RESEARCH & REVIEWS: DISCRETE MATHEMATICAL STRUCTURES
ISSN: 2394-1979 Scientific Journal Impact Factor (SJIF): 6.366 Journal DOI no.: 10.37591/RRDMS
STM JOURNALS ( Consortium eLearning Network Pvt Ltd)
Riemannian Metric based Smart Senor for Testing Corona Virus
This paper proposes an electronic device based on Riemannian metric for testing corona virus. Hamilton’s evolution equation based sensor presents the nodes of corona virus through the Riemannian metric. The initial position and terminal position of corona virus based interval is defined with respect to the time. This is measured by Riemannian metric. There exists the functional F, which is the probable structure of corona virus and its variant over the manifold M (included variants also), f is the functional change of the structure of the corona virus, fe is the variants in the structure of the corona virus. The classified testing range of corona virus is installed in the sensor system but its dynamic tracing is performed by the Riemannian metric. There is applied the tensor and curvature concept so that the deviation in the result to be minimized.
Riemannian metric, Evolution equation, Corona virus
Copyright (c) 2024 RESEARCH & REVIEWS: DISCRETE MATHEMATICAL STRUCTURES
Creating a New Coding and Decoding Algorithm Based on Violin Notes
Throughout history, people have sent their messages using encryption so that only the person they want can read them. Today, thanks to the internet and other communication networks, data can be transmitted and accessed worldwide. For this reason, it is of great importance that digital data is transmitted correctly to the other party and in a stronger, higher quality and more secure way during transmission. In this context, developing different decoding algorithms by constantly approaching coding and decoding studies from different perspectives is of great importance in the world of encryption. Based on this purpose, in this study, we presented a different, not obvious approach to the world of encryption and discussed the coding and decoding study on the violin, one of the musical instruments. In this study, a different perspective was brought to the previous coding and decoding studies and coding and decoding algorithms were created with the help of frets and notes on the violin strings, which are musical instruments. In the coding processes here, the American Standard Code for Information Interchange system was used to find the numerical code values of the letters. In this way, not only the reliability of the coding and decoding process was increased, but also the verification of information was tried to be kept at a high rate.
Graph Theory: Its Importance and Application in Recent development of Science and Engineering
Graph Theory is widely used in all the basic sciences including social science, due to its intrinsic simplicity it has got a lot of application in computer science, computer networks, engineering, genetics, data structure etc. Graphs are nothing but mathematical structure that helps in representing our everyday problem in a precise graphical manner. Graph Theory is the area of applied mathematics that deals with the study of objects and the relationship about them. The issue of finding the shortest path is extensively researched within network flow optimization problems. It holds significant relevance in multiple scientific and engineering domains, particularly in areas like transportation and traffic control, where solving various types of shortest path problems becomes essential. The transportation scientist requires a very flexible and efficient shortest path methods and applications, to solve the real world problem both from the time point of view, and also for the memory requirements applied for the same. Also, graph theoretical reasoning can cast light on such problems, providing tools which help in making decisions and help in finding answers to a variety of specific problems or questions which arises in the attempt to tackle the broader issues. Every real world problem can be converted into a mathematical problem and can be solved with the help of graph theory. Due to its intrinsic nature, graph theory is one of the important tool to study various mathematical structures and to analysis its insights.
Graph Theory, Transportation Problem, Traffic Control problem, Eulerian Graph, Dijkstra Algorithm, Data mining
Detecting Electronic Circuitboard Thermal Abnormalities with Dynamic Hypernumbers
This paper introduces the algebra of data acquisition using dynamic hypernumbers used in predicting thermal abnormalities with a rapid forecasting algorithm. Algebraic rules are defined for monitoring data from multiple sources. Each data source is considered a dynamic receptive synchronized hypernumber for building the resulting dynamic hypernumber. The algebraic dynamic hypernumber definition is used to present a low-complexity algorithm for identifying the thermal state of an electronic circuit board. The algorithm is implemented by translating the pixel colors from an infrared camera into a dynamic hypernumber temperature index. The dynamic hypernumber algorithm output is used to dynamically predict the thermal process. This prediction allows the detection of thermal abnormalities. It is shown that the method of building a dynamic hypernumber from the set of the circuit board temperatures has less electronic and computational complexity in comparison with looking for elevated temperature areas. Considering dynamic hypernumbers are acquired from the stochastic process, regression polynomials are used with best criteria fit theory to find the continuous mathematical expression of such numbers. The solution of non-linear operator equations for computing the coefficients of the polynomial is obtained using the constructive hypernumber.
