Throughout this paper, the entire ring will be treated as commutative ring with non-zero identity and all the
modules will be treated as unitary modules. In this paper we shall discuss some remarks on prime submodules in
weak co-multiplication modules.
Fixed Point Theorem in Fuzzy Metric SpaceIJERA Editor
In this present paper on fixed point theorems in fuzzy metric space . we extended to Fuzzy Metric space
generalisation of main theorem .
Mathematics Subject Classification: 47H10, 54A40
PROBABILISTIC INTERPRETATION OF COMPLEX FUZZY SETIJCSEIT Journal
The innovative concept of Complex Fuzzy Set is introduced. The objective of the this paper to investigate
the concept of Complex Fuzzy set in constraint to a traditional Fuzzy set , where the membership function
ranges from [0, 1], but in the Complex fuzzy set extended to a unit circle in a complex plane, where the
member ship function in the form of complex number. The Compressive study of mathematical operation of
Complex Fuzzy set is presented. The basic operation like addition, subtraction, multiplication and division
are described here. The Novel idea of this paper to measure the similarity between two fuzzy relations by
evaluating δ -equality. Here also we introduce the probabilistic interpretation of the complex fuzzy set
where we attempted to clarify the distinction between Fuzzy logic and probability.
The polyadic integer numbers, which form a polyadic ring, are representatives of a fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers, the polyadic Euler totient function, polyadic division with a remainder, etc. are defined. Secondary congruence classes of polyadic integer numbers, which become ordinary residue classes in the binary limit, and the corresponding finite polyadic rings are introduced. Further, polyadic versions of (prime) finite fields are considered. These can be zeroless, zeroless and nonunital, or have several units; it is even possible for all of their elements to be units. There exist non-isomorphic finite polyadic fields of the same arity shape and order. None of the above situations is possible in the binary case. It is conjectured that any finite polyadic field should contain a certain canonical prime polyadic field as a smallest finite subfield, which can be considered a polyadic analogue of GF (p).
Toeplitz Hermitian Positive Definite (THPD) matrices play an important role in signal processing and computer graphics and circular models, related to angular / periodic data, have vide applications in various walks of life. Visualizing a circular model through THPD matrix the required computations on THPD matrices using single bordering and double bordering are discussed. It can be seen that every tridiagonal THPD leads to Cardioid model.
Random Matrix Theory and Machine Learning - Part 1Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning. Part 1 covers: 1. A brief history of Random Matrix Theory, 2. Classical Random Matrix Ensembles (basic building blocks)
IOSR Journal of Electronics and Communication Engineering(IOSR-JECE) is an open access international journal that provides rapid publication (within a month) of articles in all areas of electronics and communication engineering and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in electronics and communication engineering. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.
On The Transition Probabilities for the Fuzzy States of a Fuzzy Markov ChainIJERA Editor
In this paper the theory of fuzzy logic is mixed with the theory of Markov systems and the abstraction of a
Markov system with fuzzy states introduced. The notions such as fuzzy transient, fuzzy recurrent etc., were
introduced. The results based on these notions are introduced.
Design of Microcontroller Based Multi-Frequency Ultrasonic Pulser ReceiverIJERA Editor
Ultrasonic technique is widely accepted as a characterizing tool because of its non destructive nature. Great
technical advancement in ultrasonic apparatuses over the last few years has permitted their increased use. In the
present work, an attempt has been made to develop an ultrasonic pulser receiver for pulse echo technique which
gives pulsed RF of desired frequency on selection from PC without shifting to another oscillator. A
microcontroller (ATMega16) controlled oscillator generates required frequencies on receiving a command. A
pulse repetition frequency (PRF) is generated by microcontroller in such a way to give fixed number of RF
cycles for each selected frequency. The receiver section receives the echoes from the same transducer, rectifies
and amplifies it and is projected to oscilloscope for further study. The designed multi frequency ultrasonic pulser
receiver system has been tested in standard liquids for velocity measurements.
Study of Surface Roughness Characteristics of Drilled Hole in Glass Fiber Rei...IJERA Editor
Now-a-days glass fiber-reinforced plastics (GFRP) are always applicable in variety of engineering applications.
It is used in aerospace, automotive and aircraft industries due to their well known properties. Milling of GFRP
composite materials is very difficult due to its heterogeneity and the number of troubles like as surface
delamination during machining affects uniqueness of the material and the machining parameters. Present paper
focuses the experimental details to find out delamination factor on GFRP composite laminates by using
Taguchi‟s DOE L9 orthogonal array. The main objective of the present work is to optimize the process
parameters in the drilling of GFRP composite using Taguchi DOE and to find the significance of each process
parameter using ANOVA. As far as the effect of input factors are considered, the factors drilled material and
spindle speed both have nearly predominant influence on the delamination factor of drilled holes on GFRP
composite by using CNC milling process.
