Quadrature formula error is the least

Area is the quantity that expresses the extent of a two-dimensional figure or shape or planar laminain the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object.

Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat.

The area of a shape can be measured by comparing the shape to squares of a fixed size. In mathematicsthe unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

There are several well-known formulas for the areas of simple shapes such as trianglesrectanglesand circles.

Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus. For a solid shape such as a spherecone, or cylinder, the area of its boundary surface is called the surface area.

Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebraand is a basic property of surfaces in differential geometry.

quadrature formula error is the least

Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.

quadrature formula error is the least

An approach to defining what is meant by "area" is through axioms. It can be proved that such an area function actually exists. Every unit of length has a corresponding unit of area, namely the area of a square with the given side length.

Thus areas can be measured in square metres m 2square centimetres cm 2square millimetres mm 2square kilometres km 2square feet ft 2square yards yd 2square miles mi 2and so forth. The SI unit of area is the square metre, which is considered an SI derived unit. This is equivalent to 6 million square millimetres. Other useful conversions are:. In non-metric units, the conversion between two square units is the square of the conversion between the corresponding length units.

There are several other common units for area. The are was the original unit of area in the metric systemwith:. Though the are has fallen out of use, the hectare is still commonly used to measure land: [12]. Other uncommon metric units of area include the tetradthe hectadand the myriad. The acre is also commonly used to measure land areas, where. On the atomic scale, area is measured in units of barnssuch that: [12]. The barn is commonly used in describing the cross-sectional area of interaction in nuclear physics.

In the 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk the region enclosed by a circle is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates[13] but did not identify the constant of proportionality. Eudoxus of Cnidusalso in the 5th century BCE, also found that the area of a disk is proportional to its radius squared.

Subsequently, Book I of Euclid's Elements dealt with equality of areas between two-dimensional figures.The Department of Mathematics provides an environment of research and comprehensive instruction in mathematics and applied mathematics at both undergraduate and graduate levels. Both a BA and a BS program in mathematics are offered, including a BS degree in applied mathematics and a BS degree in mathematics with a specialization in economics. Students in other fields of study may also complete a minor in mathematics; information follows the description of the major.

The requirements for a degree in mathematics or in applied mathematics express the educational intent of the Department of Mathematics; they are drawn with an eye toward the cumulative character of an education based in mathematics, the present emerging state of mathematics, and the scholarly and professional prerequisites of an academic career in mathematics.

Requirements for each bachelor's degree look to the advancement of students' general education in modern mathematics and their knowledge of its relation with the other sciences BS or with the other arts BA.

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Descriptions of the detailed requirements that give meaning to these educational intentions follow. Students should understand that any particular degree requirement can be modified if persuasive reasons are presented to the department; petitions to modify requirements are submitted in person to the director of undergraduate studies or to one of the departmental counselors.

Students should note that only one undergraduate degree may be earned from the Department of Mathematics.

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At what level does an entering student begin mathematics at the University of Chicago? The College and the Department of Mathematics offer several placement exams to help determine the correct starting point for all entering students.

During the summer and through Orientation Week, there are three such exams:. The Online Mathematics Placement Test must be taken once by all entering students in the summer prior to matriculation. The other two exams are offered during Orientation Week, and students are invited to take one or the other on the basis of their success on the Online Mathematics Placement Test.

Solely on the basis of the Online Mathematics Placement Test, the following mathematics courses are the possible placements for each student:. On the basis of the Online Mathematics Placement Test results, students may also be invited to take one of the on-campus exams. On the basis of this exam, a student may receive placement into:. Currently no course credit or placement is offered in the Department of Mathematics at the University of Chicago for British A-level or O-level examinations.

Four bachelor's degrees are available in the Department of Mathematics: the BA in mathematics, the BS in mathematics, the BS in applied mathematics, and the BS in mathematics with specialization in economics.

Programs qualifying students for the degree of BA provide more elective freedom. Programs qualifying students for the degrees of BS require more emphasis in the physical sciences, while the BS in mathematics with specialization in economics has its own set of specialized courses with more electives in economics in place of electives in the physical sciences.