Number sequences have been studied by many scientists for many years because they find applications in nature and many branches of science. The most well-known number sequence among these integer sequences is the Fibonacci sequence. Later, these integer sequences were generalized and many number sequences were defined and their properties were examined. One of these is Pell number sequences. Pell numbers can be calculated in a way similar to the recurrence relation of Fibonacci numbers. The Pell numbers are defined by with and . The first few terms of Pell numbers are 1,2,5,12,29,… . In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. In addition to being used to approximate the square root of two, Pell numbers can also be used to find square triangle numbers, create integer approximations to the right isosceles triangle, and solve certain combinatorial numbering problems. In this study, we define Pell functions on real numbers . We give the relations between Pell functions and Pell numbers. We examine Pell functions using the concept of the and functions. Additionally, if is a Pell function, then which is one of the roots of equation .
A Comprehensive Review of Mathematical Modelling using Graphs
This article provides a detailed insight into the applications and methods of mathematical modeling using graphs. Graph theory has proven to be a powerful tool for representing and analyzing complex systems in diverse fields such as computer science, social networks, transportation, biology, etc. This paper covers the importance of graph theory for solving real-world problems. Mathematical modeling using graph theory includes graphical representations, algorithms, and their applications in network analysis, optimization, and decision-making. The article examines the development of graphical modeling and shows its growing importance in interdisciplinary research. This comprehensive review explores the rich landscape of mathematical modeling using graphs and is intended to serve as a valuable resource for researchers, educators, and practitioners in various fields seeking insights into innovative applications and evolving trends in the industry.
Copyright (c) 2023 RESEARCH & REVIEWS: DISCRETE MATHEMATICAL STRUCTURES
Defining Regression Polynomials with Process Similarity Criteria
Highly accurate monitoring and prediction is essential in detecting abnormality events and forecasting demands in supply chains. This investigation provides a computational procedure for defining polynomials that satisfy similarity criteria in the monitoring process for a polynomial of any degree. It provides an example of using algorithms to predict transient thermal processes in electronic devices. The second-degree polynomial defined with this algorithm is compared with a polynomial defined using a minimum least square algorithm. This regression method can be applied to a wide variety of data sets, creating actionable metrics that inform risk analysis, failure prediction, and optimization strategies. Applications include grid stabilization, battery optimization in Electric Vehicles (EVs), and failure prediction in electronic components. As an example, this paper shows superior prediction capabilities through thermal analysis of a circuit board. The proposed method of the data prediction provides an alternative to the neural network diagnostics for the processes when real-time rapid identification is required. Such diagnostics is most critical for detecting electronic devices, battery grids, toxic ingredients intrusion in water or air systems and clean technology processes. Because the method can predict future data in milliseconds, it satisfies the requirements for such processes. The choice in defining the analytical approach in predicting events should be taken by analyzing required estimation time. The forecasting demands in the supply chain do not require high computational speed. However, when the amount of the predicted item is big, using the proposed method would benefit in decreasing computational complexity. The known method of using multiple polynomials does not provide high accuracy in prediction. The computational procedure for calculating polynomial coefficients includes the solution of the nonlinear operator equations. It is proposed to use hypernumber theory for solving such equations. The method provides an efficient solution for the complex nonlinear equations.
