1) The document discusses properties of abelian categories, including that kernels and cokernels are unique up to isomorphism.
2) It proves that in an abelian category, a morphism is a monomorphism if and only if its kernel is the zero object, and it is an epimorphism if and only if its cokernel is the zero object.
3) It also proves that in an abelian category, every bimorphism is an isomorphism, and that every monomorphism is the kernel of some morphism.
The Mean Value Theorem is the Most Important Theorem in Calculus. It allows us to relate information about the derivative of a function to information about the function itself.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
This chapter discusses 4 concepts: (1) function relations involving domains and ranges, (2) absolute value functions, (3) composite functions using substitution, and (4) inverse functions by making "x" equal to "y" and solving for y. Examples are provided for each concept to demonstrate finding outputs and inverses of simple functions.
We discuss the ideas of monotonicity (increasing or decreasing) and concavity (up or down) of a function. Because of the Mean Value Theorem, we can determine these characteristics using derivatives.
The document is a lecture on derivatives and the shapes of curves. It covers the mean value theorem, testing for monotonicity using the first derivative test, and finding intervals of monotonicity. It also discusses concavity and the second derivative test. Examples are provided to demonstrate how to find the intervals where a function is increasing or decreasing using the first derivative test.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
The Mean Value Theorem states that if a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists some value c in the interval (a,b) such that the slope of the tangent line at c is equal to the average rate of change of f over the entire interval [a,b]. In other words, the derivative of f at c is equal to the slope of the secant line through points (a, f(a)) and (b, f(b)). The Mean Value Theorem provides a relationship between the derivative of a function at a point and the change in function value over an interval containing that point.
The document introduces functions and some of their properties. It defines a function as a relation where each element in the domain is mapped to exactly one element in the codomain. It discusses one-to-one, onto, and bijective functions. It also covers composition, associativity, identity functions, and the pigeonhole principle as it relates to functions between finite sets.
The Mean Value Theorem is the Most Important Theorem in Calculus. It allows us to relate information about the derivative of a function to information about the function itself.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
This chapter discusses 4 concepts: (1) function relations involving domains and ranges, (2) absolute value functions, (3) composite functions using substitution, and (4) inverse functions by making "x" equal to "y" and solving for y. Examples are provided for each concept to demonstrate finding outputs and inverses of simple functions.
We discuss the ideas of monotonicity (increasing or decreasing) and concavity (up or down) of a function. Because of the Mean Value Theorem, we can determine these characteristics using derivatives.
The document is a lecture on derivatives and the shapes of curves. It covers the mean value theorem, testing for monotonicity using the first derivative test, and finding intervals of monotonicity. It also discusses concavity and the second derivative test. Examples are provided to demonstrate how to find the intervals where a function is increasing or decreasing using the first derivative test.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
The Mean Value Theorem states that if a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists some value c in the interval (a,b) such that the slope of the tangent line at c is equal to the average rate of change of f over the entire interval [a,b]. In other words, the derivative of f at c is equal to the slope of the secant line through points (a, f(a)) and (b, f(b)). The Mean Value Theorem provides a relationship between the derivative of a function at a point and the change in function value over an interval containing that point.
The document introduces functions and some of their properties. It defines a function as a relation where each element in the domain is mapped to exactly one element in the codomain. It discusses one-to-one, onto, and bijective functions. It also covers composition, associativity, identity functions, and the pigeonhole principle as it relates to functions between finite sets.
Abstract: Password authentication is one of the simplest and the most convenient authentication mechanisms to deal with secret data over insecure networks. Every user now holds email account for different web sites or holds a username password for their office work in their computer system. In this paper, we shall present the result of our survey through basic currently available password-authentication-related schemes. We know that Apart from getting the password hacked by some intruder, a serious threats comes from the surrounding environment, specially the colleagues, friends or visitors nearby the system. Most of them have a tendency of observing keenly while you cast your password. Here we propose a simple way to provide a new way of password through clicks that would provide a secure mechanism from both the intruders and the people nearby while you type your password in the application.
Abstract: File replication is an effective way to enhance file availability and reduce file querying delay. It creates replicas for a file to improve its probability of being encountered by requests. Here, we introduce a new concept of resource for file replication, which considers both node storage and node meeting ability. We theoretically study the influence of resource allocation on the average querying delay and derive an optimal file replication rule (OFRR) that allocates resources to each file based on its popularity and size. We then propose a file replication protocol based on the rule, which approximates the minimum global querying delay in a fully distributed manner.
Abstract: The main communication methods used by deaf people are sign language, but opposed to common thought, there is no specific universal sign language: every country or even regional group uses its own set of signs. The use of sign language in digital systems can enhance communication in both directions: animated avatars can synthesize signals based on voice or text recognition; and sign language can be translated into various text or sound forms based on different images, videos and sensors input. The ultimate goal of this research, but it is not a simple spelling of spoken language, so that recognizing different signs or letters of the alphabet (which has been a common approach) is not sufficient for its transcription and automatic interpretation. Here proposes an algorithm and method for an application this would help us in recognising the various user defined signs. The palm images of right and left hand are loaded at runtime. Firstly these images will be seized and stored in directory. Then technique called Template matching is used for finding areas of an image that match (are similar) to a template image (patch). Our goal is to detect the highest matching area. We need two primary components- A) Source image (I): In the template image in which we try to find a match. B) Template image (T): The patch image which will be compared to the template image. In proposed system user defined patterns will be having 60% accuracy while default patterns will be provided with 80% accuracy.
Privacy Preserving Data Leak Detection for Sensitive Datapaperpublications3
Abstract: Number of data leaks in the organization, research institutions and security firms have grown rapidly in recent years. The data leakage occurs if there is no proper protection. The common approach is to monitor the data that are stored in the organization local network. The existing method require the plaintext sensitive data. However, this requirement is undesirable, as it may threaten the confidentiality of the sensitive information. A privacy preserving data-leak detection solution is proposed which can be outsourced and be deployed in a semi-honest detection environment. Fuzzy fingerprint technique is designed and implemented that enhances data privacy during data-leak detection operations. The DLD provider computes fingerprints from network traffic and identifies potential leaks in them. To prevent the DLD provider from gathering exact knowledge about the sensitive data, the collection of potential leaks is composed of real leaks and noises. The evaluation results show that this method can provide accurate detection.
Using Bandwidth Aggregation to Improve the Performance of Video Quality- Adap...paperpublications3
1. The document discusses using bandwidth aggregation to improve video streaming quality over multiple wireless access networks. It formulates the multi-link video streaming process as a reinforcement learning task using a Markov Decision Process to determine optimal actions.
