Multivariate Polynomial Factorization by Interpolation Method

Factorization of polynomials arises in numerous areas in symbolic computation. It is an important capability in many symbolic and algebraic computation. There are two type of factorization of polynomials. One is convention polynomial factorization, and the other approximate polynomial factorization. Conventional factorization algorithms use symbolic methods to get exact factors of a polynomial while approximate factorization algorithms use numerical methods to get approximate factors of a polynomial. Symbolic computation often confront intermediate expression swell problem, which lower the efficiency of factorization. The numerical computation is famous for its high efficiency, but it only gives approximate results. In this paper, we present an algorithm which use approximate method to get exact factors of a multivariate polynomial. Compared with other methods, this method has the numerical computation advantage of high efficiency for some class of polynomials with factors of lower degree. The experimental results show that the method is more efficient than {\it factor} in Maple 9.5 for polynomials with more variables and higher degree

A Study on Artificial Intelligence IQ and Standard Intelligent Model

Currently, potential threats of artificial intelligence (AI) to human have triggered a large controversy in society, behind which, the nature of the issue is whether the artificial intelligence (AI) system can be evaluated quantitatively. This article analyzes and evaluates the challenges that the AI development level is facing, and proposes that the evaluation methods for the human intelligence test and the AI system are not uniform; and the key reason for which is that none of the models can uniformly describe the AI system and the beings like human. Aiming at this problem, a standard intelligent system model is established in this study to describe the AI system and the beings like human uniformly. Based on the model, the article makes an abstract mathematical description, and builds the standard intelligent machine mathematical model; expands the Von Neumann architecture and proposes the Liufeng - Shiyong architecture; gives the definition of the artificial intelligence IQ, and establishes the artificial intelligence scale and the evaluation method; conduct the test on 50 search engines and three human subjects at different ages across the world, and finally obtains the ranking of the absolute IQ and deviation IQ ranking for artificial intelligence IQ 2014.Comment: 16 pages, 8 figure

Coherent state path integral approach to correlated electron systems with deformed Hubbard operators: from Fermi liquid to Mott insulator

In strongly correlated electron systems the constraint which prohibits the double electron occupation at local sites can be realized by either the infinite Coulomb interaction or the correlated hopping interaction described by the Hubbard operators, but they both render the conventional field theory inapplicable. Relaxing such the constraint leads to a class of correlated hopping models based on the deformed Hubbard operators which smoothly interpolate the locally free and strong coupling limits by a tunable interaction parameter $0\leq \lambda\leq 1$. Here we propose a coherent state path integral approach appropriate to the deformed Hubbard operators for {\it arbitrary} $\lambda$. It is shown that this model system exhibits the correlated Fermi liquid behavior characterized by the enhanced Wilson ratio for all $\lambda$. It is further found that in the presence of on-site Coulomb interaction a finite Mott gap appears between the upper and lower Hubbard bands, with the upper band spectral weight being heavily reduced by $\lambda$. Our approach stands in general spatial dimensions and reveals an unexpected interplay between the correlated hopping and the Coulomb repulsion.Comment: 9 pages, 5 figures (including several appendices

Nonlinear W^∞\hat{W}_{\infty} Current Algebra in the SL(2,R)/U(1) Coset Model

Previously we have established that the second Hamiltonian structure of the KP hierarchy is a nonlinear deformation, called $\hat{W}_{\infty}$, of the linear, centerless $W_{\infty}$ algebra. In this letter we present a free-field realization for all generators of $\hat{W}_{\infty}$ in terms of two scalars as well as an elegant generating function for the $\hat{W}_{\infty}$ currents in the classical conformal $SL(2,R)/U(1)$ coset model. After quantization, a quantum deformation of $\hat{W}_{\infty}$ appears as the hidden current algebra in this model. The $\hat{W}_{\infty}$ current algebra results in an infinite set of commuting conserved charges, which might give rise to $W$-hair for the 2d black hole arising in the corresponding string theory at level $k=9/4$.Comment: 11