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All degree programs, whether qualifying students for a degree in mathematics or in applied mathematics, require fulfillment of the College's general education requirements.Numerical analysis is the study of algorithms that use numerical approximation as opposed to symbolic manipulations for the problems of mathematical analysis as distinguished from discrete mathematics.

Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. The growth in computing power has revolutionized the use of realistic mathematical models in science and engineering, and subtle numerical analysis is required to implement these detailed models of the world.

For example, ordinary differential equations appear in celestial mechanics predicting the motions of planets, stars and galaxies ; numerical linear algebra is important for data analysis; [2] [3] [4] stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology. Before the advent of modern computers, numerical methods often depended on hand interpolation formulas applied to data from large printed tables.

Since the mid 20th century, computers calculate the required functions instead, but many of the same formulas nevertheless continue to be used as part of the software algorithms. The numerical point of view goes back to the earliest mathematical writings. A tablet from the Yale Babylonian Collection YBCgives a sexagesimal numerical approximation of the square root of 2the length of the diagonal in a unit square.

Numerical analysis continues this long tradition: rather than exact symbolic answers, which can only be applied to real-world measurements by translation into digits, it gives approximate solutions within specified error bounds. The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following:.

The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than years ago. Many great mathematicians of the past were preoccupied by numerical analysis, [5] as is obvious from the names of important algorithms like Newton's methodLagrange interpolation polynomialGaussian eliminationor Euler's method.

To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical estimates of some functions.

The canonical work in the field is the NIST publication edited by Abramowitz and Steguna plus page book of a very large number of commonly used formulas and functions and their values at many points. The function values are no longer very useful when a computer is available, but the large listing of formulas can still be very handy.

The mechanical calculator was also developed as a tool for hand computation.

quadrature formula error is the least

These calculators evolved into electronic computers in the s, and it was then found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of numerical analysis, [5] since now longer and more complicated calculations could be done.

From this table it can be concluded that the solution is between 1. The algorithm might return any number in that range with an error less than 0.

In a two-hour race, the speed of the car is measured at three instants and recorded in the following table. A discretization would be to say that the speed of the car was constant from tothen from to and finally from to This would allow us to estimate the total distance traveled as Note that f 1. Direct methods compute the solution to a problem in a finite number of steps.

These methods would give the precise answer if they were performed in infinite precision arithmetic. Examples include Gaussian eliminationthe QR factorization method for solving systems of linear equationsand the simplex method of linear programming. In practice, finite precision is used and the result is an approximation of the true solution assuming stability. In contrast to direct methods, iterative methods are not expected to terminate in a finite number of steps.

Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit. A convergence test, often involving the residualis specified in order to decide when a sufficiently accurate solution has hopefully been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps in general.

Examples include Newton's method, the bisection methodand Jacobi iteration.Some bloggers did notice, however. But before that, let us look at the big traders or the warren buffets of this world.

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Interestingly, you can easily see which sites you are in competition with in the SERPs by looking for similar ranking keywords. Take for example Hobo-Web. Quality content drives the organic link building end-results Google claims it wants to reward with visibility in its SERPs.

High-quality seo copywriting has never been so important. Google often still relies on links to find, index, categorise and rate websites (and pages) in 2017. Content will always be king (especially in the future) but without links, content can still be a moot point.

Links are like lasers.

Links have the potential to make a site hot, and send that site to the top of the results. Some links get hotter in time, some get colder, and some disappear as each site is affected by those linking to it and from it.

In the diagram below, you can see in any collection of pages, there is natural heat, naturally hot and cold sites, because of the natural interlinking going on between pages. In this model, popular sites are hotter than unpopular sites, and this is a reflection of the real webHot sites are well known and well linked to and more likely to rank for any term if a page is properly optimised on-page and via the internal navigation structure of the website.

Hello, Google Supplemental Index. Or whatever Google calls it in 2017. Google hates paid links because it is an obvious way to generate heat signature, and ranking ability, a site might not deserve. One thing is for sure, though.