Soot Formation in Diesel Engines By Using CfdIJERA Editor
In order to meet the stringent emission standards significant efforts have been imparted to the research and
development of cleaner IC engines. Diesel combustion and the formation of pollutants are directly influenced by
spatial and temporal distribution of the fuel injected. The development and validation of computational fluid
dynamics (CFD) models for diesel engine combustion and emissions is described. The complexity of diesel
combustion requires simulations with many complex interacting sub models in order to have a success in
improving the performance and to reduce the emissions. In the present work an attempt has been made to
develop a multidimensional axe-symmetric model for CI engine combustion and emissions. Later simulations
have been carried out. Commercial validation tool FLUENT was used for simulation. The tool solves basic
governing equations of fluid flow that is continuity, momentum, species transport and energy equation. Using
finite volume method turbulence was modeled by using RNG K-ɛ model. Injection was modeled using La
Grangian approach and reaction was modeled using non premixed combustion which considers the effects of
turbulence and detailed chemical mechanism into account to model the reaction rates. The specific heats were
approximated using piecewise polynomials. Subsequently the simulated results have been validated with the
existing experimental values
Theoretical and experimental spectroscopic analysis by FTIR in the effect of ...IJERA Editor
The development of new fibrous composites with specific properties has attracted a big interest in the
development of new technologies. One of the biggest problems in this area is the improvement of the
fiber/matrix interface to increase the mechanical properties in the final composite. In this work, surface chemical
modifications of the rice husk (by-product of the rice industry) were carried out to achieve a better compatibility
with diverse polymeric matrices. These modifications include the use of three different silanes: 3-
(trimethoxysilyl) propyl methacrylate (TMS), dichlorodimethylsilane (DDS) and trichlorovinylsilane (TVS). The
natural fibers and their changes after each treatment were studied experimentally by Fourier Transformed
Infrared Spectroscopy (FTIR). This experimental spectroscopic information was compared with a theoretical
analysis of molecular vibrations using the HyperChem Release 7.0 software as molecular modeling tool. The
result of this comparison confirmed the chemical modification. It was concluded that the chemical modification
was carried out and after an experimental spectroscopic analysis (FTIR) the molecular vibrations data were agree
with those calculated by theoretical analysis being verified in this way the chemical modification of the rice-husk
fiber surface.
Fixed Point Theorem in Fuzzy Metric SpaceIJERA Editor
In this present paper on fixed point theorems in fuzzy metric space . we extended to Fuzzy Metric space
generalisation of main theorem .
Mathematics Subject Classification: 47H10, 54A40
PROBABILISTIC INTERPRETATION OF COMPLEX FUZZY SETIJCSEIT Journal
The innovative concept of Complex Fuzzy Set is introduced. The objective of the this paper to investigate
the concept of Complex Fuzzy set in constraint to a traditional Fuzzy set , where the membership function
ranges from [0, 1], but in the Complex fuzzy set extended to a unit circle in a complex plane, where the
member ship function in the form of complex number. The Compressive study of mathematical operation of
Complex Fuzzy set is presented. The basic operation like addition, subtraction, multiplication and division
are described here. The Novel idea of this paper to measure the similarity between two fuzzy relations by
evaluating δ -equality. Here also we introduce the probabilistic interpretation of the complex fuzzy set
where we attempted to clarify the distinction between Fuzzy logic and probability.
The polyadic integer numbers, which form a polyadic ring, are representatives of a fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers, the polyadic Euler totient function, polyadic division with a remainder, etc. are defined. Secondary congruence classes of polyadic integer numbers, which become ordinary residue classes in the binary limit, and the corresponding finite polyadic rings are introduced. Further, polyadic versions of (prime) finite fields are considered. These can be zeroless, zeroless and nonunital, or have several units; it is even possible for all of their elements to be units. There exist non-isomorphic finite polyadic fields of the same arity shape and order. None of the above situations is possible in the binary case. It is conjectured that any finite polyadic field should contain a certain canonical prime polyadic field as a smallest finite subfield, which can be considered a polyadic analogue of GF (p).
Toeplitz Hermitian Positive Definite (THPD) matrices play an important role in signal processing and computer graphics and circular models, related to angular / periodic data, have vide applications in various walks of life. Visualizing a circular model through THPD matrix the required computations on THPD matrices using single bordering and double bordering are discussed. It can be seen that every tridiagonal THPD leads to Cardioid model.
Random Matrix Theory and Machine Learning - Part 1Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning. Part 1 covers: 1. A brief history of Random Matrix Theory, 2. Classical Random Matrix Ensembles (basic building blocks)
IOSR Journal of Electronics and Communication Engineering(IOSR-JECE) is an open access international journal that provides rapid publication (within a month) of articles in all areas of electronics and communication engineering and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in electronics and communication engineering. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.
On The Transition Probabilities for the Fuzzy States of a Fuzzy Markov ChainIJERA Editor
In this paper the theory of fuzzy logic is mixed with the theory of Markov systems and the abstraction of a
Markov system with fuzzy states introduced. The notions such as fuzzy transient, fuzzy recurrent etc., were
introduced. The results based on these notions are introduced.
Design of Microcontroller Based Multi-Frequency Ultrasonic Pulser ReceiverIJERA Editor
Ultrasonic technique is widely accepted as a characterizing tool because of its non destructive nature. Great
technical advancement in ultrasonic apparatuses over the last few years has permitted their increased use. In the
present work, an attempt has been made to develop an ultrasonic pulser receiver for pulse echo technique which
gives pulsed RF of desired frequency on selection from PC without shifting to another oscillator. A
microcontroller (ATMega16) controlled oscillator generates required frequencies on receiving a command. A
pulse repetition frequency (PRF) is generated by microcontroller in such a way to give fixed number of RF
cycles for each selected frequency. The receiver section receives the echoes from the same transducer, rectifies
and amplifies it and is projected to oscilloscope for further study. The designed multi frequency ultrasonic pulser
receiver system has been tested in standard liquids for velocity measurements.
