Search results for: Commutative Γ-semiring

Authors: H. V. Chen, A. Y. M. Chin, S. Sharmini

Abstract:

Let n be an integer. We show the existence of at least three non-isomorphic non-commutative Latin squares of order n which are embeddable in groups when n ≥ 5 is odd. By using a similar construction for the case when n ≥ 4 is even, we show that certain non-commutative Latin squares of order n are not embeddable in groups.

Keywords: group, Latin square, embedding.

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Authors: Khalid Abdelmoumen, Mustapha Najmeddine, Hussain Ben-Azza

Abstract:

We present linear codes over finite commutative rings which are not necessarily Frobenius. We treat the notion of syndrome decoding by using Pontrjagin duality. We also give a version of Delsarte-s theorem over rings relating trace codes and subring subcodes.

Keywords: Codes, Finite Rings, Pontrjagin Duality, Trace Codes.

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Authors: Jianmin Xing Wei Shao

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Let R be a ring and n a fixed positive integer, we investigate the properties of n-strongly Gorenstein projective, injective and flat modules. Using the homological theory , we prove that the tensor product of an n-strongly Gorenstein projective (flat) right R -module and projective (flat) left R-module is also n-strongly Gorenstein projective (flat). Let R be a coherent ring ,we prove that the character module of an n -strongly Gorenstein flat left R -module is an n-strongly Gorenstein injective right R -module . At last, let R be a commutative ring and S a multiplicatively closed set of R , we establish the relation between n -strongly Gorenstein projective (injective , flat ) R -modules and n-strongly Gorenstein projective (injective , flat ) S−1R-modules. All conclusions in this paper is helpful for the research of Gorenstein dimensions in future.

Keywords: Commutative ring, n-strongly Gorenstein projective, n-Strongly Gorenstein injective, n-strongly Gorenstein flat, S-ring.

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Authors: Amir Hadi Ziaie

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In the present work, we revisit the collapse process of a spherically symmetric homogeneous scalar field (in FRW background) minimally coupled to gravity, when the phase-space deformations are taken into account. Such a deformation is mathematically introduced as a particular type of noncommutativity between the canonical momenta of the scale factor and of the scalar field. In the absence of such deformation, the collapse culminates in a spacetime singularity. However, when the phase-space is deformed, we find that the singularity is removed by a non-singular bounce, beyond which the collapsing cloud re-expands to infinity. More precisely, for negative values of the deformation parameter, we identify the appearance of a negative pressure, which decelerates the collapse to finally avoid the singularity formation. While in the un-deformed case, the horizon curve monotonically decreases to finally cover the singularity, in the deformed case the horizon has a minimum value that this value depends on deformation parameter and initial configuration of the collapse. Such a setting predicts a threshold mass for black hole formation in stellar collapse and manifests the role of non-commutative geometry in physics and especially in stellar collapse and supernova explosion.

Keywords: Gravitational collapse, non-commutative geometry, spacetime singularity, black hole physics.

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Authors: S.Krishnamoorthy, R.Arul Doss

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We introduce the notion of commuting regular Γ- semiring and discuss some properties of commuting regular Γ- semiring. We also obtain a necessary and sufficient condition for Γ-semiring to possess commuting regularity.

Keywords: Commutative Γ-semiring, Idempotent Γ-semiring, Rectangular Γ-band, Commuting regular Γ-semiring, Clifford Γ-semiring.

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Authors: Jianmin Xing, Rufeng Xing

Abstract:

In this paper we study some properties of GC-projective, injective and flat modules, where C is a semidualizing module and we discuss some connections between GC-projective, injective and flat modules , and we consider these properties under change of rings such that completions of rings, Morita equivalences and the localizations.

Keywords: Semidualizing module, C-projective(injective, flat), GC-projective (injective, flat), Commutative ring; Localizations .

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Authors: Z. Altawallbeh

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In this paper, an explicit homotopic function is constructed to compute the Hochschild homology of a finite dimensional free k-module V. Because the polynomial algebra is of course fundamental in the computation of the Hochschild homology HH and the cyclic homology CH of commutative algebras, we concentrate our work to compute HH of the polynomial algebra, by providing certain homotopic function.

Keywords: Exterior algebra, free resolution, free and projective modules, Hochschild homology, homotopic function, symmetric algebra.

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Authors: Jamal Laaouine, Mohammed Elhassani Charkani

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Let p be a prime and let b be an integer. MDS b-symbol codes are a direct generalization of MDS codes. The γ-constacyclic codes of length pˢ over the finite commutative chain ring Fₚm [u]/ < u² > had been classified into four distinct types, where is a nonzero element of the field Fₚm. Let C₃ be a code of Type 3. In this paper, we obtain the b-symbol distance db(C₃) of the code C₃. Using this result, necessary and sufficient conditions under which C₃ is an MDS b-symbol code are given.

Keywords: constacyclic code, repeated-root code, maximum distance separable, MDS codes, b-symbol distance, finite chain rings

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Authors: Praloy Kumar Biswas, Prof. Dipanwita Roy Chowdhury

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Grobner basis calculation forms a key part of computational commutative algebra and many other areas. One important ramification of the theory of Grobner basis provides a means to solve a system of non-linear equations. This is why it has become very important in the areas where the solution of non-linear equations is needed, for instance in algebraic cryptanalysis and coding theory. This paper explores on a parallel-distributed implementation for Grobner basis calculation over GF(2). For doing so Buchberger algorithm is used. OpenMP and MPI-C language constructs have been used to implement the scheme. Some relevant results have been furnished to compare the performances between the standalone and hybrid (parallel-distributed) implementation.

