3 figure

Insight into the Microscopic Structure of an AdS Black Hole from Thermodynamical Phase Transition

Comparing with an ordinary thermodynamic system, we investigate the possible microscopic structure of a charged anti-de Sitter black hole completely from the thermodynamic viewpoint. The number density of the black hole molecules is introduced to measure the microscopic degrees of freedom of the black hole. We found that the number density suffers a sudden change accompanied by a latent heat when the black hole system crosses the small-large black hole coexistence curve, while when the system passes the critical point, it encounters a second-order phase transition with a vanishing latent heat due to the continuous change of the number density. Moreover, the thermodynamic scalar curvature suggests that there is a weak attractive interaction between two black hole molecules. These phenomena might cast new insight into the underlying microscopic structure of a charged anti-de Sitter black hole.Comment: 5 pages, 4 figures, accepted for publication in Phys. Rev. Let