www.shoup.net/ntb.
Acknowledgments. I would like to thank a **number** of people who volunteered their time and energy in reviewing parts of the book at various stages: Joël Alwen, Siddhartha Annapureddy, John Black, Carl Bosley, Joshua Brody, Jan Camenisch, David Cash, Sherman Chow, Ronald Cramer, Marisa Debowsky, Alex Dent, Nelly Fazio, Rosario Gennaro, Mark Giesbrecht, Stuart Haber, Kristiyan Haralambiev, Gene Itkis, Charanjit Jutla, Jonathan Katz, Eike Kiltz, Alfred Menezes, Ilya Mironov, Phong Nguyen, Antonio Nicolosi, Roberto Oliveira, Leonid Reyzin, Louis Salvail, Berry Schoenmakers, Hovav Shacham, Yair Sovran, Panos Toulis, and Daniel Wichs. A very special thanks goes to George Stephanides, who trans- lated the first edition of the book into Greek and reviewed the entire book in prepa- ration for the second edition. I am also grateful to the National Science Foundation for their support provided under grants CCR-0310297 and CNS-0716690. Finally, thanks to David Tranah for all his help and advice, and to David and his colleagues at Cambridge University Press for their progressive attitudes regarding intellectual property and open access.

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4 = 3 4 < 1.
4 Diophantine Equations
Finally, once we’ve developed this **algebra**, we can begin applying it to Diophantine equations, central ob- jects of study in **number** **theory**. Diophantine equations ask us to find integer solutions, which are often significantly harder than finding real or complex solutions. We will grapple with a few famous examples in the coming sections, relying on the structure of the rings studied earlier.

The orbit of a Lagrangian under this nonlinear action of the ultraviolet group is in general infinite-dimensional. It can sometimes be cut down to a finite-dimensional space as follows. As in Example 26, we cut down to the group of renormalizations of mass dimension at most 0, which acts on the space of Lagrangians whose cou- pling constants all have mass dimension at least 0. If we also add the condition that the Lagrangian is Lorentz invariant, then we sometimes get finite-dimensional spaces of Lagrangians. The point is that the classical fields themselves tend to have positive mass dimension, so if the coupling constants all have nonnegative mass dimension then the fields appearing in any term of the Lagrangian have total mass at most d (canceling out the −d coming from the density) which severely limits the possibilities. At one time the Lagrangians with all coupling constants of nonnegative mass dimension were called renormalizable Lagrangians, though now all Lagrangians are regarded as renormalizable in a more general sense where one allows an infinite **number** of terms in the Lagrangian.

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1. **Introduction**
Harris, Shepherd-Barron, and Taylor have proved in [Harris et al. 2010] a potential modularity theorem, showing that certain Galois representations become automor- phic after a sufficiently large totally real base change. In their argument, a key role is played by certain families of hypersurfaces, called Dwork families — in particular, by the part of the cohomology of the family which is invariant under a certain group action. (We will write F for the motive given by this part of the cohomology.) The importance of F to their argument is reflected in the statement of the theorem they prove: in order to prove an l-adic Galois representation r is potentially modular using their theorem, one requires, among other conditions, that the restriction of the residual representation of r to inertia at primes above l be isomorphic to the restriction of the residual representation of some element of the family F.

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Notice that apart from the identity element, no rotation can fix any arrangement, so when g 6= ι, Fix G (g) = ∅, while Fix G (ι) = X. Hence the **number** of indistinguishable seating plans is
7!/7 = 6! = 720.
Example 2.41 . Find the **number** of distinguishable ways there are to colour the edges of an equilateral triangle using four different colours, where each colour can be used on more than one edge.

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Preface
First Edition
This text is an **introduction** to **algebra** for undergraduates who are interested in careers which require a strong background in mathematics. It will benefit stu- dents studying computer science and physical sciences, who plan to teach math- ematics in schools, or to work in industry or finance. The book assumes that the reader has a solid background in linear **algebra**. For the first 12 chapters el- ementary operations, elementary matrices, linear independence and rank are im- portant. In the second half of the book abstract vector spaces are used. Students will need to have experience proving results. Some acquaintance with Euclidean geometry is also desirable. In fact I have found that a course in Euclidean geom- etry fits together very well with the **algebra** in the first 12 chapters. But one can avoid the geometry in the book by simply omitting chapter 7 and the geometric parts of chapters 9 and 18.

