Poincare duality in Morava K-theory for classifying spaces of orbifolds

Greenlees and Sadofsky showed that the classifying spaces of finite groups are self-dual with respect to Morava K-theory K(n). Their duality map was constructed using a transfer map. We generalize their duality map and prove a K(n)-version of Poincare duality for classifying spaces of orbifolds. By regarding these classifying spaces as the homotopy types of certain differentiable stacks, our construction can be viewed as a stack version of Spanier-Whitehead type construction. Some examples and calculations will be given at the end.Comment: 36 page

The description of phase transition of Bardeen black hole in the Ehrenfest scheme

The phase transition of a Bardeen black hole is studied by considering Ehrenfest's equations. The thermodynamic variables such as entropy, potential and heat capacity are calculated from the first law of thermodynamics for black holes. That no discontinuity in entropy and potential of the black holes means that the first order phase transition will not generate for the Bardeen black holes. However, the divergence of heat capacity at constant potential and satisfaction of the Ehrenfest's equations indicates that the second order phase transition of Bardeen black hole will appear.Comment: 9 pages, 8 figure

The time delay in strong gravitational lensing with Gauss-Bonnet correction

The time delay between two relativistic images in the strong gravitational lensing governed by Gauss-Bonnet gravity is studied. We derive and calculate the expression of time delay due to the Gauss-Bonnet coupling. It is shown that the time delay for two images with larger space each other is longer. We also find that the ratio of Gauss-Bonnet coefficient and the mass of gravitational source changes in the region like $\frac{\alpha}{M}\in[0,2)$. The time delay is divergent with $\frac{\alpha}{M}\longrightarrow 2$.Comment: 9 pages, 1 figur

The calculation of the thermodynamic quantities of the Bardeen black hole

In this work we research on the thermodynamical properties of the Bardeen black holes. We compute the series of new thermodynamic quantities such as local temperature, heat capacity, off-shell free energy of this kind of black hole in detail. We further analyze the thermodynamical characteristics of the Bardeen black hole by varying its charge $q$ to check the existence and stability of the black hole.Comment: 7 pages, 6 figure

The greybody factor for scalar fields in the Schwarzschild spacetime with an f(R)f(R) global monopole

The greybody factor of massless scalar fields in the four-dimensional Schwarzschild spacetime involving an $f(R)$ global monopole is derived. We show how the monopole parameter and the deviation from the standard general relativity adjust the greybody factor. We also demonstrate that the effects from the global monopole and $f(R)$ gravity theory are manifest in the energy emission rate and the generalized absorption cross section of the scalar fields.Comment: 15 psges, 12 figure

Representation spaces for central extensions and almost commuting unitary matrices

Let $\Gamma$ denote a central extension of the form $1\to \mathbb{Z}^r\to\Gamma\to \mathbb{Z}^n\to 1$. In this paper we describe the topology of the spaces of homomorphisms $\text{Hom}(\Gamma, U(m))$ and the associated moduli spaces $\text{Rep}(\Gamma, U(m))$, where $U(m)$ is the group of $m\times m$ unitary matrices.Comment: 23 pages. Minor typos fixed. To appear in the Journal of the London Mathematical Societ

Anti-Forging Quantum Data: Cryptographic Verification of Quantum Cloud Computing

Quantum cloud computing is emerging as a popular model for users to experience the power of quantum computing through the internet, enabling quantum computing as a service (QCaaS). The question is, when the scale of the computational problems becomes out of reach of classical computers, how can users be sure that the output strings sent by the server are really from a quantum hardware? In 2008, Shepherd and Bremner proposed a cryptographic verification protocol based on a simplified circuit model called IQP (instantaneous quantum polynomial-time), which can potentially be applied to most existing quantum cloud platforms. However, the Shepherd-Bremner protocol has recently been shown to be insecure by Kahanamoku-Meyer. Here we present a generalized model of cryptographic verification protocol, where the Shepherd-Bremner model can be regarded as a special case. This protocol not only can avoid the attack by Kahanamoku-Meyer but also provide several additional security measures for anti-forging quantum data. In particular, our protocol admits a simultaneous encoding of multiple secret strings, strengthening significantly the hardness for classical hacking. Furthermore, we provide methods for estimating the correlation functions associated with the secret strings, which are the key elements in our verification protocol.Comment: 7 pages, 2 figures. Comments are welcom

Denoising a Point Cloud for Surface Reconstruction

Surface reconstruction from an unorganized point cloud is an important problem due to its widespread applications. White noise, possibly clustered outliers, and noisy perturbation may be generated when a point cloud is sampled from a surface. Most existing methods handle limited amount of noise. We develop a method to denoise a point cloud so that the users can run their surface reconstruction codes or perform other analyses afterwards. Our experiments demonstrate that our method is computationally efficient and it has significantly better noise handling ability than several existing surface reconstruction codes.Comment: 13 pages, 6 figure

Implicit Manifold Reconstruction

Let ${\cal M} \subset \mathbb{R}^d$ be a compact, smooth and boundaryless manifold with dimension $m$ and unit reach. We show how to construct a function $\varphi: \mathbb{R}^d \rightarrow \mathbb{R}^{d-m}$ from a uniform $(\varepsilon,\kappa)$-sample $P$ of $\cal M$ that offers several guarantees. Let $Z_\varphi$ denote the zero set of $\varphi$. Let $\widehat{{\cal M}}$ denote the set of points at distance $\varepsilon$ or less from $\cal M$. There exists $\varepsilon_0 \in (0,1)$ that decreases as $d$ increases such that if $\varepsilon \leq \varepsilon_0$, the following guarantees hold. First, $Z_\varphi \cap \widehat{\cal M}$ is a faithful approximation of $\cal M$ in the sense that $Z_\varphi \cap \widehat{\cal M}$ is homeomorphic to $\cal M$, the Hausdorff distance between $Z_\varphi \cap \widehat{\cal M}$ and $\cal M$ is $O(m^{5/2}\varepsilon^{2})$, and the normal spaces at nearby points in $Z_\varphi \cap \widehat{\cal M}$ and $\cal M$ make an angle $O(m^2\sqrt{\kappa\varepsilon})$. Second, $\varphi$ has local support; in particular, the value of $\varphi$ at a point is affected only by sample points in $P$ that lie within a distance of $O(m\varepsilon)$. Third, we give a projection operator that only uses sample points in $P$ at distance $O(m\varepsilon)$ from the initial point. The projection operator maps any initial point near $P$ onto $Z_\varphi \cap \widehat{\cal M}$ in the limit by repeated applications.Comment: A preliminary version appears in Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, 2014, 161--17

The strong field gravitational lensing in the Schwarzschild black hole pierced by a cosmic string

In this work the gravitational lensing in the strong field limit around the Schwarzschild black hole pierced by a cosmic string is studied. We find that the deflection angle and the time delay of the relativistic images depend on the tension of cosmic string. It is interesting that the deflection angle is greater when the tension of cosmic string is stronger. The time delay between two images is more obvious in the case of weaker tension.Comment: 10 pages, 4 figure