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The notable increase in the size and dimensions of data have presented challenges for data storage and retrieval. The Bloom filter and its generations, due to efficient space overheads and constant query delays, have been broadly applied to querying memberships of a big data set. However, the Bloom filter and most of the variants regard each element as a 1-dimensional string and adopt multiple different string hashes to project the data. The interesting problem is when the inputs are numerical vectors with high dimensions, it remains unknown whether they can be projected into the Bloom filter in their original format. Furthermore, we investigate whether the projection is random and uniform. To address these problems, this paper presents a new uniform Prime-HD-BKDERhash family and a new Bloom filter (P-HDBF) to retrieve the membership of a big data set with the numerical high dimensions. Since the randomness and uniformity of data mapping determines the performance of the Bloom filter, to verify these properties, we first introduce information entropy. Our theoretical and experimental results show that the P-HDBF can randomly and uniformly map the data in their native formats. Moreover, the P-HDBF provides an efficient solution alternative to implement membership search with space-time overheads. This advantage may be suitable for engineering applications that are resource-constrained or identification of the nuances of the graphics and images. [ABSTRACT FROM AUTHOR]

In this paper we introduce n -fold (positive) implicative basis logic and the related algebras called n -fold (positive) implicative BL-algebras. Also we define n -fold (positive) implicative filters and we prove some relations between these filters and construct quotient algebras via these filters. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a sharpness dependent filter design based on the fairing of surface normal, whereby the filtering algorithm automatically selects a filter. This may be a mean-filter, a min-filter, or a filter ranked between these two, depending on the local sharpness value and the sharpness dependent weighting function selected. To recover the original shape of a noisy model, the algorithm selects a mean-filter for flat regions and a min-filter for distinguished sharp regions. The selected sharpness dependent weighting function has a Gaussian, Laplacian, or El Fallah Ford form that approximately fits the sharpness distribution found in all tested noisy models. We use a sharpness factor in the weighting function to control the degree of feature preserving. The appropriate sharpness factor can be obtained by sharpness analysis based on the Bayesian classification. Our experiment results demonstrate that the proposed sharpness dependent filter is superior to other approaches for smoothing a polygon mesh, as well as for preserving its sharp features. [Copyright &y& Elsevier]

In this paper, we show that the filters and local filters are equivalent in pseudo-effect algebras. Ideals and local ideals and generalized ideals are equivalent in the pseudo-effect algebras, too. [ABSTRACT FROM AUTHOR]

We make a start on the investigation of “small” solutions to systems of homogeneous linear equations over non-commutative division algebras. In this paper we prove some upper and lower bounds for the sizes of solutions to such systems. To measure solutions and coefficient matrices we define heights which satisfy natural invariance and finiteness properties. [Copyright &y& Elsevier]

This paper presents a new method for designing orthonormal wavelet filter banks using the Remez exchange algorithm. It is well known that orthonormal wavelet bases can be generated by paraunitary filter banks. Thus, synthesis of orthonormal wavelet bases can be reduced to the design of paraunitary filter banks. From the orthonormality and regularity of wavelets, we derive some conditions that must be satisfied for FIR paraunitary filter banks, and investigate the relationship between the filter coefficients and their zeros. According to the relationship, we apply the Remez exchange algorithm in the z-domain directly and formulate the design problem in the form of a linear problem. Therefore, we can easily get a set of filter coefficients by solving the linear equations. Optimal solutions with an arbitrary regularity can be obtained after using an iteration procedure. In the proposed method, the main advantage is that less computational complexity is required as compared with the conventional methods. © 1998 Scripta Technica. Electron Comm Jpn Pt 3, 81(6): 1–8, 1998 [ABSTRACT FROM AUTHOR]