# DEGREE THEORY E.N. DANCER

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 DEGREE THEORY E.N. DANCER References: N. Lloyd, Degree theory, Cambridge University Press. K. Deimling, Nonlinear functional analysis, Springer Verlag. L. Nirenberg, Topics in nonlinear functional analysis, Courant Institute lecture notes, New York University. R.F. Brown, A topological introduction to nonlinear analysis, Birkhauser. K.C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhauser. M. Matzeu and A. Vignoli, Topological nonlinear analysis, Birkhauser. (The first three are general references on degree theory) LECTURE 1 The idea of degree theory is to give a "count" of the number of solutions of nonlinear equations but to count solutions in a special way so that the count is stable to changes in the equations. To see why the obvious count does not work well consider a family of maps ft( x) on R defined by ft( x) = x 2 - t. As we vary t,ft changes smoothly. Fort< 0, it is easy to see that ft(x) = 0 has no solution, f 0 (x) = 0 has zero as its only solution while fort> 0, there are two solutions ±Vt. Hence the numbers of solutions changes as we vary t. Hence, to obtain something useful, we need a more careful count. A clue is that 0:: has different signs at the Typeset by AMS-'IEX 183

2 two solutions ±vt. Before, we proceed further, we need some notation. AssumeD is a bounded open set in Rn and f : D ---+ Rn is C 1. We say p is a regular value off if det f' (X) -I- 0 (equivalently f' ( x) is invertible) whenever x E D and f( x) = p. One thinks of the regular points as the nice points. Note that pis a regular value if p rf. f(d). Now suppose f : D ---+ Rn is smooth, p rf. f(8d) and pis a regular value of f. Since pis a regular value off, the inverse function theorem implies that the solutions of.f(x) =pin Dare isolated in D. On the other hand {xed: f(x) = p} is compact since it is a closed subset of D. Since a compact metric space which consists of isolated points is easily seen to be finite, it follows that {xed: f(x) = p} is finite. We then define the degree of.f deg (f,p, D) to be L sign det.f'(x;) where x; are the solutions of.f(x) =pin D. I stress that we are assuming f is smooth, p rf. f( ad) and p is a regular value of f. Here sign y = { 1-1 if y > 0 if y < 0 and sign 0 is not defined. You might ask why we assume p rf. f(8d). The reason is that, otherwise, a solution might move from inside D to out of D as we make small perturbations of f or p. In this case, we would expect that any "count" of the number of solutions might well change. Suppose pis not a regular value off (but.f is smooth and p rf..f(od)). We try to define deg (f,p, D)= lim deg (f,p;, D) ---oo where p; are regular values off approaching p. There are two problems with this. Firstly, we need to know that there are such regular values and secondly that the limit exists (and is independent of the choice of p;). The first problem is resolved by the following. 184

3 Sard's Theorem. Assume f : D Rn is C 1 where D is open in Rn (not necessarily bounded). Then the set of regular values off are dense in Rn. This is proved by showing that the complement of the set of regular values of f has measure zero. We only sketch the main idea. The main idea in the proof is to show that if.to tfc D is and f' ( x 0) is not invertible, then, for B a small ball center xo, f(b) is squashed close to the hyperplane f( x 0 )+ f' ( x 0 )Rn (since f is well approximated by its derivative near xo). Note that a hyperplane has zero measure. One uses this to show f(b) has much smaller measure than B if jj has small radius. To overcome the second problem is more difficult. (Note that the set of regular values off need not be connected). It turns out that the shortest proof is indirect. We consider the integral where Jf denotes the Jacobian off (where f is as above), > in coo on Rn, support of > is contained in an n- dimensional cube C, p E int C, C II f( 8D) is empty and L = 1. Such a > is said to be admissible. It is easy to find at least one admissible > exists by choosing C a small cube center p. The advantage of using integrals is that they are much more regular under perturbations. We will show (rather sketch) that d( >) is in fact the same as de g. To see this, note that, if > 1 and are admissible with support in the same cube C, then One proves the right hand side is zero by proving that the integrand is the eli vergence of a (vector) function vanishing near 8 D and applying the eli vergence theorem. This is the messy part of the proof. This implies that d( >) is independent of admissible >. Secondly, note that p does not appear explicitly in the integral defining d( >) and hence we deduce that d( >) is locally constant in p. Thirdly, if pis a regular value of j, then d( >) = deg(f,p,d). To see this note that, by the implicit function theorem, f( x) = p has only a finite number of solutions {x;}7=l in D and we can choose disjoint neighbourhoods D; of x; in D 185

