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From Wikipedia
In computer programming, particularly functional programming and type theory, an algebraic data type (sometimes also called a variant type) is a datatype each of whose values is data from other datatypes wrapped in one of the constructors of the datatype. Any wrapped datum is an argument to the constructor. In contrast to other datatypes, the constructor is not executed and the only way to operate on the data is to unwrap the constructor using pattern matching. The most common algebraic data type is a list with two constructors: Nil or [] for an empty list, and Cons (an abbreviation of construct), ::, or : for the combination of a new element with a shorter list (for example Cons 1 [2, 3, 4] or 1:[2,3,4]). Special cases of algebraic types are product types i.e. tuples and records (only one constructor), sum types or tagged unions (many constructors with a single argument) and enumerated types (many constructors with no arguments). Algebraic types are one kind of composite type (i.e. a type formed by combining other types). An algebraic data type may also be an abstract data type (ADT) if it is exported from a module without its constructors. Values of such a type can only be manipulated using functions defined in the same module as the type itself. In set theory the equivalent of an algebraic data type is a disjoint union â€“ a set whose elements are pairs consisting of a tag (equivalent to a constructor) and an object of a type corresponding to the tag (equivalent to the constructor arguments). Examples For example, in Haskell we can define a new algebraic data type, Tree: data Tree = Empty  Leaf Int  Node Tree Tree Here, Empty, Leaf and Node are called data constructors. Tree is a type constructor (in this case a nullary one). In the rest of this article constructor shall mean data constructor. Similarly, in OCaml syntax the above example may be written: type tree = Empty  Leaf of int  Node of tree * tree In most languages that support algebraic data types, it's possible to define polymorphic types. Examples are given later in this article. Somewhat similar to a function, a data constructor is applied to arguments of an appropriate type, yielding an instance of the data type to which the type constructor belongs. For instance, the data constructor Leaf is logically a function Int > Tree, meaning that giving an integer as an argument to Leaf produces a value of the type Tree. As Node takes two arguments of the type Tree itself, the datatype is recursive. Operations on algebraic data types can be defined by using pattern matching to retrieve the arguments. For example, consider a function to find the depth of a Tree, given here in Haskell: depth :: Tree > Int depth Empty = 0 depth (Leaf n) = 1 depth (Node l r) = 1 + max (depth l) (depth r) Thus, a Tree given to depth can be constructed using any of Empty, Leaf or Node and we must match for any of them respectively to deal with all cases. In case of Node, the pattern extracts the subtrees l and r for further processing. Algebraic data types are particularly wellsuited to the implementation of abstract syntax. For instance, the following algebraic data type describes a simple language representing numerical expressions: data Expression = Number Int  Add Expression Expression  Minus Expression  Mult Expression Expression  Divide Expression Expression An element of such a data type would have a form such as Mult (Add (Number 4) (Minus (Number 1))) (Number 2). Writing an evaluation function for this language is a simple exercise; however, more complex transformations also become feasible. For instance, an optimization pass in a compiler might be written as a function taking an abstract expression as input and returning an optimized form. Explanation What is happening is that we have a datatype, which can be â€œone of several types of things.â€� Each â€œtype of thingâ€� is associated with an identifier called a constructor, which can be thought of as a kind of tag for that kind of data. Each constructor can carry with it a different type of data. A constructor could carry no data at all (e.g. "Empty" in the example above), carry one piece of data (e.g. â€œLeafâ€� has one Int value), or multiple pieces of data (e.g. â€œNodeâ€� has two Tree values). When we want to do something with a value of this Tree algebraic data type, we deconstruct it using a process known as pattern matching. It involves matching the data with a series of patterns. The example function "depth" above patternmatches its argument with three patterns. When the function is called, it finds the first pattern that matches its argument, performs any variable bindings that are found in the pattern, and evaluates the expression corresponding to the pattern. Each pattern has a form that resembles the structure of some possible value of this datatype. The first pattern above simply matches values of the constructor Empty. The second pattern above matches values of the constructor Leaf. Patterns are recursive, so then the data that is associated with that constructor is matched with the pattern "n". In this case, a lowercase identifier represents a pattern that matches any value, which then is bound to a variable of that name â€” in this case, a variable â€œnâ€� is bound to the integer value stored in the data type â€” to be used in the expression to be evaluated. The recursion in patterns in this example are trivial, but a possible more complex recursive pattern would be something like Node (Node (Leaf 4) x) (Node y (Node Empty z)). Recursive patterns several layers deep are used for example in balancing redblack trees, which involve cases that require looking at colors several layers deep. The example above is operationally equivalent to the following pseudocode: switch on (data.constructor) case Empty: return 0 case Leaf: let n = data.field1 return 1 case Node: let l = data.field1 