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16 CHAPTER 2. BASIC COMMUTATIVE **ALGEBRA** generated abelian groups can be presented as quotients of finite rank free abelian groups, and such a presentation can be reinterpreted in terms of matrices over the integers. Next we describe how to use row and column operations over the integers to show that every matrix over the integers is equivalent to one in a canonical diagonal form, called the Smith normal form. We obtain a proof of the theorem by reinterpreting Smith normal form in terms of groups.

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3. John Cannon, Allan Steel, et al., Magma **Computational** **Algebra** System, http:
//magma.maths.usyd.edu.au/magma/.
4. Burcin Erocal, William Stein, The Sage Project: Unifying Free Mathemat- ical Software to Create a Viable Alternative to Magma, Maple, Mathe- matica and Matlab, http://wstein.org/papers/icms/icms 2010.pdf, http://www.

Historical Note
Joseph-Louis Lagrange (1736–1813), born in Turin, Italy, was of French and Italian descent. His talent for mathematics became apparent at an early age. Leonhard Euler recognized Lagrange’s abilities when Lagrange, who was only 19, communicated to Euler some work that he had done in the calculus of variations. That year he was also named a professor at the Royal Artillery School in Turin. At the age of 23 he joined the Berlin Academy. Frederick the Great had written to Lagrange proclaiming that the “greatest king in Europe” should have the “greatest mathematician in Europe” at his court. For 20 years Lagrange held the position vacated by his mentor, Euler. His works include contributions to **number** **theory**, group **theory**, physics and mechanics, the calculus of variations, the **theory** of equations, and differential equations. Along with Laplace and Lavoisier, Lagrange was one of the people responsible for designing the metric system. During his life Lagrange profoundly influenced the development of mathematics, leaving much to the next generation of mathematicians in the form of examples and new problems to be solved.

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A good argument could be made that linear **algebra** is the most useful subject in all of mathematics and that it exceeds even courses like calculus in its signiﬁcance. It is used extensively in applied mathematics and engineering. Continuum mechanics, for example, makes use of topics from linear **algebra** in deﬁning things like the strain and in determining appropriate constitutive laws. It is fundamental in the study of statistics. For example, principal component analysis is really based on the singular value decomposition discussed in this book. It is also fundamental in pure mathematics areas like **number** **theory**, functional analysis, geometric measure **theory**, and diﬀerential geometry. Even calculus cannot be correctly understood without it. For example, the derivative of a function of many variables is an example of a linear transformation, and this is the way it must be understood as soon as you consider functions of more than one variable.

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On higher functional levels, digital systems exhibit features that are related to **number** **theory**, graph **theory**, and discrete mathematics in general. Then, it is natural that many parts of these (and other) fields have been tied together to form the theoretical foundations of design and analysis of digital systems and collectively constituting the Switching **theory**.
In this paper, we present some historical observations about the early development of switching **theory**, especially the application of Boolean **algebra**. We want to draw attention to the contributions of scholars from different parts of the World many of which cannot be found in the standard literature.

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Excellent, but when do we stop? Let us think about this for a moment. We have given up the power of ran- domness and are forcing ourselves to choose sequen- tially among small numbers for the trial bases a. Can we argue heuristically that they continue to behave as if they were random choices? Well, there are some connections among them. For example, if taking a = 2 does not result in a proof that n is composite, then neither will taking any power of 2. It is theoretically possible for 2 and 3 not to give proofs that n is com- posite but for 6 to work just ﬁne, but this turns out not to be very common. So let us amend the heuristic and assume that we have independence for prime values of a. Up to log n log log n there are about log n primes (via the prime **number** theorem discussed later in this article); so, heuristically, the probability that n is composite, but that none of these primes help us to prove it, is about 4 − log n < n −4/3 . Since the inﬁnite sum