Applications of Mathematical Modelling for Sustainable Development
Sustainable development is an urgent global challenge that requires innovative solutions to address complex socio-environmental challenges while ensuring the well-being of current and future generations. Mathematical modelling has evolved into an indispensable tool, providing valuable insights for comprehending, evaluating, and enhancing systems across diverse domains. By harnessing the power of mathematical modelling, we can support the transformative changes needed to build a sustainable future. In this paper, I examine the role of mathematical modelling in achieving the sustainable development goals and highlight its potential to provide insights, guide decisions and promote a more resilient and just world. We discuss the key elements of sustainability and show how mathematical modeling can help to achieve these goals. Mathematical modelling plays a pivotal role in addressing the multifaceted challenges associated with sustainable development. This abstract provides an overview of the diverse applications of mathematical modelling in the context of sustainable development, highlighting its indispensable contributions to environmental preservation, resource management, and societal well-being.
Lucas Divisor Cordial labeling of various graphs
In this we found six new graphs families satisfying the condition of Lucas divisor cordial labeling. We prove that Shadow of star graph is Lucas divisor cordial. We prove that Degree splitting of star graph is Lucas divisor cordial. We prove that Jewel graph Jn is Lucas divisor cordial. And we prove that Globe graph Gln is Lucas divisor cordial. Further we prove that K1, 1, n is Lucas divisor cordial. And we also prove that Degree splitting of path graph is Lucas divisor cordial labeling.
Integer Cordial Labeling in context of Ring sum of Graphs
An Integer Cordial Labeling of a graph G*(p, q) is an one-one map g: V * or as p is even or odd, which induces an edge labeling defined by such that the number of edges labeled 1 and the number of edges labeled 0 differ by at most 1. If a graph has integer cordial labeling (I.C.L.), then it is called integer cordial graph (I.C.G.). In this paper, I have investigated six new results in context of ring sum of graphs which admits integer cordial labeling.
Catalyzing Scientific Exploration: Essential Algorithms in Python for Computational Excellence
This research work delves into the foundational algorithms that underlie scientific computing, leveraging the Python programming language. It navigates through essential numerical techniques, optimization approaches, and linear algebra operations, highlighting their critical importance in diverse scientific fields. By offering illustrative examples and concise Python code snippets; the study emphasizes Python's role as a versatile and accessible platform for researchers and practitioners engaged in scientific computation. Python's widespread use in scientific computing is attributed to its rich library ecosystem, including NumPy, SciPy, and scikit-learn, which provide efficient implementations of these algorithms. The study effectively bridges the gap between theoretical concepts and practical application, facilitating a deeper understanding for readers. In essence, this study illuminates the core algorithms driving scientific computing via Python, making valuable insights and tools accessible for a wide range of researchers and professionals.
In agriculture, a minimum distance often needs to be maintained while planting trees in an orchard or a field, so that each plant gets sufficient nutrients, water and sunlight. A common and obvious way of planting trees in a field, with each tree at a minimum distance (say 1 m) from its neighbors, would be along the corners of a square grid.
A Novel Binary Arithmetic Computational Method
The proposed algorithm for arithmetic operations presents an innovative and intriguing approach, primarily centered around the use of counters and the manipulation of '1's in binary representations. This algorithm promises to introduce significant advancements in computational efficiency and accuracy, making it a potential game-changer in the field of arithmetic calculations. At the core of this method lies the reliance on the number of '1's in each location within a binary number. The user's task is to count these '1's accurately, as the entire algorithm pivots on this crucial factor. If the count of '1's at a specific location is odd, the resulting sum output is '1'; otherwise, it is '0'. Moreover, to ensure continuity in the calculations, half of the '1's will be carried over to the subsequent level, contributing to a streamlined and consistent computational process. One of the standout features of this approach is its adaptability to handle signed binary computations with ease. The use of the 1's and 2's complement methodology further enhances its capability to deal with negative numbers efficiently, demonstrating its robustness and versatility. With its focus on optimizing the utilization of '1's in binary numbers, this algorithm showcases a fresh perspective in arithmetic operations. By leveraging this fundamental element, it opens up new possibilities for enhancing the speed and accuracy of calculations, potentially revolutionizing diverse applications, such as data processing, encryption, and digital signal processing. Furthermore, the reliance on counters as a foundational concept in this algorithm introduces an element of parallel processing, leading to potential opportunities for harnessing the power of parallel computing architectures, thereby optimizing its implementation on modern computational platforms.