let r = data.field2 return 1 + max (depth l) (depth r) The comparison of this with pattern matching will point out some of the advantages of algebraic data types and pattern matching. First is type safety. The pseudocode above relies on the diligence of the programmer to not access field2 when the constructor is a Leaf, for example. Also, the type of field1 is different for Leaf and Node (for Leaf it is Int; for Node it is Tree), so the type system would have difficulties assigning a static type to it in a safe way in a traditional record data structure. However, in pattern matching, the type of each extracted value is checked based on the types declared by the relevant constructor, and how many values you can extract is known based on the constructor, so it does not face these problems. Second, in pattern matching, the compiler statically checks that all cases are handled. If one of the cases of the â€œdepthâ€� function above were missing, the compiler would issue a warning, indicating that a case is not handled. This task may seem easy for the simple patterns above, but with many complicated recursive patterns, the task becomes difficult for the average human (or compiler, if it has to check arbitrary nested ifelse constructs) to handle. Similarly, there may be patterns which never match (i.e. it is already covered by previous patterns), and the compiler can also check and issue warnings for these, as they may indicate an error in reasoning. Do not confuse these patterns with regular expression patterns used in string pattern matching. The purpose is similar â€” to check whether a piece of data matches certain constraints, and if so, extract relevant parts of it for processing â€” but the mechanism is very different. This kind of pattern matching on algebraic data types matches on the structural properties of an object rather than on the character sequence of strings. Theory A general algebraic data type is a possibly recursive sum type of product types. Each constructor tags a product type to separate it from others, or if there is only one constructor, the data type is a product type. Further, the parameter types of a constructor are the factors of the product type. A parameterless constructor corresponds to the empty product. If a dat
The algebraic connectivity of a graph G is the secondsmallest eigenvalue of the Laplacian matrix of G. This eigenvalue is greater than 0 if and only if G is a connected graph. This is a corollary to the fact that the number of times 0 appears as an eigenvalue in the Laplacian is the number of connected components in the graph. The magnitude of this value reflects how well connected the overall graph is, and has been used in analysing the synchronizability of networks. Properties The algebraic connectivity of a graph G is greater than 0 if and only if G is a connected graph. Furthermore, the value of the algebraic connectivity is bounded above by the traditional (vertex) connectivity of the graph. If the number of vertices of a connected graph is n and the diameter is D, the algebraic connectivity is known to be bounded below by 1/nD, and in fact (in a result due to Brendan McKay) by 4/nD. For the example shown above, for example, 4/18 = 0.222 â‰¤ 0.722 â‰¤ 1, but for many large graphs the algebraic connectivity is much closer to the lower bound than the upper. Unlike the traditional connectivity, the algebraic connectivity is dependent on the number of vertices, as well as the way in which vertices are connected. In random graphs, the algebraic connectivity decreases with the number of vertices, and increases with the average degree. The exact definition of the algebraic connectivity depends on the type of Laplacian used. Fan Chung has developed an extensive theory using a rescaled version of the Laplacian, eliminating the dependence on the number of vertices, so that the bounds are somewhat different. In models of synchronization on networks, such as the Kuramoto model, the Laplacian matrix arises naturally, and so the algebraic connectivity gives an indication of how easily the network will synchronize. However, other measures, such as the average distance (characteristic path length) can also be used, and in fact the algebraic connectivity is closely related to the (reciprocal of the) average distance. The algebraic connectivity also relates to other connectivity attributes, such as the isoperimetric number, which is bounded below by half the algebraic connectivity. The Fiedler vector The original theory related to algebraic connectivity was produced by Miroslav Fiedler. In his honor the eigenvector associated with the algebraic connectivity has been named the Fiedler vector. The Fiedler vector can be used to partition a graph. For the example graph in the introductory section, the Fiedler vector is <0.415, 0.309, 0.069, âˆ’0.221, 0.221, âˆ’0.794>. The negative values are associated with the poorly connected vertex 6, and the neighbouring articulation point, vertex 4; while the positive values are associated with the other vertices. The signs of the values in the Fiedler vector can therefore be used to partition this graph into two components: {1, 2, 3, 5} and {4, 6}. Alternatively, the value of 0.069 (which is close to zero) can be placed in a class of its own, partitioning the graph into three components: {1, 2, 5}, {3}, and {4, 6}.
In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection ofvertices or 'nodes' and a collection of edges that connect pairs of vertices. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another; see graph (mathematics) for more detailed definitions and for other variations in the types of graphs that are commonly considered. The graphs studied in graph theory should not be confused with "graphs of functions" and other kinds of graphs.
Graphs are one of the prime objects of study in Discrete Mathematics. Refer to Glossary of graph theory for basic definitions in graph theory.
