An Algorithm for Fast Multiplication of Kaluza Numbers

This paper presents a new algorithm for multiplying two Kaluza numbers. Performing this operation directly requires 1024 real multiplications and 992 real additions. We presented in a previous paper an effective algorithm that can compute the same result with only 512 real multiplications and 576 real additions. More effective solutions have not yet been proposed. Nevertheless, it turned out that an even more interesting solution could be found that would further reduce the computational complexity of this operation. In this article, we propose a new algorithm that allows one to calculate the product of two Kaluza numbers using only 192 multiplications and 384 additions of real numbers.

Optimal fast Johnson–Lindenstrauss embeddings for large data sets

Johnson–Lindenstrauss embeddings are widely used to reduce the dimension and thus the processing time of data. To reduce the total complexity, also fast algorithms for applying these embeddings are necessary. To date, such fast algorithms are only available either for a non-optimal embedding dimension or up to a certain threshold on the number of data points. We address a variant of this problem where one aims to simultaneously embed larger subsets of the data set. Our method follows an approach by Nelson et al. (New constructions of RIP matrices with fast multiplication and fewer rows. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1515-1528, 2014): a subsampled Hadamard transform maps points into a space of lower, but not optimal dimension. Subsequently, a random matrix with independent entries projects to an optimal embedding dimension. For subsets whose size scales at least polynomially in the ambient dimension, the complexity of this method comes close to the number of operations just to read the data under mild assumptions on the size of the data set that are considerably less restrictive than in previous works. We also prove a lower bound showing that subsampled Hadamard matrices alone cannot reach an optimal embedding dimension. Hence, the second embedding cannot be omitted.

A Fast Multiplication Approach Using a Tree-Based Structure

This paper presents a technique for integer number multiplication using a tree-based structure. In the proposed method, both the generation of the partial products and the addition of partial products are completed in the tree structure. The proposed multiplication approach has been designed in two steps: Firstly, the partial products are generated in a tree-based structure using the fewest numbers of gates. Secondly, diagonal partial products additions have been done by the partial products residing in the diagonal partial product nodes to get a faster multiplication result, where two partial product nodes Pi , and Pk,l are diagonal only if |i - k| = |j - l|where i and k are the multiplicand bits; and j and l are the multiplier bits. The comparative study shows that the proposed multiplication algorithm outperforms the existing techniques; e.g., the proposed 4 × 4 multiplication algorithm improves 50% on the worst case running time complexity over the best known existing ones. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 6(1), Dec 2019 P 20-26

Stability of the 7-3 Compressor Circuit for Wallace Tree. Part I

SummaryTo evaluate our formal verification method on a real-size calculation circuit, in this article, we continue to formalize the concept of the 7-3 Compressor (STC) Circuit [6] for Wallace Tree [11], to define the structures of calculation units for a very fast multiplication algorithm for VLSI implementation [10]. We define the circuit structure of the tree constructions of the Generalized Full Adder Circuits (GFAs). We then successfully prove its circuit stability of the calculation outputs after four and six steps. The motivation for this research is to establish a technique based on formalized mathematics and its applications for calculation circuits with high reliability, and to implement the applications of the reliable logic synthesizer and hardware compiler [5].

Detection methods for influenza A H1N1 virus with special reference to biosensors: a review

H1N1 (Swine flu) is caused by influenza A virus, which is a member of Orthomyxoviridae family. Transmission of H1N1 occurs from human to human through air or sometimes from pigs to humans. The influenza virus has different RNA segments, which can reassert to make new virus strain with the possibility to create an outbreak in unimmunized people. Gene reassortment is a process through which new strains are emerging in pigs, as it has specific receptors for both human influenza and avian influenza viruses. H1N1 binds specifically with an α-2,6 glycosidic bond, which is present in human respiratory tract cells as well as in pigs. Considering the fact of fast multiplication of viruses inside the living cells, rapid detection methods need an hour. Currently, WHO recommended methods for the detection of swine flu include real-time PCR in specific testing centres that take 3–4 h. More recently, a number of methods such as Antigen–Antibody or RT-LAMP and DNA biosensors have also been developed that are rapid and more sensitive. This review describes the various challenges in the diagnosis of H1N1, and merits and demerits of conventional vis-à-vis latest methods with special emphasis on biosensors.