Keywords: Grobner basis, Buchberger Algorithm, Distributed- Parallel Computation, OpenMP, MPI.

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Authors: Hazem M. El-Bakry, Qiangfu Zhao

Abstract:

Recently, fast neural networks for object/face detection were presented in [1-3]. The speed up factor of these networks relies on performing cross correlation in the frequency domain between the input image and the weights of the hidden layer. But, these equations given in [1-3] for conventional and fast neural networks are not valid for many reasons presented here. In this paper, correct equations for cross correlation in the spatial and frequency domains are presented. Furthermore, correct formulas for the number of computation steps required by conventional and fast neural networks given in [1-3] are introduced. A new formula for the speed up ratio is established. Also, corrections for the equations of fast multi scale object/face detection are given. Moreover, commutative cross correlation is achieved. Simulation results show that sub-image detection based on cross correlation in the frequency domain is faster than classical neural networks.

Keywords: Conventional Neural Networks, Fast Neural Networks, Cross Correlation in the Frequency Domain.

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Authors: Areej M. Abduldaim, Nadia M. G. Al-Saidi

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Algebra is one of the important fields of mathematics. It concerns with the study and manipulation of mathematical symbols. It also concerns with the study of abstractions such as groups, rings, and fields. Due to the development of these abstractions, it is extended to consider other structures, such as vectors, matrices, and polynomials, which are non-numerical objects. Computer algebra is the implementation of algebraic methods as algorithms and computer programs. Recently, many algebraic cryptosystem protocols are based on non-commutative algebraic structures, such as authentication, key exchange, and encryption-decryption processes are adopted. Cryptography is the science that aimed at sending the information through public channels in such a way that only an authorized recipient can read it. Ring theory is the most attractive category of algebra in the area of cryptography. In this paper, we employ the algebraic structure called skew -Armendariz rings to design a neoteric algorithm for zero knowledge proof. The proposed protocol is established and illustrated through numerical example, and its soundness and completeness are proved.

Keywords: Cryptosystem, identification, skew π-Armendariz rings, skew polynomial rings, zero knowledge protocol.

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Authors: Ajay Kumar Pandey, Mayank Kumar

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White grub (Coleoptera: Scarabaeidae) is a major destructive pest in western Himalayan region of Uttarakhand. Beetles feed on apple, apricot, plum, walnut etc. during night while, second and third instar grubs feed on live roots of cultivated as well as non-cultivated crops. Collection and identification of scarab beetles through light trap was carried out at Crop Research Centre, Govind Ballab Pant University Pantnagar, Udham Singh Nagar (Uttarakhand) during 2018. Field trials were also conducted in 2018 to evaluate pre and post sown application of different insecticides against the white grub infesting soybean. The insecticides like Carbofuran 3 Granule (G) (750 g a.i./ha), Clothianidin 50 Water Dispersal Granule (WG) (120 g a.i./ha), Fipronil 0.3 G (50 g a.i./ha), Thiamethoxam 25 WG (80 g a.i./ha), Imidacloprid 70 WG (300 g a.i./ha), Chlorantraniliprole 0.4% G(100 g a.i./ha) and mixture of Fipronil 40% and Imidacloprid 40% WG (300 g a.i./ha) were applied at the time of sowing in pre sown experiment while same dosage of insecticides were applied in standing soybean crop during (first fortnight of July). Commutative plant mortality data were recorded after 20, 40, 60 days intervals and compared with untreated control. Total 23 species of white grub beetles recorded on the light trap and Holotrichia serrata Fabricious (Coleoptera: Melolonthinae) was found to be predominant species by recording 20.6% relative abundance out of the total light trap catch (i.e. 1316 beetles) followed by Phyllognathus sp. (14.6% relative abundance). H. rosettae and Heteronychus lioderus occupied third and fourth rank with 11.85% and 9.65% relative abundance, respectively. The emergence of beetles of predominant species started from 15th March, 2018. In April, average light trap catch was 382 white grub beetles, however, peak emergence of most of the white grub species was observed from June to July, 2018 i.e. 336 beetles in June followed by 303 beetles in the July. On the basis of the emergence pattern of white grub beetles, it may be concluded that the Peak Emergence Period (PEP) for the beetles of H. serrata was second fortnight of April for the total period of 15 days. In May, June and July relatively low population of H. serrata was observed. A decreasing trend in light trap catch was observed and went on till September during the study. No single beetle of H. serrata was observed on light trap from September onwards. The cumulative plant mortality data in both the experiments revealed that all the insecticidal treatments were significantly superior in protection-wise (6.49-16.82% cumulative plant mortality) over untreated control where highest plant mortality was 17.28 to 39.65% during study. The mixture of Fipronil 40% and Imidacloprid 40% WG applied at the rate of 300 g a.i. per ha proved to be most effective having lowest plant mortality i.e. 9.29 and 10.94% in pre and post sown crop, followed by Clothianidin 50 WG (120 g a.i. per ha) where the plant mortality was 10.57 and 11.93% in pre and post sown treatments, respectively. Both treatments were found significantly at par among each other. Production-wise, all the insecticidal treatments were found statistically superior (15.00-24.66 q per ha grain yields) over untreated control where the grain yield was 8.25 & 9.13 q per ha. Treatment Fipronil 40% + Imidacloprid 40% WG applied at the rate of 300 g a.i. per ha proved to be most effective and significantly superior over Imidacloprid 70WG applied at the rate of 300 g a.i. per ha.

Keywords: Bio efficacy, insecticide, Holotrichia, soybean, white grub.

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