2. It proposes a quality-adaptive streaming approach that uses an adaptive search algorithm to obtain sub-optimal solutions in real time. The approach aims to achieve smooth, high quality video streaming while avoiding interruptions.
3. The approach considers metrics like startup latency, playback fluency, average playback quality, playback smoothness, and wireless service cost in its reward function to balance video quality of service with cost. It was evaluated using a realistic testbed with an Android phone.
Abstract: The project involves creation of 1. Website which will help an NGO named Juvenile Diabetic Parent Association of India (JDPAI) to reach out to the larger audiences and help them out in managing their diabetes and 2. As an Android powered Mobile device is the gadget found commonly in the common man we will develop an Android app which will help in managing diabetes by helping in counting the daily calorie intake, giving suggestions on different situations faced by the diabetics in their daily lives etc. The website has all the information about JDPAI and its work throughout the years. The android app will have all the information about the organization as well as extra features like calorie counting, sugar level manager, etc. This app will help the diabetics (Majorly type 1) manage their lives without any stress. The entire result given on each query fired by the user would be stored on the database of the app.
Tree Based Proactive Source Routing Protocol for MANETspaperpublications3
bstract: A mobile adhoc network (MANET) is a wireless communication network and the node that does not lie within the direct transmission range of each other depends on the intermediate nodes to forward data. Opportunistic data forwarding has not been widely utilized in mobile adhoc networks (MANETs) and the main reason is the lack of an efficient lightweight proactive routing scheme with strong source routing capability. PSR protocol facilitates opportunistic data forwarding in MANETs. In PSR, each node maintains a breadth-first search spanning tree of the network rooted at it-self. This information is periodically exchanged among neighboring nodes for updated network topology information. Here added a Mobile sink to reduce the overhead in case of number of child node increases and also to reduce the delay.
Abstract: In today’s commercial, industrial and domestic world, Automation plays an important role, it is actually an arrangement of different elements in order to regulate, direct, sense and command itself to achieve a desired result. “Automatic Fire Fighting Robot” project employs the electrical thermostat technology for the controlling the fire 24 hrs. The system is cost effective, has a wide applications which when implement can show good and effective result. It can be use deliberately in industrial applications, commercial and in domestic sectors where the requirement of automatic work demands. Synchronization of various equipment involve in the system i.e Thermostat Sensor, water jet, wireless remote and wireless android device WiFi enabled Camera. This is mean to simulate the real world operation of Robot performing a fire extinguishing function. Fuzzy logic provided an appropriate solution to the otherwise complex task of mathematically deriving an exact model for the non-linear control system upon which conventional control techniques could then be applied. Making Robot wireless increases the effective area of operation, thereby making it possible to control the robot from remote location. Keeping all above factors in mind the Robot is capable of being remotely controlled and live video buffering i.e possessing a multimedia interface was convinced and developed.
Supervised Multi Attribute Gene Manipulation For Cancerpaperpublications3
Abstract: Data mining, the extraction of hidden predictive information from large databases, is a powerful new technology with great potential to help companies focus on the most important information in their data warehouses. Data mining tools predict future trends and behaviours, allowing businesses to make proactive, knowledge-driven decisions. The automated, prospective analyses offered by data mining move beyond the analyses of past events provided by retrospective tools typical of decision support systems.
They scour databases for hidden patterns, finding predictive information that experts may miss because it lies outside their expectations. Data mining techniques are the result of a long process of research and product development. This evolution began when business data was first stored on computers, continued with improvements in data access, and more recently, generated technologies that allow users to navigate through their data in real time. Data mining takes this evolutionary process beyond retrospective data access and navigation to prospective and proactive information delivery.
Folder Security Using Graphical Password Authentication Schemepaperpublications3
Abstract: Now a day most of the user are facing problem for providing the security to the folder, so that it will not be accesses by the unauthorized user. Taking in action all these problems I have designed a model which will provide a best security to your folders using graphical password authentication model. Graphical passwords are an alternative to alphanumeric passwords in which users click on images to authenticate themselves rather than type alphanumeric strings. We have developed one such system, called Pass Points, and evaluated it with human users. Beginning around 1999, a multitude of graphical based password scheme which have been proposed as alternative to text based password scheme, motivated by the promise of improved password memorability and thus usability. This paper presents a detailed evaluation of the Pass Points and pattern matching password scheme which provides high level of security and provides security to your folder.
This document summarizes research on fuzzy-closed sets and fuzzy separation axioms in fuzzy topological spaces. Some key points:
- It introduces the concept of fuzzy α-closed sets and studies their basic properties.
- It discusses properties of fuzzy separation axioms like fuzzy αT-spaces and fuzzy αT1-spaces, where α is a natural number from 0 to 3 or 0 to 4.
- Theorems are provided about fuzzy α-closed sets and their relationships to other concepts like fuzzy α-neighborhoods and fuzzy α-interiors/closures.
- Properties of fuzzy topological spaces satisfying different fuzzy separation axioms are characterized, such as characterizing when a space
Fingerprint Feature Extraction, Identification and Authentication: A Reviewpaperpublications3
Abstract: In the modern computerized world, due to high demand on fingerprint identification system, a lot of challenges keep arising in each phase of system, which include fingerprint image enhancement, feature extraction, features matching and fingerprint classification. Applications such as online banking and online shopping use techniques that depend on personal identification numbers, keys, or passwords. But there is the risk of data being forgotten, lost, or even stolen. One of the solutions to it may be biometric authentication methods which provide a unique way to identify, recognize and authenticate people. Fingerprints being the oldest methods of biometric authentication, are being explored at large. The main focus of the paper is to review fingerprint feature extraction, identification and authentication in different image/pattern based and minutiae-based fingerprints.
Analysis of Fungus in Plant Using Image Processing Techniquespaperpublications3
Abstract: The present work proposes a methodology for analysis of fungus in plant, using image processing techniques. The fungus kills the young seedling; it spread by air and can also infect plant. Therefore it is very important to monitor the leaf at regular intervals so as to keep track on quality of growth of plant. For analysis of fungus is focused on technique using MATLAB 7.0. The image are captured by digital camera mobile and processed using image growing, then the part of the leaf spot has been used for the classification purpose of the trait and test. The acquired image are in jpeg format and are converted to gray scale image. The gray scale images are enhanced and make noise free. The Ostu algorithm is applied to get threshold image. The pixel neighborhood is applied to enhance the pixel of leaf to show clearly the fungus area. Clustering is applied to get infected part of the leaf. RGB image is then segmented for analysis of fungus in plant. Comparative analyses of Image Edge Detection techniques are presented. It has been observed that the Canny Edge Detection algorithm is computationally more expensive compared to Sobel Edge Detection technique. The fungus infected area is 24.5951%.
