The included figure shows a part of the data set collected in 1997, which analyz
ID: 1719683 • Letter: T
Question
The included figure shows a part of the data set collected in 1997, which analyzed the structure of a representative segment of the Internet. The graph is displayed using the so-called log-log plot (see axis labels), which is used when large ranges of data need to be displayed simultaneously, without losing clarity. Quantities that appear linearly related (as approximated by the dashed line) when displayed in a log-log plot are of strong interest to those interested in analyzing the large scale effects coming from a small number of sources, such as the small number of websites driving most traffic in the internet (think of how much traffic daily goes through facebook.com, google.com, amazon.com, and only a handful of others, compared to the rest of the internet).
Figure 1: The number of web pages with a given number of links, as found in a 1997 web crawl of about 200 million web pages. Two double-arrows mark the outliers discussed in the last sub-question. In this exercise you will analyze this graph.
(a) Start by assigning variables to relevant quantities involved here, and symbolically write the number of web pages as a function of the number of links. (As an example, x(t) may be understood as a symbol for “distance x depends on time t”.)
(b) By using the data set and the grid behind it, estimate the slope and the intercept of the line that connects the logarithms of variables defined in part (a). In your report, describe the procedure precisely: label the point(s) on the graph that you used to estimate the coefficients (slope and intercept), write out any calculations that you performed, and give the final expression of the line in terms of logarithms of quantities. You can cut-out (or copy-paste) the graph from the problem and use it in your report.
(c) Use the equation of the line from (b) to find the functional relationship between variables, by performing symbolic operations that result in elimination of any logarithms. The end result should not look like an equation of a line.1 Congratulations: you have just derived a model for your data set!
(d) When you create a new web site, it may have no links to it. Describe how one may use your model to predict the number of freshly-connected websites (#links = 0) using limits. Interpret the number in the context of the problem, discuss whether it is a realistic assumption, and what that means for the validity of your model equation. Make sure your explanation is succinct, but complete.
(e) The two outliers labeled by arrows are both approximately 3 boxes above the values predicted by the model, which would lie roughly on the continuation of the dashed line. Compute the number of websites that is incorrectly predicted at each of the outliers. How do you explain the large difference between these two numbers when the error arrows on the graph are roughly the same length?
Explanation / Answer
In mathematics, an equation is an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. In this situation, variables are also known as unknowns and the values which satisfy the equality are known as solutions. An equation differs from an identity in that an equation is not necessarily true for all possible values of the variable.[1][2]
There are many types of equations, and they are found in many areas of mathematics. The techniques used to examine them differ according to their type.
Algebra studies two main families of equations: polynomial equations and, among them, linear equations. Polynomial equations have the form P(x) = 0, where P is a polynomial. Linear equations have the form a(x) + b = 0, where a is a linear function and b is a vector. To solve them, one uses algorithmic or geometric techniques, coming from linear algebra or mathematical analysis. Changing the domain of a function can change the problem considerably. Algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory. These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions.
In geometry, equations are used to describe geometric figures. As equations that are considered, such as implicit equations or parametric equations have infinitely many solutions, the objective is now different: instead of given the solutions explicitly or counting them, which is impossible, one uses equations for studying properties of figures. This is the starting idea of algebraic geometry, an important area of mathematics.
Differential equations are equations involving one or more functions and their derivatives. They are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model real-life processes in areas such as physics, chemistry, biology, and economics.
The "=" symbol, which appear in every equation, was invented in 1557 by Robert Recorde, who considered that nothing could be more equal than parallel straight lines with the same length.
Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers,[1] algebra introduces quantities without fixed values, known as variables.[2] This use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.
The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Most quantitative results in science and mathematics are expressed as algebraic equations.
Here's a collection of resources that I started on Mathgroup (a collection of Mathematica learning resources) and updated here at Stack Overflow. As this site is dedicated to Mathematica it makes more sense to maintain it here. This represents a huge amount of information; of course it's not exhaustive so feel free to improve it! Also, don't hesitate to share it and suggest other interesting links! Remember, you can always search the online Documentation Center of Mathematica, that is identical to the built-in help of the latest software version.
Links to more advanced aspects of the program that you can start to appreciate once you understand the basics are provided in separate answers (below) as this post became too large.
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