If Google were confident they could via algorithmic calculation clearly identify paid links, the whole internet marketing industry would not have been talking about the war on paid links. The hotter a site, or rather page, the better the link for your site. If a keyword is in the anchor text of the link, even better, but only if it is an EDITORIAL LINK, in 2017. The problem is, the easiest links to find out there are on pages with very cold, or at least very diluted heat signatures.

I prefer focusing on improving those signals above, rather than just manipulating link popularity.

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Google calls that spam, and will punish you for it if it detects an intent to deceive its algorithms. Not that that stops a good spammer of course. If you have, you need to concentrate on dissociating yourself first from unnatural links).

Two of the 3 major (UK) search engines (recently) encouraged you to ask websites for links to improve the ranking of your site, and the biggest, Google, tells you not to do it at all. Actionable Advice From Yahoo and Bing: Get links from related sites. Ask for links to increase the number of links to your site. That said, techniques used are often quite revealing of intent. Back To Table Of ContentsGetting other sites to link to yours is easy when you have content worth linking to.

It is that simple in some cases. The more content you add to your site, the more likely people will link to it, today or in the future. So the first thing you should be doing is adding high-quality content to your site. I think How-To articles, or articles that discuss a common problem often discussed in forums, etc. See that little FAQ on your site.Scheduling, simulation, and related modeling processes are used to optimize business processes and management challenges.

Learn more about current and historical statisticians: Ask a statistician videos cover current uses and future trends in statistics. SAS loves stats profiles statisticians working at SAS.

Celebrating statisticians commemorates statistics practitioners from history. Statistics Procedures Community Join our statistics procedures community, where you can ask questions and share your experiences with SAS statistical products. Big Data Get more insights on big data including articles, research and other hot topics. Marketing Explore insights from marketing movers and shakers on a variety of timely topics.

All Rights Reserved Feedback jQuery( document ). Please use the links on the left to access a variety of content available to you or to learn more about getting involved with the division. The ASQ Statistics Division has been publishing a newsletter with the latest in Division news since winter 1980.

For much of that time, there have been Mini-Papers and the Youden Address with technical content. However, this has changed with the 2015 rebranding as Statistics Digest, which contains more technical content such as regular columns by some of the leading names in the field of statistics for quality.

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The trip was superb and the redoing of our itinerary because of the weather was seamless which we appreciated greatly.

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I would highly recommend Nordic Visitor to anyone traveling - they were professional, and so very very helpful. I had a lot of last minute requests that Kristin was able to take care of seamlessly. As a solo female traveler Nordic Visitor gave me peace of mind while traveling in a new place - I don't know if I could have done this trip without them.

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The second day you even sent me to a massage, which was greatly appreciated. Kolbrun arranged for us to have an extra night in Oslo before beginning our tour. The train ride from Oslo to Bergen was terrific. We had time in each city to see the sites and thoroughly enjoyed the cruise aboard the Hurtigruten MS Nordkapp.

Two of the highlights were seeing a spectacular display of the Northern Lights and experiencing the dog sledding in Tromso. We definitely recommend Nordic Visitor to others. The combined organisation of your staff has been incredible. When a snow storm hit Iceland we were contacted by consultants and alternative accomodation was arranged without additional costs. We had zero stress our whole trip as a result of excellent forethought, planning and distribution of the information needed for our trip.

Also the tips marked on the map about food, sightseeing things etc were a gold mine. We used the travel guide with sightseeing locations and suggestions from all the staff on our entire drive throughout Iceland and it added to our experience tenfold.

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The tours were amazing and the ease of planning the trip was wonderful. You gave us a trip if a lifetime and memories to cherish. We were blown away by the service. The champagne and chocolates was a nice touch, too.

Everything was communicated very efficiently, and with total transparency. Nothing to regret, even if we had to cut it short by a day of driving, we wouldn't have changed a thing about our visit. This was my second time to Iceland, first time with Nordic Visitor, and it most definitely won't be our last (on both accounts). We were actually looking at other tours from Nordic Visitor, while driving back into Reykjavik. We both want to do the ring road, eventually, and we will definitely be booking with Nordic Visitor again.

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