Efficient and Optimal Routing Scheme for Wireless Sensor Networkspaperpublications3
Abstract: The Wireless Sensor Networks (WSNs) have emerged as a new category of networking systems with limited computing, communication, and storage resources. In many sensing applications source nodes deliver packets to sink nodes via multiple hops, leading to the problem on how to find routes that enable all packets to be delivered in required time frames, while simultaneously taking into account factors such as energy efficiency and load balancing. To solve this problem one data collection protocol is developed called EDAL, which stands for Energy-efficient Delay-aware Lifetime-balancing data collection. Methods used are centralized heuristic and ant colony gossiping to find best energy efficient path. Then integrate EDAL with compressive sensing to reduce the amount of traffic generated and to reduce delay in the network.
Comparative Study of Data Mining Classification Algorithms in Heart Disease P...paperpublications3
Abstract: This research paper intends to provide an accuracy of the best algorithm from the classification algorithms. The main objective of this research work is to predict more accurately the presence of heart disease in patients. The dataset used here is from the UCI Machine Learning Repository based Hungarian-14-heart disease with 294 instances. In this research paper, we use weka tool with five classifiers like Naïve Bayes, Logistic function, RBF Network function, Decision Table rule, SMO function and their performance on the diagnosis has been compared. From that Naïve Bayes provides 86% accurate result in 0 seconds and RBF Network provides 86% of accuracy in 0.17 seconds. The research result shows both the Naïve Bayes outperforms with a time duration of 0 seconds to build the model.
Avoiding Anonymous Users in Multiple Social Media Networks (SMN)paperpublications3
Abstract: The main aim of this project is secure the user login and data sharing among the social networks like Gmail, Facebook and also find anonymous user using this networks. If the original user not available in the networks, but their friends or anonymous user knows their login details means possible to misuse their chats. In this project we have to overcome the anonymous user using the network without original user knowledge. Unauthorized user using the login to chat, share images or videos etc This is the problem to be overcome in this project .That means user first register their details with one secured question and answer. Because the anonymous user can delete their chat or data In this by using the secured questions we have to recover the unauthorized user chat history or sharing details with their IP address or MAC address. So in this project they have found out a way to prevent the anonymous users misuse the original user login details.
Abstract: We are making one android application for our college. This application gives the complete details about the college like Admission procedure, various departments’ information etc. In that special job hunting section we are going to provide a new job opening in every one hour duration. The best advantage of these application will be messaging system that we are going to provide in the application. Every student will receive an SMS every hour about the job opening. If teacher wants send any kind of message any group of student, they can directly send the SMS notification to all the students with our messaging system. These help the student of college to be updated always and also attract all students to get admission in our college.
Abstract: The dramatic growth of smartphones in recent years, the challenge of limited energy capacity of these devices has not been solved satisfactorily. However, in the era of cloud computing, the limitation on energy capacity can be eased off in an efficient way by offloading heavy tasks to the cloud. It is important for smartphone and cloud computing developers to have insights into the energy cost of smartphone applications before implementing the offloading techniques.
In this paper, we evaluate the energy cost of multimedia applications on smartphones that are connected to Multimedia Cloud Computing (MCC). In other words, we investigate the feasibility of MCC. Specifically, we compared the energy costs for uploading and downloading a audio file to and from MCC with the energy costs of encoding the same audio file on a smartphone. The aforementioned comparison was performed by using HTTP and FTP Internet protocols with 3G and Wi-Fi network interfaces.
Enhancing Data Security in Cloud Storage Auditing With Key Abstractionpaperpublications3
Abstract: Auditing is an important service to verify the data in the cloud. Most of the auditing protocols are based on the assumption that the client’s secret key for auditing is secure. The security is not fully achieved, because of the low security parameters of the client. If the auditing protocol is not secured means the data of the client will exposed inevitably. In this paper a new mechanism of cloud auditing is implemented. And investigate to reduce the damage of the client key exposure in cloud storage auditing. Here the designing is built upon to overcome the week key auditing process. The auditing protocol is designed with the help of key exposure resilience. In the proposed design, the binary tree structure and the pre-order traversal technique is used to update the secret keys of the client. The security proof and the performance shows the cloud storage auditing with key exposure resilience is very efficient.
Abstract: Data Mining has wide applications in many areas such as banking, medicine, scientific research and among government agencies. Classification is one of the commonly used tasks in data mining applications. The cloud computing, users have the opportunity to outsource their data, in encrypted form, as well as the data mining tasks to the cloud. Since the data on the cloud is in encrypted form, existing privacy preserving classification techniques are not applicable. On solving the classification problem over encrypted data. A secure k-NN classifier over encrypted data in the cloud. The k-NN protocol protects the confidentiality of the data, user’s input query, and data access patterns. To develop a secure k-NN classifier over encrypted data under the standard semi-honest model. Also, we empirically analyze the efficiency of our solution through various experiments.
Kahler Differential and Application to Ramification - Ryan Lok-Wing PangRyan Lok-Wing Pang
This document discusses the application of Kähler differentials to the study of ramification in algebraic number theory. It defines Kähler differentials and constructs the module of relative differentials. Properties like exact sequences are proven. The concept of the different ideal is introduced, which encodes ramification data in field extensions. It is shown that the different ideal is the annihilator of the module of Kähler differentials, providing a geometric characterization of ramification.
This document discusses using holographic soliton automata to model discrete spacetime. It proposes that spacetime is fundamentally discrete and composed of causal sets. This allows the examination of nonlocal phenomena through soliton cellular automata which behave as localized excitations on the discrete spacetime background. Tensor calculus is applied to model the automata as crystals undergoing time evolution according to affine quantum group symmetries and box-ball soliton dynamics. The goal is to understand if the dynamics of a combinatorial metric crystal respects the laws of physics on a discrete spacetime substrate.
Abstract: Password authentication is one of the simplest and the most convenient authentication mechanisms to deal with secret data over insecure networks. Every user now holds email account for different web sites or holds a username password for their office work in their computer system. In this paper, we shall present the result of our survey through basic currently available password-authentication-related schemes. We know that Apart from getting the password hacked by some intruder, a serious threats comes from the surrounding environment, specially the colleagues, friends or visitors nearby the system. Most of them have a tendency of observing keenly while you cast your password. Here we propose a simple way to provide a new way of password through clicks that would provide a secure mechanism from both the intruders and the people nearby while you type your password in the application.