4 such that f maps D; diffeomorphically on to a neighbourhood of p and f' ( x) has fixed sign on D; (by shrinking D; is necessary). vve than choose admissible cp such that the cube C <;;;; n.f(d;). By the change of variable theorem for integrals { cf;(.f(x))jj(x)dx =sign JJ(x;) r cf;(z)dz lv, Jf(Di) =sign J1(x;) since support cp <;;;; f(d;) Thus h d(cf;) =!, cf;(.f(x))jj(x)dx = L!, cf;(.f(x)jj(x)dx D i=l D; since the integrand is zero outside Uf= 1 D; k = L sign JJ(xi) i=l = deg (.f,p, D) Note that we have used that d( cp) is independent of cp to be able to choose a special and the reason the argument works is the close relation between d( cp) and the change of variable theorem. We now easily complete the proof that the degree is defined. If p is not a regular value for f. Then d( cjj) is the same for all q near p. However, if q is a regular value, d( ) = deg(f,q,d). Hence we see that for all regular values p; off near p, deg (f,p;,d) = d(cf;) and hence lim deg (f,p;,d) exists, as required Note that we could use the interval d( ) to define the degree but it is harder to deduce properties of the degree from this definition. 186

5 LECTURE 2 Assume f : Rn Rn is smooth, Dis bounded open in Rn and p tf:_ f( ad). Our proof that.lim deg (!,pi, D) has a limit if Pi are regular values off approaching Z--+00 p was rather indirect (by way of d( )). There is another proof which considers two regular values Pi, Pi of f near p and studies how the solutions off( x) = tp; + (1- t)pj changes as t varies from 0 to 1. This proof while more intuitive is rather more technical. Note that the degree can also be constructed by more topological arguments. Note that by its construction deg (!, p, D) is integer valued. We now consider properties of the degree we have constructed. The basic properties are the following where we assume D is bounded open in Rn, f : D Rn is continuous and p t/:- f( ad). (i) If deg (!, p, D) "I 0, there exists x E D such that f( x) = p. (ii) (excision) If D;, i = 1,, m, are disjoint open subsets of D and f( x) "I p if x ED\ U ~ 1 D;, then m deg (.f,p,d) =I: deg (f,p,d;) i=l (iii) (products) If D 1 is bounded open in Rm, g : D Rm is continuous and q t/:- g(8d1), then deg ((f,g),(p,q),d x DI) = deg (f,p,d) deg (g,q,d1 ). Here (!,g) is the function on Rm+n defined by (f,g)(x,y) = (f(x),g(y)) for X E Rn, y E Rm. (iv) homotopy invariance. IfF : D X [a, b] Rn is continuous and iff( x, t) -:j:; p for x E 8D and t E [a, b], then deg (F1,p, D) is defined and independent of t forte [a, b]. Here F1 is the map of D into Rn defined by F1(x) = F(x, t). Note that we have not actually defined the degree of maps f which are only continuous. The above 4 properties are first proved for smooth functions. Properties (i) - (iii) are proved first for p a regular value and the general case is proved by approximating p by regular values. In the smooth case, (iv) is proved by using the integral formula for the degree (i.e. d( )) to show that deg ( Ft, p, D) is continuous 187

