A Heuristic Description of Fast Fourier Transform

Fast Fourier Transform (FFT) is an efficient algorithm to compute the Discrete Fourier Transform (DFT) and its inverse. In this paper, we pay special attention to the description of complex-data FFT. We analyze two common descriptions of FFT and propose a new presentation. Our heuristic description is helpful for students and programmers to grasp the algorithm entirely and deeply

Improvement Of Barreto-Voloch Algorithm For Computing rrth Roots Over Finite Fields

Root extraction is a classical problem in computers algebra. It plays an essential role in cryptosystems based on elliptic curves. In 2006, Barreto and Voloch proposed an algorithm to compute $r$th roots in ${F}_{q^m}$ for certain choices of $m$ and $q$. If $r\,||\,q-1$ and $(m, r)=1,$ they proved that the complexity of their method is $\widetilde{\mathcal {O}}(r(\log m+\log\log q)m\log q)$. In this paper, we extend the Barreto-Voloch algorithm to the general case that $r\,||\,q^m-1$, without the restrictions $r\,||\,q-1$ and $(m, r)=1$. We also specify the conditions that the Barreto-Voloch algorithm can be preferably applied

A Convergence Theorem for the Graph Shift-type Algorithms

Graph Shift (GS) algorithms are recently focused as a promising approach for discovering dense subgraphs in noisy data. However, there are no theoretical foundations for proving the convergence of the GS Algorithm. In this paper, we propose a generic theoretical framework consisting of three key GS components: simplex of generated sequence set, monotonic and continuous objective function and closed mapping. We prove that GS algorithms with such components can be transformed to fit the Zangwill's convergence theorem, and the sequence set generated by the GS procedures always terminates at a local maximum, or at worst, contains a subsequence which converges to a local maximum of the similarity measure function. The framework is verified by expanding it to other GS-type algorithms and experimental results

Characterizing A Database of Sequential Behaviors with Latent Dirichlet Hidden Markov Models

This paper proposes a generative model, the latent Dirichlet hidden Markov models (LDHMM), for characterizing a database of sequential behaviors (sequences). LDHMMs posit that each sequence is generated by an underlying Markov chain process, which are controlled by the corresponding parameters (i.e., the initial state vector, transition matrix and the emission matrix). These sequence-level latent parameters for each sequence are modeled as latent Dirichlet random variables and parameterized by a set of deterministic database-level hyper-parameters. Through this way, we expect to model the sequence in two levels: the database level by deterministic hyper-parameters and the sequence-level by latent parameters. To learn the deterministic hyper-parameters and approximate posteriors of parameters in LDHMMs, we propose an iterative algorithm under the variational EM framework, which consists of E and M steps. We examine two different schemes, the fully-factorized and partially-factorized forms, for the framework, based on different assumptions. We present empirical results of behavior modeling and sequence classification on three real-world data sets, and compare them to other related models. The experimental results prove that the proposed LDHMMs produce better generalization performance in terms of log-likelihood and deliver competitive results on the sequence classification problem

Martin points on open manifolds of non-positive curvature

The Martin boundary of a Cartan-Hadamard manifold describes a fine geometric structure at infinity, which is a sub-space of positive harmonic functions. We describe conditions which ensure that some points of the sphere at infinity belong to the Martin boundary as well. In the case of the universal cover of a compact manifold with Ballmann rank one, we show that Martin points are generic and of full harmonic measure. The result of this paper provides a partial answer to an open problem of S. T. Yau

Non-parametric Power-law Data Clustering

It has always been a great challenge for clustering algorithms to automatically determine the cluster numbers according to the distribution of datasets. Several approaches have been proposed to address this issue, including the recent promising work which incorporate Bayesian Nonparametrics into the $k$-means clustering procedure. This approach shows simplicity in implementation and solidity in theory, while it also provides a feasible way to inference in large scale datasets. However, several problems remains unsolved in this pioneering work, including the power-law data applicability, mechanism to merge centers to avoid the over-fitting problem, clustering order problem, e.t.c.. To address these issues, the Pitman-Yor Process based k-means (namely \emph{pyp-means}) is proposed in this paper. Taking advantage of the Pitman-Yor Process, \emph{pyp-means} treats clusters differently by dynamically and adaptively changing the threshold to guarantee the generation of power-law clustering results. Also, one center agglomeration procedure is integrated into the implementation to be able to merge small but close clusters and then adaptively determine the cluster number. With more discussion on the clustering order, the convergence proof, complexity analysis and extension to spectral clustering, our approach is compared with traditional clustering algorithm and variational inference methods. The advantages and properties of pyp-means are validated by experiments on both synthetic datasets and real world datasets