Study and Analysis of the Results in Fixed Point Fuzzy Metric Spaces
The main objective of this study is to use rational inequalities to demonstrate several purpose theorems mounted on fuzzy metric domains. The findings of numerous writers in the body of literature already exist, and our results expand and generalize them. We also provide several applications to back up our findings. In this study, the topic of fixed points in fuzzy metric space is the main focus. It inspires us to create various mappings in a symmetric fuzzy metric space because a tuple can be represented in a fuzzy metric space. One of the most significant fields of basic research for mathematicians is metric spaces. Different experimenters developed the idea of metric space by adding additional variables in metric space or by creating various compressions in various disciplines. It motivates us to develop various mappings in a symmetric-fuzzy metric space because of how a tuple is represented in a fuzzy metric space. For mathematicians, metric spaces are one of the key fundamental topics of study. In order to introduce new contractions in various domains or to increase the number of variables in the metric space, several academics first established the idea of metric space. To make fixed points in metric spaces easier, many mapping types have been proposed. The fixed-point theorem, which is original and based on fuzzy metric space, is the subject of this study as we show several significant conclusions on it. In this study, metric spaces are discussed. Due to their similar function to the real line R in calculus, these are essential to functional analysis. In actuality, they generalize R and were developed to serve as a foundation for an integrated approach to significant issues from diverse areas of research.
Common Fixed Point Theorem for Rational Inequality in Fuzzy Metric Space Under C-Class Function and Implicit Function
The purpose of this study is to establish some fixed point results for fuzzy metric space satisfying certain rational inequality under C-Class function along with Implicit function. The idea of C-Class functions in introduced in the process of building a fuzzy metric space that associate with implicit functions. In this way, our results extend and generalize and improve results of fuzzy metric space of previously published.
Evolutionary Perspective for Developing Beyond Abacus
Currently, all over the world, teaching math means calculations, crunching numbers mentally, and solving huge number equations in the least time, and not. This research is based on bringing about a change in mathematics education by challenging the existing model of mathematics where being good at calculation or computation is treated as being good at math. In other words, mathematics is currently treated as merely a computational tool. We propose to treat math as a science of appreciating, creating, and recognizing patterns in environment/nature, sciences, business and day-to-day life, and all other endeavors and going beyond the abacus to make math a living subject. It aspires to be established as a full-fledged industry to make math teaching and learning more active and interesting today and in the future through a diverse range of courses, services, and assistance provided for various math skill sets.
Copyright (c) 2022 RESEARCH & REVIEWS: DISCRETE MATHEMATICAL STRUCTURES
Galois Theory and Ricci Flow Curvature Applications for Generating the Mandelbrot Set
In this paper, an application of Galois theory is presented through Ricci flow curvature in Mandelbrot set. The Mandelbrot set is analyzed over Ricci flow equation for generating the fractal set simply. Although, the fractal set is generated as the points in the complex plane connected to the Julia set and approaches to infinity. This paper proposes an efficient method of fractal set generation based on onto-one correspondence between Ricci flow manifold and the Mandelbrot set over the Riemanninan curvature. First, the Mandelbrot set transforms into the vector and then each element of the Mandelbrot vector corresponds to the n-dimension curvature tensor. The Mandelbrot set is formed by _finite interval and sequence of time instead of infinite interval and without sequence of time. Riemannian metric approach generates the fractal set faster than the Euclidean metric.
A New Potential on b-Rectangular Metric Spaces
As a generalization of rectangular metric space and b-metric space, the idea of b-rectangular metric space is introduced. An effort of this article is to identify some common fixed-point results in brectangular metric space for generalized almost contraction of two mapping. The obtained results expand and generalize some previously published data in the literature, and we provide some examples. In the line of present works as obtain results in this paper one may extend different types of metric spaces such as probabilistic metric space, quasi metric space, quasi semi metric space, partial metric space, cone metric space under the flavor of rational conditions. This may be regarded an active area of research to the future workers in the different branches.