6 in t and then using that a continuous integer valued function on [a, b] is constant. Finally, iff is only continuous, deg (f, p, D) is defined to be deg (.f, p, D) where.f is a smooth function uniformly close to f on D. To show that this is independent of the choice of f one uses that, if h is another smooth approximation to j, then one can apply homotopy invariance to the smooth homotopy t.f(x) + (1 - t)h(x) for xed, t E [0, 1]. Lastly, one uses an approximation argument to extend properties (i) - (iv) from the smooth case to the continuous case. This is actually all straightforward, albeit a little tedious. This completes the construction of the degree. It is easy to deduce from the definition a number of other properties of the degree. In particular, deg (f,p, D)= deg (f(x)- p, 0, D) whenever deg (f,p, D) is defined and that deg (A, 0, D) = sign det A if A is an invertible n x n m.atrix and D is a bounded open set containing zero (and where we are identifying a matrix and the corresponding linear map). In general, one can prove many properties of the degree by first proving it for f smooth and p a regular value off and then using limit arguments. The homotopy invaria.nce property is a very important property of the degree because it often enables us to calculate the degree by deforming our map to a much simpler map. This enables us to calculate the degree of some complicated maps. As a simple application, we prove the very useful Brouwer fixed point theorem. Assume S is closed bounded and convex in Rn and f : S --+ S is continuous. Then f has a fixed point i.e. there exists xes such than x = f(x). This theorem is used in many places. For example, it is frequently used to prove the existence of periodic solutions of ordinary differential equations. To prove the theorem, we assume that int S is non-empty and 0 E int S. (It is not difficult to reduce the general case to this case using the geometry of convex sets). We assume.4(x) f:. x if X E as. (Otherwise we are finished). We use the homotopy (x, t)--+ X- ta(x). The convexity and that 0 E int S imply that x f:. ta( x) if x E S and t E [0, 1] (since if x - ta(x) = 0, x = (1 - t)o + ta(x) E int S). Thus, by homotopy invariance, 188

7 deg (I- A, 0, int S) = deg (J, 0 int S) = 1 # 0 and hence there exists x E int S such that x - A( x) = 0 i.e. x is a fixed point. I mention one more useful property of the degree. Firstly if D ~ Rn is bounded open and symmetric (that is x E D x E D), 0 E D, f : D Rn is continuous and odd and if f(x) # 0 for x E 8D, then deg (f,o,d) is odd. (The most important point is that it is non-zero.) If f is smooth and zero is a regular value of f, then this is easy to prove because (f(o) = 0 and if f(x) = 0 then f( -x) = 0 and J 1 (x) = J 1 (-x) (since f is odd). The main difficulty in the general case is to prove that we can approximate a smooth odd map by a smooth odd map which has zero as a regular value. This is rather technical. This result has many uses. for example, it easily implies that a continuous odd mapping of Rm into Rn with m > n has a zero on each sphere llxll = r. We will return to uses for differential equations later. It has many other uses on the geometry of Rn. For example, it can be used to prove that iff: Rn Rn is 1-1 and continuous then it is an open map (that is,.f maps open sets to open sets). There are many other results on the computation of the degree but do not assume the degree is always easy to evaluate! 189

