Optimal function spaces for the weak continuity of the distributional kk-Hessian

In this paper we introduce the notion of distributional $k$-Hessian associated with Besov type functions in Euclidean $n$-space, $k=2,\ldots,n$. Particularly, inspired by recent work of Baer and Jerison on distributional Hessian determinant, we show that the distributional $k$-Hessian is weak continuous on the Besov space $B(2-\frac{2}{k},k)$, and the result is optimal in the framework of the space $B(s,p)$, i.e., the distributional $k$-Hessian is well defined in $B(s,p)$ if and only if $B(s,p)\subset B_{loc}(2-\frac{2}{k},k)$.Comment: 20 page

Currents carried by the subgradient graphs of semi-convex functions and applications to Hessian measure

In this paper we study integer multiplicity rectifiable currents carried by the subgradient (subdifferential) graphs of semi-convex functions on a $n$-dimensional convex domain, and show a weak continuity theorem with respect to pointwise convergence for such currents. As an application, the $k$-Hessian measures are calculated by a different method in terms of currents.Comment: 15 page

Emergence of space and cosmic evolution based on entropic force

In this paper, we propose a model in which an additional pressure due to the effects of the entropic force is added to the ideal fluid. Furthermore, we obtain the dynamic equation in the FRW universe which contains the quantum gravitational effects based on the description of entropic force and emergence of space. Our model can well explain the age of the universe and the effect of the current accelerating expansion. We give the relation between the luminosity distance and the redshift factor, and compare this relation with that of lambda cold dark matter model($\Lambda CDM$ model).Comment: 10 pages, 2 figure

Thermodynamics of the universe bounded by the cosmological event horizon and dominated by the tachyon fluid

Our aim is to investigate the thermodynamic properties of the universe bounded by the cosmological event horizon and dominated by the tachyon fluid. We give two different laws of evolution of our universe. Further, we show the first law and the generalized second law of thermodynamics (GSLT) are both satisfied in two cases, but their properties of the thermodynamic equilibrium are totally different. Besides, under our solutions, we find the validity of the laws of thermodynamics is irrelevant with the parameters of the tachyon fluid. Finally, we conclude that the universe bounded by the cosmological event horizon and dominated by the tachyon fluid has a good thermodynamic description. In turn, the thermodynamic description can provide a good physical interpretation for the dynamic evolution of our universe due to the equivalence between the first law of thermodynamics and the Friedmann equation to some extent.Comment: 12 pages, 1 figur

Generalized Content-Preserving Warps for Image Stitching

Local misalignment caused by global homography is a common issue in image stitching task. Content-Preserving Warping (CPW) is a typical method to deal with this issue, in which geometric and photometric constraints are imposed to guide the warping process. One of its essential condition however, is colour consistency, and an elusive goal in real world applications. In this paper, we propose a Generalized Content-Preserving Warping (GCPW) method to alleviate this problem. GCPW extends the original CPW by applying a colour model that expresses the colour transformation between images locally, thus meeting the photometric constraint requirements for effective image stitching. We combine the photometric and geometric constraints and jointly estimate the colour transformation and the warped mesh vertexes, simultaneously. We align images locally with an optimal grid mesh generated by our GCPW method. Experiments on both synthetic and real images demonstrate that our new method is robust to colour variations, outperforming other state-of-the-art CPW-based image stitching methods

Threshold Trapdoor Functions and Their Applications

We introduce a cryptographic primitive named threshold trapdoor functions (TTDFs), from which we give generic constructions of threshold and revocation encryptions under adaptive corruption model. Then, we show TTDF can be instantiated under the decisional Diffie-Hellman (DDH) assumption and the learning with errors (LWE) assumption. By combining the instantiations of TTDF with the generic constructions, we obtain threshold and revocation encryptions which compare favorably over existing schemes. The experimental results show that our proposed schemes are practical.Comment: 23 pages, 1 figures, 4 table

