Search results for: fourier multiplier operators

Authors: Fethi Soltani, Adel Almarashi, Idir Mechai

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Tikhonov regularization and reproducing kernels are the most popular approaches to solve ill-posed problems in computational mathematics and applications. And the Fourier multiplier operators are an essential tool to extend some known linear transforms in Euclidean Fourier analysis, as: Weierstrass transform, Poisson integral, Hilbert transform, Riesz transforms, Bochner-Riesz mean operators, partial Fourier integral, Riesz potential, Bessel potential, etc. Using the theory of reproducing kernels, we construct a simple and efficient representations for some class of Fourier multiplier operators Tm on the Paley-Wiener space Hh. In addition, we give an error estimate formula for the approximation and obtain some convergence results as the parameters and the independent variables approaches zero. Furthermore, using numerical quadrature integration rules to compute single and multiple integrals, we give numerical examples and we write explicitly the extremal function and the corresponding Fourier multiplier operators.

Keywords: fourier multiplier operators, Gauss-Kronrod method of integration, Paley-Wiener space, Tikhonov regularization

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Authors: Xianwei Zheng, Yuan Yan Tang

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Classical window Fourier transform has been widely used in signal processing, image processing, machine learning and pattern recognition. The related Gabor transform is powerful enough to capture the texture information of any given dataset. Recently, in the emerging field of graph signal processing, researchers devoting themselves to develop a graph signal processing theory to handle the so-called graph signals. Among the new developing theory, windowed graph Fourier transform has been constructed to establish a time-frequency analysis framework of graph signals. The windowed graph Fourier transform is defined by using the translation and modulation operators of graph signals, following the similar calculations in classical windowed Fourier transform. Specifically, the translation and modulation operators of graph signals are defined by using the Laplacian eigenvectors as follows. For a given graph signal, its translation is defined by a similar manner as its definition in classical signal processing. Specifically, the translation operator can be defined by using the Fourier atoms; the graph signal translation is defined similarly by using the Laplacian eigenvectors. The modulation of the graph can also be established by using the Laplacian eigenvectors. The windowed graph Fourier transform based on these two operators has been applied to obtain time-frequency representations of graph signals. Fundamentally, the modulation operator is defined similarly to the classical modulation by multiplying a graph signal with the entries in each Fourier atom. However, a single Laplacian eigenvector entry cannot play a similar role as the Fourier atom. This definition ignored the relationship between the translation and modulation operators. In this paper, a new definition of the modulation operator is proposed and thus another time-frequency framework for graph signal is constructed. Specifically, the relationship between the translation and modulation operations can be established by the Fourier transform. Specifically, for any signal, the Fourier transform of its translation is the modulation of its Fourier transform. Thus, the modulation of any signal can be defined as the inverse Fourier transform of the translation of its Fourier transform. Therefore, similarly, the graph modulation of any graph signal can be defined as the inverse graph Fourier transform of the translation of its graph Fourier. The novel definition of the graph modulation operator established a relationship of the translation and modulation operations. The new modulation operation and the original translation operation are applied to construct a new framework of graph signal time-frequency analysis. Furthermore, a windowed graph Fourier frame theory is developed. Necessary and sufficient conditions for constructing windowed graph Fourier frames, tight frames and dual frames are presented in this paper. The novel graph signal time-frequency analysis framework is applied to signals defined on well-known graphs, e.g. Minnesota road graph and random graphs. Experimental results show that the novel framework captures new features of graph signals.

Keywords: graph signals, windowed graph Fourier transform, windowed graph Fourier frames, vertex frequency analysis

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Authors: Hafiz Md. Hasan Babu, Khandaker Mohammad Mohi Uddin, Nitish Biswas, Sarreha Tasmin Rikta, Nuzmul Hossain Nahid

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Nanomedicine and bioengineering use biological systems that can perform computing operations. In a biocomputational circuit, different types of biomolecules and DNA (Deoxyribose Nucleic Acid) are used as active components. DNA computing has the capability of performing parallel processing and a large storage capacity that makes it diverse from other computing systems. In most processors, the multiplier is treated as a core hardware block, and multiplication is one of the time-consuming and lengthy tasks. In this paper, cost-effective DNA multipliers are designed using algorithms of molecular DNA operations with respect to conventional ones. The speed and storage capacity of a DNA multiplier are also much higher than a traditional silicon-based multiplier.

