Take one book. These are not propositions as they are not declarative in nature, that is, they do not declare a definite truth value T or F. Propositional Calculus is. In a book he was reading around , Fermat claimed to have a proof but not enough space This story is the subject of the popular book, Fermat's Enigma by . This text explains how to use mathematical models and methods to analyze prob lems that arise in computer science. Proofs play a central role.
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Simplifying Statement. (Optional) See textbook for more identities. DeMorgan. Distributive law. The DeMorgan's Law allows us to always “move the NOT inside” . Foundation of Computer Science Notes pdf files - MFCS Notes pdf Note: These notes are according to the R09 Syllabus book of JNTU. Download Mathematical Foundation of Computer Science Notes Pdf. We provide bartlocawinlo.ml Mathematical Foundation of Computer Science MFCS lecture notes.
Algebraic graph theory has been applied to many areas including dynamic systems and complexity.
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Other topics[ edit ] A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or weighted graphs , are used to represent structures in which pairwise connections have some numerical values. For example, if a graph represents a road network, the weights could represent the length of each road. There may be several weights associated with each edge, including distance as in the previous example , travel time, or monetary cost.
Such weighted graphs are commonly used to program GPS's, and travel-planning search engines that compare flight times and costs. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy  and L'Huilier ,  and represents the beginning of the branch of mathematics known as topology. The techniques he used mainly concern the enumeration of graphs with particular properties.
These were generalized by De Bruijn in Cayley linked his results on trees with contemporary studies of chemical composition. Many incorrect proofs have been proposed, including those by Cayley, Kempe , and others. The study and the generalization of this problem by Tait , Heawood , Ramsey and Hadwiger led to the study of the colorings of the graphs embedded on surfaces with arbitrary genus. The four color problem remained unsolved for more than a century.
In Heinrich Heesch published a method for solving the problem using computers. A simpler proof considering only configurations was given twenty years later by Robertson , Seymour , Sanders and Thomas. Another important factor of common development of graph theory and topology came from the use of the techniques of modern algebra.
The first example of such a use comes from the work of the physicist Gustav Kirchhoff , who published in his Kirchhoff's circuit laws for calculating the voltage and current in electric circuits. Main article: Graph drawing Graphs are represented visually by drawing a point or circle for every vertex, and drawing a line between two vertices if they are connected by an edge.
If the graph is directed, the direction is indicated by drawing an edge. A graph drawing should not be confused with the graph itself the abstract, non-visual structure as there are several ways to structure the graph drawing.
All that matters is which vertices are connected to which others by how many edges and not the exact layout. In practice, it is often difficult to decide if two drawings represent the same graph. Depending on the problem domain some layouts may be better suited and easier to understand than others.
The pioneering work of W. Tutte was very influential on the subject of graph drawing. Among other achievements, he introduced the use of linear algebraic methods to obtain graph drawings. Graph drawing also can be said to encompass problems that deal with the crossing number and its various generalizations. The crossing number of a graph is the minimum number of intersections between edges that a drawing of the graph in the plane must contain.
For a planar graph , the crossing number is zero by definition. Drawings on surfaces other than the plane are also studied. Graph-theoretic data structures[ edit ] Main article: Graph abstract data type There are different ways to store graphs in a computer system. The data structure used depends on both the graph structure and the algorithm used for manipulating the graph.
Theoretically one can distinguish between list and matrix structures but in concrete applications the best structure is often a combination of both. List structures are often preferred for sparse graphs as they have smaller memory requirements.
Matrix structures on the other hand provide faster access for some applications but can consume huge amounts of memory. List structures include the incidence list , an array of pairs of vertices, and the adjacency list , which separately lists the neighbors of each vertex: Much like the incidence list, each vertex has a list of which vertices it is adjacent to. Matrix structures include the incidence matrix , a matrix of 0's and 1's whose rows represent vertices and whose columns represent edges, and the adjacency matrix , in which both the rows and columns are indexed by vertices.
In both cases a 1 indicates two adjacent objects and a 0 indicates two non-adjacent objects. The Laplacian matrix is a modified form of the adjacency matrix that incorporates information about the degrees of the vertices, and is useful in some calculations such as Kirchhoff's theorem on the number of spanning trees of a graph.
The distance matrix , like the adjacency matrix, has both its rows and columns indexed by vertices, but rather than containing a 0 or a 1 in each cell it contains the length of a shortest path between two vertices. Enumeration[ edit ] There is a large literature on graphical enumeration : the problem of counting graphs meeting specified conditions.
Some of this work is found in Harary and Palmer Subgraphs, induced subgraphs, and minors[ edit ] A common problem, called the subgraph isomorphism problem , is finding a fixed graph as a subgraph in a given graph. One reason to be interested in such a question is that many graph properties are hereditary for subgraphs, which means that a graph has the property if and only if all subgraphs have it too. Unfortunately, finding maximal subgraphs of a certain kind is often an NP-complete problem.
For example: Finding the largest complete subgraph is called the clique problem NP-complete. One special case of subgraph isomorphism is the graph isomorphism problem. It asks whether two graphs are isomorphic. It is not known whether this problem is NP-complete, nor whether it can be solved in polynomial time.
