Algebra: Structure And Method, Book 1 Free Download
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Each method should be documented and provide a doc test (we are not givingexamples here). In addition, any method defined for the objects or elementsof a category should be supported by a test method, that is executed whenrunning the test suite.
Inheritance from UniqueRepresentationautomatically provides our class with pickling, preserving the uniqueparent condition. If we had defined the class in some external module orin an interactive session, pickling would work immediately.
In the following sections, we will successively add or change details ofMyFrac. Rather than giving a full class definition in each step, wedefine new versions of MyFrac by inheriting from the previouslydefined version of MyFrac. We believe this will help the reader tofocus on the single detail that is relevant in each section.
The string representation is returned by the single--underscore method_repr_. In order to make our fraction field elements distinguishablefrom those already present in Sage, we use a different string representation.
since \(a\) is defined as an element of \(P\). However, we cannot verify yet thatthe integers are contained in the fraction field of the ring of integers. Itdoes not even give a wrong answer, but results in an error:
Sometimes the base classes do not reflect the mathematics: The set of \(m\timesn\) matrices over a field forms, in general, not more than a vectorspace. Hence, this set (called MatrixSpace)is not implemented on top of sage.rings.ring.Ring. However, if\(m=n\), then the matrix space is an algebra, thus, is a ring.
One can provide default methods for all objects of a category, andfor all elements of such objects. Hence, the category framework is a wayto inherit useful stuff that is not present in the base classes. Thesedefault methods do not rely on implementation details, but on mathematicalconcepts.
So, there is no immediate gain for our fraction fields, but additional methodsbecome available to our fraction field elements. Note that some of thesemethods are place-holders: There is no default implementation, but it isrequired (respectively is optional) to implement these methods:
All elements of \(P\) should use the element class. In order to make surethat this also holds for the result of arithmetic operations, we createdthem as instances of self.__class__ in the arithmetic methods ofMyElement.
P.zero() defaults to returning P(0) and thus returns aninstance of P.element_class. Since P.sum([...]) starts the summation withP.zero() and the class of the sum only depends on the firstsummand, by our implementation, we have:
The category framework is sometimes blamed for speed regressions, as ingithub issue #9138 and github issue #11900. But if the category framework is usedproperly, then it is fast. For illustration, we determine the time needed toaccess an attribute inherited from the element class. First, we consider anelement that uses the class that we implemented above, but does not use thecategory framework properly:
A coercion happens implicitly, without being explicitly requested by theuser. Hence, coercion must be based on mathematical rigour. In our example,any element of \(P_2\) can be naturally interpreted as an element of \(P_1\). Wethus have:
We have seen above that some conversions into our fraction fields becameavailable after providing the attribute Element. However, we cannotconvert elements of a fraction field into elements of another fraction field,yet:
Recall that above, the test \(1 \in P\) failed with an error. We try again andfind that the error has disappeared. This is because we are now able toconvert the integer \(1\) into \(P\). But the containment test still yields awrong answer:
By the method above, a parent coercing into the base ring will also coerceinto the fraction field, and a fraction field coerces into another fractionfield if there is a coercion of the corresponding base rings. Now, we have:
The category framework does not only provide functionality but also a testframework. This section logically belongs to the section on categories, butwithout the bits that we have implemented in the section on coercion, ourimplementation of fraction fields would not have passed the tests yet.
We have already seen above that a category can require/suggest certain parentor element methods, that the user must/should implement. This is in order tosmoothly blend with the methods that already exist in Sage.
We have implemented all abstract methods (or inherit them from base classes),we use the category framework, and we have implemented coercions. So, we areconfident that the test suite runs without an error. In fact, it does!
Unfortunately, the list of elements that is returned by the default method isof length one, and that single element could also be a bit more interesting.The method an_element relies on a method _an_element_(), so, we implementthat. We also override the some_elements method.
Last, we observe that the new test has automatically become part of the testsuite. We remark that the existing tests became more serious as well, since wemade sage.structure.parent.Parent.an_element() return something moreinteresting.
Mathematics is essential to science and engineering, and is a fascinating field in its own right. Scientific and engineering problems have often inspired new developments in mathematics, and, conversely, mathematical results have frequently had an impact on business, engineering, the sciences, and technology. At Stevens, we think that an undergraduate program in mathematics should be broad enough to prepare you for a job in industry, while giving you the background to continue your education at the graduate level, should you choose to do so.
The standard program for a concentration in mathematics includes the courses listed below, although not necessarily in exactly the order listed. If these courses do not meet your needs and goals, your program can be changed with the consent of your advisor. For example, you may wish to write a senior thesis, or you may be eligible for advanced placement or the honors calculus sequence. Alternatively, you may want to strengthen your grasp of fundamental concepts by taking MA 134 Discrete Mathematics. See the Department of Mathematics web page for information on when particular courses are offered.
Minor in Mathematical Sciences
A minor in mathematical sciences can be a valuable qualification for students concentrating in other areas. A minor consists of the courses MA 115, MA 116, MA 134, MA 221, MA 222, MA 227, MA 232, MA 234 and one other course chosen with the consent of the Department. The 300-level mathematics courses are typical choices. A student with sufficient background and the consent of the Department may substitute another course for a required course. The average grade in the nine courses must be at least 2.50 to be awarded the Minor in Mathematical Sciences.
Interdisciplinary Program in Computational Science
For students interested in interdisciplinary science and engineering, Stevens offers an undergraduate computational science program. Computational science is a new field in which techniques from mathematics and computer science are used to solve scientific and engineering problems. See the description of the Program in Computational Science in the Interdisciplinary Programs section.
Adequate undergraduate preparation for admission to any masters degree or certificate program, except Financial Engineering, includes analytic geometry and calculus, elementary differential equations, one semester of linear algebra, and one semester of probability or probability and statistics. It is possible to be admitted with the requirement that you make up a deficiency in preparation. For Financial Engineering see below.
Master of Science - Applied Mathematics
This program provides a background in mathematical techniques which are useful in solving practical problems in science and engineering. You are encouraged to include courses from other departments in your program of study.
The program requires 30 credits (10 courses) of coursework. You may transfer up to one third of this amount from outside Stevens. If you know the material in one of the required courses, you may substitute another course. In both cases, you will need the approval of a department advisor. All elective courses must be chosen with the consent of a department advisor.
Master of Science - Stochastic Systems
This program focuses on analysis and optimal decision-making for complex systems involving uncertain data and risk. The program integrates courses in statistics, stochastic processes, stochastic optimization, and stochastic optimal control theory. The application of these mathematical methods to financial systems, network design and routing, supply-chain management, telecommunication systems, pattern recognition, and other areas is outlined. Students are encouraged to apply the techniques they learn to problems derived from their professional work and interests.
Graduate Certificate Programs The Mathematical Science department offers a number of graduate certificate programs. Each program consists of four courses, including one elective chosen with the consent of the departmental advisor. Most courses may be used toward a master's degree, as well as for the certificate. Admission requirements are the same as for the corresponding master's program. Requirements for the Applied Statistics Certificate Program are the same as those listed above for all programs, except Financial Engineering.
Doctoral Program
The primary requirement for a doctoral degree in mathematics is that you produce a dissertation containing an original and significant result in mathematics and its application. You will work under the guidance of a faculty advisor who is an expert in your area of research.
Preparation for dissertation work includes both courses in mathematical fundamentals and practice in communicating mathematics orally and in writing. The courses you take will not necessarily include everything you will need to know. As a doctoral student you will be expected to learn mathematics on your own outside of class when necessary.
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