Keywords: biological systems, DNA multiplier, large storage, parallel processing

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Authors: Milad Kaboli

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A new CMOS low voltage current-mode four-quadrant analog multiplier based on the squarer circuit with voltage output is presented. The proposed circuit is composed of a pair of current subtractors, a pair differential-input V-I converters and a pair of voltage squarers. The circuit was simulated using HSPICE simulator in standard 0.18 μm CMOS level 49 MOSIS (BSIM3 V3.2 SPICE-based). Simulation results show the performance of the proposed circuit and experimental results are given to confirm the operation. This topology of multiplier results in a high-frequency capability with low power consumption. The multiplier operates for a power supply ±1.2V. The simulation results of analog multiplier demonstrate a THD of 0.65% in 10MHz, a −3dB bandwidth of 1.39GHz, and a maximum power consumption of 7.1mW.

Keywords: analog processing circuit, WTA, LTA, low voltage

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Authors: A. S. Issa, S. K. Q. AL-Omari

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In this paper, we investigate certain spaces of generalized functions for the Fourier and Fourier type integral transforms. We discuss convolution theorems and establish certain spaces of distributions for the considered integrals. The new Fourier type integral is well-defined, linear, one-to-one and continuous with respect to certain types of convergences. Many properties and an inverse problem are also discussed in some details.

Keywords: Boehmian, Fourier integral, Fourier type integral, generalized quotient

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Authors: Mohammed Husein Mohammad Rashid

Abstract:

For a bounded linear operator T acting on a complex infinite dimensional Hilbert space ℋ, we say that T is ∗-class A operator (abbreviation T∈A*) if |T²|≥ |T*|². In this article, we prove the following assertions:(i) we establish some conditions which imply the normality of ∗-class A; (ii) we consider ∗-class A operator T ∈ ℬ(ℋ) with reducing kernel such that TX = XS for some X ∈ ℬ(K, ℋ) and prove the Fuglede-Putnam type theorem when adjoint of S ∈ ℬ(K) is dominant operators; (iii) furthermore, we extend the asymmetric Putnam-Fuglede theorem the class of ∗-class A operators.

Keywords: fuglede-putnam theorem, normal operators, ∗-class a operators, dominant operators

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Authors: Mohamad Baqer Heidari, Hefzollah.Mohammadian

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This brief presents two original implementations of improved accuracy current-mode multiplier/divider circuits. Besides the advantage of their simplicity, these original multiplier/divider structures present the advantage of very small linearity errors that can be obtained as a result of the proposed design techniques (0.75% and 0.9%, respectively, for an extended range of the input currents). The original multiplier/divider circuits permit a facile reconfiguration, the presented structures representing the functional basis for implementing complex function synthesizer circuits. The proposed computational structures are designed for implementing in 0.18-µm CMOS technology, with a low-voltage operation (a supply voltage of 1.2 V). The circuits’ power consumptions are 60 and 75 µW, respectively, while their frequency bandwidths are 79.6 and 59.7 MHz, respectively.

Keywords: analog signal processing, current-mode operation, functional core, multiplier, reconfigurable circuits, industrial package systems

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Authors: Tulin Coskun

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We investigate the approximation of analytic functions of several variables in polydisc by the sequences of linear k-positive operators in Gadjiev sence. The approximation of analytic functions of complex variable by linear k-positive operators was tackled, and k-positive operators and formulated theorems of Korovkin's type for these operators in the space of analytic functions on the unit disc were introduced in the past. Recently, very general results on convergence of the sequences of linear k-positive operators on a simply connected bounded domain within the space of analytic functions were proved. In this presentation, we extend some of these results to the approximation of analytic functions of several complex variables by sequences of linear k-positive operators.