History
The paper written by Leonhard Euler on the Seven Bridges of KÃ¶nigsbergand published in 1736 is regarded as the first paper in the history of graph theory. This paper, as well as the one written byVandermonde on the knight problem, carried on with the analysis situs initiated byLeibniz. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy and L'Huillier, and is at the origin of topology.
More than one century after Euler's paper on the bridges of KÃ¶nigsberg and while Listing introduced topology, Cayley was led by the study of particular analytical forms arising from differential calculus to study a particular class of graphs, the trees. This study had many implications in theoreticalchemistry. The involved techniques mainly concerned the enumeration of graphs having particular properties. Enumerative graph theory then rose from the results of Cayley and the fundamental results published by PÃ³lya between 1935 and 1937 and the generalization of these by De Bruijn in 1959. Cayley linked his results on trees with the contemporary studies of chemical composition. The fusion of the ideas coming from mathematics with those coming from chemistry is at the origin of a part of the standard terminology of graph theory.
In particular, the term "graph" was introduced by Sylvester in a paper published in 1878 in Nature, where he draws an analogy between "quantic invariants" and "covariants" of algebra and molecular diagrams:
 "[...] Every invariant and covariant thus becomes expressible by a graph precisely identical with a KekulÃ©an diagram or chemicograph. [...] I give a rule for the geometrical multiplication of graphs, i.e. for constructing a graph to the product of in or covariants whose separate graphs are given. [...]" (italics as in the original).
One of the most famous and productive problems of graph theory is the four color problem: "Is it true that any map drawn in the plane may have its regions colored with four colors, in such a way that any two regions having a common border have different colors?" This problem was first posed by Francis Guthrie in 1852 and its first written record is in a letter of De Morgan addressed to Hamilton the same year. 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. Tait's reformulation generated a new class of problems, the factorization problems, particularly studied by Petersen and KÅ‘nig. The works of Ramsey on colorations and more specially the results obtained by TurÃ¡n in 1941 was at the origin of another branch of graph theory, extremal graph theory.
The four color problem remained unsolved for more than a century. In 1969 Heinrich Heesch published a method for solving the problem using computers. A computeraided proof produced in 1976 by Kenneth Appel and Wolfgang Haken makes fundamental use of the notion of "discharging" developed by Heesch. The proof involved checking the properties of 1,936 configurations by computer, and was not fully accepted at the time due to its complexity. A simpler p
From Encyclopedia
A graph is a pictorial representation of the relationship between two quantities. A graph can be anything from a simple bar graph that displays the measurements of various objects to a more complicated graph of functions in two or three dimensions. The former shows the relationship between the kind of object and its quantity; the latter shows the relationship between input and output. Graphing is a way to make information easier for a viewer to absorb. The simplest graphs show the number of many objects. For example, a bar graph might name the months of the year along a horizontal axis and show numbers (for the number of days in each month) along a vertical axis. Then a rectangle (or bar) is drawn above each month. The height of the bar might indicate the number of days in that month on which it rained, or on which a person exercised, or on which the temperature rose above 90 degrees. See the generic example of a bar graph below (top right). Another simple kind of graph is a circle graph or pie graph, which shows fractions or percentages. In this kind of graph, a circle is divided into pieshaped sectors. Each sector is given a label and indicates the fraction of the total area that goes with that label. See the generic example of a pie graph below (top middle). A pie chart might be used to display the percentages of a budget that are allotted to various expenditures. If the sector labeled "medical bills" takes up twotenths of the area of the circle, that means that twotenths, or 20 percent, of the budget is devoted to medical expenses. Usually the percentages are written on the sectors along with their labels to make the graph easier to read. In both bar graphs and pie graphs, the reader can immediately pick out the largest and smallest categories without having to search through a chart or list, making it easy to compare the relative sizes of many objects simultaneously. Often the two quantities being graphed can both be represented numerically. For example, student scores on examinations are often plotted on a graph, especially if there are many students taking the exam. In such a graph, the numbers on the horizontal axis represent the possible scores on the exam, and the numbers on the vertical axis represent numbers of students who earned that score. The information could be plotted as a simple bar graph. If only the top point of each bar is plotted and a curve is drawn to connect these points, the result is a line graph. See the generic example of a line graph on the pevious page (top left). Although the points on the line between the plotted points do not correspond to any pieces of information, a smooth line can be easier to understand than a large collection of bars or dots. Graphs become slightly more complicated when one (or both) of the quantities in the graph can have continuous values rather than a discrete set. A common example of this is a quantity that changes over time. For example, a scientist might be observing the rate of growth of bacteria. The rates could be plotted so that the horizontal axis displays units of time and the vertical axis displays numbers (how many bacteria exist). Then, for instance, the point (3,1000) would mean that at "time 3" (which could mean three o'clock, or