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IX. Conclusion
Although Clifford algebras are primarily known for their use in constructing representations of spin groups, they also provide a useful algebraic structure for calculations involving vectors. For example, Clifford **algebra** not only generalizes vector calculus operations such as divergence and curl to n-dimensional vector spaces, but also allows for calculations involving non-Euclidean vectors to be easily handled. The applications of Clifford **algebra** are numerous, and the scope of this work greatly limited the **number** of applications of Clifford **algebra** that could be discussed; in particular, the ability of Clifford **algebra** to describe conformal transformations and the use of Clifford **algebra** in projective geometry were both completely neglected. Its usefulness in calculations involving relativistic phenomena were mentioned only in passing. Any reader interested in learning relativistic electrodynamics should consider reading Baylis’ book Electrodynamics: A Modern Geometric Approach, which provides a good **introduction** to Clifford **algebra** before applying it to physics usually addressed using cumbersome space-time tensors. In closing, the author highly recommends Clifford Algebras and Spinors by Lounesto as an excellent reference which greatly assisted in the production of this work.

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The second semester of the ICT course is titled “Applying **Computational**
Thinking to Solve Problems” and demands a higher complexity of skills. Some example problems include drawing a clock with accurate measurements, applying
transformations in the correct order to intentionally preserve or manipulate an object’s properties, or extracting data from a table in order to calculate discounts and sales taxes of a fictional sock company. In order to manage the complexity of the activities, students learn all about creating and using functions with multiple parameters. These CodeWorld functions use the “function notation” of mathematics which is introduced for the first time in the Common Core State Standards in **Algebra** I (F.IF.A) 13 . The ICT curriculum continues to gain speed as more abstract concepts are introduced to the students such as using random numbers, experimenting with the simple user interface, and using lists to store homogeneous data 14 or tuples to store heterogeneous data 15 collections.

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[4] Jingguo Bi and Qi Cheng. Lower bounds of shortest vector lengths in random ntru lattices. In The 9th annual conference on **Theory** and Applications of Models of Com- putation (TAMC), volume 7287 of Lecture Notes in Computer Science. Springer-Verlag, 2012.

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Remark 13.3.3. Before giving the proof, we pause with a brief remark about Os- trowski. According to
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Ostrowski.html
Ostrowski was a Ukrainian mathematician who lived 1893–1986. Gautschi writes about Ostrowski as follows: “... you are able, on the one hand, to emphasise the abstract and axiomatic side of mathematics, as for example in your **theory** of general norms, or, on the other hand, to concentrate on the concrete and constructive aspects of mathematics, as in your study of numerical methods, and to do both with equal ease. You delight in finding short and succinct proofs, of which you have given many examples ...” [italics mine]

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2.1 **Introduction**
Historically, experience with unique prime factorization of integers led mathe- maticians in the early days of algebraic **number** **theory** to a general intuition that factorization of algebraic integers into primes should also be unique. A likely reason for this misconception is the actual definition of what is a prime **number**. The familiar definition is that a prime **number** is a **number** which is divisible only by 1 and itself. Since units in Z are ± 1, this definition can be rephrased as: if p = ab, then one of a or b must be a unit. Equivalently over Z, a prime **number** p satisfies that if p | ab, then p | a or p | b. However, these two definitions are not equivalent anymore over general rings of integers. In fact, the second property is actually stronger, and if one can get a factorization with

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substantive definition. For covariance and correlation matrices, the determinant is a **number** that is sometimes used to express the “generalized variance” of the matrix. That is, covariance matrices with small determinants denote variables that are redundant or highly correlated. Matrices with large determinants denote variables that are independent of one another. The determinant has several very important properties for some

The study of Lie **Algebra** requires a thorough understanding of linear **algebra**, group, and ring **theory**. The following provides a cursory review of these subjects as they will appear within the scope of this paper. Unless otherwise stated, our refresher for group **theory** was derived from Aigli Papantonopoulou’s text **Algebra** Pure and Applied [Aig02].

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In the above notation, the scalar 1 refers to the correlation of the Y variable with itself. The row vector r Y x refers to the set of correlations between the variable Y and the set of p random variables in x. Rxx is the p × p matrix of correlations of the predictor variables. We will refer to the “order” of the “partitioned form” as the **number** of rows and columns in the partitioning, which is distinct from the **number** of rows and columns in the matrix being represented. For example, suppose there were p = 5 predictor variables in Example 4.2. Then the matrix R is a 6 × 6 matrix, but the example shows a “2 × 2 partitioned form.”

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