Keywords: analytic functions, approximation of analytic functions, Linear k-positive operators, Korovkin type theorems

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Authors: M. H. M. Rashid

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A Banach space operator T obeys property (gm) if the isolated points of the spectrum σ(T) of T which are eigenvalues are exactly those points λ of the spectrum for which T − λI is a left Drazin invertible. In this article, we study the stability of property (gm), for a bounded operator acting on a Banach space, under perturbation by finite rank operators, by nilpotent operators, by quasi-nilpotent operators, or more generally by algebraic operators commuting with T.

Keywords: Weyl's Theorem, Weyl Spectrum, Polaroid operators, property (gm)

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Authors: Rishikesh Yadav, Ramakanta Meher, Vishnu Narayan Mishra

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The main aim of this article is to investigate the statistical convergence of the summation of integral type operators and to obtain the weighted statistical convergence. The rate of statistical convergence by means of modulus of continuity and function belonging to the Lipschitz class are also studied. We discuss the convergence of the defined operators by graphical representation and put a better rate of convergence than the Szasz-Mirakjan-Kantorovich operators. In the last section, we extend said operators into bivariate operators to study about the rate of convergence in sense of modulus of continuity and by means of Lipschitz class by using function of two variables.

Keywords: The Szasz-Mirakjan-Kantorovich operators, statistical convergence, modulus of continuity, Peeters K-functional, weighted modulus of continuity

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Authors: G. S. Thakur

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In this paper, four new operators (O1, O2, O3, O4) are proposed, defined and considered to study the new properties and identities on hesitant fuzzy sets. These operators are useful for different operation on hesitant fuzzy sets. The various theorems are proved using the new operators. The study of the proposed new operators has opened a new area of research and applications.

Keywords: vague sets, hesitant fuzzy sets, intuitionistic fuzzy set, fuzzy sets, fuzzy multisets

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Authors: Chiou-Yng Lee, Wen-Yo Lee, Chieh-Tsai Wu, Cheng-Chen Yang

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Shifted polynomial basis (SPB) is a variation of polynomial basis representation. SPB has potential for efficient bit-level and digit-level implementations of multiplication over binary extension fields with subquadratic space complexity. For efficient implementation of pairing computation with large finite fields, this paper presents a new SPB multiplication algorithm based on Karatsuba schemes, and used that to derive a novel scalable multiplier architecture. Analytical results show that the proposed multiplier provides a trade-off between space and time complexities. Our proposed multiplier is modular, regular, and suitable for very-large-scale integration (VLSI) implementations. It involves less area complexity compared to the multipliers based on traditional decomposition methods. It is therefore, more suitable for efficient hardware implementation of pairing based cryptography and elliptic curve cryptography (ECC) in constraint driven applications.

Keywords: digit-serial systolic multiplier, elliptic curve cryptography (ECC), Karatsuba algorithm (KA), shifted polynomial basis (SPB), pairing computation

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Authors: Shams Alyusof

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This work is to enrich the studies of the frames due to their prominent role in pure mathematics as well as in applied mathematics and many applications in computer science and engineering. Recently, there are remarkable studies of operators that preserve frames on some spaces, and this research could be considered as an extension of such studies. Indeed, this paper is to we characterize weighted composition operators that preserve frames in weighted Hardy spaces on the open unit disk. Moreover, it shows that this characterization does not apply to generalized weighted composition operators on such spaces. Nevertheless, this study could be extended to provide more specific characterizations.

Keywords: frames, generalized weighted composition operators, weighted Hardy spaces, analytic functions

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Authors: Nithiya C., Komathi B. J., Praveena N. G., Samuda Prathima

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At present, the ultimate aim in digital system designs, especially at the gate level and lower levels of design abstraction, is power optimization. Adders are a nearly universal component of today's integrated circuits. Most of the research was on the design of high-speed adders to execute addition based on various adder structures. This paper discusses the ideal path for selecting an arithmetic unit for IoT applications. Based on the analysis of eight types of 16-bit adders, we found out Carry Look-ahead (CLA) produces low power. Additionally, multiplier and accumulator (MAC) unit is implemented with the Booth multiplier by using the low power adders in the order of preference. The design is synthesized and verified using Synopsys Design Compiler and VCS. Then it is implemented by using Cadence Encounter. The total power consumed by the CLA based booth multiplier is 0.03527mW, the total area occupied is 11260 um², and the speed is 2034 ps.