Complete Solution for the Rainbow Numbers of Matchings

For a given graph $H$ and $n\geq 1$, let $f(n,H)$ denote the maximum number $c$ for which there is a way to color the edges of the complete graph $K_n$ with $c$ colors such that every subgraph $H$ of $K_n$ has at least two edges of the same color. Equivalently, any edge-coloring of $K_n$ with at least $rb(n,H)=f(n,H)+1$ colors contains a rainbow copy of $H$, where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number $rb(n,H)$ is called the {\it rainbow number of $H$}. Erd\H{o}s, Simonovits and S\'{o}s showed that $rb(n,K_3)=n$. In 2004, Schiermeyer used some counting technique and determined the rainbow numbers $rb(n,kK_2)$ for $k\geq 2$ and $n\geq 3k+3$. It is easy to see that $n$ must be at least $2k$. So, for $2k \leq n<3k+3$, the rainbow numbers remain not determined. In this paper we will use the Gallai-Edmonds structure theorem for matchings to determine the exact values for rainbow numbers $rb(n,kK_2)$ for all $k\geq 2$ and $n\geq 2k$, giving a complete solution for the rainbow numbers of matchings.Comment: 20 page

General Spectrum Sensing in Cognitive Radio Networks

The successful operation of cognitive radio (CR) between CR transmitter and CR receiver (CR link) relies on reliable spectrum sensing. To network CRs requires spectrum sensing at CR transmitter and further information regarding the spectrum availability at CR receiver. Redefining the spectrum sensing along with statistical inference suitable for cognitive radio networks (CRN), we mathematically derive conditions to allow CR transmitter forwarding packets to CR receiver under guaranteed outage probability, and prove that the correlation of localized spectrum availability between a cooperative node and CR receiver determines effectiveness of the cooperative scheme. Applying our novel mathematical model to potential hidden terminals in CRN, we illustrate that the allowable transmission region of a CR, defined as neighborhood, is no longer circular shape even in a pure path loss channel model. This results in asymmetric CR links to make bidirectional links generally inappropriate in CRN, though this challenge can be alleviated by cooperative sensing. Therefore, spectrum sensing capability determines CRN topology. For multiple cooperative nodes, to fully utilize spectrum availability, the selection methodology of cooperative nodes is developed due to limited overhead of information exchange. Defining reliability as information of spectrum availability at CR receiver provided by a cooperative node and by applying neighborhood area, we can compare sensing capability of cooperative nodes from both link and network perspectives. In addition, due to lack of centralized coordination in dynamic CRN, CRs can only acquire local and partial information within limited sensing duration, robust spectrum sensing is therefore proposed. Limits of cooperative schemes and their impacts on network operation are also derived.Comment: 26 pages, 7 figures, 47 references, submitted to IEEE Trans. on Information Theor

Clustering Kinetics of Granular Media in Three Dimensions

Three-dimensional molecular dynamics simulations of dissipative particles (~ 10^6) are carried out for studying the clustering kinetics of granular media during cooling. The inter-connected high particle density regions are identified, showing tube-like structures. The energy decay rates as functions of the particle density and the restitution coefficient are obtained. It is found that the probability density function of the particle density obeys an exponential distribution at late stages. Both the fluctuation of density and the mean cluster size of the particle density have power law relations against time during the inelastic coalescing process.Comment: 4 pages, 5 figures, revte

Sketch-pix2seq: a Model to Generate Sketches of Multiple Categories

Sketch is an important media for human to communicate ideas, which reflects the superiority of human intelligence. Studies on sketch can be roughly summarized into recognition and generation. Existing models on image recognition failed to obtain satisfying performance on sketch classification. But for sketch generation, a recent study proposed a sequence-to-sequence variational-auto-encoder (VAE) model called sketch-rnn which was able to generate sketches based on human inputs. The model achieved amazing results when asked to learn one category of object, such as an animal or a vehicle. However, the performance dropped when multiple categories were fed into the model. Here, we proposed a model called sketch-pix2seq which could learn and draw multiple categories of sketches. Two modifications were made to improve the sketch-rnn model: one is to replace the bidirectional recurrent neural network (BRNN) encoder with a convolutional neural network(CNN); the other is to remove the Kullback-Leibler divergence from the objective function of VAE. Experimental results showed that models with CNN encoders outperformed those with RNN encoders in generating human-style sketches. Visualization of the latent space illustrated that the removal of KL-divergence made the encoder learn a posterior of latent space that reflected the features of different categories. Moreover, the combination of CNN encoder and removal of KL-divergence, i.e., the sketch-pix2seq model, had better performance in learning and generating sketches of multiple categories and showed promising results in creativity tasks