Study of Surface Roughness Characteristics of Drilled Hole in Glass Fiber Rei...IJERA Editor
Now-a-days glass fiber-reinforced plastics (GFRP) are always applicable in variety of engineering applications.
It is used in aerospace, automotive and aircraft industries due to their well known properties. Milling of GFRP
composite materials is very difficult due to its heterogeneity and the number of troubles like as surface
delamination during machining affects uniqueness of the material and the machining parameters. Present paper
focuses the experimental details to find out delamination factor on GFRP composite laminates by using
Taguchi‟s DOE L9 orthogonal array. The main objective of the present work is to optimize the process
parameters in the drilling of GFRP composite using Taguchi DOE and to find the significance of each process
parameter using ANOVA. As far as the effect of input factors are considered, the factors drilled material and
spindle speed both have nearly predominant influence on the delamination factor of drilled holes on GFRP
composite by using CNC milling process.
Soot Formation in Diesel Engines By Using CfdIJERA Editor
In order to meet the stringent emission standards significant efforts have been imparted to the research and
development of cleaner IC engines. Diesel combustion and the formation of pollutants are directly influenced by
spatial and temporal distribution of the fuel injected. The development and validation of computational fluid
dynamics (CFD) models for diesel engine combustion and emissions is described. The complexity of diesel
combustion requires simulations with many complex interacting sub models in order to have a success in
improving the performance and to reduce the emissions. In the present work an attempt has been made to
develop a multidimensional axe-symmetric model for CI engine combustion and emissions. Later simulations
have been carried out. Commercial validation tool FLUENT was used for simulation. The tool solves basic
governing equations of fluid flow that is continuity, momentum, species transport and energy equation. Using
finite volume method turbulence was modeled by using RNG K-ɛ model. Injection was modeled using La
Grangian approach and reaction was modeled using non premixed combustion which considers the effects of
turbulence and detailed chemical mechanism into account to model the reaction rates. The specific heats were
approximated using piecewise polynomials. Subsequently the simulated results have been validated with the
existing experimental values
Theoretical and experimental spectroscopic analysis by FTIR in the effect of ...IJERA Editor
The development of new fibrous composites with specific properties has attracted a big interest in the
development of new technologies. One of the biggest problems in this area is the improvement of the
fiber/matrix interface to increase the mechanical properties in the final composite. In this work, surface chemical
modifications of the rice husk (by-product of the rice industry) were carried out to achieve a better compatibility
with diverse polymeric matrices. These modifications include the use of three different silanes: 3-
(trimethoxysilyl) propyl methacrylate (TMS), dichlorodimethylsilane (DDS) and trichlorovinylsilane (TVS). The
natural fibers and their changes after each treatment were studied experimentally by Fourier Transformed
Infrared Spectroscopy (FTIR). This experimental spectroscopic information was compared with a theoretical
analysis of molecular vibrations using the HyperChem Release 7.0 software as molecular modeling tool. The
result of this comparison confirmed the chemical modification. It was concluded that the chemical modification
was carried out and after an experimental spectroscopic analysis (FTIR) the molecular vibrations data were agree
with those calculated by theoretical analysis being verified in this way the chemical modification of the rice-husk
fiber surface.
Optimizing Hybrid Wind/Diesel Generator System Using BAT AlgorithmIJERA Editor
Hybrid system comprising of Wind/Diesel generation system for a practical standalone application considers
Wind turbine generators and diesel generator as primary power sources for generating electricity. Battery banks
are considered as a backup power source. The total value of cost is reduced by meeting energy demand required
by the customers. Bat optimization technique is implemented to optimize wind and battery modules. Wind and
battery banks are considered as primary sources and diesel generator as a secondary power source for the system
Experimental Study on Two-Phase Flow in Horizontal Rectangular Minichannel wi...IJERA Editor
An experimental study was conducted to investigate two-phase air-water flow characteristics, in horizontal
rectangular minichannel with Y-junction. The width (W), the height (H) and the hydraulic diameter (DH) of the
rectangular cross section for the upstream side of the junction are 4.60 mm, 2.50 mm and 3.24 mm, while those
for the downstream side are 2.36 mm, 2.50 mm and 2.43 mm. The entire test section was machined from
transparent acrylic block, so that the flow structure could be visualized. Liquid single-phase and air-liquid twophase
flow experiments were conducted at room temperature. The flow pattern, the bubble velocity, the bubble
length, and the void fraction were measured with a high-speed video camera. Pressure profile upstream and
downstream from the junction was also measured for the respective flows, and the pressure loss due to the
contraction at the junction was determined from the pressure profiles. Two flow patterns, i.e., slug and annular
flows, were observed in the fully-developed region apart from the junction. In the analysis, the frictional pressure
drop data, the two-phase frictional multiplier data, bubble velocity data, bubble length data and void fraction data
were compared with calculations by some correlations in literatures. In addition, new pressure loss coefficient
correlations for the pressure drop at the junction has been proposed. Results of such experiment and analysis are
described in the present paper.