Keywords: carry look-ahead, carry select adder, CSA, internet of things, ripple carry adder, design rule check, power delay product, multiplier and accumulator

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Authors: Preeti Sharma

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This paper deals with the Stancu type generalization of Lupaş-Durrmeyer operators. We establish some direct results in the polynomial weighted space of continuous functions defined on the interval [0, 1]. Also, Voronovskaja type theorem is studied.

Keywords: Lupas-Durrmeyer operators, polya distribution, weighted approximation, rate of convergence, modulus of continuity

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Authors: Hyun-Ho Lee, Kee-Won Kim

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The arithmetic operations over GF(2m) have been extensively used in error correcting codes and public-key cryptography schemes. Finite field arithmetic includes addition, multiplication, division and inversion operations. Addition is very simple and can be implemented with an extremely simple circuit. The other operations are much more complex. The multiplication is the most important for cryptosystems, such as the elliptic curve cryptosystem, since computing exponentiation, division, and computing multiplicative inverse can be performed by computing multiplication iteratively. In this paper, we present a parallel computation algorithm that operates Montgomery multiplication over finite field using redundant basis. Also, based on the multiplication algorithm, we present an efficient semi-systolic multiplier over finite field. The multiplier has less space and time complexities compared to related multipliers. As compared to the corresponding existing structures, the multiplier saves at least 5% area, 50% time, and 53% area-time (AT) complexity. Accordingly, it is well suited for VLSI implementation and can be easily applied as a basic component for computing complex operations over finite field, such as inversion and division operation.

Keywords: finite field, Montgomery multiplication, systolic array, cryptography

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Authors: A. Cherifi, B. S. Bouazza, A. O. Dahman, B. Yagoubi

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Transmission over Optical channels will introduce inter-symbol interference (ISI) as well as inter-channel (or inter-carrier) interference (ICI). To decrease the effects of ICI, this paper proposes equalizer for the Optical OFDM system based on the fractional Fourier transform (FrFFT). In this FrFT-OFDM system, traditional Fourier transform is replaced by fractional Fourier transform to modulate and demodulate the data symbols. The equalizer proposed consists of sampling the received signal in the different time per time symbol. Theoretical analysis and numerical simulation are discussed.

Keywords: OFDM, fractional fourier transform, internet and information technology

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Authors: Anna Sikharulidze, Gia Sirbiladze

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For the evaluation of unreliability levels of movement on the closed routes in the vehicle routing problem, the fuzzy operators family is constructed. The interactions between routing factors in extreme conditions on the roads are considered. A multi-criteria decision-making model (MCDM) is constructed. Constructed aggregations are based on the Choquet integral and the associated probability class of a fuzzy measure. Propositions on the correctness of the extension are proved. Connections between the operators and the compositions of dual triangular norms are described. The conjugate connections between the constructed operators are shown. Operators reflect interactions among all the combinations of the factors in the fuzzy MCDM process. Several variants of constructed operators are used in the decision-making problem regarding the assessment of unreliability and possibility levels of movement on closed routes.

Keywords: vehicle routing problem, associated probabilities of a fuzzy measure, choquet integral, fuzzy aggregation operator

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Authors: Neha Bhardwaj

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In this paper, we consider positive linear operators and study the Voronovskaya type result of the operator then obtain an error estimate in terms of the higher order modulus of continuity of the function being approximated and its A-statistical convergence. Also, we compute the corresponding rate of A-statistical convergence for the linear positive operators.