Achieving Multi-objectives Using a Single Neural Network
Achieving multi-objectives using a single neural network is a research area that aims to develop techniques to optimize multiple objectives simultaneously using a single neural network. This approach can be applied in various fields, such as image and speech recognition, robotics, and natural language processing, to achieve better performance and reduce the complexity of the models. The key challenge in this area is to balance the trade-off between different objectives and to develop appropriate algorithms for optimizing multiple objectives. Various techniques have been proposed in the literature, including evolutionary algorithms, reinforcement learning, and multi-objective optimization algorithms. The success of this approach depends on the ability of the neural network to learn to balance different objectives and to generalize to new tasks.
Fuzzifying Topology Generated by Fuzzy Metric
The purpose of this study is to present a new method to generate a fuzzifying topology with fuzzy metrics in Kramosil and Michalek sense. For this, with the help of fuzzy metric, we obtained parameter-dependent crisp topology bases and thus a family of topologies. This family consists of the level topologies of the fuzzifying topology to be generated. Then, we studied the continuity of transformations between generated fuzzifying topologies and the convergence of sequences on the generated fuzzifying topology.
All-in-One Focus image using Guided Filters
Machine Vision System plays a vital role in manufacturing industries for the automatic inspection of quality of the finished product. The success of machine vision system for real time application is purely depends on the response of the hardware system in capturing the image of the object and the image processing algorithm which is used to extract the features of the object from the captured image. For the extraction of features, the image should contain the relevant information about all the objects in the scene and it should be noise free. Due to the finite focal length of the lens used in machine vision camera, it is possible to capture the clear image of objects in the scene which are in the particular range of distance from the lens. The other objects in the scene will be blurred. In this situation, image fusion is used to combine more than one source images with different focus to form the all-in-one focus image. In all-in-one focus image, all objects in the scene are in clear focus with greater visibility. One of the most widely used filters in image fusion to generate all-in- one focus image without noise while preserving edge information of object of interest is Standard Guided Filters. The time complexity of guided filters is O (N), where N is the image&#39;s pixel count.. Fast guided filters were proposed in the literature to reduce the time complexity of standard guided filters. This paper evaluates the performance of standard guided filters and fast guided Filters in fusing multi focused images to generate all-in-one focus image. Also this paper enhances the visibility of input images before image fusion process to improve the quality of all-in-focus image.
Image Fusion, Depth of field, All-in-one Focus image, Guided Filters and Fast Guided Filters.
Novel Distance Measure for Linear Diophantine Fuzzy Sets
Unlike other fuzzy set theories, the theory of Linear Diophantine fuzzy set (LDFS) aims to overcome uncertainties via the reference parameters. The Parameters provide flexibility and efficiency to LDFS in handling the uncertain data. In this study, we introduce new distance measure for linear Diophantine fuzzy sets. Then, we studied the properties of proposed distance measure. Finally, we established a new similarity measure for LDFSs based on distance measure.