Abstract: File replication is an effective way to enhance file availability and reduce file querying delay. It creates replicas for a file to improve its probability of being encountered by requests. Here, we introduce a new concept of resource for file replication, which considers both node storage and node meeting ability. We theoretically study the influence of resource allocation on the average querying delay and derive an optimal file replication rule (OFRR) that allocates resources to each file based on its popularity and size. We then propose a file replication protocol based on the rule, which approximates the minimum global querying delay in a fully distributed manner.
Abstract: The main communication methods used by deaf people are sign language, but opposed to common thought, there is no specific universal sign language: every country or even regional group uses its own set of signs. The use of sign language in digital systems can enhance communication in both directions: animated avatars can synthesize signals based on voice or text recognition; and sign language can be translated into various text or sound forms based on different images, videos and sensors input. The ultimate goal of this research, but it is not a simple spelling of spoken language, so that recognizing different signs or letters of the alphabet (which has been a common approach) is not sufficient for its transcription and automatic interpretation. Here proposes an algorithm and method for an application this would help us in recognising the various user defined signs. The palm images of right and left hand are loaded at runtime. Firstly these images will be seized and stored in directory. Then technique called Template matching is used for finding areas of an image that match (are similar) to a template image (patch). Our goal is to detect the highest matching area. We need two primary components- A) Source image (I): In the template image in which we try to find a match. B) Template image (T): The patch image which will be compared to the template image. In proposed system user defined patterns will be having 60% accuracy while default patterns will be provided with 80% accuracy.
Privacy Preserving Data Leak Detection for Sensitive Datapaperpublications3
Abstract: Number of data leaks in the organization, research institutions and security firms have grown rapidly in recent years. The data leakage occurs if there is no proper protection. The common approach is to monitor the data that are stored in the organization local network. The existing method require the plaintext sensitive data. However, this requirement is undesirable, as it may threaten the confidentiality of the sensitive information. A privacy preserving data-leak detection solution is proposed which can be outsourced and be deployed in a semi-honest detection environment. Fuzzy fingerprint technique is designed and implemented that enhances data privacy during data-leak detection operations. The DLD provider computes fingerprints from network traffic and identifies potential leaks in them. To prevent the DLD provider from gathering exact knowledge about the sensitive data, the collection of potential leaks is composed of real leaks and noises. The evaluation results show that this method can provide accurate detection.
Using Bandwidth Aggregation to Improve the Performance of Video Quality- Adap...paperpublications3
1. The document discusses using bandwidth aggregation to improve video streaming quality over multiple wireless access networks. It formulates the multi-link video streaming process as a reinforcement learning task using a Markov Decision Process to determine optimal actions.
2. It proposes a quality-adaptive streaming approach that uses an adaptive search algorithm to obtain sub-optimal solutions in real time. The approach aims to achieve smooth, high quality video streaming while avoiding interruptions.
3. The approach considers metrics like startup latency, playback fluency, average playback quality, playback smoothness, and wireless service cost in its reward function to balance video quality of service with cost. It was evaluated using a realistic testbed with an Android phone.
Abstract: The project involves creation of 1. Website which will help an NGO named Juvenile Diabetic Parent Association of India (JDPAI) to reach out to the larger audiences and help them out in managing their diabetes and 2. As an Android powered Mobile device is the gadget found commonly in the common man we will develop an Android app which will help in managing diabetes by helping in counting the daily calorie intake, giving suggestions on different situations faced by the diabetics in their daily lives etc. The website has all the information about JDPAI and its work throughout the years. The android app will have all the information about the organization as well as extra features like calorie counting, sugar level manager, etc. This app will help the diabetics (Majorly type 1) manage their lives without any stress. The entire result given on each query fired by the user would be stored on the database of the app.
Tree Based Proactive Source Routing Protocol for MANETspaperpublications3
bstract: A mobile adhoc network (MANET) is a wireless communication network and the node that does not lie within the direct transmission range of each other depends on the intermediate nodes to forward data. Opportunistic data forwarding has not been widely utilized in mobile adhoc networks (MANETs) and the main reason is the lack of an efficient lightweight proactive routing scheme with strong source routing capability. PSR protocol facilitates opportunistic data forwarding in MANETs. In PSR, each node maintains a breadth-first search spanning tree of the network rooted at it-self. This information is periodically exchanged among neighboring nodes for updated network topology information. Here added a Mobile sink to reduce the overhead in case of number of child node increases and also to reduce the delay.
Abstract: In today’s commercial, industrial and domestic world, Automation plays an important role, it is actually an arrangement of different elements in order to regulate, direct, sense and command itself to achieve a desired result. “Automatic Fire Fighting Robot” project employs the electrical thermostat technology for the controlling the fire 24 hrs. The system is cost effective, has a wide applications which when implement can show good and effective result. It can be use deliberately in industrial applications, commercial and in domestic sectors where the requirement of automatic work demands. Synchronization of various equipment involve in the system i.e Thermostat Sensor, water jet, wireless remote and wireless android device WiFi enabled Camera. This is mean to simulate the real world operation of Robot performing a fire extinguishing function. Fuzzy logic provided an appropriate solution to the otherwise complex task of mathematically deriving an exact model for the non-linear control system upon which conventional control techniques could then be applied. Making Robot wireless increases the effective area of operation, thereby making it possible to control the robot from remote location. Keeping all above factors in mind the Robot is capable of being remotely controlled and live video buffering i.e possessing a multimedia interface was convinced and developed.
Supervised Multi Attribute Gene Manipulation For Cancerpaperpublications3
Abstract: Data mining, the extraction of hidden predictive information from large databases, is a powerful new technology with great potential to help companies focus on the most important information in their data warehouses. Data mining tools predict future trends and behaviours, allowing businesses to make proactive, knowledge-driven decisions. The automated, prospective analyses offered by data mining move beyond the analyses of past events provided by retrospective tools typical of decision support systems.
They scour databases for hidden patterns, finding predictive information that experts may miss because it lies outside their expectations. Data mining techniques are the result of a long process of research and product development. This evolution began when business data was first stored on computers, continued with improvements in data access, and more recently, generated technologies that allow users to navigate through their data in real time. Data mining takes this evolutionary process beyond retrospective data access and navigation to prospective and proactive information delivery.