Keywords: Poisson distribution, Voronovskaya, modulus of continuity, a-statistical convergence

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Authors: Ahmad Zahran, Ahmed Herzallah, Ahmad Ahmad, Mahran Quraan

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Generation of high DC voltages is necessary for testing the insulation material of high voltage AC transmission lines with long lengths. The harmonic and ripple contents of the output DC voltage supplied by high voltage DC circuits require the use of costly capacitors to smooth the output voltage after rectification. This paper proposes a new modular multiplier high voltage DC generator with embedded Cockcroft-Walton circuits that achieve a negligible harmonic and ripple contents of the output DC voltage without the need for costly filters to produce a nearly constant output voltage. In this new topology, Cockcroft-Walton modules are connected in series to produce a high DC output voltage. The modules are supplied by low input AC voltage sources that have the same magnitude and frequency and shifted from each other by a certain angle to eliminate the harmonics from the output voltage. The small ripple factor is provided by the smoothing column capacitors and the phase shifted input voltages of the cascaded modules. The constituent harmonics within each module are determined using Fourier analysis. The viability of the proposed DC generator for testing purposes and the effectiveness of the cascaded connection are confirmed by numerical simulations using MATLAB/Simulink.

Keywords: Cockcroft-Walton circuit, harmonics, ripple factor, HVDC generator

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Authors: Safeer Hussain Khan

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In this paper, we prove a strong convergence result using a recently introduced iterative process with contractive-like operators. This improves and generalizes corresponding results in the literature in two ways: the iterative process is faster, operators are more general. In the end, we indicate that the results can also be proved with the iterative process with error terms.

Keywords: contractive-like operator, iterative process, fixed point, strong convergence

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Authors: Safeer Hussain Khan, H. Fukhar-ud-Din

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The concept of quasi-contractive type operators was given by Berinde and extended by Imoru and Olatinwo. They named this new type as contractive-like operators. On the other hand, Xu and Noo introduced a three-step-one-mappings iterative process which can be seen as a generalization of Mann and Ishikawa iterative processes. Approximating common fixed points has its own importance as it has a direct link with minimization problem. Motivated by this, in this paper, we first extend the iterative process of Xu and Noor to the case of three-step-three-mappings and then prove a strong convergence result using contractive-like operators for this iterative process. In general, this generalizes corresponding results using Mann, Ishikawa and Xu-Noor iterative processes with quasi-contractive type operators. It is to be pointed out that our results can also be proved with iterative process involving error terms.

Keywords: contractive-like operator, iterative process, common fixed point, strong convergence

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Authors: A. Cherifi, B. Bouazza, A. O. Dahmane, B. Yagoubi

Abstract:

Transmission over Optical channels will introduce inter-symbol interference (ISI) as well as inter-channel (or inter-carrier) interference (ICI). To decrease the effects of ICI, this paper proposes equalizer for the Optical OFDM system based on the fractional Fourier transform (FrFFT). In this FrFT-OFDM system, traditional Fourier transform is replaced by fractional Fourier transform to modulate and demodulate the data symbols. The equalizer proposed consists of sampling the received signal in the different time per time symbol. Theoretical analysis and numerical simulation are discussed.

Keywords: OFDM, (FrFT) fractional fourier transform, optical OFDM, equalization algorithm

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Authors: Özlem Öksüzer

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In this paper, we introduce variation detracting property of Kantorovich-Stancu type operators in the space of functions of bounded variation. These problems are studied with respect to the variation seminorm.

Keywords: Kantorovich-Stancu type operators, variation seminorm, variation detracting property, absolutely continuous function

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Authors: Nina Ponikvar, Katja Zajc Kejžar

