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8606 Federal Register /Vol 89 No 27/Thursday February 8

5 days ago · the EA; see EA sections 4 3 3 1 4 3 3 2 and 5 3 4 5 EPA also solicits comment on whether the potential impacts of this rulemaking may be affected by the availability of other authorities that program implementers might rely on to satisfy corrective action requirements to address PFAS at RCRA facilities including other RCRA authorities such

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Linear Algebra and Its Applications (Fourth Edition)

This book begins with the central problem of linear algebra:solving linear equations The most important ease and the simplest is when the number of unknowns equals the number of equations We havenequations innunknowns starting withn=2: Two equations 1x+ 2y= 3 Two unknowns 4x+ 5y= 6: (1) The unknowns arexandy

PDF	Algèbre 4 Lfma2 Hedi Regeiba Départements de Mathématiques Algèbre 4 Lfma2 2019-2020 présenté par Hedi Regeiba Faculté des Sciences de Gabès cité Erriadh 6072 Zrig Gabès Tunisie Page

PDF	Algebre4 cours Algèbre 4 STRUCTURES ALGEBRIQUES AZZOUZ CHERRABI ELMOSTAFA JABBOURI Année 2007-2008 Page 2 ii Page 3 Table des matières 1 Arithmétique 1 1 1 Les

Ce cours est appliqué à la formalisation mathématique en sciences économiques, politiques et sociales en général, avec un accent particulier vers les applications de gestion.

1.2.1 Operations on sets

For background on sets, see the section A.2 in the appendix. We're familiar with many operations on the real numbers Raddition, subtraction, mul-tiplication, division, negation, reciprocation, powers, roots, etc. Addition, subtraction, and multiplication are examples of binary operations, that is, functions R R R which take two real numbers as t

x x

operation that has an identity element is said to have inverses if for each element there is an inverse element such that when combined by the operation they yield the identity element for the operation. Addition has inverses, and multiplication has inverses of nonzero elements. Finally, there is a particular relation between the operations of addi

(y + z)x = yx + zx:

Multiplication distributes over addition, that is, when multiplying a sum by x we can dis-tribute the x over the terms of the sum. Exercise 1. On properties of operations. xy (a) Is the binary operation x y = for positive x and y a commutative operation? x + y That is, is it true that x y = y x for all positive x and y? Is it associative? Explain y

1.3.3 Monomorphisms and epimorphisms

Two common kinds of homomorphisms are monomorphisms and epimorphisms, often called monos and epis for short. When a homomorphism f : A B is an injective function, it's called a monomorphism; and when it a surjective function, it's an epimorphism (but, in the category of rings, we'll see there are more epimorphisms than just the surjective ring ho

d = am + bn:

Now that we have the major theorems on gcds, there are a few more fairly elementary proprieties of gcds that are straightforward to prove, such as these. mathcs.clarku.edu

b = tnb + uab:

Now, if n ab, then n divides the right hand side of the equation, but that equals the left hand side, so n b. mathcs.clarku.edu

2.1.4 Sub elds

Frequently we'll nd one eld contained in another eld. For instance, the eld of rational numbers Q is part of the eld of real numbers R, and R is part of the eld of complex numbers C. They're not just subsets, Q R C, but they have the same operations. Here's the precise de nition of sub eld. mathcs.clarku.edu

n (x y), then n (y x).

For transitivity, suppose that x y (mod n) and y z (mod n). Then n (x y) and n (y z), so there exist k and m such that nk = x y and nm = y z. Therefore n(k + m) = x z, showing that n (x z). Hence x z (mod n). q.e.d. mathcs.clarku.edu

Proof. Let x < y

x and y. If x is negative while m in an Archimedean eld. We're looking for a rational number between n y is positive, then the rational number 0 lies between them. We can reduce the case where they're both negative to the case where they're both positive mathcs.clarku.edu

m m

by noting that if lies between x and y, then lies between x and y. mathcs.clarku.edu

2.5.2 The quaternions H

We're not going to study skew elds, but one is of particular importance, the quaternions, denoted H. The letter H is in honor of Hamilton, their inventor. We can de ne a quaternion a as an expression mathcs.clarku.edu