Folder Security Using Graphical Password Authentication Schemepaperpublications3
Abstract: Now a day most of the user are facing problem for providing the security to the folder, so that it will not be accesses by the unauthorized user. Taking in action all these problems I have designed a model which will provide a best security to your folders using graphical password authentication model. Graphical passwords are an alternative to alphanumeric passwords in which users click on images to authenticate themselves rather than type alphanumeric strings. We have developed one such system, called Pass Points, and evaluated it with human users. Beginning around 1999, a multitude of graphical based password scheme which have been proposed as alternative to text based password scheme, motivated by the promise of improved password memorability and thus usability. This paper presents a detailed evaluation of the Pass Points and pattern matching password scheme which provides high level of security and provides security to your folder.
This document summarizes research on fuzzy-closed sets and fuzzy separation axioms in fuzzy topological spaces. Some key points:
- It introduces the concept of fuzzy α-closed sets and studies their basic properties.
- It discusses properties of fuzzy separation axioms like fuzzy αT-spaces and fuzzy αT1-spaces, where α is a natural number from 0 to 3 or 0 to 4.
- Theorems are provided about fuzzy α-closed sets and their relationships to other concepts like fuzzy α-neighborhoods and fuzzy α-interiors/closures.
- Properties of fuzzy topological spaces satisfying different fuzzy separation axioms are characterized, such as characterizing when a space
Fingerprint Feature Extraction, Identification and Authentication: A Reviewpaperpublications3
Abstract: In the modern computerized world, due to high demand on fingerprint identification system, a lot of challenges keep arising in each phase of system, which include fingerprint image enhancement, feature extraction, features matching and fingerprint classification. Applications such as online banking and online shopping use techniques that depend on personal identification numbers, keys, or passwords. But there is the risk of data being forgotten, lost, or even stolen. One of the solutions to it may be biometric authentication methods which provide a unique way to identify, recognize and authenticate people. Fingerprints being the oldest methods of biometric authentication, are being explored at large. The main focus of the paper is to review fingerprint feature extraction, identification and authentication in different image/pattern based and minutiae-based fingerprints.
Analysis of Fungus in Plant Using Image Processing Techniquespaperpublications3
Abstract: The present work proposes a methodology for analysis of fungus in plant, using image processing techniques. The fungus kills the young seedling; it spread by air and can also infect plant. Therefore it is very important to monitor the leaf at regular intervals so as to keep track on quality of growth of plant. For analysis of fungus is focused on technique using MATLAB 7.0. The image are captured by digital camera mobile and processed using image growing, then the part of the leaf spot has been used for the classification purpose of the trait and test. The acquired image are in jpeg format and are converted to gray scale image. The gray scale images are enhanced and make noise free. The Ostu algorithm is applied to get threshold image. The pixel neighborhood is applied to enhance the pixel of leaf to show clearly the fungus area. Clustering is applied to get infected part of the leaf. RGB image is then segmented for analysis of fungus in plant. Comparative analyses of Image Edge Detection techniques are presented. It has been observed that the Canny Edge Detection algorithm is computationally more expensive compared to Sobel Edge Detection technique. The fungus infected area is 24.5951%.
Efficient and Optimal Routing Scheme for Wireless Sensor Networkspaperpublications3
Abstract: The Wireless Sensor Networks (WSNs) have emerged as a new category of networking systems with limited computing, communication, and storage resources. In many sensing applications source nodes deliver packets to sink nodes via multiple hops, leading to the problem on how to find routes that enable all packets to be delivered in required time frames, while simultaneously taking into account factors such as energy efficiency and load balancing. To solve this problem one data collection protocol is developed called EDAL, which stands for Energy-efficient Delay-aware Lifetime-balancing data collection. Methods used are centralized heuristic and ant colony gossiping to find best energy efficient path. Then integrate EDAL with compressive sensing to reduce the amount of traffic generated and to reduce delay in the network.
Comparative Study of Data Mining Classification Algorithms in Heart Disease P...paperpublications3
Abstract: This research paper intends to provide an accuracy of the best algorithm from the classification algorithms. The main objective of this research work is to predict more accurately the presence of heart disease in patients. The dataset used here is from the UCI Machine Learning Repository based Hungarian-14-heart disease with 294 instances. In this research paper, we use weka tool with five classifiers like Naïve Bayes, Logistic function, RBF Network function, Decision Table rule, SMO function and their performance on the diagnosis has been compared. From that Naïve Bayes provides 86% accurate result in 0 seconds and RBF Network provides 86% of accuracy in 0.17 seconds. The research result shows both the Naïve Bayes outperforms with a time duration of 0 seconds to build the model.
Avoiding Anonymous Users in Multiple Social Media Networks (SMN)paperpublications3
Abstract: The main aim of this project is secure the user login and data sharing among the social networks like Gmail, Facebook and also find anonymous user using this networks. If the original user not available in the networks, but their friends or anonymous user knows their login details means possible to misuse their chats. In this project we have to overcome the anonymous user using the network without original user knowledge. Unauthorized user using the login to chat, share images or videos etc This is the problem to be overcome in this project .That means user first register their details with one secured question and answer. Because the anonymous user can delete their chat or data In this by using the secured questions we have to recover the unauthorized user chat history or sharing details with their IP address or MAC address. So in this project they have found out a way to prevent the anonymous users misuse the original user login details.
Abstract: We are making one android application for our college. This application gives the complete details about the college like Admission procedure, various departments’ information etc. In that special job hunting section we are going to provide a new job opening in every one hour duration. The best advantage of these application will be messaging system that we are going to provide in the application. Every student will receive an SMS every hour about the job opening. If teacher wants send any kind of message any group of student, they can directly send the SMS notification to all the students with our messaging system. These help the student of college to be updated always and also attract all students to get admission in our college.
Abstract: The dramatic growth of smartphones in recent years, the challenge of limited energy capacity of these devices has not been solved satisfactorily. However, in the era of cloud computing, the limitation on energy capacity can be eased off in an efficient way by offloading heavy tasks to the cloud. It is important for smartphone and cloud computing developers to have insights into the energy cost of smartphone applications before implementing the offloading techniques.
In this paper, we evaluate the energy cost of multimedia applications on smartphones that are connected to Multimedia Cloud Computing (MCC). In other words, we investigate the feasibility of MCC. Specifically, we compared the energy costs for uploading and downloading a audio file to and from MCC with the energy costs of encoding the same audio file on a smartphone. The aforementioned comparison was performed by using HTTP and FTP Internet protocols with 3G and Wi-Fi network interfaces.
