Ten commandments for good default expression simplification

This article provides goals for the design and improvement of default computer algebra expression simplification. These goals can also help users recognize and partially circumvent some limitations of their current computer algebra systems. Although motivated by computer algebra, many of the goals are also applicable to manual simplification, indicating what transformations are necessary and sufficient for good simplification when no particular canonical result form is required.After motivating the ten goals, the article then explains how the Altran partially factored form for rational expressions was extended for Derive and for the computer algebra in Texas Instruments products to help fulfill these goals. In contrast to the distributed Altran representation, this recursive partially factored semi-fraction form:

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does not unnecessarily force common denominators,

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discovers and preserves significantly more factors,

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can represent general expressions, and

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can produce an entire spectrum from fully factored over a common denominator through complete multivariate partial fractions, including a dense subset of all intermediate forms.

     

A local construction of the Smith normal form of a matrix polynomial

We present an algorithm for computing a Smith form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and then combines them into a global Smith form, whereas other algorithms apply a sequence of unimodular row and column operations to the original matrix. The performance of the algorithm in exact arithmetic is reported for several test cases.     

On the decoding of binary cyclic codes with the Newton identities

We revisit in this paper the concept of decoding binary cyclic codes with Gröbner bases. These ideas were first introduced by Cooper, then Chen, Reed, Helleseth and Truong, and eventually by Orsini and Sala. We discuss here another way of putting the decoding problem into equations: the Newton identities. Although these identities have been extensively used for decoding, the work was done manually, to provide formulas for the coefficients of the locator polynomial. This was achieved by Reed, Chen, Truong and others in a long series of papers, for decoding quadratic residue codes, on a case-by-case basis. It is tempting to automate these computations, using elimination theory and Gröbner bases.Thus, we study in this paper the properties of the system defined by the Newton identities, for decoding binary cyclic codes. This is done in two steps, first we prove some facts about the variety associated with this system, then we prove that the ideal itself contains relevant equations for decoding, which lead to formulas.Then we consider the so-called online Gröbner basis decoding, where the work of computing a Gröbner basis is done for each received word. It is much more efficient for practical purposes than preprocessing and substituting into the formulas. Finally, we conclude with some computational results, for codes of interesting length (about one hundred).     

An explicit duality for quasi-homogeneous ideals

Given r≥n$r\ge n$ quasi-homogeneous polynomials in n$n$ variables, the existence of a certain duality is shown and made explicit in terms of generalized Morley forms. This result, that can be seen as a generalization of [Corollary 3.6.14, Jouanolou, J.-P., 1996. Résultant anisotrope, compléments et applications. Electron. J. Combin. 3 (2), Research Paper 2, approx. 91 pp. (electronic), the Foata Festschrift] where this duality is proved in the case r=n$r=n$, was observed by the author at the same time. We will actually closely follow the proof of the above-cited corollary in this paper.     

基于相互认证和3DES加密的智能卡远程支付系统认证方案

针对现有基于智能卡支付系统的安全方案存在密码暴露、信息泄露和身份认证等问题,提出一种新的基于相互认证和3DES加密的智能卡远程支付系统认证方案。分析基于二次剩余的支付认证方案的不足,在注册、登录、身份认证和密码更改阶段对其进行改进,避免密码暴露攻击,提高密码更改阶段的安全性,同时结合3DES加密算法对支付信息进行加密处理。性能分析表明,该方案能有效抵御多种攻击,且用户能够自由地修改密码,同时可对用户信息进行匿名保护。与现有智能卡支付认证方案相比,该方案提高了支付系统的安全性能且具有较小的计算复杂度。     

Imaging and Doppler parameter estimation for maneuvering target using axis mapping based coherently integrated cubic phase function