8 LECTURE 3 In this lecture, we extend the degree to infinite dimensions. It turns out that this is important for many applications. First note that we cannot expect to be able to do this for all maps. The simplest way to see this is to note that the analogue of Brouwer's fixed point theorem fails in some infinite dimensional Banach spaces. We consider the Banach space c0 of sequences (x;)~ 1 with norm ll(x;)ll = sup;;::::1lx;1. We then consider the map T: co--+ co defined by T(x1,x2, ) = (1,x1,x2 ). Note that Tis an affine map. It is easy to check that T is continuous and that T maps the closed unit ball in co into itself. However, T has no fixed point in c 0 because if T(x;) = (x;), then x1 = 1 and Xi+l = x; for i 2:: L Thus x; = 1 for i 2:: 1, which contradicts that (xi) E c0. It turns out that in every infinite- dimensional Banach space there is always a closed bounded set S and a continuous map of S into itself without a fixed point. Hence to proceed further, we need a restricted class of maps. Assume that W is a closed subset of a Banach space I<. We say that A : W --+ E is completely continuous if A is continuous and if A( S) is a compact subset of E for each bounded subset S of VV. We will construct a degree for maps I- A where A is completely continuous. It turns out that many (but far from all) of the mappings occurring in applications are completely continuous. The reason that we can construct a degree for such maps is that we can approximate completely continuous maps by maps whose range lie in a finite- dimensional space. Lemma. If }( is a compact convex subset of a Banach space E and E > 0, there is a continuous map Pc : }( --+ }( such that II P,( x) - x II :::; E for x E K and the range of Pc is contained in a finite-dimensional subspace of E. 1 This is proved by noting that a compact set has a finite "2E net and using some form of partition of unity. The details are not difficult. In using this in our applications, it is useful to note that the closed convex hull of a compact set is compact. If D is bounded and open in E and A : D --+ E is completely continuous, then 190

9 P,A = P,oA is a continuous map such that JIP,A(x)- A(x)JI ::::; eon D and the range of P,A is contained in a finite-dimensional subspace of E. (We define I< to be the closed convex hull of A( D). If p rt (I- A)( an) and e is small, we define deg (I -A,p,D) = deg ((I -A,)JM,p,DnM) where A,= P,A,M is a finite-dimensional subspace of E containing p and the range of A,. Note that I- A, will map D n M into M. There is quite a bit to be checked here. The right hand side is defined by our earlier construction. We need to prove that the finite-dimensional degree is defined and is independent of e and the choice of M. The details are tedious but not difficult. Note that a simple compactness argument ensures that there is an a> 0 such that Jlx- A(x)ll 2: a if X E an, that our finite-dimensional degree is a degree on finite dimensional normed spaces rather than just Rn because it is easy to check that our original degree on Rn is independent of the choice of basis in Rn and that we need to prove a lemma on the finite- dimensional degree to prove that the right hand side of the definition is independent of the choice of M. By using finite-dimensional approximations and compactness arguments, it 1s not difficult to check that the four basic properties of the finite-dimensional degree have analogues here. Assume that D is bounded and open in E and A : D ~ E is completely continuous such that Xi= A(x) + p for X E an. The following hold. (i) If deg (I- A,p,D) =/= 0, there exists xed such that x = A(x) + p. (ii) If Di, i = 1,, m, are disjoint open subsets of D such that x =/= A( x) + p m if X E D\ u~l Di, then deg (I- A,p, D) = L deg (I- A,p, Di) i=l (iii) products. If D 1 is bounded open in a Banach space F, G : Di ~ F is completely continuous and q E F such that x =/= G( x) + q for x E an1, then deg (I- (A, G), (p, q), D x Dl) = deg (I- A,p, D) deg (I- G, q, DI). (iv) homotopy invariance. If H : D x [a, b] ~ E is completely continuous and X- H(x, t) i= p for X E an, t E [a, b], then deg (I- Ht,p, D) is independent oft forte [a,b]. Note that in (iv) it is not sufficient to assume that H is continuous and each H, is completely continuous. However this is sufficient if we also assume that Ht is uniformly continuous in T (uniformly for xed). 191