Transportation Problem with Pentagonal Intuitionistic Fuzzy Numbers Solved Us...IJERA Editor
This paper presents a solution methodology for transportation problem in an intuitionistic fuzzy environment in
which cost are represented by pentagonal intuitionistic fuzzy numbers. Transportation problem is a particular
class of linear programming, which is associated with day to day activities in our real life. It helps in solving
problems on distribution and transportation of resources from one place to another. The objective is to satisfy
the demand at destination from the supply constraints at the minimum transportation cost possible. The problem
is solved using a ranking technique called Accuracy function for pentagonal intuitionistic fuzzy numbers and
Russell’s Method
Cryptographic Technique Used Lower and Upper Triangular Decomposition MethodIJERA Editor
In this paper, the main cryptographic technique we will use affine cipher used for encryption and also
decryption by using one of the linear algebra technique lower and upper triangular technique
Effect of Rotation on a Layer of Micro-Polar Ferromagnetic Dusty Fluid Heated...IJERA Editor
This paper deals with the theoretical investigation of effect of rotation on micro-polar ferromagnetic dusty fluid
layer heated from below in a porous medium. Linear stability analysis and normal mode analysis methods are
used to find an exact solution for a flat micro-polar ferromagnetic fluid layer contained between two free
boundaries . In case of stationary convection, the effect of various parameters like medium permeability
parameter, non-buoyancy magnetization parameter, micro-polar coupling parameter, spin-diffusion parameter,
micro-polar heat conduction parameter, dust particles parameter and rotation parameter has been analyzed and
results are depicted graphically. In the absence of dust particles, rotation, micro-viscous effect and micro-inertia,
the sufficient condition is obtained for non-oscillatory modes
Interactive E-Lecture Using Video Annotation in Learning GroupsIJERA Editor
Now day‘s users are interested in distance learning as there is rapid growth in digital data due to day today
development in information as well as computer technology. Also its applications or usage have tremendous
response in market. Peoples are attracted towards interactivity in each thing, we found that for e-learning is a
very interactive way to learn and understand things. Currently, YouTube is the global way of video sharing. It is
having certain limitations such as, it having inactivity in online learning. In online study students expecting
some extra guidelines from given resources. In this project we developed video annotation system for foster
active learning. In this project, we achieved active participation of students. There is certain kind of technologies
that extracts some important keywords from textual information. MOOC‘s model is another technology to solve
interaction problem of users in active learning. It also has limitations that it suffered from the problem of
gamification. Our system is interactive as it provides real-time annotations to the video. In our system user can
give their active participation as they have direct interaction to our system. As part of our contribution in this
project we did SVM analysis to provide recommended videos for end users. SVM is Support Vector Machine
algorithm; it classifies the things according to user interest. So, in our system user can search for video and they
get recommended video list for their study.
Solving Transportation Problems with Hexagonal Fuzzy Numbers Using Best Candi...IJERA Editor
In this paper, we introduce a Fuzzy Transportation Problem (FTP) in which the values of transportation costs are
represented as hexagonal fuzzy numbers. We use the Best candidate method to solve the FTP. The Centroid
ranking technique is used to obtain the optimal solution.
A Fast Floating Point Double Precision Implementation on FpgaIJERA Editor
In the modern day digital systems, floating point units are an important component in many signal and image
processing applications. Many approaches of the floating point units have been proposed and compared with
their counterparts in recent years. IEEE 754 floating point standard allows two types of precision units for
floating point operations, single and double. In the proposed architecture double precision floating point unit is
used and basic arithmetic operations are performed. A parallel architecture is proposed along with the high
speed adder, which is shared among other operations and can perform operations independently as a separate
unit. To improve the area efficiency of the unit, carry select adder is designed with the novel resource sharing
technique which allows performing the operations with the minimum usage of the resources while computing
the carry and sum for „0‟ and „1‟. The design is implemented using the Xilinx Spartan 6 FPGA and the results
show the 23% improvement in the speed of the designed circuit
Weldability of Friction Welding Process for AA2024 Alloy and SS304 Stainless ...IJERA Editor
The objective of this work was to assess the weldability of AA2024 alloy and SS304 stainless steel. The process
parameters were frictional time, frictional pressure, rotational speed and forging pressure. The joints were
evaluated for their strength, bulk deformation, penetration and flange formation employing finite element
analysis software code. For friction welding of AA2024 alloy and SS304 stainless steel, the ratio of forging
pressure / frictional pressure should be optimum for good welding.
The Role Played by the Advance Maintenance Practice in manufacturing pеrforma...IJERA Editor
It is not obvious how firms should mеasurе thеir manufacturing pеrformancеs. Various approachеs, most of
thеm with a largе numbеr of mеasurеs on diffеrеnt hiеrarchical lеvеls, еxist. Many of thе mеasurеs usеd arе
considеrеd obsolеtе and inconsistеnt for various rеasons. Thе usеfulnеss of most cost accounting systеms,
individual mеasurеs as wеll as morе comprеhеnsivе activity-basеd costing systеms, arе frеquеntly quеstionеd
sincе thеy do not covеr manufacturing pеrformancеs rеlativе to thе compеtitivе capabilitiеs (е.g. Dixon еt al.,
1990, Whitе, 1996). Anothеr sеrious problеm with most pеrformancе mеasurеmеnt systеms usеd in firms is that
thеy oftеn includе too many diffеrеnt mеasurеs, which makеs it difficult to undеrstand thе "big picturе" (Kееgan
еt al., 1989).