On the KP Hierarchy, W^∞\hat{W}_{\infty} Algebra, and Conformal SL(2,R)/U(1) Model --- The Classical and Quantum Cases

We give a unified description of our recent results on the the inter-relationship between the integrable infinite KP hierarchy, nonlinear $\hat{W}_{\infty}$ current algebra and conformal noncompact $SL(2,R)/U(1)$ coset model both at the classical and quantum levels. In particular, we present the construction of a quantum version of the KP hierarchy by deforming the second KP Hamiltonian structure through quantizing the $SL(2,R)_k/U(1)$ model and constructing an infinite set of commuting quantum $\hat{W}_{\infty}$ charges (at least at $k$=1).Comment: (Invited talk given the second at the XXI International Conference on Differential Geometric Methods in Theoretical Physics, Nankai Institute of Mathematics, Tianjin, China; June 5-9, 1992; to appear in Proceedings.), 12p, Latex fil

A Note on Gradually Varied Functions and Harmonic Functions

Any constructive continuous function must have a gradually varied approximation in compact space. However, the refinement of domain for $\sigma-$-net might be very small. Keeping the original discretization (square or triangulation), can we get some interesting properties related to gradual variation? In this note, we try to prove that many harmonic functions are gradually varied or near gradually varied; this means that the value of the center point differs from that of its neighbor at most by 2. It is obvious that most of the gradually varied functions are not harmonic.This note discusses some of the basic harmonic functions in relation to gradually varied functions.Comment: 7 pages and 2 figure

Exact Bivariate Polynomial Factorization in Q by Approximation of Roots

Factorization of polynomials is one of the foundations of symbolic computation. Its applications arise in numerous branches of mathematics and other sciences. However, the present advanced programming languages such as C++ and J++, do not support symbolic computation directly. Hence, it leads to difficulties in applying factorization in engineering fields. In this paper, we present an algorithm which use numerical method to obtain exact factors of a bivariate polynomial with rational coefficients. Our method can be directly implemented in efficient programming language such C++ together with the GNU Multiple-Precision Library. In addition, the numerical computation part often only requires double precision and is easily parallelizable

A Short Note on Zero-error Computation for Algebraic Numbers by IPSLQ

The PSLQ algorithm is one of the most popular algorithm for finding nontrivial integer relations for several real numbers. In the present work, we present an incremental version of PSLQ. For some applications needing to call PSLQ many times, such as finding the minimal polynomial of an algebraic number without knowing the degree, the incremental PSLQ algorithm is more efficient than PSLQ, both theoretically and practically.Comment: 4 page

p-norm-like Constraint Leaky LMS Algorithm for Sparse System Identification

In this paper, we propose a novel leaky least mean square (leaky LMS, LLMS) algorithm which employs a p-norm-like constraint to force the solution to be sparse in the application of system identification. As an extension of the LMS algorithm which is the most widely-used adaptive filtering technique, the LLMS algorithm has been proposed for decades, due to the deteriorated performance of the standard LMS algorithm with highly correlated input. However, both ofthem do not consider the sparsity information to have better behaviors. As a sparse-aware modification of the LLMS, our proposed Lplike-LLMS algorithm, incorporates a p-norm-like penalty into the cost function of the LLMS to obtain a shrinkage in the weight update, which then enhances the performance in sparse system identification settings. The simulation results show that the proposed algorithm improves the performance of the filter in sparse system settings in the presence of noisy input signals.Comment: 3 pages, 1 table, 4 figures, 10 equations, 10 references. arXiv admin note: substantial text overlap with arXiv:1503.0133

Non-Fermi liquid behavior in Bose-Fermi mixtures at two dimensions

In this paper we study the low temperature behaviors of a system of Bose-Fermi mixtures at two dimensions. Within a self-consistent ladder diagram approximation, we show that at nonzero temperatures $T\rightarrow0$ the fermions exhibit non-fermi liquid behavior. We propose that this is a general feature of Bose-Fermi mixtures at two dimensions. An experimental signature of this new state is proposed.Comment: 4 pages, 2 figures. supplementary materia