three seconds after starting, or various other times, depending on the units being used and the starting point of the experiment) there were one thousand bacteria in the sample. The rise and fall of the graph show the increases and decreases in the number of bacteria. In this case, even though only a finite set of points represent actual data, the remaining points do have a natural interpretation. For instance, suppose that in addition to the point (3,1000), the graph also contains the point (4,1500) and that both of these points correspond to actual measurements. If the scientist joins all of the points on the graph by a line, then the point (3.5,1200) might lie on the graph, or perhaps the point (3.5,1350). There are many different lines that can be drawn through a collection of points. Looking at the overall shape of the data points helps the scientist decide which line is the most reasonable fit. In the previous example, the scientist could estimate that at time 3.5, there were 1200 (or 1350) bacteria in the sample. Thus graphing can be helpful in making estimates and predictions. Sometimes the purpose for drawing a graph may not be to view the data already known but to construct a mathematical model that will allow one to analyze data and make predictions. One of the simplest models that can be constructed from a set of data is called a bestfit line. Such a line is useful in situations in which the data are roughly linearâ€”that is, they are increasing or decreasing at a roughly constant rate but do not fall precisely on a line. (See graph on the previous page, bottom right.) A bestfit line can be a very useful tool for analyzing data because lines have very simple formulas describing their behavior. If, for instance, one has collected data up to time 5 and wishes to predict what the value will be at time 15, the value 15 can be inserted into the formula for the line to derive an estimation. One can also determine how good an estimate is likely to be by computing the correlation factor for the data. The correlation factor is a quantity that measures how close the set of data is to being linear; that is, how good a "fit" the bestfit line actually is. One of the most common uses of graphs is to display the information encoded in a function. A function, informally speaking, is an operation or rule that can be applied to numbers. Functions are usually graphed in the cartesian plane (that is, the x,y plane) with the horizontal or x axis representing the input variable and the vertical or y axis representing the output variable. The graph of a function differs from the other types of graphs described so far in that all the points on the graph represent actual information. A concrete relationship, usually given by a mathematical formula, connects the two objects being analyzed. For example, the "squaring" function takes numbers and squares them. Thus an input of the number 1 corresponds to an output of 1; an input of 2 corresponds to an output of 4; an input of âˆ’7 corresponds to an output of 49; and so on. Therefore, the graph of this function contains the points (1, 1), (2, 4), (âˆ’7, 49), and infinitely many others. Does the point (10, 78) lie on this graph? To determine the answer, examine which characteristics all the points on the graph have in common. Any point on the graph of a function represents an inputoutput pair, with the x coordinate representing input and the y coordinate representing output. With the squaring function, each output value is the square of the corresponding input value, so on the graph of the squaring function, each y coordinate must be the square of the corresponding x coordinate. Because 78 is not the square of 10, the point (10, 78) does not lie on the graph of the squaring function. It is traditional to name graphs with an equation rather than with words. The equation of any graph, regardless of whether it is the graph of a function, is meant to be a perfect description of the graphâ€”it should tell the viewer the relationship between the x  and y coordinates of the numbers being graphed. For example, the equation of the graph of the squaring function is y = x Â² because the y coordinate of any point on the graph is the square of the x coordinate. The line that passes through the point (0, 3) and slants upwards with slope 4 (that is, at a rate of four units up for every one unit to the right) has equation y = 4x + 3. This indicates that for every point on the graph, the y coordinate is 3 more than 4 times the x coordinate. An equation of a graph has many uses: it is not only a description of the graph but also a mechanism for finding points on the graph and a test for determining whether a given point lies on the graph. For example, to find out whether the point (278, 3254) lies on the line y = 4x + 3, simply insert (278, 3254), resulting in the inequality 3254 â‰ 4(278) + 3. Because t
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Answers:ti83 can also use for algebra 1, algebra 2, trigonometry, and calculus. The TI89 Titanium's advanced functionality and 3D graphing make problemsolving for AP*, advanced mathematics, and engineering courses infinitely easier. It's the most powerful TI graphing calculator allowed for use on the AP* Calculus, AP Statistics, AP Physics, AP Chemistry, PSAT/NMSQT**, SAT I , SAT II, and Math IC & IIC exams. So, If you are in algebra 1 graph and linear equations , you should use ti83...or not, accept to you But when you are in algebra two, you will need ti83 or ti84(they are the same)
Answers:http://nces.ed.gov/nceskids/CreateAGraph/default.aspx
Answers:Types of line graph Sometimes line graphs will represent different sets of data on the same line graph, so comparisons can be made. This is sometimes called a comparison line graph. If information can be subdivided into two (or more) types of data  eg total population growth, female population growth and male population growth  then all three can be drawn on the graph. This type of line graph is called a compound line graph. Where the points on a graph do not map into a neat line, a line is drawn through the middle of the points to show the trend or correlation. This is called a scatter graph.
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