Enhancing Data Security in Cloud Storage Auditing With Key Abstractionpaperpublications3
Abstract: Auditing is an important service to verify the data in the cloud. Most of the auditing protocols are based on the assumption that the client’s secret key for auditing is secure. The security is not fully achieved, because of the low security parameters of the client. If the auditing protocol is not secured means the data of the client will exposed inevitably. In this paper a new mechanism of cloud auditing is implemented. And investigate to reduce the damage of the client key exposure in cloud storage auditing. Here the designing is built upon to overcome the week key auditing process. The auditing protocol is designed with the help of key exposure resilience. In the proposed design, the binary tree structure and the pre-order traversal technique is used to update the secret keys of the client. The security proof and the performance shows the cloud storage auditing with key exposure resilience is very efficient.
Abstract: Data Mining has wide applications in many areas such as banking, medicine, scientific research and among government agencies. Classification is one of the commonly used tasks in data mining applications. The cloud computing, users have the opportunity to outsource their data, in encrypted form, as well as the data mining tasks to the cloud. Since the data on the cloud is in encrypted form, existing privacy preserving classification techniques are not applicable. On solving the classification problem over encrypted data. A secure k-NN classifier over encrypted data in the cloud. The k-NN protocol protects the confidentiality of the data, user’s input query, and data access patterns. To develop a secure k-NN classifier over encrypted data under the standard semi-honest model. Also, we empirically analyze the efficiency of our solution through various experiments.
Kahler Differential and Application to Ramification - Ryan Lok-Wing PangRyan Lok-Wing Pang
This document discusses the application of Kähler differentials to the study of ramification in algebraic number theory. It defines Kähler differentials and constructs the module of relative differentials. Properties like exact sequences are proven. The concept of the different ideal is introduced, which encodes ramification data in field extensions. It is shown that the different ideal is the annihilator of the module of Kähler differentials, providing a geometric characterization of ramification.
This document discusses using holographic soliton automata to model discrete spacetime. It proposes that spacetime is fundamentally discrete and composed of causal sets. This allows the examination of nonlocal phenomena through soliton cellular automata which behave as localized excitations on the discrete spacetime background. Tensor calculus is applied to model the automata as crystals undergoing time evolution according to affine quantum group symmetries and box-ball soliton dynamics. The goal is to understand if the dynamics of a combinatorial metric crystal respects the laws of physics on a discrete spacetime substrate.
The document discusses finding the right abstractions for reasoning problems. It describes Andreas Blass' insight about a category called PV that models problems and reductions between them. PV objects are binary relations representing problems, with morphisms describing reductions. The talk discusses using this framework and Dialectica categories to model Kolmogorov's theory of problems from 1932 and Veloso's theory. It provides examples of modeling geometry and tangent plane problems as Kolmogorov problems and reductions between them.
(1) The document discusses products of LF-topologies and separation concepts in LF-topological spaces.
(2) It defines what a GL-monoid is and introduces uniform structures on GL-monoids to characterize arbitrary products of elements in a GL-monoid.
(3) The paper then builds the LF-topology product of a family of LF-topological spaces and shows that Kolmogoroff and Hausdorff properties are inherited by the product LF-topology from the factor spaces.
This document summarizes three applications of commutative algebra to string theory. The first two applications involve interpreting certain products in topological field theory as Ext computations for sheaves on a Calabi-Yau manifold or in terms of matrix factorizations, which can be analyzed using computer algebra tools. The third application relates monodromy in string theory to solutions of differential equations, showing how monodromy can be described in terms of a computed ring.
1) The document discusses fiber integrals of differential forms along oriented submersions. It defines the pushforward or fiber integral f* of a differential form along a submersion f, which maps forms on the total space to forms on the base space.
2) Properties of the fiber integral are proven, including normalization, the Fubini theorem, base change formulas, and Stokes' theorem.
3) The document also discusses orientations of vector bundles and submersions, which are needed to define the fiber integral.
This document discusses the concept of a basis for a topology. A basis is a collection of subsets of a set X that generates a topology on X in a specific way. It provides several examples of bases, including the standard basis for the real numbers using open intervals. It also introduces some alternative topologies on the real line generated by different bases, such as the lower limit topology and K-topology, and compares their relative fineness.
This document discusses the definition and properties of topological bases. A basis for a topology is a collection of subsets that satisfies two properties: (1) every point is contained in at least one basis element, and (2) intersections of basis elements can be refined by another basis element. The topology generated by a basis B consists of all unions of basis elements. Important results shown include that the collection of basis elements forms a basis for the generated topology, and two bases generate the same topology if one refines the other. Examples of bases for standard topologies on R and RxR are given.
The document discusses variations on Higman's Lemma, which states that a set of finite sequences ordered by embedding forms a well quasi-order if and only if the underlying set ordered does. The document defines well quasi-orders and related concepts. It then examines properties of products, coproducts, equalizers, coequalizers and exponentiation in categories of well quasi-orders and related categories. It proves Higman's Lemma using categorical concepts like equalizers and coequalizers. It also examines properties of categories of descending chains of a quasi-order and how these relate to properties of the original quasi-order.
Similar to Some Aspects of Abelian Categories (12)
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
Batteries -Introduction – Types of Batteries – discharging and charging of battery - characteristics of battery –battery rating- various tests on battery- – Primary battery: silver button cell- Secondary battery :Ni-Cd battery-modern battery: lithium ion battery-maintenance of batteries-choices of batteries for electric vehicle applications.
Fuel Cells: Introduction- importance and classification of fuel cells - description, principle, components, applications of fuel cells: H2-O2 fuel cell, alkaline fuel cell, molten carbonate fuel cell and direct methanol fuel cells.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
Comparative analysis between traditional aquaponics and reconstructed aquapon...bijceesjournal
The aquaponic system of planting is a method that does not require soil usage. It is a method that only needs water, fish, lava rocks (a substitute for soil), and plants. Aquaponic systems are sustainable and environmentally friendly. Its use not only helps to plant in small spaces but also helps reduce artificial chemical use and minimizes excess water use, as aquaponics consumes 90% less water than soil-based gardening. The study applied a descriptive and experimental design to assess and compare conventional and reconstructed aquaponic methods for reproducing tomatoes. The researchers created an observation checklist to determine the significant factors of the study. The study aims to determine the significant difference between traditional aquaponics and reconstructed aquaponics systems propagating tomatoes in terms of height, weight, girth, and number of fruits. The reconstructed aquaponics system’s higher growth yield results in a much more nourished crop than the traditional aquaponics system. It is superior in its number of fruits, height, weight, and girth measurement. Moreover, the reconstructed aquaponics system is proven to eliminate all the hindrances present in the traditional aquaponics system, which are overcrowding of fish, algae growth, pest problems, contaminated water, and dead fish.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, Serbia
1. ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Vol. 2, Issue 1, pp: (33-41), Month: April 2015 – September 2015, Available at: www.paperpublications.org
Page | 33
Paper Publications
Some Aspects of Abelian Categories
Dhanjit Barman
Department of MathematicsGauhati University, Guwahati, Assam
Abstract: Here we discuss how zero objects and zero morphisms behave in an abelian category. We also provide
proofs of isomorphism between the kernels and between the cokernels of a morphism. We discuss some properties
of abelian category.