Performance Analysis of Multi-QoS Model of OCDMA System by Adopting OPPM Sign...IJERA Editor
In this paper, optical CDMA which combines the large bandwidth of the fibre medium with the flexibility of the
CDMA technique to achieve high speed connectivity has been used. For achieving this purpose, OPPM
signalling and switching techniques have been employed. This helps in the achievement of high tolerance to
Multiple Access Interference, further resulting in improvement of both the Bit Error Rate and optical channel
capacity without the need to decrease the light pulse width. An OPPM scheme to support multimedia services
with different transmission rates and Quality of Service requirements is proposed. Packet Switching technique
has been employed to help in the transfer of data in the form of packets from the source to the destination via a
specified route. The results are calculated by using PPM signalling and switching technology.But switching
technology has been found to be far better than the OOK-OCDMA if the average power has been considered as
the restraining factor.
Supply Chain Coordination with Buy-back and Wholesale-price Contracts under R...IJERA Editor
In order to increase supply chain total profit, we design a supply chain collaboration mechanism between one
supplier and one retailer. Throughout the paper we present one supplier and one retailer set up with random
demand in a decentralized supply chain. An incentive function on the buy-back cost and wholesale price cost is
presented and scheme through the buy-back contract and wholesale-price contract has been developed to enrich
the retailer take part in the collaboration ways. Furthermore, the paper shows that, how corporation between
buy-back contract and wholesale contract could coordinate the supply chain in both a decentralized and
centralized supply chain. Lastly, we set up numerical analyses and the result shows the implied collaboration
mechanism is not permit the decentralized system to achieve the same performance as the centralized decision
however it allows both members in the supply chain gains profit sharing by setting up the contract parameters.
Physico-Chemical Evaluation of Wastewater from Abattoir, Brewery, Soap and Oi...IJERA Editor
The discharge of industrial wastewater in the city of Moundou deteriorates the quality of surface and
underground water and soils. In this study the physicochemical quality of industrial effluents was investigated in
different seasons (summer, winter and rainy). Three sampling sites were used (Central Abattoir discharge,
Cotontchad (soap and oil factory) discharge, and Brewery discharge), for sampling from July 2013 to December
2014. The following physico-chemical parameters were determined: pH, Temperature, EC, dissolved oxygen,
COD, BOD5, NO3,PO4,SO4. Also, the heavy metals: Cu, Cd, Mn, Ni, Pb, As, Zn, Cr, Fe, Al, was analyzed on
spectrophotometers and results were compared with World Health Organization (WHO) permissible limits.
This study revealed that most parameters were much higher than the permissible limit for wastewater
discharges:some parameters were to higher: pH (12,6), Temperature (37,8 °C), C.E (4270 μS/cm), organic
matters: COD (1200 mg/l), SO4 (1280 mg/l), PO4(4460 mg/l), NO3 (63,6 mg/l), (Fe (63,34 mg/l), Zn (13,27
mg/l), Pb (4,0 mg/l), Cu (25,34 mg/l), Cd (31,78 mg/l), Cr (5,9 mg/l), Ni (39,5 mg/l. The study concludes that
discharge of effluents by the companies; factory and materials from other anthropogenic sources severely
pollute the Logone River with heavy metals and other pollutants. We recommended that each industry recycle
its wastewater and put in place specific treatment plants, because pollutants to eliminate vary depending on the
industry.
Ergonomic Evaluation of the Angle of Abduction in Laptops EnvironmentIJERA Editor
Laptops in 21st century are an integral part of every professional in vivid fields. Off late there has been
emergence of several ergonomic injuries such as repetitive strain injuries (RSIs) due to extensive usage of
laptops, which can be closely linked with applied force and postures. This study investigated the effect of
various angles of keyboard on the applied force and motor action plus response time while performing five
distinct tasks. On the basis of literature two different laptops were selected for performing different tasks. For
each case the three levels of platform angle were considered as 0°, 5°, and 10°. Male subjects were selected to
perform five distinct tasks for each platform angle for both laptops. The force applied (in milli-volts) and the
motor action plus response time (milli-seconds) were recorded using an oscilloscope. The data collected were
analyzed through ANOVA using MINITAB software. The abduction angle with the least mean response time
and applied force were considered as the best from ergonomics viewpoint. The ANOVA results showed that the
angle of abduction for both laptops (small and large) do have significant effect on applied force but not on motor
action plus response time. The analysis of results indicate that 10° angle of abduction in case of small laptops
should be applied to minimize musculoskeletal disorder and repetitive strain injuries.
Research relevance: This work suggests that those responsible for the function and operation of laptops would
have to redesign the system to reduce injuries, as far as musculoskeletal disorder, repetitive strain injuries and
other related problems are concerned. The present work can be quite useful for the system designers of
tomorrow.
A Fuzzy Inventory Model with Perishable and Aging ItemsIJERA Editor
A parametric multi-period inventory model for perishable items considered in this paper. Each item in the stock perishes in a given period of time with some uncertainty. A model derived for recursive unnormalized conditional distributions of {} given the information accumulated about the inventory level- surviving items processes.
In this paper , some results about the concept of completely coretractable ring are studied and also another results about
the coretractable module are recalled and investigated.
Abstract: Throughout this paper, all rings have identities and all modules are unitary right modules.
Let R be a ring and M an R-module. A module M is called coretractable if for each proper submodule
N of M, there exists a nonzero homomorphism f from M/N into M. Our concern in this paper is to
develop basic properties of coretractable modules and to look for any relations between coretractable
modules and other classes of modules.
Keyword:. coretractable modules, multiplication modules, quasi-Dedekind modules, rationally–
injective modules, prime module
ABSTRACT--- Let R be a ring with identity and M be an R-module with unitary . In this paper we introduce the
notion of Y-coretractable module . Some basic properties of this class of modules are investigated and some
relationships between these modules and other related concepts are introduced .