In this paper, we propose a novel imaging and Doppler parameter estimation algorithm for ground maneuvering targets. Since the cross-track acceleration will induce the quadratic chirp rate (third-order phase) in the phase history, it may cause the maneuvering target severely smeared in the Doppler domain. To obtain a well-focused target imaging result, the quadratic chirp rate must be estimated accurately. Though cubic phase function (CPF) is efficient in estimating the parameters of a single maneuvering target, it may suffer from the identifiability problem when dealing with multiple maneuvering targets. To address these issues, an axis mapping (AM) based coherently integrated cubic phase function (CICPF) algorithm is proposed. This algorithm consists of two stages. Firstly, the linear chirp rate migration (i.e. quadratic chirp rate) of target in the time and chirp-rate domain is corrected by AM. After that, a dechirping technique is utilized to coherently integrate the auto-terms, and suppress the cross-terms and spurious peaks. Compared with several existing quadratic chirp rate estimation approaches, AM based CICPF (AMCICPF) algorithm can acquire lower signal-to-noise ratio threshold and estimate the centroid frequency, chirp rate and quadratic chirp rate of maneuvering target simultaneously. By compensating the chirp rate and quadratic chirp rate, a finely focused maneuvering target imaging can be obtained. Both simulated and real data processing results show that the AMCICPF algorithm serves as a good candidate for maneuvering target Doppler parameter estimation and imaging.     

Time-relevant stability of 2D systems

For many 2D systems, one of the independent variables plays a distinct role in the evolution of the trajectories; since often this special independent variable is time, we call such systems ‘time-relevant’. In this paper, we introduce a stability notion for time-relevant systems described by higher-order difference equations. We give algebraic tests in terms of the location of the zeros of the determinant of a polynomial matrix describing the system. We also give an LMI characterization of time-relevant stability involving only constant matrices.     

Identification and data-driven model reduction of state-space representations of lossless and dissipative systems from noise-free data

We illustrate procedures to identify a state-space representation of a lossless or dissipative system from a given noise-free trajectory; important special cases are passive systems and bounded-real systems. Computing a rank-revealing factorization of a Gramian-like matrix constructed from the data, a state sequence can be obtained; the state-space equations are then computed by solving a system of linear equations. This idea is also applied to perform model reduction by obtaining a balanced realization directly from data and truncating it to obtain a reduced-order model.     

Estimating life-form cover fractions in California sage scrub communities using multispectral remote sensing

The California sage scrub (CSS) community type in California's Mediterranean-type ecosystems is known for its high biodiversity and is home to a large number of rare, threatened, and endangered species. Because of extensive urban development in the past fifty years, this ecologically significant community type is highly degraded and fragmented. To conserve endangered CSS communities, monitoring internal conditions of communities is as crucial as monitoring distributions of the community type in the region. Vegetation type mapping and field sampling of individual plants provide ecologically meaningful information about CSS communities such as spatial distribution and species compositions, respectively. However, both approaches only provide spatially comprehensive information but no information about internal conditions or vice versa. Therefore, there is a need for monitoring variables which fill the information gap between vegetation type maps and field-based data. A number of field-based studies indicate that life-form fractional cover is an effective indicator of CSS community health and habitat quality for CSS-obligated species. This study investigates the effectiveness of remote sensing approaches for estimating fractional cover of true shrub, subshrub, herb, and bare ground in CSS communities of southern California. Combinations of four types of multispectral imagery ranging from 0.15 m resolution scanned color infrared aerial photography to 10 m resolution SPOT 5 multispectral imagery and three image processing models - per-pixel, object-based, and spectral mixture models - were tested.An object-based image analysis (OBIA) routine consistently yielded higher accuracy than other image processing methods for estimating all cover types. Life-form cover was reliably predicted, with error magnitudes as low as 2%. Subshrub and herb cover types required finer spatial resolution imagery for more accurate predictions than true shrub and bare ground types. Positioning of sampling grids had a substantial impact on the reliability of accuracy assessment, particularly for cover estimates predicted using multiple endmember spectral mixture analysis (MESMA) applied to SPOT imagery. Of the approaches tested in this study, OBIA using pansharpened QuickBird imagery is one of the most promising approaches because of its high accuracy and processing efficiency and should be tested for more heterogeneous CSS landscapes. MESMA applied to SPOT imagery should also be examined for effectiveness in estimating factional cover over more extensive habitat areas because of its low data cost and potential for conducting retrospective studies of vegetation community conditions.