Keywords: Zero object, Zero Morphism Kernel, Cokernel, Image, Coimage, Parallel Morphism, Abelian category,
Preadditive category Additive Category, Preabelian Category.
1. INTRODUCTION
Here we discuss the behaviour of zero object and zero morphism in an abelian category. We also discuss how kernels of a
morphism are isomorphic and that of cokernels of a morphism are also isomorphic. We have proved some properties of
abelian categories. We have given proof of some properties of exact sequences. We also try to prove that in an abelian
category every bimorphism is an isomorphism.
2. PRELIMINARIES
For notions of category theory we shall in general follow the notation and terminology of Popescu [6]. However, we do
deviate somewhat.
For C a category and A, B objects of C, Mor(A, B) denotes the set of morphisms from A to B.
We will also follow Popescu [6] for the definition of Preadditive, Additive and Preabelian and abelian category.
For kernel and cokernel we follow MacLane[2].
We follow the definition terminology of zero object and zero morphism from Popescu[6] and Pareigis[11].
We shall use the definition of Balanced category from Mitchel [3] Monomorphism from Schubert[5] and epimorphism
and isomorphism from Pareigis[11].
3. MAIN RESULTS
1. PROPOSITION: There is a unique isomorphism between each pair of zero objects in a category.
Proof : Let us suppose that Z band X are two zero objects.Then Mor (Z,X) contains exactly one morphism and Mor (X,Z)
consists only one morphism.Therefore the composition morphism Z→X→Z is unique and it will be 1Z, as Mor(Z,Z) has
only 1Z..
Similarly the compositon morphism X→Z→X is also unique and it is 1X.
Hence Z and X are isomorphic.
2. PROPOSITION: Let C be a category with zero objects. Then for objects A,B ∈ ob C , the map 0(A,B) : A → B is
independent of the choice of our zero object Z.
2. ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Vol. 2, Issue 1, pp: (33-41), Month: April 2015 – September 2015, Available at: www.paperpublications.org
Page | 34
Paper Publications
Proof : Let Z and X be zero objects in C. Then by definition of zero objects and by the definition of 0(A,B) : A → B
we have the following diagram.....
Now gof = 0(A,B),where f,g are unique. Since X is zero object so f’ and g’ are also unique. Thus k is unique as both Z
and X are zero objects. We need to show that 0(A,B) = g’ o f’
By the uniqueness of g’ and f’ we have
g’ok = g and kof = f’.
now g’o(kof) = gof’
= > (g’ok)of =g’of’
= > gof = g’of’
= > 0(A,B) = g’of’
Thus 0(A,B) is independent of the choice of our zero object.
Note : This proposition shows that 0(A,B) is unique for objects A and B.
Also in a category with zero objects , the zero morphism exists for any pair of objects (A,B) by its definition.
From now onwards we will denote zero morphism by just 0.
3. Composition with zero morphisms yields zero morphism.
Proof : It is obvious from the definition of zero morphism.
4 . Kernels are unique up to isomorphism.
Proof : Let C be a category with zero object 0. Let f: A→B is a morphism and k: S →A and g : D →A are two kernels
of f.
As g : D →A is kernel of f: A→B and fok = 0, therefore by definition of kernel there is a unique p: S → D such that gop
= k………………………………….(i)
The following diagram shows—
Next , as k: S → A is a kernel of f: A→B and fog = 0,
then there exists a unique morphism h: D → S such that
koh = g …………………………………(ii)
3. ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Vol. 2, Issue 1, pp: (33-41), Month: April 2015 – September 2015, Available at: www.paperpublications.org
Page | 35
Paper Publications
(i) = > (gop)oh = koh
= > go(poh) = g [from (ii)]
= > poh = 1D [as kernel (g) is monomorphism ] ………(iii)
(ii) = > (koh)op = gop
= > ko(hop) = k [ from (i) ]
= > hop = 1S [since kernel (k) is monomorphism]……….(iv)
From (iii) and (iv) we have that both the kernels are isomorphic.
5. Cokernels are unique up to isomorphism.
Proof : Let C be a category with zero object 0. Let f: A→B be a morphism and m: B → C and g : B → D are
cokernels of f: A→B
As g : B → D is a cokernel of f: A→B and mof = 0
Therefore by definition of cokernel we have a unique morphism p : D →C such that the following diagram........
Also m: B → C is a cokernel of f: A→B and gof = 0, therefore by definition we have a unique morphism h : C →D
such that the following diagram ......
(ii) = > po(hom) = pog
= > (poh)om = m [from (i)]
= > poh = 1C ………………………………….(iii)
(i) = > ho(pog) =hom
= > (hop)og = g [from (ii)]
4. ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Vol. 2, Issue 1, pp: (33-41), Month: April 2015 – September 2015, Available at: www.paperpublications.org
Page | 36
Paper Publications
= > hop = 1D ……………………………………(iv)
From (iii) and (iv) we have that the two cokernels are isomorphic.
Note that kernels and cokernels need not exist for all morphisms.
6. In an abelian category C ,the following are equivalent-
(i) f: A→ B is monomorphism
(ii) 0 is kernel of f
Proof:- Let f: A→ B be a monomorphism and g: C → A be
its kernel.
So fog = 0.
= > g = 0 (since f is monomorphism)
= > 0 is kernel of f
Conversely suppose that 0: 0 → A is kernel of f: A→B.
Let g,h :C→A be such that fog =foh
=.> fo(g-h)=0 (since C is abelian category)
= > there exists a unique morphism k:C →0 such that 0ok = g-h
= > g-h = 0 .(since 0ok is zero morphism)
= >g = h
= > f is monomorphism.
7. In an abelian category ,the following are equivalent-
(i) f: A→ B is epimorphism
(ii) 0 is cokernel of f
Proof:- Let f: A→ B be a epimorphism and g:B→C be its cokernel.
So gof =0.
= > g=0 (since f is epimorphism)
= > 0 is cokernel of f
Conversely suppose that 0: B → 0 is cokernel of f: A→B.