Abstract: Let R be a ring with identity and M be an R-module with unity. In this paper we introduce the notion of C-coretractable modules. Some basic properties of this class of modules are investigated and some relationships between these modules and other related concepts are introduced. Also, we give the notion of strongly C-coretractable and study it comparison with C-coretractable.
Keywords and phrases:(strongly) coretractable modules, (strongly) C-coretractable, mono-coretractable, mono-c-coretractable, simple closed, C-Rickart, multiplication module and quasi-Dedekind module.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Relative superior mandelbrot and julia sets for integer and non integer valueseSAT Journals
Abstract
The fractals generated from the self-squared function,
2 zz c where z and c are complex quantities have been studied
extensively in the literature. This paper studies the transformation of the function , 2 n zz c n and analyzed the z plane and
c plane fractal images generated from the iteration of these functions using Ishikawa iteration for integer and non-integer values.
Also, we explored the drastic changes that occurred in the visual characteristics of the images from n = integer value to n = non
integer value.
Keywords: Complex dynamics,
Relative Superior Julia set, Relative Superior Mandelbrot set.
The polyadic integer numbers, which form a polyadic ring, are representatives of a fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers, the polyadic Euler totient function, polyadic division with a remainder, etc. are defined. Secondary congruence classes of polyadic integer numbers, which become ordinary residue classes in the binary limit, and the corresponding finite polyadic rings are introduced. Further, polyadic versions of (prime) finite fields are considered. These can be zeroless, zeroless and nonunital, or have several units; it is even possible for all of their elements to be units. There exist non-isomorphic finite polyadic fields of the same arity shape and order. None of the above situations is possible in the binary case. It is conjectured that any finite polyadic field should contain a certain canonical prime polyadic field as a smallest finite subfield, which can be considered a polyadic analogue of GF (p).
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Some Remarks on Prime Submodules in Weak Co-Multiplication Modules
1. Arvind Kumar Sinha Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 2, (Part - 1) February 2016, pp.110-114
www.ijera.com 110 | P a g e
Some Remarks on Prime Submodules in Weak Co-Multiplication
Modules
Arvind Kumar Sinha*
, Dibyendu Sasmal**
*
Assistant Professor, Department of Mathematics, National Institute of Technology Raipur, Chhattisgarh,
INDIA.
**
Project Fellow, Department of Mathematics, National Institute of Technology Raipur, Chhattisgarh, INDIA.
ABSTRACT
Throughout this paper, the entire ring will be treated as commutative ring with non-zero identity and all the
modules will be treated as unitary modules. In this paper we shall discuss some remarks on prime submodules in
weak co-multiplication modules.
Mathematical Subject classification: 13C05, 13C13, 13A15, 13C99, 13E05, 16D10, 16D70.
Keywords - Multiplication Modules, Co-multiplication Modules, Weak Multiplication Modules, Weak co-
multiplication Modules, Prime Submodules.
I. INTRODUCTION
In the theory of modules, prime submodules of
modules are very important concept in an algebraic
structure and have obtained a very important
attention in the society of research of this field.
In 1981, Barnad [26] has given the concept of
multiplication modules. After using the concept
prime submodules of modules, the concept of weak
multiplication module has been given by Azizi Shiraz
[6] in the year 2003. Ansari-Toroghy and Farshadifer
[8] has introduced the notion co-multiplication
module as a dual notion of multiplication module in
2007. Using the concept of prime submodule of
modules the concept of weak co-multiplication
module has been given by Atani and Atani [7] in the
year 2009. So in that way we can say that the notion
co-multiplication module is the dual of the notion
weak multiplication module. The aim of this paper is
to provide more information in this field and to
discuss some results on prime submodules in weak
co-multiplication modules.
II. PRELIMINARIES
Throughout this paper, we consider all rings as
commutative rings with identities and all modules as
unital modules. The notation Z will denote the ring of
integers and Q will denote the ring of rational
numbers. In this section, we give some basic
definitions and results for further understanding.
If N and K are submodules of R-module M then
the residual ideal N by K is defined as an ideal
(N :R K) = {r ϵ R | rK N}.
In a special case in which N= 0, the ideal
(0 :R K) is called annihilator of K and it is denoted
by Ann R (K). Also the submodule (0 :M I) is called
the annihilator of I in M and it is denoted by Ann n
(I).
If R is a ring and N is a submodule of an
R-module M, the ideal {r ϵ R | rM N} will be
denoted by [N : M]. Then [0 : M] is the annihilator
of M, Ann (M).
Definition 2.1 [1]: An R-module M is called a
multiplication module if for each submodule N of
M, N = IM, for some ideal I of R. In this case we
can take I = [N : M].
It is easy to see that in this case N = [N: M] M,
where [N : M] = Ann (M/N). (see [17])
Clearly, M is a multiplication module if and only if
for each mM, Rm = [Rm : M] M. (see [2])
For an R-module M, we define the ideal
(M) = Mm
M]:[Rm .
If M is multiplication then M = Mm
Rm
= Mm
M]M:[Rm
= ( Mm
M]:[Rm ) M
= (M) M.
Moreover, if N is a submodule of M, then
N = [N : M] M
= [N : M] (M)M
= (M) [N : M] M
= (M) N. (See [3])
Example 2.2 [4]: Examples of multiplication ideals
(i.e. ideals of a rings R that are multiplication R-
modules) include invertible ideals, principal ideals
and ideals generated by idempotent.