distance measure, Linear diophantine fuzzy sets
Machine Learning for Solving Satisfiability Problems and Applications in Cryptanalysis
Over the past two decades, we have seen sensational changes within the effectiveness of conflictbased Clause-Learning Boolean Fulfillment (CDCLSAT) solvers on mechanical issues from an assortment of areas. The accessibility of effective general-purpose look devices such as SAT Solver has driven numerous analysts to propose SAT-based strategies for cryptanalysis, such as identifying clashes in hash capacities and breaking symmetric encryption plans. Most of the as of now proposed SAT-based cryptanalysis strategies are Black Box innovations, i.e. the cryptanalysis issue is encoded as SAT, and CDCLSAT Solver is utilized to illuminate that illustration. The drawback of this approach is that the encoding produced in this way can be as well expansive for any cutting edge arrangement to unravel viably. Another noteworthy shortcoming of this approach is that the solver is in no way arranged to address a particular or given illustration. The drawback of this approach is that the encoding created in this way can be as well expansive for any advanced arrangement to unravel successfully. Another noteworthy shortcoming of this approach is that the solver is in no way arranged to address a particular or given case. To fathom these issues, we propose a technique called CDCL (crypto) (Hypothesis in Immersion Module motivated by the CDCL(T) show in Salver). Plan with SAT solvers permits us to distinguish or demonstrate powerless key squares in differential and straight assaults of total circular piece ciphers based on the Worldwide Information Encryption Calculation (Thought), i.e. WIDEA for nn and MESH-64. In layout: (i) we found an assortment of defenseless keys for WIDEA-n and (ii) illustrated that specific course powerless keys do not exist in MESH-64. SAT Solvers have given answers to two complementary open issues (displayed in Fast Program Encryption 2009): the nearness and non-existence of such slight keys. In spite of the fact that these issues could seem troublesome to reply to, SAT resolvers have given a successful arrangement. In spite of the fact that these issues could seem troublesome to reply, SAT resolvers have given an effective course of action. We'll additionally report on the test that comes around for advanced SAT Solver execution as encoded cryptanalytic errands finished up dynamically troublesome. Assignments are less comparable to pre-image attacks on the MD4 calculation.
Hyperspaces and Superspaces: Linearity and Boundedness
Continuing the previous research, we analyze constructions of hyperspaces and superspaces elaborating classifications of such spaces. The main emphasis is on linear hyperspaces, which are formed over linear spaces with additional structures such as norms and seminorms. Linear hyperspaces are constructed over multiseminormed linear spaces and their properties are studied. In particular, it is demonstrated that a linear hyperspace over a multiseminormed linear space is a multihyperseminormed linear space. This research is connected with problems of contemporary quantum physics solving some of these problems related to divergence in a variety of theoretical formulas. The direction originated with the elaboration of distribution theory was further developed by creation of the theory of hypernumbers and extrafunctions. Theory of linear hyperspaces is the next stage in this direction.
Vector space, hypervector, linear hyperspace, superspace, sequential hyperspace, seminorm, norm, bounded hypervector
A New Approach to Hypersoft Sets and their Application in Decision Making
An uncertainty based model, parallel to the soft set (SS) model introduced by Molodtsov, was proposed by Smarandache in the form of hypersoft sets (HSS). The basic characteristic of both these models is that these are parametrized family of sets which are induced by the parameters involved. However, Smarandache had left open the basic operation on HSS. In this paper, we first redefine HSS using the characteristic function based approach, and through it standardize the basic operations like union, intersection and complementation, which are logical and uniform besides being easy to use. We further extend HSS to propose the notions of extended hypersoft set (EHSS) and fuzzy hypersoft set (FHSS). As an application, we propose a decision making algorithm on FHSS model and illustrate the process through an example. This is the first such algorithm using FHSS.
Soft set, hypersoft set, extended hypersoft set, fuzzy hypersoft set, decision making
This paper presents a new approach of graph theory based on natural numbers and matrix only. The natural number is used for vertex and matrix for the edge. The fast graph generation is an advantage of this approach by the successor function. The connected and non-connected graph both is defined by the axiom. There is the arithmetic based on the additive and multiplicative operation over the matrix with the set of natural number. Basically, the graph is defined here without any structure or geometry. This new approach lies with the set of natural numbers and its matrix representation. The proposed graph theory is based on axioms. The geometrical phenomenon is the key inspiration of this theory. In conventional graph theory, the distance between two vertices is defined only by the straight line. The curve distance is ignored. In the proposed graph theory, the straight line as well as curve are also considered. The matrix representation is sufficient for both the edges. If there will be knot instead the curve, then the defined representation is formed the graph. Thus, it is associated with smaller domain and range than the existed graph theory.