Critical behaviors and local transformation properties of wave function

We investigate crossing behavior of ground state entanglement Renyi entropies of quantum critical systems. We find a novel property that the ground state in one quantum phase cannot be locally transferred to that of another phase, that means a global transformation is necessary. This also provides a clear evidence to confirm the long standing expectation that entanglement Renyi entropy contains more information than entanglement von Neumann entropy. The method of studying crossing behavior of entanglement Renyi entropies can distinguish different quantum phases well. We also study the excited states which still give interesting results.Comment: 4 page

Poker-CNN: A Pattern Learning Strategy for Making Draws and Bets in Poker Games

Poker is a family of card games that includes many variations. We hypothesize that most poker games can be solved as a pattern matching problem, and propose creating a strong poker playing system based on a unified poker representation. Our poker player learns through iterative self-play, and improves its understanding of the game by training on the results of its previous actions without sophisticated domain knowledge. We evaluate our system on three poker games: single player video poker, two-player Limit Texas Hold'em, and finally two-player 2-7 triple draw poker. We show that our model can quickly learn patterns in these very different poker games while it improves from zero knowledge to a competitive player against human experts. The contributions of this paper include: (1) a novel representation for poker games, extendable to different poker variations, (2) a CNN based learning model that can effectively learn the patterns in three different games, and (3) a self-trained system that significantly beats the heuristic-based program on which it is trained, and our system is competitive against human expert players.Comment: 8 page

Gaussian quantum steering and its asymmetry in curved spacetime

We study Gaussian quantum steering and its asymmetry in the background of a Schwarzschild black hole. We present a Gaussian channel description of quantum state evolution under the influence of the Hawking radiation. We find that thermal noise introduced by Hawking effect will destroy the steerability between an inertial observer Alice and an accelerated observer Bob who hovers outside the event horizon, while it generates steerability between Bob and a hypothetical observer anti-Bob inside the event horizon. Unlike entanglement behaviors in curved spacetime, here the steering from Alice to Bob suffers from a "sudden death" and the steering from anti-Bob to Bob experiences a "sudden birth" with increasing Hawking temperature. We also find that the Gaussian steering is always asymmetric and the maximum steering asymmetry cannot exceed $\ln 2$, which means the state never evolves to an extremal asymmetry state. Furthermore, we obtain the parameter settings that maximize steering asymmetry and find that (i) $s=arccosh(\frac{\cosh^2 r}{1-\sinh^2r})$ is the critical point of steering asymmetry, and (ii) the attainment of maximal steering asymmetry indicates the transition between one-way steerability and both-way steerability for the two-mode Gaussian state under the influence of Hawking radiation.Comment: 7 pages, 3 figures, to appear in Phys. Rev.

Tunable Band Topology Reflected by Fractional Quantum Hall States in Two-Dimensional Lattices

Two-dimensional lattice models subjected to an external effective magnetic field can form nontrivial band topologies characterized by nonzero integer band Chern numbers. In this Letter, we investigate such a lattice model originating from the Hofstadter model and demonstrate that the band topology transitions can be realized by simply introducing tunable longer-range hopping. The rich phase diagram of band Chern numbers is obtained for the simple rational flux density and a classification of phases is presented. In the presence of interactions, the existence of fractional quantum Hall states in both |C|=1 and |C|>1 bands is confirmed, which can reflect the band topologies in different phases. In contrast, when our model reduces to a one-dimensional lattice, the ground states are crucially different from fractional quantum Hall states. Our results may provide insights into the study of new fractional quantum Hall states and experimental realizations of various topological phases in optical lattices.Comment: published version (6 pages, 6 figures, including a supplemental material