RESEARCH ARTICLE OPEN ACCESS
2. Arvind Kumar Sinha Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 2, (Part - 1) February 2016, pp.110-114
www.ijera.com 111 | P a g e
Sang Cheol Lee, Sunah Kim and Sang-Cho
Chung [5] has defined multiplication module in the
term of extended as follows:
Definition 2.3 [5]: Let R be a commutative ring and
let M be an R-module. Then a submodule N of M is
said to be extended if N = IM, for same ideal I of R.
Definition 2.4 [5]: M is called multiplication module
if every submodule of M is extended. For example,
every proper submodule of the Z-module Z(p∞
) is a
multiplication module but the Z-module Z(p∞
) itself
is not a multiplication module.
In H. Ansari - Toroghy and F. Farshadifar [8]
the notion of a co-multiplication module was
introduced as a dual of the concepts of a
multiplication module.
Definition 2.5 [5]: An R-module m is defined to be a
co-multiplication module if for each submodule N
of M, N = (0 :M I) for same ideal I of R.
In this care we can take I = Ann (N).
Example 2.6: The Z-module Z2
∞
is co-
multiplication module since all of its proper
submodules are of the form (0 :M 2k
Z) for K =
0,1,2,3,4,……
It is clear that M is co multiplication module if and
only if for every submodule N of M, we have Ann M
(Ann R (N)) = N. (see[9])
Example 2.7: Z4 is a co-multiplication Z-module.
Consider the Z-module M = Z / 4Z and set N = 2Z /
4Z. Then N and M/N are co-multiplication
Z-modules. (see [20])
An R-module M is called co-multiplication
module if for any submodule N of M, there exists an
ideal I of such that N = Ann M (I). (see [11] [12])
An R-module M is co-multiplication module if and
only if for any submodule N of M, N = (0 :M Ann
(N)) (see [8] )
Every proper submodule of a co-multiplication
module is co-multiplication module. (see [8])
Example 2.8: If V is a two dimensional vector space
over a filed K then V cannot be co-multiplication
module but every proper submodule as everyone-
dimensional vector space is co-multiplication
module. (see [11])
Example 2.9 [8]: Let p be any prime number. Let
M = Z (p∞
). Then M is a co-multiplication Z-module
where Z is the ring of integers.
Proof: Fix a prime integer p. Define a set
Qp = { r / pt
| r, t Z}
Then, Qp is additive abelian group containing Z.
Define a set M = Z (p∞
) = Qp / Z.
Then, M is a Z- module. Let N be any submodule of
M. Then N = Z(1/pi
+ Z) , for some integer i.( i ≥ 0 ).
Set I = pi
Z.
Now, Ann M (pi
Z) = N. Hence M is a co-
multiplication Z-module.
Now here is an example which shows that not
every R-module is co-multiplication module.
Example 2.10 [11]: Consider Z as a Z-module. Now,
2Z is a submodule of Z, we have
Ann Z (2Z) = {mZ | 2Zm = 0} = (0).
Now, if Z is a co-multiplication module then by [8],
we have
2Z = ( 0 : Z AnnZ (2Z))
But (0 :Z AnnZ (2Z)) = (0 :Z 0 ) = Z ≠ 2Z
Therefore, Z is not a co-multiplication module.
As in [8] the notion of co-multiplication module
was introduced as a dual of the concept of a
multiplication module. An R-module M is called co-
multiplication (co-m for short) if for every
submodule N of M, there exists an Ideal I of R such
that N = (0 :M I).
It is clear that M is co-m if and only if for every
submodule N of M, we have Ann M (Ann R (N)) = N.
Definition 2.11 [10]: A proper submodule N of M is
said to be prime submodule if from r m N for r
R and m M, We can deduce that r (N : M) or
m N. (see for example in [10])
Definition 2.12 [6]: An R-module M is called weak
multiplication module if M doesn’t have any prime
submodule or every prime submodule N of M We
have N = IM where I is an ideal of R.
Example 2.13 [6]: Q is a weak multiplication
Z-module, which is not a multiplication Z-module.
Every finitely generated weak multiplication module
is a multiplication module. (see [6]).
In 2009 Reza Ebrahimi Atani and Shahabaddin
Ebrahimi Atani [7] have given the notion weak co-
multiplication modules as the dual notion of weak
multiplication modules and studied on weak co-
multiplication modules.
Definition 2.14 [7]: Let R be a commutative ring. An
R-module M is defined to be a weak co-
multiplication module if M doesn’t have any prime
submodule or for every prime sub module N of M,
N = (0 :M I) for same ideal I of R.
One can easily show that if M is a weak co-
multiplication module, then N = (0 :M Ann(N)) for
every prime submodule N of M.
We note that every co-multiplication module is weak
co-multiplication module. (see [19])
It should be mentioned that another dual notion of
weak multiplication module is defined in [19] as
second submodule.
3. Arvind Kumar Sinha Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 2, (Part - 1) February 2016, pp.110-114
www.ijera.com 112 | P a g e
Definition 2.15 [19]: A submodule N of M is said to
be a second submodule if rN = N or rN = 0 for
each r R.
Definition 2.16 [19]: An R-module M is a weak co-
multiplication module if M does not have any
second submodule or for every second submodule or
for every second submodule S of M, we have
S = (0 :M I), where I is an ideal of R.