Let g,h :B→C be such that gof=hof
=.> (g-h)of = 0 (since C is abelian category)
= > there exists a unique morphism k:0 →C such
that ko0 =g-h
= > g-h = 0 .(since ko0 is zero morphism)
= > g=h
= > f is epimorphism.
8. In an abelian category, if f:A→ B is monomorphism
then ker(cokerf)= imf coincides with f:A→ B.
5. ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Vol. 2, Issue 1, pp: (33-41), Month: April 2015 – September 2015, Available at: www.paperpublications.org
Page | 37
Paper Publications
Proof:- Let us consider the diagram-
As f is monomorphism so kerf = 0 and cokrnel of 0: 0→A is the identity morphism 1A : A→A. Thus the above diagram
becomes
Since C is abelian category we have f’’ is isomorphism. So A is isomorphic to ker(cokerf).
For example, in the category Ab of abelian groups, let f: A→B be a monomorphism. Then f(A)≌A.
Also p: B→ B/f(A is cokernel of f: A→B .
Consider the diagram
such that pog = 0.Let us define h: C→A such that h(c)= g(c) which is well defined.
Now foh(c) =f(h(c))
= h(c)
= g(c)
= > foh =g. Also his unique i.e. the diagram
Hence kernel of p: B→ B/f(A) ≌ A is f: A→B .
6. ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Vol. 2, Issue 1, pp: (33-41), Month: April 2015 – September 2015, Available at: www.paperpublications.org
Page | 38
Paper Publications
9. In an abelian category, if f:A→ B is epimorphism then coker(kerf) coincides with f:A→ B.
Proof : Let us consider the diagram-
As f is epimorphism so cokerf = 0 and kernel of 0: B→0 is the identity morphism 1B : B→B. Thus the above diagram
becomes
Since C is abelian category we have f’’ is isomorphism. So B is isomorphic to coker(kerf).
10. A morphism f : A → B is an epimorphism iff imf= B.
Proof:- f is epimorphism iff coker(kerf)=B
Since C is abelian so coimf ≌ imf i.e. coker(kerf) ≌ ker(cokerf)
= > B≌ker(cokerf)
= > B≌imf.
11. Generally in any category C every isomorphism is bimorphism but the converse may not be true.But in case of
an abelian category every bimorphism is isomorphism.
Proof: Let C be an abelian category and f : A → B be a bimorphism i.e. f is monomorphism as well as epimorphism. Let
us consider the diagram
Now f is monomorphism = > 0 is kernel of f
And cokernel of 0→A is the identity morphism 1A: A→A.
Also fis epimorphism = > 0 is cokernel of f
And kernel of B→0 is the identity morphism 1B :B→B.
7. ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Vol. 2, Issue 1, pp: (33-41), Month: April 2015 – September 2015, Available at: www.paperpublications.org
Page | 39
Paper Publications
Thus the above diagram becomes
Which gives f = 1B o f’’ o 1A
= > f= f’’
= > f is isomorphism as f’’ is isomorphism.
12. Let the category C has a zero object 0.
(a) A ∈ obC is a zero object iff 1A is zero morphism.
(b) if m : A →0 is a monomorphism, then A is a zero object.
Proof: (a) Let A be a zero object. Then 1A is the only morphism from A→ A as A is zero object. And hence 1A is zero
morphism.
Conversely suppose that 1A is zero morphism. Then the following diagram commutes
Let m,n : B → A. Then 1Aog = g and 1A o f = f. As 0 is zero object we have a unique morphism k: B →0 such that from
the diagram
We have fom= k = fon.
Now gofom = gofon
= > 1A om = 1A on
= > m = n.
Thus A is terminal object.
Similarly it can be shown that A is initial object .
Hence A is zero object.
8. ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Vol. 2, Issue 1, pp: (33-41), Month: April 2015 – September 2015, Available at: www.paperpublications.org
Page | 40
Paper Publications
(b) Let m: A→0 be monomorphism
Let us consider B f,g A m 0 such that
mof = mog
= > f = g [ as m is monomorphism]
Thus A is terminal object.
Let f,g : A→B be two morphisms. Then we have the following diagram
Such that nom = f, nom = g
= > f = g
Therefore A is initial object.
Hence A is zero object.
13. If f : A →B is monomorphism in an abelian category then it is kernel of some other morphism.
Proof: Let us consider the diagram
As f is monomorphism so kerf =0 , coker(0) = A and p = 1ASo the above diagram becomes
Also C is abelian so f’’ is isomorphism ,by definition.
Therefore fof’’ = q. Which shows that f : A→ B is kernel of j: B →cokerf.
9. ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Vol. 2, Issue 1, pp: (33-41), Month: April 2015 – September 2015, Available at: www.paperpublications.org
Page | 41
Paper Publications
14. Any epimorphism in an abelian category is the cokernel of a morphism.
Proof : Let us consider the diagram
As f is epimorphism so cokerf is 0. Therefore ker(0)= B
And q = 1B. Now the above diagram becomes
As C is abelian category so f’’ is isomorphism, by definition. Thus f’’ o f = p.Which shows that f: A →B is cokernel of i :
kerf →A.
ACKNOWLEDGEMENT
Dr. Khanindra Chandra Chowdhury, Deptt. of Mathematics, Gauhati University.
REFERENCES
[1] Anderson, Frank W. & Fuller, Kent R., Rings and Categories of Modules, Springer-Verlag New York berlin
Heidelberg London paris Tokyo Hong Kong Barcelona Budapast.
[2] Mac Lane, S., 1971: Categories for the Working Mathematician, Springer-Verlag New York Berlin.
[3] Mitchel, Barry.1965: Theory of Categories, Academic Press New York and London.
[4] Krishnan, V.S., 1981: An introduction to Category Theory,North Holland NewYork Oxford.
[5] Schubert, Horst, 1972: Categories, Springer-Verlag BerlinHeidelberg New York.
[6] Popescu, N., 1973: Abelian Categories with Applications to Rings and Modules, Academic Press, London & New
York.
[7] Awodey, Steve., 2006: Category Theory, Second Edition, Clarendon Press, Oxford.
[8] Borceux, Francis.,1994: Hand Book of Categorical Algebra,Cambridge University Press
[9] Simmons, Harold., 2011:An Introduction to Category Theory ,Cambridge University Press.
[10] Freyd, P., 1965: Abelian Categories, An Introduction totheTheory of Functors, A Harper International Edition, 0
jointly published by Harper & Row,NewYork,Evaston & London and JOHN WEATHERHILL INC. TOKYO.
[11] Pareigis, Bodo., 1970: Categories and Functors, Academic Press New York, London.