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A similar problem is finding induced subgraphs in a given graph. Again, some important graph properties are hereditary with respect to induced subgraphs, which means that a graph has a property if and only if all induced subgraphs also have it.
Finding maximal induced subgraphs of a certain kind is also often NP-complete. For example: Finding the largest edgeless induced subgraph or independent set is called the independent set problem NP-complete.
Still another such problem, the minor containment problem, is to find a fixed graph as a minor of a given graph. A minor or subcontraction of a graph is any graph obtained by taking a subgraph and contracting some or no edges.
Many graph properties are hereditary for minors, which means that a graph has a property if and only if all minors have it too. For example, Wagner's Theorem states: A graph is planar if it contains as a minor neither the complete bipartite graph K3,3 see the Three-cottage problem nor the complete graph K5.
A similar problem, the subdivision containment problem, is to find a fixed graph as a subdivision of a given graph. A subdivision or homeomorphism of a graph is any graph obtained by subdividing some or no edges.
The voltage decreased gradually with time as the supply of organic waste and sewage sludge were degraded and used by the microbes. In anaerobic condition with the total weight of the sample mixture 10 kg, at its 60th day, the final output of voltage was mV. This voltage variation reading was taken over 60 days until it stopped showing a minimum output of voltage.
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However, the voltage did not increase more after the 21th day because of the bacterial activity decreased for the shortage of food. The sample III produces a fair amount of voltage yield though the total amount of waste is less than any other samples.
In this cell also, the voltage generation were increased sharply within the second day of the operation of the microbial fuel cell. It increased gradually and peaked after 13th days.
Peak voltage was around mV using the ratio of sewage sludge and organic waste in anaerobic condition and the total weight of the mixture was same as sample I and sample II. This voltage variation reading was taken until it stopped showing a minimum of output of voltage. However, voltage did not increase after the 53th day because of the bacterial activity was decreased for the shortage of food. Bioelectricity from Pure Samples In sample IV, the experiment was conducted only with the organic solid wastes to observe the variation of voltage generation than other MFCs with the time which is shown in Figure 5.
Only a total of 10 kg organic waste used to ensure the volume equivalent level of the designed MFC with this sample. This cell observation continued for 60 days and the final voltage gained of mV. Strick et al. On the other hand, Khare and Bundela  reported mV voltage using waste water from food industries.
In contrast with above two results, the maximum voltage generation was recorded from in this study using only kitchen wastes in a one-chamber MFC. In Figure 5 , sample V contained pure sewage sludge to observe the variation of voltage generation with the time.
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From the second day, the voltage generation was high; but within the time, the voltage output decreased gradually like sample IV. Jati  and Hamzah  reported due to the presence of nutrients sewage sludge which supports the growth of microorganisms and accelerate the electricity production in very short time.
But it lowered from mV to mV 2 - 7 days. This sample showed much higher values of voltage generation compared with the other four samples.
This way shows us an effective solution to manage the vast nutrient rich sewage sludge from the commercial and industrial areas and convert it constantly something beneficial to the human kind.
In addition, the use of these sewage sludge helps from the drainage problems in urban areas also.
However, using only the sewage sludge and the peak voltage gained mV at the 13th day from the microbial fuel cell. The final voltage of this cell was also plentiful which presented mV voltage output at 60th day.
Relationship between Voltage Output and Current Density Always there is definite relationship between voltage output and current density.
Figure 6 represents the relationship between voltage generation current in the MFCs at the 2nd week of operation. It shows that the relationship is almost linear.
This linear curve indicates how stable the MFCs continue voltage generation as a function of current production with time  and the relationship stated that a stable voltage generation. These values indicate that mixed samples voltage production rate was initially low whereas the rate for pure samples was high at initial stage.
Effect of Mixing Ratio on Voltage Generation Sewage sludge and organic wastes were mixed with a ratio of the same volume to produce bioelectricity in a one chambered microbial fuel cell MFC under anaerobic condition. It means sewage sludge has the potential for bioelectricity production as a resource recovery option.
Green energy production is presented as one of the ways of fulfilling alternative fuel demands of the future and to overcome the global energy crisis . Similarly, sample IV had a sustaining voltage from 1st week to 6th week after sample V. And other samples voltage generation was low and stable. From the sample I, II and III, it shows that co-digestion of organic waste and sewage sludge produce less amount of voltage and more production days required for microbial activity.Also, "the Feynman graphs and rules of calculation summarize quantum field theory in a form in close contact with the experimental numbers one wants to understand.
At 28th day, the voltages were started to decrease gradually with time as the supply of food was used up by the bacteria.
Electrical conductivity can be even more enhanced by mixing carbon black with AC during the preparation of the cathode . The organic waste and sewage sludge can be utilized as a power generation source as well as compost production for using as an alternative soil conditioner. Plain AC had the lowest performances 1. This voltage reading was taken until it stopped showing a better output.
Bodlaender and A. Traveling Salesman Problems in Temporal Graphs. Parameterized Approximations via d - Skew-Symmetric Multicut.
Electrical conductivity can be even more enhanced by mixing carbon black with AC during the preparation of the cathode .
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