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While the positive impacts of heritage on a broad societal spectrum have long been recognized and measured, the economic effects of the heritage sector are often less visible and frequently underestimated. At macro level, economic effects are usually studied based on one of the two mainstream approach, i.e. either the multiplier approach or the value chain approach. Consequently, there is limited comparability of the empirical results due to the use of different methodological approach in the literature. Furthermore, it is also not clear on which criteria the used approach was selected. Our aim is to bring the attention to the difference in the scope of effects that are encompassed by the two most frequent methodological approaches to valuation of economic effects of cultural heritage on macroeconomic level, i.e. the multiplier approach and the value chain approach. We show that while the multiplier approach provides a systematic, theory-based view of economic impacts but requires more data and analysis, the value chain approach has less solid theoretical foundations and depends on the availability of appropriate data to identify the contribution of cultural heritage to other sectors. We conclude that the multiplier approach underestimates the economic impact of cultural heritage, mainly due to the narrow definition of cultural heritage in the statistical classification and the inability to identify part of the contribution of cultural heritage that is hidden in other sectors. Yet it is not possible to clearly determine whether the value chain method overestimates or underestimates the actual economic impact of cultural heritage since there is a risk that the direct effects are overestimated and double counted, but not all indirect and induced effects are considered. Accordingly, these two approaches are not substitutes but rather complementary. Consequently, a direct comparison of the estimated impacts is not possible and should not be done due to the different scope. To illustrate the difference of the impact assessment of the cultural heritage, we apply both approaches to the case of Slovenia in the 2015-2022 period and measure the economic impact of cultural heritage sector in terms of turnover, gross value added and employment. The empirical results clearly show that the estimation of the economic impact of a sector using the multiplier approach is more conservative, while the estimates based on value added capture a much broader range of impacts. According to the multiplier approach, each euro in cultural heritage sector generates an additional 0.14 euros in indirect effects and an additional 0.44 euros in induced effects. Based on the value-added approach, the indirect economic effect of the “narrow” heritage sectors is amplified by the impact of cultural heritage activities on other sectors. Accordingly, every euro of sales and every euro of gross value added in the cultural heritage sector generates approximately 6 euros of sales and 4 to 5 euros of value added in other sectors. In addition, each employee in the cultural heritage sector is linked to 4 to 5 jobs in other sectors.

Keywords: economic value of cultural heritage, multiplier approach, value chain approach, indirect effects, slovenia

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Authors: Rana Gill, Chandandeep Kaur

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Segmentation of images can be implemented using different fundamental algorithms like edge detection (discontinuity based segmentation), region growing (similarity based segmentation), iterative thresholding method. A comprehensive literature review relevant to the study gives description of different techniques for vehicle number plate detection and edge detection techniques widely used on different types of images. This research work is based on edge detection techniques and calculating threshold on the basis of five edge operators. Five operators used are Prewitt, Roberts, Sobel, LoG and Canny. Segmentation of characters present in different type of images like vehicle number plate, name plate of house and characters on different sign boards are selected as a case study in this work. The proposed methodology has seven stages. The proposed system has been implemented using MATLAB R2010a. Comparison of all the five operators has been done on the basis of their performance. From the results it is found that Canny operators produce best results among the used operators and performance of different edge operators in decreasing order is: Canny>Log>Sobel>Prewitt>Roberts.

Keywords: segmentation, edge detection, text, extracting characters

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Authors: Mehdi Jafari Matehkolaee

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In this paper, we have obtained the adjoint of an arbitrary operator (linear and nonlinear) in Hilbert space by introducing an n-dimensional Riemannian manifold. This general formalism covers every linear operator (non – differential) in Hilbert space. In fact, our approach shows that instead of using the adjoint definition of an operator directly, it can be obtained directly by relying on a suitable generalized space according to the action of the operator in question. For the case of nonlinear operators, we have to change the definition of the linear operator adjoint. But here, we have obtained an adjoint of these operators with respect to the definition of the derivative of the operator. As a matter of fact, we have shown one of the straight applications of the ''Frechet derivative'' in the algebra of the operators.

Keywords: adjoint operator, non-linear operator, differentiable operator, manifold

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Authors: Giulio Nisoli, Jonas Brüngger, Karin Hostettler, Nicole Stoller, Katrin Fischer

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Technical operators in the railway domain are experts responsible for the supervisory control of the railway power grid as well as of the railway tunnels. The technical systems used to master these demanding tasks are constantly increasing in their degree of automation. It becomes therefore difficult for technical operators to maintain the control over the technical systems and the processes of their job. In particular, the operators must have the necessary experience and knowledge in dealing with a malfunction situation or unexpected event. For this reason, it is of growing importance that the skills relevant for the execution of the job are maintained and further developed beyond the basic training they receive, where they are educated in respect of technical knowledge and the work with guidelines. Training methods aimed at improving the cognitive skills needed by technical operators are still missing and must be developed. Goals of the present study were to identify which are the relevant cognitive skills of technical operators in the railway domain and to define which topics should be addressed by the training of these skills. Observational interviews were conducted in order to identify the main tasks and the organization of the work of technical operators as well as the technical systems used for the execution of their job. Based on this analysis, the most demanding tasks of technical operators could be identified and described. The cognitive skills involved in the execution of these tasks are those, which need to be trained. In order to identify and analyze these cognitive skills a cognitive task analysis (CTA) was developed. CTA specifically aims at identifying the cognitive skills that employees implement when performing their own tasks. The identified cognitive skills of technical operators were summarized and grouped in training topics. For every training topic, specific goals were defined. The goals regard the three main categories; knowledge, skills and attitude to be trained in every training topic. Based on the results of this study, it is possible to develop specific training methods to train the relevant cognitive skills of the technical operators.