10 There is analogue of Brouwer's fixed point theorem, known as Schauder's fixed point theorem. Assume that D is closed and convex in a Banach space E, A is continuous, A( D) ~ D and A( D) is compact. Then A has a fixed point. The easiest way to prove this to use the lemma to find a finite dimensional approximation to which we can apply Brouwer's theorem. Nearly all the extra properties of the degree in finite dimensions have natural analogues here. The only one that is a little different is the formula for the degree of a linear map. Assume that B : E ----t E is linear and compact and I - B has trivial kernel. (Note that for linear maps complete continuity is equivalent to compactness.) The spectrum of B consists of zero plus a finite or countable sequence of eigenvalues with 0 as it only limit point. If.\; is a non-zero eigenvalue of B, the algebraic multiplicity m(.\;) of.\; is defined to be the dimension of the kernel of (.\;I- B)q for large enough q. It can be shown that m( A;) is finite and independent of q for large q. The result is then that deg (I- B,O,Eo) = (-l)i:;m(.>.;) where the summation is over the (real) eigenvalues of B in (1, oo) and E 0 is the open ball of radius 8 in E. Note that the sum is finite. The proof of this is by using linear operator theory and constructing homotopies to reduce to the finite dimensional case. As a final comment, there has been a good deal of work on extending the degree to mappings only defined on closed convex subsets of Banach spaces and to evaluating degree of critical points obtained by variational methods (as in Chabrowski's lectures or Chang's book). One major difference in the extension to closed convex sets is that the formula for the degree of isolated solutions becomes rather more complicated. This is discussed in my chapter in the book of Matzeu and Vignoli. 192

11 LECTURE 4 In this lecture, we consider some very simple applications of the degree. We do not try to obtain the most general results. Firstly, we consider an application of Brouwer's fixed point theorem to prove the existence of periodic solutions of ordinary differential equations. Assume f : Rn+l ~ Rn is C 1 and T periodic in the last variable, i.e., f(x, t + T) = f(x, t) if x E Rn, t E R. We are interested in the existence of T periodic solutions of the equation x 1 (t) = f(x(t), t) (1) To do this, we let U(t,x 0 ) denote the solution of (1) with U(O,x 0 ) = x 0 (for x 0 E Rn). Standard results ensure that (1) has a unique solution satisfying the initial condition. Vl!e need to place an assumption on f which ensures that U(t, x 0 ) is defined for 0 :::; t :::; T. (The only way this could fail is that the solution blows up before t = T). A sufficient condition ensuring this is true is that there is a ]{ > 0 such that llf(x, t)ll :::; Kllxll fort E [0, T] and for llxlllarge (where II II is one of the standard norms on Rn). In this case U(t,x 0 ) is continuous in t and x 0. Since f is T periodic in t, it is not difficult to show that U ( t, x 0 ) is a T periodic solution of (1) if and only if U(T, x 0 ) = x0 i.e. x 0 is a fixed point of the map x ~ U(T, x ). Hence we can hope to our earlier degree results (to the map f(x) = x- U(T,x)). The simplest rest of the this type is that if we can find a closed ball Br in Rn such that U(T,x) E B,. if x E Br then Brouwer's theorem (applied on Br) implies that the map x ~ U(T, x) has a fixed point and hence the original equation has a T periodic solution. For example, the inclusion condition on Br holds if < f(x, t), x > < 0 for llxll = r and 0 :::; t :::; T (where <, > is the usual scalar product on Rn and we must use the norm induced by the scalar product). Before we look at the second example, I need one very useful consequence of degree theory. Assume that E is a Banach space and H : E X [0, 1] ~ E is completely continuous, the map x ~ H( x, 0) is odd and there is an R > 0 such that x #- H(x, t) if llxll = Rand t E [0, 1]. Then the equation x = H(x, 1) has a 193