Analysis Of Gradient Descent In Back Propagation Neural Network For Maximum And Efficient Utilization
The back-propagation is a successfully established algorithm for multi-layered perceptron neural networks, which is usually with successfully for tiny network architectures or small tasks. In this paper we have highlighted the important libraries for the purpose of implementation of neural networks. After that we have given the process of feed forward in neural network and how we optimize this process by updating the weights by going backward towards the previous layers. The Author will explain how to calculate gradient descant and role of activation functions and their importance in neural networks. Author will provide the deep insights of pytorch and keras library. Author provides an overview of the fundamental theory of backpropagation neural network architecture, including architectural design, performance assessment, function approximation capabilities (and learning), and other aspects of neural network architecture. The survey contains information that has already been published as well as some new findings, such as a formulation of the backpropagation neural network design that allows it to be considered a legitimate neural network. Learn about gradient descent, the role of cost functions especially as a barometer within Machine Learning, different forms of gradient descents, learning rates, and other topics in depth in this article on Gradient Descent in Machine Learning.
ANN, GD, Backpropagation, Deep Learning, Calculus
Sum Divisor Cordial Labeling of Wheel Related Graph Families
In this paper, we prove that the graphs obtained by switching of a rim in wheel Wn, switching of a vertex of degree 2 and 3 in gear graph Gn, switching of a vertex of degree 2 and 4 in flower graph fln, switching of a vertex of degree 2 and 3 in shell graph Sn are sum divisor cordial (SDC). A sum divisor cordial labeling (SDCL) of a graph G with the vertex set V is a bijection from V to {1, 2, 3, . . . , |V (G) |} such that if each an edge xy is assigned the label 1 if 2 divides [g(x) + g(y)] and 0 otherwise, then the number of edges having label 0 and the number of edges having label 1 under the condition |eg(0)−eg(1)| ≤ 1. If G has a sum divisor cordial labeling (SDCL), then it is called a sum divisor cordial (SDC) graph.
The basic relations between universal algebras are homomorphisms and their special cases – monomorphisms, epimorphisms, and isomorphisms (cf., for example, (Kurosh, 1963; Cohn, 1965)). In this paper, we introduce and study a novel relation between universal algebras, which is called projectivity and allows obtaining properties of universal algebras from properties of those universal algebras to which they are weakly or finally projective. The key property of projectivity relations is that they transfer operations from one prearithmetic to another. In addition, given a universal algebra, the projectivity relation allows building universal algebras of the same type on arbitrary sets.
universal algebra, projectivity, arithmetic, prearithmetic, operation, grouppoid, ring
In this Paper I consider a minimal vertex covering sets and minimal strong vertex covering sets. I define Hypergraph, Sub-hypergraph, Partial sub-hypergraph, Vertex covering set, Minimal Vertex covering set, upper vertex covering set and upper vertex covering number in this paper and I consider the effect of removing a vertex on the upper vertex covering number of the hypergraph. Then I prove that the upper vertex covering number does not increase when a vertex is removed from the hypergraphs and I define upper strong vertex set and upper strong vertex covering number. Here again I consider the effect of removing a vertex from the hypergraph on the strong vertex covering number of the hypergraph. Let G be a hypergraph and S⊆V(G) be a vertex covering set of G. Then S is a minimal vertex covering set if and only if for every vertex v in S there is an edge f such that S∩f={v}. And I Prove Let G be a hypergraph and v∈V(G). If αb (G\v)<αb (G) then there is a αb-set of G say T such that v∈T. Let G be a hypergraph and v∈V(G). Then αb (G\v)=αb (G) if and only if every αb-set of G \ v is a αb-set of G. Let G be a hypergraph and v∈V(G). Now I prove that if a vertex is removed from the hypergraph and if the upper vertex covering number decrease, then it is exactly 1 decrease. It means Suppose there is an αb-set T of G such that v∈T. If αb (G\v)<αb (G) then αb (G\v)=αb (G)-1.
Vertex covering set, Vertex covering number, Minimal vertex covering set, Upper vertex covering set, Upper vertex covering number.