Keywords: cognitive skills, cognitive task analysis, technical operators in the railway domain, training topics

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Authors: Natália Maria Rego

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ALeibnizn-algebraGis aK-vector space endowed whit a n-linearbracket operation [-,…-] : GG … G→ Gsatisfying the fundamental identity, which can be expressed saying that the right multiplication map Ry2, …, ᵧₙ: Gn→ G, Rᵧ₂, …, ᵧₙn(ˣ¹, …, ₓₙ) = [[ˣ¹, …, ₓₙ], ᵧ₂, …, ᵧₙ], is a derivation. This structure, together with its skew-symmetric version, named as Lie n-algebra or Filippov algebra, arose in the setting of Nambumechanics, an n-ary generalization of the Hamiltonian mechanics. Thefirst goal of this work is to provide a characterization of various classes of central extensions of Leibniz n-algebras in terms of homological properties. Namely, Commutator extension, Quasi-commutator extension, Stem extension, and Stem cover. These kind of central extensions are characterized by means of the character of the map *(E): nHL1(G) → M provided by the five-term exact sequence in homology with trivial coefficients of Leibniz n-algebras associated to an extension E : 0 → M → K → G → 0. For a free presentation 0 →R→ F →G→ 0of a Leibniz n-algebra G,the term M(G) = (R[F,…n.., F])/[R, F,..n-1..,F] is called the Schur multiplier of G, which is a Baer invariant, i.e., it does not depend on the chosen free presentation, and it is isomorphic to the first Leibniz n-algebras homology with trivial coefficients of G. A central extension of Leibniz n-algebras is a short exact sequenceE : 0 →M→K→G→ 0such that [M, K,.. ⁿ⁻¹.., K]=0. It is said to be a stem extension if M⊆[G, .. n.., G]. Additionally, if the induced map M(K) → M(G) is the zero map, then the stem extension Eis said to be a stem cover. The second aim of this work is to analyze the interplay between stem covers of Leibniz n-algebras and the Schur multiplier. Concretely, in the case of finite-dimensional Leibniz n-algebras, we show the existence of coverings, and we prove that all stem covers with finite-dimensional Schur multiplier are isoclinic. Additionally, we characterize stem covers of perfect Leibniz n-algebras.

Keywords: leibniz n-algebras, central extensions, Schur multiplier, stem cover

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Authors: Shiksha Jain, Ramesh Mishra

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One of the most widely used complex signals processing operation is filtering. The most important FIR digital filter are widely used in DSP for filtering to alter the spectrum according to some given specifications. Power consumption and Area complexity in the algorithm of Finite Impulse Response (FIR) filter is mainly caused by multipliers. So we present a multiplier less technique (DA technique). In this technique, precomputed value of inner product is stored in LUT. Which are further added and shifted with number of iterations equal to the precision of input sample. But the exponential growth of LUT with the order of FIR filter, in this basic structure, makes it prohibitive for many applications. The significant area and power reduction over traditional Distributed Arithmetic (DA) structure is presented in this paper, by the use of slicing of LUT to the desired length. An architecture of 16 tap FIR filter is presented, with different length of slice of LUT. The result of FIR Filter implementation on Xilinx ISE synthesis tool (XST) vertex-4 FPGA Tool by using proposed method shows the increase of the maximum frequency, the decrease of the resources as usage saving in area with more number of slices and the reduction dynamic power.

Keywords: multiplier less technique, linear phase symmetric FIR filter, FPGA tool, look up table

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