Rings

Rings are things like Z that have the three operations of addition, subtraction, and multi-plication, but they don't need division. The lack of a division operation makes them more complicated and more interesting. The concept of prime, for example, is uninteresting for elds, but very interesting for Z and other rings. Most of our rings will have c

3.2 Factoring Zn by the Chinese remainder theorem

We'll look at the structure of the cyclic ring Zn when n is composite in more detail. In particular, when n is not a power of a prime number, then Zn is a product of smaller cyclic rings. mathcs.clarku.edu

Zn = Y Zpei :

i i=1 Proof. The third statement is a special case of the second. The second follows from the rst by induction on r. That leaves us with the rst statement. In one direction, Zn Zk Zm, the function giving the isomorphism is fairly obvious; it's built of the two functions Zn Zk and Zn Zm that are easy to describe. There is an obvious candidate

Zm

[x]n 7 ([x]k; [x]m) In order to show that this is an isomorphism, all we need to do is to show that it's a bijection, and for that, all we need to do is to show that it's an injection since the sets Zn and Zk Zm have the same cardinality. Suppose that [x]n and [y]n are sent to the same element in Zk Zm. Then [x]k = [y]k and [x]m = [y]m, that is, k

A 7 (A \\ S1; A \\ S2; : : : ; A \\ Sn)

> q _ p q p q p :p _ :q p q q $ p p $ :q :q :p q ^ p p ^ :q mathcs.clarku.edu

3.3.3 A partial order on a Boolean ring

If we de ne x y to mean xy = y, then our Boolean ring will have a partial ordering. Recall that a partial ordering on a set is a re exive, antisymmetric, and transitive relation. Re exive: x x, since x2 = x. Antisymmetric: x y and y x imply x = y, since xy = x and yx = y imply x = y. 3. Transitive: x y and y z imply x z, since xy = x and yz = y imp

m k

pairs (m; n) and (k; l) can name the same integer = if ml = nk. That suggests if we mathcs.clarku.edu

n l l n

That's easily veri ed since Associativity of addition. You can easily show it, but it's a big mess. Associativity of multiplication. Pretty easy. mathcs.clarku.edu

3.5.1 The formal de nition of categories

Unlike elds, rings, and groups, we won't require that categories build on sets. In a category the collection of all its objects won't be a set because the collection is larger than any set. That's not a problem since theories don't have to be built on set theory. Indeed, set theory itself is not built on set theory. mathcs.clarku.edu

De nition 3.17. A category C consists of

objects often denoted with uppercase letters, and morphisms (also called maps or arrows) often denoted with lowercase letters. Each morphism f has a domain which is an object and a codomain which is also an object. If the domain of f is A and the codomain is B, then we write f : A B or A B. The collection of all morphisms from A to B is denoted

m m m m m

g g(n) = g n = g(m) = h(m) = h n = h h(n) = h g(n): mathcs.clarku.edu

n n

Therefore, the ring homomorphism : Z Q is an epimorphism in R, the category of rings. It is also a monomorphism. But it is not an isomorphism. In many categories, if a morphism is both monic and epic, then it's also an isomorphism. That's true in the category S of setsand in the category G of groups, but not in the category R of rings. This examp

3.6 Kernels, ideals, and quotient rings

These three concepts are closely related. For a ring homomorphism f : R S, the inverse image of 0 is a subset of R called the kernel of f and denoted Ker f. It can't be just any subset, as we'll see, since it's closed under addition and multiplication by elements of R. A subset with those properties we'll call an ideal of R. Every ideal I of R is

3.6.3 Quotient rings, R=I

As mentioned above the kernel of a ring homomorphism f tells us when two elements are sent to the same element: f(x) = f(y) if and only if x y 2 Ker f. We can use Ker f to construct a \\quotient ring" R= Ker f by identifying two elements x and y in R if their di erence lies in Ker f. In fact, we can do this not just for kernels of homomorphisms, but

R R= Ker f = f(R) S

The rst is the projection from R onto its quotient ring R= Ker f, the second is the isomor-phism R= Ker f = f(R), and the third is the inclusion of the image f(R) as a subring of S. mathcs.clarku.edu

q.e.d.

This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu This principle implies that there is some well-ordering of the real numbers R. It's not the usual order, of course, since the usual order does not well order R. In fact, no particular well-ordering of R can ever be described. mathcs.clarku.edu

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