12 solution. This is proved by noting that by homotopy invariance deg (I- H(, 1),0,ER) = deg (I- H(,O),O,ER)-=/= 0 by the result on the degree of odd mappings. This is a very nice theorem because the statement is very simple and it is very widely applied. In practice, one usually establishes for a larger that x-=/= H(x, t) if t E [0, 1] and llxll 2: R. In other words, one is proving that any solution of x = H(x, t) satisfies llxll :SR. In practice, this is usually by far the most difficult step in verifying the assumption of the result above. Such a result is called an a priori bound. In some of the lectures on partial differential equations, I am sure you have discussed the problem of obtaining a priori bounds. In fact, the above result often reduces the proof of the existence of a solution of a partial differential equation to the proof of a priori bounds in a suitable Banach space. (The choice of a suitable Banach space is not always simple). vve give a very simple application of the result above which requires few technicalities. Assume Q is a bounded domain in Rn with smooth boundary and g : R----+ R is continuous such that y- 1 g(y)----+ a as IYI----+ oo where a is not an eigenvalue of -tl under Dirichlet boundary conditions on D. We prove that the equation -tlu(x) = g(u(x)) inn u(x) = 0 on on (2) has a solution. By a solution we mean a function in W 2'P(Q) n C(D) which satisfies the equation almost everywhere. (It will be a classical solution if g is a little more 1 n (and p > 1 ). regular). It is convenient to work in the space W 2,P(fl) with p > 2 By the Sobolev embedding theorem, this ensures that W 2,P(Q) ~ C(TI). We write K to denote the inverse of -6 under Dirichlet boundary conditions. We define G: C(TI)----+ C(TI) by (G(v.))(x) = g(u(x)). Then (2) is equivalent to the equation u = KG( tl ). Thus (2) becomes a fixed point problem. KG is easily seen to be completely continuous since we can think of KG as the composite J( o go i where i is the natural inclusion of W 2,P(fl) into C(TI), since i is compact, since G is continuous and since J{ is a continuous map of LP(fl) into W 2,P(Q). (The latter is 194

13 a regularity result for the Laplacian.) Lastly, if we define H: W 2 P(SZ) x [0, 1] ~ W 2 P(SZ) by H(u, t) = K(tG(u) + (1 - t)au) it is fairly easy to prove that the assumption of the above result hold and thus (2) has a solution. (The a priori bound corresponds to proving a bound for solutions of -.6-u = tg(u) + (1- t)au inn u = 0 on an.) We omit the details. In general in partial differential equations, one often has to choose the spaces carefully especially if the equations are nonlinear in derivatives in u. (In fact, these methods run into severe difficulties if the equations are highly nonlinear.) The methods also tend to run into difficulties if n is not bounded because the complete continuity fends to fail. There have been many attempts to define degrees for mappings which are not completely continuous to try to overcome this last problem. Some of these are discussed in Ize's chapter in the book of Matzeu and Vignoli. Thirdly, I obtain a result on Banach spaces though it could be very easily applied to ordinary and partial differential equations. In many applications, there is a rather trivial solution of the problem and we want to look for other solutions. For simplicity, assume that the trivial solution is zero. Hence, we assume that E is a Banach space, A : E ~ E is completely continuous, A(O) = 0 and we look at the equation x = AA( x ), (A might correspond to some physical parameter.) Our assumptions ensure 0 is a solution of the equation for all A and we look for other solutions. We assume that there is a linear mapping B on E such that IIA(x)- Bxll/llxll ~ 0 as x ~ 0. (B is in fact the derivative of A at zero in a certain sense. One can in fact define a calculus on Banach spaces). It is not difficult to prove that B is compact (since A is completely continuous). Assume Il-l is a non-zero eigenvalue of B of odd algebraic multiplicity. One can then prove that there are solutions (x,a) of x = AA(x) with x =/:- 0 and llxll + la -Ill arbitrarily small. One usually says that ( 0, 1-l) is a bifurcation point. The interest here is that the main assumption (that 1-l has odd multiplicity) is purely on the linear part of A. 195

14 I omit the proof but the key ideas are the formula. for the degree of a. linear map and that if T =f. 0 is small and 8 is very small compared with T, deg (I- (f.-l + T )A, 0, E 0 ) = deg (I- (f.-l + r)b, 0, Eo) and we can evaluate the right hand side. It turns out that much more is true. There is a connected set of solutions of x = >.A( x) in E x R with x =f. 0 which starts at (0, f.-l) (as above) and continues rather globally (that is to points where xis not small or>. is not close to f.-l). This can be found in Brown's book. Lastly, in many applications, the problem may require that we only look for nonnegative solutions of our equation. Thus it might be natural to look for solutions u of an elliptic equation which satisfy u( x) ~ 0 on n (for example if u represents a population). Thus it is often natural to look at problems on convex sets such as {u E C(f!): u(x) ~ 0 on n} rather than the whole space. 196

### 1 Local Brouwer degree

1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### 8.1 Examples, definitions, and basic properties

8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A k-form ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)-form σ Ω k 1 (M) such that dσ = ω.