Remark in H. Ansari-Toroghy and F. Farshadifar
[19] it is clear that every co-multiplication R-module
is a weak co-multiplication R-module. However the
converse is not true in general.
For example, the Z-module Q (Here Q denotes the
field of rational numbers) is a weak co-multiplication
Z-module while it is not a co-multiplication
Z-module.
In H. Ansari-Toroghy and F. Farshadifar [19] M
is called weak co-multiplication module when for
each second submodule N of M, there is an ideal I of
R, such that N = (0 :M I) but here we are the term
weak co-multiplication module exclusively in sense
of [7] only.
Definition 2.17 [13]: Let R be a commutative ring.
An R-module L is said to be co-cyclic if L E (R|P)
for some maximal ideal P of R. (See [13] and [14])
Proposition 2.18 [8]: Every co-cyclic module over a
commutative complete Noetherian ring is a co-
multiplication module.
Proof: (See [8])
Definition 2.19 [15]: An R-module M is said to be
semi simple module (resp. co-semisimple module) in
case every submodule of M is the sum (resp.
intersection) of minimal (resp. maximal) submodule.
(Also See [16])
III. MAIN RESULTS
There is a large body of researches
concerning with multiplication modules. It is natural
to ask the following question: to what extent does the
dual of their results hold for co-multiplication
modules. Similar questions arise for weak
multiplication modules and weak co-multiplication
modules. The purpose of this paper is to obtain more
information about this class of modules.
Proposition 3.1 [8]: every co-cyclic module over a
commutative complete Noetherian ring is a co-
multiplication module.
Remark 3.2: Since every co-multiplication module
is weak co-multiplication module therefore every co-
cyclic module over a commutative complete
Noetherian ring is a weak co-multiplication module.
Proposition 3.3 [15]: Let M be an R-module and if
M is co-semi simple module such that Ann R (N) ≠
Ann R (M) for every maximal submodule N of M,
then M is co-multiplication module.
Remark 3.4: Since every co-multiplication module
is weak co-multiplication module. Therefore, let M
be an R-module and if M is co-semisimple module
such that Ann (N) ≠ Ann (M) for every maximal
submodule N of M then M is weak co-multiplication
module.
Proposition 3.5 [15]: Let M be an R-module if R is
Noetherian ring then every injective multiplication
R-module is co-multiplication module.
Remark 3.6: Since every co-multiplication module is
weak co-multiplication module. Therefore, Let M be
an R-module if R is Noetherian ring then every
injective multiplication R-module is weak co-
multiplication module.
As multiplication modules have been studied by
many mathematicians from different point of view. A
most suitable reference is Z. A. EI-Bast and P. F.
Smith [2]. Particularly in H. Ansari - Toroghy and F.
Farshadifar [8] have introduced co-multiplication
modules as a dual to multiplication modules. Under
various conditions co-multiplication modules are
cyclic. (See [18])
Lemma 3.7 [18]:
(1) If a submodule N of M equals (0 :M I) for same
ideal I of R then (N : M) = (Ann (M) :R I)
(2) M is a weak co-multiplication R-module if and
only if it is a weak co-multiplication ( R/Ann
(M) ) - module.
(3) Suppose that M is an R-module and R = R1×R2,
where R1 and R2 are non- trivial rings. Then M =
M1 M2 where M1 is an R1 module and M2 is an
R2-module. Also in this case M is weak co-
multiplication module if and only if both M1 and
M2 are weak co-multiplication modules.
Proof: (see [18])
Thus the results for weak co-multiplication
modules follow the similar proof of co-multiplication
modules.
According to above lemma 3.7, when M is a co-
multiplication module, it may be reduced to a faithful
co-multiplication module. (See [18])
Similarly when M is a weak co-multiplication
module, it may be reduced to a faithful weak co-
multiplication module.
Theorem 3.8 [18]: If M is a faithful weak co-
multiplication R-module with a maximal submodule
4. Arvind Kumar Sinha Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 2, (Part - 1) February 2016, pp.110-114
www.ijera.com 113 | P a g e
N and R is a reduced ring with decomposition as a
finite direct product of indecomposable ring, then M
R and R is semi-simple.
Proof: See [18]
Corollary 3.9 [18]: Assume that M is a weak co-
multiplication module having a maximal submodule
(for example, if M is finitely generated) and Y =
Ann (M). Then Y is a prime ideal if and only if Y is a
maximal ideal and M R / Y is a simple module.
Corollary 3.10 [18]: If M is a finitely generated co-
multiplication module with Ann (M) a radical ideal,
then M is cyclic and R / Ann (M) is a semisimple
ring.
Proof: see [18].
Definition 3.11 [18]: A chained ring is a ring in
which every two ideals are comparable. For example,
localization of Z at any prime ideal or more generally
every valuation domain is a chained ring.
Lemma 3.12 [18]: If R is a chained ring and M is a
co-multiplication module having a maximal
submodule N, then M is cyclic.
Proof: see [18].
Remark 3.13: Since every co-multiplication module
is weak co-multiplication module. (see [19]), so for
the above results, under the various conditions, weak
co-multiplication modules are also cyclic.
Thus since every co-multiplication module
is weak co-multiplication module, so for the above
results, under the various conditions, weak co-
multiplication modules are also cyclic.
IV. Acknowledgement
This research was carried out by grant received
from the funding agency Chhattisgarh Council of
Science and Technology (CCOST) Raipur 492001
INDIA. First and second author would like to thanks
to CCOST Raipur for the financial support.
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