### Reference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.

5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### 1 Norms and Vector Spaces

008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points

Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a

### MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

### ORIENTATIONS. Contents

ORIENTATIONS Contents 1. Generators for H n R n, R n p 1 1. Generators for H n R n, R n p We ended last time by constructing explicit generators for H n D n, S n 1 by using an explicit n-simplex which

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

### Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand

Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such

### On the Product Formula for the Oriented Degree for Fredholm Maps of Index Zero between Banach Manifolds

On the Product Formula for the Oriented Degree for Fredholm Maps of Index Zero between Banach Manifolds Pierluigi Benevieri Massimo Furi Maria Patrizia Pera Dipartimento di Matematica Applicata, Università

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### Math 5311 Gateaux differentials and Frechet derivatives

Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.

University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

### 16.3 Fredholm Operators

Lectures 16 and 17 16.3 Fredholm Operators A nice way to think about compact operators is to show that set of compact operators is the closure of the set of finite rank operator in operator norm. In this

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### Fixed Point Theorems

Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation

### (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties

Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

### Math 312 Homework 1 Solutions

Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

### ( ) which must be a vector

MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

### Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

### The Ideal Class Group

Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

### Math 231b Lecture 35. G. Quick

Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

### The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line

The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni,

### The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

### Chapter 5. Banach Spaces

9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on

### Example 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x

Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

### FIXED POINT SETS OF FIBER-PRESERVING MAPS

FIXED POINT SETS OF FIBER-PRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 e-mail: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics

### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### Finite dimensional topological vector spaces

Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

### ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction

ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZ-PÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that

### Surface bundles over S 1, the Thurston norm, and the Whitehead link

Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3-manifold can fiber over the circle. In

### Separation Properties for Locally Convex Cones

Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

### Fuzzy Differential Systems and the New Concept of Stability

Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida

### 1 Fixed Point Iteration and Contraction Mapping Theorem

1 Fixed Point Iteration and Contraction Mapping Theorem Notation: For two sets A,B we write A B iff x A = x B. So A A is true. Some people use the notation instead. 1.1 Introduction Consider a function

### No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

### RINGS WITH A POLYNOMIAL IDENTITY

RINGS WITH A POLYNOMIAL IDENTITY IRVING KAPLANSKY 1. Introduction. In connection with his investigation of projective planes, M. Hall [2, Theorem 6.2]* proved the following theorem: a division ring D in

### T ( a i x i ) = a i T (x i ).

Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### 9 More on differentiation

Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......

### Orthogonal Projections

Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors

### Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### CHAPTER 1 BASIC TOPOLOGY

CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is

### The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

### THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING

THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The Black-Scholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages

### 1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain

Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is

### 24. The Branch and Bound Method

24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### Vector Spaces II: Finite Dimensional Linear Algebra 1

John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

### TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

### Introducing Functions

Functions 1 Introducing Functions A function f from a set A to a set B, written f : A B, is a relation f A B such that every element of A is related to one element of B; in logical notation 1. (a, b 1

### Section 13.5 Equations of Lines and Planes

Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.

### THREE DIMENSIONAL GEOMETRY

Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

### CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí

Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian Pasquale

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### Ri and. i=1. S i N. and. R R i

The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

### MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

### Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

### Lecture 7: Finding Lyapunov Functions 1

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1

### Lecture 18 - Clifford Algebras and Spin groups

Lecture 18 - Clifford Algebras and Spin groups April 5, 2013 Reference: Lawson and Michelsohn, Spin Geometry. 1 Universal Property If V is a vector space over R or C, let q be any quadratic form, meaning

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty