A UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS


 Erick Holmes
 2 years ago
 Views:
Transcription
1 A UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS RICHARD A. TAPIA Appendix E: Differeniaion in Absrac Spaces I should be no surprise ha he differeniaion of funcionals (real valued funcions) defined on absrac spaces plays a fundamenal role in coninuous opimizaion heory and applicaions. Differenial ools were he major par of he early calculus of variaions and acually i was heir use ha moivaed he erminology calculus of variaions. The ools ha we develop ha allow us o work in he full generaliy of vecor spaces are he socalled onesided direcional variaion, he direcional variaion, and he direcional derivaive. Their developmen will be our firs ask. In some heory and applicaions we need more powerful differeniaion ools. Towards his end our second ask will be he presenaion of he Gâeaux and Fréche derivaives. Since hese derivaives offer more powerful ools, heir definiion and use will require sronger srucure han jus vecor space srucure. The price ha we have o pay is ha we will require a normed linear space srucure. While normed linear spaces are no as general as vecor spaces, he noion of he Fréche derivaive in a normed linear space is a useful heoreical ool and carries wih i a very saisfying heory. Hence, i is he preferred differeniaion noion in heoreical mahemaics. Moreover, a imes in our opimizaion applicaions we will have o urn o his elegan heory. However, we sress ha he less elegan differeniaion noions will ofen lead us o surprisingly general and useful applicaions; herefore, we promoe and embrace he sudy of all hese heories. We sress ha he various differeniaion noions ha we presen are no made one bi simpler, or proofs shorer, if we resric our aenion o finie dimensional spaces, indeed o IR n. Dimension of spaces is no an inheren par of differeniaion when properly presened. However, we quickly add ha we will ofen urn o IR n for examples because of is familiariy and rich base of examples. The reader ineresed in consuling references on differeniaion will find a hos of reamens in elemenary analysis books in he lieraure. However, we have been guided mosly by Orega and Rheinbold [2]. Our reamen is no he same as heirs bu i is similar. In consideraion of he many sudens ha will read our presenaion, we have included numerous examples hroughou our presenaion and have given hem in unusually complee deail. 1. Inroducion. The noion of he derivaive of a funcion from he reals o he reals has been known and undersood for cenuries. I can be raced back o Leibniz and Newon in he lae 1600 s and, a leas in undersanding, o Ferma and Descares in he early 1600 s. However, he exension from f : IR IR o f : IR n IR. and hen o f : IR n IR m was no handled well. A brief presenaion of his hisorical developmen follows. We moivae his discussion wih he leading quesion: Wha comes firs, he derivaive or he gradien? Keep in mind ha he derivaive should be a linear form, and no a number, as was believed incorrecly for so many years. I should be even more embarrassing ha his belief in he number inerpreaion persised even in he presence of he noions of he direcional derivaive, he Gâeaux derivaive, and he Fréche derivaive, all linear forms ha we sudy in his appendix. I mus have been ha hese laer noions were all viewed as differen animals, whose naural habia was he absrac worlds of vecor spaces and no IR n. Consider a funcion f : IR IR. From elemenary calculus we have he following noion of he derivaive of f a x: f (x) = lim x 0 f(x + x) f(x). (1.1) x If insead we consider f : IR n IR, hen (1.1) no longer makes sense, since we canno form x + x, i.e., we canno add scalars and vecors and we canno divide by vecors. So, hisorically as is so ofen done in mahemaics he problem a hand is reduced o a problem wih known soluion, i.e., o he known one dimensional case, by considering he parial derivaives of f a x i f(x) = lim x 0 f(x 1,..., x i + x,..., x n ) f(x 1,..., x n ), i = 1,..., n. (1.2) x 0
2 The parial derivaives are hen used o form f(x), he socalled gradien of f a x, i.e., 1 f(x) 2 f(x) f(x) =. (1.3). n f(x) Then he gradien is used o build he socalled oal derivaive or differenial of f a x as f (x)(η) = f(x), η. (1.4) These ideas were readily exended o F : IR n IR m by wriing F (x) = (f 1 (x 1,..., x n ),..., f m (x 1,..., x n )) T and defining F (x)(η) = f 1 (x), η. f m (x), η. (1.5) If we inroduce he socalled Jacobian marix, 1 f 1 (x)... n f 1 (x) JF (x) =..., (1.6) 1 f m (x)... n f m (x) hen we can wrie F (x)(η) = JF (x)η. (1.7) This sandard approach is acually backwards and is flawed. I is flawed because i depends on he noion of he parial derivaives which in urn depends on he noion of a naural basis for IR n, and he dependence becomes raher problemaic once we leave IR n. The backwardness follows from he realizaion ha he gradien and hence he Jacobian marix follow from he derivaive and no he oher way around. This will become readily apparen in 2 and 3. Hisorically, he large obsacle o producing saisfacory undersanding in he area of differeniaion came from he failure o view f (x) in (1.1) as he linear form f (x) η evaluaed a η = 1. In IR he number a and he linear form a η are easily idenified, or a leas no separaed. I followed ha hisorically a derivaive was always seen as a number and no a linear form. This led o he noion of parial derivaive, and in urn o he linear form f (x) given in (1.3). This linear form was no seen as a number, hence i was no seen as a derivaive. This moivaed he erminology differenial and a imes oal derivaive, for he linear form defined in (1.3). Much confusion resuled in erms of he usage derivaive, differenial, and oal derivaive. We reinforce our own criical saemens by quoing from he inroducory page o Chaper??, he differenial calculus chaper, in Dieudonne []. Tha presenaion, which hroughou adheres sricly o our general geomeric oulook on analysis, aims a keeping as close as possible o he fundamenal idea of calculus, namely he local approximaion of funcions by linear funcions. In he classical eaching of Calculus, his idea is immediaely obscured by he accidenal fac ha, on a onedimensional vecor space, here is a oneoone correspondence beween linear forms and numbers, and herefore he derivaive a a poin is defined as a number insead of a linear form. This slavish subservience o he Shibboleh of numerical inerpreaion a any cos becomes worse when dealing wih funcions of several variables. 2. Basic Differenial Noions. We begin wih a formal definiion of our differenial noions excep for he noion of he Fréche derivaive, i will be presened in 4. 1
3 Definiion 2.1. Consider f : X Y where X is a vecor space and Y is a opological linear space. Given x, η X, if he (onesided) limi f +(x)(η) = lim 0 f(x + η) f(x) exiss, hen we call f +(x)(η) he righ or forward direcional variaion of f a x in he direcion η. If he (wosided) limi f (x)(η) = lim 0 f(x + η) f(x) exiss, hen we call f (x)(η) he direcional variaion of f a x in he direcion η. The erminology forward direcional variaion of f a x, and he noaion f +(x), imply ha f +(x)(η) exiss for all η X wih an analogous saemen for he direcional variaion. If f (x), he direcional variaion of f a x, is a linear form, i.e., f (x)(η) is linear in η, hen we call f (x) he direcional derivaive of f a x. If, in addiion, X and Y are normed linear spaces and f (x), he direcional derivaive of f a x, exiss as a bounded linear form (f (x) L[X, Y ]), hen we call f (x) he Gâeaux derivaive of f a x. Furhermore, we say ha f is direcionally, respecively Gâeaux, differeniable in D X o mean ha f has a direcional, respecively Gâeaux, derivaive a all poins x D. Remark 2.2. Clearly he exisence of he direcional variaion means ha he forward direcional variaion f +(x)(η), and he backward direcional variaion f (x)(η) = lim 0 f(x + η) f(x) exis and are equal. Remark 2.3. In he case where f : IR n IR m he noions of direcional derivaive and Gâeaux derivaive coincide, since in his seing all linear operaors are bounded. See Theorem C.5.4. Remark 2.4. The lieraure abounds wih uses of he erms derivaive, differenial, and variaion wih qualifiers direcional, Gâeaux, or oal. In his conex, we have srongly avoided he use of he confusing erm differenial and he even more confusing qualifier oal. Moreover, we have spen considerable ime and effor in selecing erminology ha we believe is consisen, a leas in flavor, wih ha found in mos of he lieraure. Remark 2.5. I should be clear ha we have used he erm variaion o mean ha he limi in quesion exiss, and have reserved he use of he erm derivaive o mean ha he eniy under consideraion is linear in he direcion argumen. Hence derivaives are always linear forms, variaions are no necessarily linear forms. Remark 2.6. The erm variaion comes from he early calculus of variaions and is usually associaed wih he name Lagrange. In all our applicaions he opological vecor space Y will be he reals IR. We presened our definiions in he more general seing o emphasize ha he definiion of variaion only requires a noion of convergence in he range space. In erms of homogeneiy of our variaions, we have he following properies. Proposiion 2.7. Consider f : X Y where X is a vecor space and Y is a opological vecor space. Also, consider x, η X. Then (i) The forward direcional variaion of f a x in he direcion η is nonnegaively homogeneous in η, i.e., f +(x)(αη) = αf +(x)(η) for α 0. (ii) If f +(x)(η) is homogeneous in η, hen f (x)(η) exiss and he wo are equal. Hence, if f +(x)(η) is homogeneous in η, hen f +(x)(η) is he direcional variaion of f a x. (iii) The direcional variaion of f a x in he direcion η is homogeneous in η, i.e, f (x)(αη) = αf (x)(η) for all α IR. Proof. The proofs follow direcly from he definiions and he fac ha f (x)(η) = f +(x)( η). 2 (2.1) (2.2) (2.3)
4 Fig Absolue value funcion We now consider several examples. Figure E.1 shows he graph of he funcion f(x) = x. For his funcion f +(0)(1) = 1 and f (0)(1) = 1. More generally, if X is a normed linear space and we define f(x) = x, hen f +(0)(η) = η and f (0)(η) = η. This shows he obvious fac ha a forward (backward) variaion need no be a direcional variaion. The direcional variaion of f a x in he direcion η, f (x)(η), will usually be linear in η; however, he following example shows ha his is no always he case. Example 2.8. Le f : R 2 R be given by f(x) = x 1x 2 2 x 2 1 +, x = (x 1, x 2 ) 0 and f(0) = 0. Then x2 2 f (0)(η) = η 1η2 2 η η2 2 The following example shows ha he exisnce of he parial derivaives is no a sufficien condiion for he exisence of he direcional variaion. Example 2.9. Le f : R 2 R be given by f(x) = x 1x 2 x 2 1 +, x = (x 1, x 2 ) T 0 and f(0) = 0. x2 2 Then f 1 η 1 η 2 (0)(η) = lim 0 (η1 2 + η2 2 ) exiss if and only if η = (η 1, 0) or η = (0, η 2 ). In 5 we will show ha he exisence of coninuous parial derivaives is sufficien for he exisence of he Gâeaux derivaive, and more. The following observaion, which we sae formally as a proposiion, is exremely useful when one is acually calculaing variaions. Proposiion Consider f : X IR, where X is a real vecor space. Given x, η IR, le φ() = f(x+η). Then φ +(0) = f +(x)(η) and φ (0) = f (x)(η). Proof. The proof follows direcly once we noe ha φ() φ(0) = f(x + η) f(x). Example Given f(x) = x T A x, wih A IR n n and x, η IR n, find f (x)(η). Firs, we define Then, we have φ() = f(x + η)& = x T A x + x T A η + η T A x + 2 η T A η. (2.4) φ () = x T Aη + η T Ax + 2η T Aη, (2.5) 3
5 and by he proposiion φ (0) = x T Aη + η T Ax = f (x)(η). (2.6) Recall ha he ordinary meanvalue heorem from differenial calculus says ha given a, b IR if he funcion f : IR IR is differeniable on he open inerval (a, b) and coninuous a x = a and x = b, hen here exiss θ (0, 1) so ha f(b) f(a) = f (a + θ(b a))(b a). (2.7) We now consider an analog of he meanvalue heorem for funcions which have a direcional variaion. We will need his ool in proofs in 5. Proposiion Le X be a vecor space and Y a normed linear space. Consider an operaor f : X Y. Given x, y X assume f has a direcional variaion a each poin of he line {x + (y x) : 0 1} in he direcion y x. Then for each bounded linear funcional δ Y (i) δ(f(y) f(x)) = δ[f (x + θ(y x))(y x)], for some 0 < θ < 1. Moreover, (ii) f(y) f(x) sup f (x + θ(y x))(y x). 0<θ<1 (iii) If in addiion X is a normed linear space and f is Gâeaux differeniable on he given domain, hen (iv) f(y) f(x) sup f (x + θ(y x)) y x. 0<θ<1 If in addiion f has a Gâeaux derivaive a an addiional poin x 0 X, hen f(y) f(x) f (x 0 )(y x) sup f (x + θ(y x)) f (x 0 ) y x. 0<θ<1 Proof. Consider g() = δ(f(x + (y x))) for [0, 1]. Clearly g () = δ(f (x + (y x))(y x)). So he condiions of he ordinary meanvalue heorem (2.7) are saisfied for g on [0, 1] and g(1) g(0) = g (θ) for some 0 < θ < 1. This is equivalen o (i). Using Corollary C.7.3 of he HahnBanach Theorem C.7.2 choose δ Y such ha δ = 1 and δ(f(y) f(x)) = f(y) f(x). Clearly (ii) now follows from (i) and (iii) follows from (ii). Inequaliy (iv) follows from (iii) by firs observing ha g(x) = f(x) f (x 0 )(x) saisfies he condiions of (iii) and hen replacing f in (iii) wih g. While he Gâeaux derivaive noion is quie useful; i has is shorcomings. For example, he Gâeaux noion is deficien in he sense ha a Gâeaux differeniable funcion need no be coninuous. The following example demonsraes his phenomenon. Example Le f : R 2 R be given by f(x) = x3 1 x 2, x = (x 1, x 2 ) 0, and f(0) = 0. Then f (0)(η) = 0 for all η X. Hence f (0) exiss and is a coninuous linear operaor, bu f is no coninuous a 0. We accep his deficiency, bu quickly add ha for our more general noions of differeniaion we have he following limied, bu acually quie useful, form of coninuiy. Orega and Rheinbold [2] call his hemiconinuiy. Proposiion Consider he funcion f : X IR where X is a vecor space. Also consider x, η X. If f (x)(η) (respecively f +(x)(η)) exiss, hen φ() = f(x + η) as a funcion from IR o IR is coninuous (respecively coninuous from he righ) a = 0. Proof. The proof follows by wriing f(x + η) f(x) = (f(x + η) f(x)) and hen leing approach 0 in he appropriae manner. Example Consider f : IR n IR. I should be clear ha he direcional variaion of f a x in he coordinae direcions e 1 = (1, 0,..., 0) T,..., e n = (0,..., 0, 1) T are he parial derivaives i f(x), i = 1,..., n given in (1.2). Nex consider he case where f (x) is linear. Then from Theorem?? we know ha f (x) mus have a represenaion in erms of he inner produc, i.e., here exiss a(x) IR n such ha f (x)(η) = a(x), η η IR n. (2.8) 4
6 Now by he lineariy of f (x)(η) in η = (η 1,..., η n ) T we have f (x)(η) = f (x)(η 1 e η n e n ) = η 1 1 f(x) η n n f(x) = f(x), η (2.9) where f(x) is he gradien vecor of f a x given in (1.3). Hence, a(x), he represener of he direcional derivaive a x, is he gradien vecor a x and we have f (x)(η) = f(x), η η IR n. (2.10) Example Consider F : IR n IR m which is direcionally differeniable a x. We wrie F (x) = (f 1 (x 1,..., x n ),..., f m (x 1,..., x n )) T. By Theorem C 5.4 we know ha here exiss an m n marix A(x) so ha F (x)(η) = A(x)η η IR n. The argumen used in Example 2.15 can be used o show ha f 1 (x), η F (x)(η) =.. f m (x), η I now readily follows ha he marix represener of F (x) is he socalled Jacobian marix, JF (x) = given in (1.5), i.e., F (x)(η) = JF (x)η. Remark We sress ha hese represenaion resuls are sraighforward because we have made he srong assumpion ha he derivaives exis. The challenge is o show ha he derivaives exis from assumpions on he parial derivaives. We do exacly his in The Gradien Vecor. While (1.4) and (2.8) end up a he same place, i is imporan o observe ha in (1.4) he derivaive was defined by firs obaining he gradien, while in (2.8) he gradien followed from he derivaive as is represener. This disincion is very imporan if we wan o exend he noion of he gradien o infinie dimensional spaces. We reinforce hese poins by exending he noion of gradien o inner produc spaces. Definiion 3.1. Consider f : H IR where H is an inner produc space wih inner produc,. Assume ha f is Gâeaux differeniable a x. Then if here exiss f(x) H such ha f (x)(η) = f(x), η for all η H, we call his (necessarily unique) represener of f (x)( ) he gradien of f a x. Remark 3.2. If our inner produc space H is complee, i.e. a Hilber space, hen by he Riesz represenaion heorem (Theorem C.6.1) he gradien f(x) always exiss. However, while he calculaion of f (x) is almos always a sraighforward maer, he deerminaion of f(x) may be quie challenging. Remark 3.3. Observe ha while f : H H we have ha f : H H. I is well known ha in his generaliy f(x) is he unique direcion of seepes ascen of f a x in he sense ha f(x) is he unique soluion of he opimizaion problem maximize subjec o f (x)(η) η f(x). The proof of his fac is a direc applicaion of he CauchySchwarz inequaliy. The wellknown mehod of seepes descen considers ieraes of he form x α f(x) for α > 0. This saemen is made o emphasize ha an expression of he form x αf (x) would be meaningless. This noion of gradien in Hilber space is due o Golomb [ ] in Exension of he noion of he gradien o normed linear spaces is an ineresing opic and has been considered by several auhors, see Golomb and Tapia [ ] 1972 for example. This noion is discussed in Chaper 14. 5
7 4. The Fréche Derivaive. The direcional noion of differeniaion is quie general. I allows us o work in he full generaliy of realvalued funcionals defined on a vecor space. In many problems, including problems from he calculus of variaions his generaliy is exacly wha we need and serves us well. However, for cerain applicaions we need a sronger noion of differeniaion. Moreover, many believe ha a saisfacory noion of differeniaion should possess he propery ha differeniable funcions are coninuous. This leads us o he concep of he Fréche derivaive. Recall ha by L[X, Y ] we mean he vecor space or bounded linear operaors from he normed linear space X o he normed linear space Y. We now inroduce he noion of he Fréche derivaive. Definiion 4.1. Consider f : X Y where boh X and Y are normed linear spaces. Given x X, if a linear operaor f (x) L[X, Y ] exiss such ha f(x + x) f(x) f (x)( x) lim = 0, x X, (4.1) x 0 x hen f (x) is called he Fréche derivaive of f a x. The operaor f : X L[X, Y ] which assigns f (x) o x is called he Fréche derivaive of f. Remark 4.2. Conrary o he direcional variaion a a poin, he Fréche derivaive a a poin is by definiion a coninuous linear operaor. Remark 4.3. By (4.1) we mean, given ɛ > 0 here exiss δ > 0 such ha f(x + x) f(x) f (x)( x) ɛ x (4.2) for every x X such ha x δ. Proposiion 4.4. Le X and Y be normed linear spaces and consider f : X Y. If f is Fréche differeniable a x, hen f is Gâeaux differeniable a x and he wo derivaives coincide. Proof. If he Fréche derivaive f (x) exiss, hen by replacing x wih x in (4.1) we have lim 0 f(x + x) f(x) f (x)( x) = 0. Hence f (x) is also he Gâeaux derivaive of f a x. Corollary 4.5. The Fréche derivaive, when i exiss, is unique. Proof. The Gâeaux derivaive defined as a limi mus be unique, hence he Fréche derivaive mus be unique. Proposiion 4.6. If f : X Y is Fréche differeniable a x, hen f is coninuous a x. Proof. Using he lefhand side of he riangle inequaliy on (4.2) we obain f(x + x) f(x) (ɛ + f (x) ) x. The proposiion follows. Example 4.7. Consider a linear operaor L : X Y where X and Y are vecor spaces. For x, η X we have L(x + η) L(x) = L(η). Hence i is clear ha for a linear operaor L he direcional derivaive a each poin of X exiss and coincides wih L. 1 Moreover, if X and Y are normed linear spaces and L is bounded hen L (x) = L is boh a Gâeaux and a Fréche derivaive. Example 4.8. Le X be he infiniedimensional normed linear space defined in Example C.5.1. Moreover, le δ be he unbounded linear funcional defined on X given in his example. Since we saw above ha δ (x) = δ, we see ha for all x, δ has a direcional derivaive which is never a Gâeaux or a Fréche derivaive, i.e., i is no bounded. Example 2.9 describes a siuaion where f (0)(η) is a Gâeaux derivaive. Bu i is no a Fréche derivaive since f is no coninuous a x = 0. For he sake of compleeness, we should idenify a siuaion where he funcion is coninuous and we have a Gâeaux derivaive a a poin which is no a Fréche derivaive. Such a funcion is given by (6.4). 1 There is a sligh echnicaliy here, for if X is no a opological linear space, hen he limiing process in he definiion of he direcional derivaive is no defined, even hough i is redundan in his case. Hence, formally we accep he saemen. 6
8 5. Condiions Implying he Exisence of he Fréche Derivaive. In his secion, we invesigae condiions implying ha a direcional variaion is acually a Fréche derivaive and ohers implying he exisence of he Gâeaux or Fréche derivaive. Our firs condiion is fundamenal. Proposiion 5.1. Consider f : X Y where X and Y are normed linear spaces. Suppose ha for some x X, f (x), is a Gâeaux derivaive. Then he following wo saemen are equivalen: (i) f (x) is a Fréche derivaive. (ii) The convergence in he direcional variaion defining relaion (2.2) is uniform wih respec o all η such ha η = 1. Proof. Guided by (4.2) we consider he saemen f(x + x) f(x) f (x)( x) x Consider he change of variables x = η in (5.1) o obain he saemen f(x + η) f(x) ɛ whenever x δ. (5.1) f (x)(η) ɛ whenever η X, η = 1 (5.2) and 0 < δ. Our seps can be reversed so (5.1) and (5.2) are equivalen saemens, and correspond o (i) and (ii) of he proposiion. There is value in idenifying condiions which imply ha a direcional variaion is acually a Gâeaux derivaive as we assumed in he previous proposiion. The following proposiion gives such condiions. Proposiion 5.2. Le X and Y be normed linear spaces. Assume f : X Y has a direcional variaion in X. Assume furher ha (i) for fixed x, f (x)(η) is coninuous in η a η = 0, and (ii) for fixed η, f (x)(η) is coninuous in x for all x X. Then f (x) L[X, Y ], i.e., f (x) is a bounded linear operaor. Proof. By Proposiion?? f (x) is a homogeneous operaor; hence f (x)(0) = 0. By (i) above here exiss r > 0 such ha f (x)(η) 1 whenever η r. I follows ha ( ) f (x)(η) = η rη r f (x) 1 η r η ; hence f (x) will be a coninuous linear operaor once we show i is addiive. Consider η 1, η 2 X. Given ɛ > 0 from Definiion 2.1 here exiss τ > 0 such ha f (x)(η 1 + η 2 ) f (x)(η 1 ) f (x)(η 2 ) [ ] [ ] [ ] f(x + η1 + η 2 ) f(x) f(x + η1 ) f(x) f(x + η2 ) f(x) 3ɛ, whenever τ. Hence f (x)(η 1 + η 2 ) f (x)(η 1 ) f (x)(η 2 ) 1 f( + η 1 + η 2 ) f(x + η 1 ) f(x + η 2 ) f(x) + 3ɛ. By (i) of Proposiion 2.12 and Corollary C.7.3 of he HahnBanach heorem here exiss δ Y such ha δ = 1 and 7
9 f(x + η 1 + η 2 ) f(x + η 1 ) f(x + η 2 ) f(x) = δ(f(x + η 1 + η 2 ) f(x + η 1 )) δ(f(x + η 2 ) f(x)) = δ(f (x + η 1 + θ 1 η 2 )(η 2 )) δ(f (x + θ 2 η 2 )(η 2 )) = δ[f (x + η 1 + θ 1 η 2 )(η 2 ) f (x)(η 2 ) +f (x)(η 2 ) f (x + θ 2 η 2 )(η 2 )] f (x + η 1 + θ 1 η 2 )(η 2 ) f (x)(η 2 ) + f (x)(η 2 ) f (x + θ 2 η 2 ), 0 < θ 1, θ 2 < 1, 2 ɛ if is sufficienly small. I follows ha f (x)(η 1 + η 2 ) f (x)(η 1 ) f (x)η 2 5ɛ since ɛ > 0 was arbirary f (x) is addiive. This proves he proposiion. We nex esablish he wellknown resul ha a coninuous Gâeaux derivaive is a Fréche derivaive. Proposiion 5.3. Consider f : X Y where X and Y are normed linear spaces. Suppose ha f is Gâeaux differeniable in an open se D X. If f is coninuous a x D, hen f (x) is a Fréche derivaive. Proof. Consider x D a coninuiy poin of f. Since D is open here exiss ˆr > 0 saisfying for η X and η < ˆr, x + η D for (0, 1). Choose such an η. Then by par (i) of Proposiion 2.12 for any δ Y δ[f(x + η) f(x) f (x)(η)] = δ[f (x + θη)(η) f (x)(η)], for some θ (0, 1). By Corollary C.7.3 of he HahnBanach heorem we can choose δ so ha f(x + η) f(x) f (x)(η) f (x + θη) f (x) η. Since f is coninuous a x, given ɛ > 0 here exiss r, 0 < r < ˆr, such ha f(x + η) f(x) f (x)(η) ɛ η whenever η r. This proves he proposiion. Consider f : IR n IR. Example 2.9 shows ha he exisence of he parial derivaives does no imply he exisence of he direcional variaion, le alone he Gâeaux or Fréche derivaives. However, i is a wellknown fac ha he exisence of coninuous parial derivaives does imply he exisence of he Fréche derivaive. Proofs of his resul can be found readily in advanced calculus and inroducory analysis exs. We found several such proofs, hese proofs are reasonably challenging and no a all sraighforward. For he sake of compleeness of he curren appendix we include an elegan and comparaively shor proof ha we found in he book by Fleming [1]. Proposiion 5.4. Consider f : D IR n IR where D is an open se. Then he following are equivalen. (i) f has coninuous firsorder parial derivaives in D. (ii) f is coninuously Fréche differeniable in D. Proof. [(i) (ii)] We proceed by inducion on he dimension n. For n = 1 he resul is clear. Assume ha he resul is rue in dimension n 1. Our ask is o esablish he resul in dimension n. Has will be used o denoe (n 1)vecors. Hence ˆx = (x 1,..., x n 1 ) T. Le x 0 = (x 0,..., x 0 n) T be a poin in D and choose δ 0 > 0 so ha B(x 0, δ 0 ) D. Wrie φ(ˆx) = f(x 1,..., x n 1, x 0 n) = f(ˆx, x n 0 ) for ˆx such ha (ˆx, x 0 n) D. Clearly, i φ(ˆx) = i f(ˆx, x 0 n), i = 1,..., n 1 (5.3) 8
10 where i denoes he parial derivaive wih respec o he ih variable. Since i f is coninuous i φ is coninuous. By he inducion hypohesis φ is Fréche differeniable a ˆx 0. Therefore, given ɛ > 0 here exiss δ 1, saisfying 0 < δ 1 < δ 0 so ha φ(ˆx 0 + ĥ) φ(ˆx0 ) φ(ˆx 0 ), ĥ < ɛ ĥ whenever ĥ < δ 1. (5.4) Since n f is coninuous here exiss δ 2 saisfying 0 < δ 2 < δ 1 such ha n f(y) n f(x 0 ) < ɛ whenever y x 0 < δ 2. (5.5) Consider h IR n such ha h < δ 2. By he ordinary meanvalue heorem, see (2.7), for some θ (0, 1). Seing gives Since f(x 0 ) = φ(ˆx 0 ) we can wrie f(x 0 + h) φ(ˆx 0 + ĥ) = f(ˆx + ĥ, x0 n + h n ) f(ˆx 0 + ĥ, x0 n) = n f(ˆx 0 + ĥ, x0 n + θh n )h n (5.6) y = (ˆx 0 + ĥ, x0 n + θh n ) (5.7) y x 0 = θ h n < h < δ 2. (5.8) f(x 0 + h) f(x 0 ) = [f(x 0 + h) φ(ˆx 0 + ĥ) + φ(ˆx0 + ĥ) φ(ˆx0 ). (5.9) Now, f(x 0 ), h is a bounded linear funcional in h. Our goal is o show ha i is he Fréche derivaive of f a x 0. Towards his end we firs use (5.6) and (5.7) o rewrie (5.9) as f(x 0 + h) f(x 0 ) = n f(y)h n + φ(ˆx 0 + ĥ). (5.10) Now we subrac f(x 0, h) from boh sides of (5.10) and recall (5.3) o obain f(x 0 + h) f(x 0 ) f(x 0 ), h = [ n f(y)h n n f(x 0 )h n ] + [φ(ˆx 0 + ĥ) φˆx0 φ(ˆx 0 ), ĥ ]. (5.11) Observing ha h n h and ĥ h and calling on (5.4) and (5.5) for bounds we see ha (5.11) readily leads o f(x 0 + h) f(x 0 ) f(x 0 ), h < ɛ h n + ɛ ĥ 2ɛ h (5.12) whenever h < δ 2. This shows ha f is Fréche differeniable a x 0 and f (x 0 )(h) = f(x 0 ), h. (5.13) Since he parials i f are coninuous a x 0, he gradien operaor f is coninuous a x 0 and i follows ha he Fréche derivaive f is coninuous a x 0. [(ii) (i)] Consider x 0 D. Since f is a Fréche derivaive we have he represenaion (5.13) (see (2.10)). Moreover, coninuous parials implies a coninuous gradien and in urn a coninuous derivaive. This proves he proposiion. Remark 5.5. The challenge in he above proof was esablishing lineariy. The proof would have been immediae if we had assumed he exisence of a direcional derivaive. Corollary 5.6. Consider F : D IR n IR m where D is an open se. Then he following are equivalen. (i) F has coninuous firsorder parial derivaives in D. 9
11 (ii) F is coninuously Fréche differeniable in D. Proof. Wrie F (x) = (f 1 (x),..., f m (x)) T. The proof follows from he proposiion by employing he relaionship F (x)(η) = f 1 (x), η. f m (x), η = JF (x)η where JF (x) represens he Jacobian marix of F a x. See Example This proof aciviy requires familiariy wih he maerial discussed in?? of Appendix C. 6. The Chain Rule. The chain rule is a very powerful and useful ool in analysis. We now invesigae condiions which guaranee a chain rule. Proposiion 6.1. (Chain Rule). Le X be a vecor space and le Y and Z be normed linear spaces. Assume: (i) h : X Y has a forward direcional variaion, respecively direcional variaion, direcional derivaive, a x X, and (ii) g : Y Z is Fréche differeniable a h(x) Y. Then f = g h : X Z has a forward direcional variaion, respecively direcional variaion, direcional derivaive, a x X and f +(x) = g (h(x))h +(x). (6.1) If X is also a normed linear space and h (x) is a Gâeaux, respecively Fréche, derivaive, hen f (x) is also a Gâeaux, respecively Fréche, derivaive. Hence Proof. Given x consider η X. Le y = h(x) and y = h(x + η) h(x). Then f(x + η) f(x) = g(y + y) g(y) = g (y)( y) + g(y + y) g(y) g (y)( y) = g (h(x + η) h(x)) (y) + g(y + y) g(y) g (y)( y) y h(x + η) h(x). [ ] f(x + η) f(x) g (h(x))h (x)(η) [ ] g (h(x)) h(x + η) h(x) h (x)(η) + g(y + y) g(y) g (y)( y) h(x + η) h(x) y. (6.2) Observe ha if h has a forward direcional variaion a x, hen alhough h may no be coninuous a x, we do have ha h is coninuous a x in each direcion (see Proposiion 2.14), i.e., y 0 as 0. By leing 0 in (6.2) we see ha f has a forward direcional variaion a x. Clearly, he same holds for wosided limis. To see ha variaion can be replaced by derivaive we need only recall ha he composiion of wo linear forms is a linear form. Suppose ha X is also a normed linear space. From (6.1) i follows ha f (x) is a bounded linear form if g (h(x)) and h (x) are bounded linear forms. Hence, f (x) is a Gâeaux derivaive whenever h (x) is a Gâeaux derivaive. Now suppose ha h (x) is a Fréche derivaive. Our ask is o show ha f (x) is Fréche. We do his by esablishing uniform convergence in (6.2) for η such ha η = 1. From Proposiion 4.4 h is coninuous a x. Hence given ɛ > 0 δ > 0 so ha y = h(x + η) h(x) < ɛ 10
12 whenever η = < δ. So as 0, y 0 uniformly in η saisfying η = 1. Using he lefhand side of he riangle inequaliy we can wrie h(x + η) h(x) h (x)) η + h(x + η) h(x) h (x)(η). (6.3) Since h is Fréche differeniable a x he second erm on he righhand side of he inequaliy in (6.3) goes o zero as 0 uniformly for η saisfying η = 1. Hence he erm on he lefhand side of he inequaliy (6.3) is bounded for small uniformly in η saisfying η = 1. I now follows ha as 0 he expression on he lefhand side of he inequaliy (6.3) goes o zero uniformly in η saisfying η = 1. This shows by Proposiion 5.1 ha f (x) is a Fréche derivaive. We now give an example where in Proposiions 6.1 he funcion h is Fréche differeniable, he funcion g has a Gâeaux derivaive and ye he chain rule (6.1) does no hold. The requiremen ha g be Fréche differeniable canno be weakened. Example 6.2. Consider g : IR 2 IR defined by g(0) = 0 and for nonzero x = (x 1, x 2 ) T Also consider h : IR 2 IR defined by g(x) = x 2(x x 2 2) 3/2 [(x x2 2 )2 + x 2 (6.4) 2 ]. h(x) = (x 1, x 2 2) T. (6.5) We are ineresed in he composie funcion f : IR 2 IR obained as f(x) = g(h(x)). We can wrie and f(0) = g(h(0)) = g(0) = 0. To begin wih (see Example 2.16) ( ) ( ) h 1 0 η1 (x)(η) = 0 2x 2 η 2 so f(x) = x 2(x x 2 4) 3/2 [(x x2 4 )2 + x 4 2 ] x 0 (6.6) h (0)(η) = ( ) ( ) 1 0 η1 = 0 0 η 2 (6.7) ( ) η1. (6.8) 0 Moreover, h (0) in (6.8) is clearly Fréche since h (x) in (6.7) gives a bounded linear operaor which is coninuous in x (see Proposiion 5.3). Direc subsiuion shows ha Hence g (0) is a Gâeaux derivaive. I follows ha Again direc subsiuion shows ha g (g(η) g(0) g(η) (0)(η) = lim = lim = 0. (6.9) 0 0 g (h(0))(h (0)) = 0. (6.10) f (0)(η) = η3 1η2 2 η (6.11) η4 2 Since (6.10) and (6.11) do no give he same righhand side our chain rule fails. I mus be ha g (0) is no a Fréche derivaive. We now demonsrae his fac direcly. A his juncure here is value in invesigaing he coninuiy of g a x = 0. If g is no coninuous a x = 0, we will have demonsraed ha g (0) canno be a Fréche derivaive. On he oher hand, if we show ha g is coninuous a x = 0, hen our example of chain rule failure is even more compelling. We consider showing ha g(x) 0 as x 0. From he definiion of g in (6.4) we wrie g(x) = x 2δ 3 δ 4 + x 2 2 (6.12) 11
13 where x x 2 2 = δ 2. Since δ is he Euclidean norm of (x 1, x 2 ) T a resaemen of our ask is o show ha g(x) 0 as δ 0. We herefore sudy φ : IR IR defined by φ(x) = xδ3 δ 4 + x 2 for x 0. (6.13) Direc calculaions show ha φ(0) = 0, φ(x) > 0 for x > 0, φ (x) > 0 for x < δ 2, φ (x) = 0 for x = δ 2, and φ (x) < 0 for x > δ 2. I follows ha φ has a unique minimum a x = δ 2 and φ(δ 2 ) = δ 2. This means ha g(x) δ 2 x. Hence, g(x) 0 as δ 0 and g is coninuous a x = 0. We encouner his resul wih slighly mixed emoions. While i adds credibiliy o he example, i.e., we have an example where he funcion is coninuous and he Gâeaux derivaive is no a Fréche derivaive, we now have o do more work o show ha g (0) is no a Fréche derivaive. Our only remaining opion is o demonsrae ha he convergence in (6.9) is no uniform wih respec o η saisfying η = 1. Toward his end consider η = (η 1, η 2 ) T such ha η = 1, i.e., η η 2 2 = 1. Then If we choose η 2 =, hen g(η) g(η) = η η2 2. (6.14) = 1 2. (6.15) So for small we can always find an η wih norm one of he form η = (η 1, ). Hence he convergence (6.9) is no uniform wih respec o η saisfying η = 1. The choice η 2 = should follow somewha direcly by reflecing on (6.14); however, his is no he way we found i. Being so pleased wih our opimizaion analysis of he funcion φ in (6.13) we did he same hing wih he funcion given in (6.14). I urns ou ha i is maximized in η 2 wih he choice η 2 =. If any choice should demonsrae he lack of uniform convergence, he maximizer should cerainly do i. The suden should explore our approach. 7. A Schemaic Overview and an Example. We begin by presening our hierarchy of noions of differeniaion schemaically in Figure 7.1. For his purpose i suffices o work wih a funcional f : X IR where X is a vecor space. For a given x X we will consider arbirary η X. We now illusrae hese noions and implicaions wih an imporan example. Example 7.1. Le C 1 [a, b] be he vecor space of all realvalued funcions which are coninuously differeniable on he inerval [a, b]. Suppose f : R 3 R has coninuous parial derivaives wih respec o he second and hird variables. Consider he funcional J : C 1 [a, b] R defined by J(y) = b a f(x, y(x), y (x)) dx. (7.1) We denoe he parial derivaives of f by f 1, f 2, and f 3 respecively. Using he echnique described in Proposiion 2.10 for η C 1 [a, b], we define Then, φ() = J(y + η) = φ () = b a b a f(x, y(x) + η(x), y + η (x))dx. [f 2 (x, y(x) + η(x), y (x) + η (x))η(x) + f 3 (x, y(x) + η(x), y (x) + η (x))η (x)]dx, 12
14 Forward Direcional Variaion X is a vecor space f +(x)(η) Direcional Variaion (wosided limi) X is a vecor space f (x)(η) = f +(x)(η) = f (x)(η) Direcional Derivaive X is a vecor space f (x)(η), he direcional variaion is linear in η Gâeaux Derivaive X is a normed linear space f (x)(η) is a direcional derivaive f (x)(η) is bounded in η Fréche Derivaive X is a normed linear space f (x) is a Gâeaux derivaive convergence in he definiion of he Gâeaux variaion 2.2 is uniform wih respec o all η saisfying η = 1. Fig so ha J (y)(η) = φ (0) = b a [f 2 (x, y(x), y (x))η(x) + f 3 (x, y(x), y (x))η (x)]dx. (7.2) I is readily apparen ha he direcional variaion J (y)(η) is acually a direcional derivaive, i.e., J (y)(η) is linear in η. Before we can discuss Gâeaux or Fréche derivaives of J we mus inroduce a norm on C 1 [a, b]. Toward his end le η = max η(x) + max a x b a x b η (x) for each η C 1 [a, b]. Recall ha we have assumed f has coninuous parial derivaives wih respec o he second and hird variables. Also y and y are coninuous on [a, b]. Hence, he composie funcions 13
15 f i (x, y(x), y (x)) i = 2, 3 are coninuous on he compac se [a, b] and herefore aain heir maxima on [a, b]. I follows ha [ ] J (y)(η) (b a) max f 2(x, y(x), y (x)) + max f 3(x, y(x), y (x)) η ; a x b a x b hence J (y) is a bounded linear operaor, and is herefore a Gâeaux derivaive. We now show ha J is he Fréche derivaive. Using he ordinary meanvalue heorem for real valued funcions of several real variables allows us o wrie J(y + η) J(y) = = Z b a Z b where 0 < θ(x) < 1. Using (7.2) and (7.3) we wrie J(y + η) J(y) J (y)(η) = Z b a a [f(x, y(x) + η(x), y (x) + η (x)) f(x, y(x), y (x))]dx [f 2(x, y(x) + θ(x)η(x), y (x) + θ(x)η (x)) + f 3(x, y(x) + θ(x)η(x), y (x) + θ(x)η (x), y (x))]dx (7.3) {f 2(x, y(x) + θ(x)η(x), y (x) + θ(x)η(x)) f 2(x, y(x), y (x))]η(x) + [f 3(x, y(x) + θ(x)η(x), y (x) + θ(x)η(x)) f 3(x, y(x), y (x))η (x)]}dx. (7.4) For our given y C 1 [a, b] choose a δ 0 and consider he se D IR 3 defined by D = [a, b] [ y δ 0, y + δ 0] [ y δ 0, y + δ 0]. (7.5) Clearly D is compac and since f 2 and f 3 are coninuous, hey are uniformly coninuous on D. So, given ɛ > 0, by uniform coninuiy of f 2 and f 3 on D here exiss δ > 0 such ha if η < δ hen all he argumens of he funcion f 2 and f 3 in (7.4) are conained in D and J(y + η) J(y) J (y)(η) 2ɛ(b a) η. (7.6) Hence, J (y) is he Fréche derivaive of J a y according o he definiion of he Fréche derivaive, see Definiion 4.1 and Remark 4.3. Alernaively, we can show ha J (y) is he Fréche derivaive by demonsraing ha he map J defined from C 1 [a, b] ino is opological dual space (see C.7) is coninuous a y and hen appealing o Proposiion 5.3. Since we are aemping o learn as much as possible from his example we pursue his pah and hen compare he wo approaches. Familiariy wih C.5 will faciliae he presenaion. From he definiions and (7.2) we obain for y, h C 1 [a, b] J (y + h) J (y) = sup J (y + h)(η) J (y)(η) η =1» (b a) max a x b y (x) + h (x)) f 2(x, y(x), y (x)) + max a x b y (x) + h (x)) f 3(x, y(x), y (x)). (7.7) Observe ha he uniform coninuiy argumen used o obain (7.4) and (7.6) can be applied o (7.7) o obain J (y + h) J (y) 2ɛ(b a) whenever h δ. Hence J is coninuous a y and J (y) is a Fréche derivaive. We now commen on wha we have learned from our wo approaches. Clearly, he second approach was more direc. Le s explore why his was so. For he purpose of adding undersanding o he role ha uniform coninuiy played in our applicaion we offer he following observaion. Consider he funcion g(y) = f i(x, y(x), y (x)) (for i = 1 or i = 2) viewed as a funcion of y C 1 [a, b] ino C 0 [a, b], he coninuous funcions on [a, b] wih he sandard max norm. I is exacly he uniform coninuiy of f i on he compac se D IR 3 ha enables he coninuiy of g a y. 14
16 Observe ha in our firs approach we worked wih he definiion of he Fréche derivaive which enailed working wih he expression J(y + η) J(y) J (y)(η). (7.8) The expression (7.8) involves funcions of differing differenial orders; hence can be problemaic. However, in our second approach we worked wih he expression J (y + h) J (y) (7.9) which involves only funcions of he same differenial order. This should be less problemaic and more direc. Now, observe ha i was he use of he ordinary meanvalue heorem ha allowed us o replace (7.8) wih (J (y + θη) J (y))(η). (7.10) Clearly (7.10) has he same flavor as (7.9), So i should no be a surprise ha he second half of our firs approach and our second approach were essenially he same. However, he second approach gave us more for our money; in he process we also esablished he coninuiy of he derivaive. Our summarizing poin is ha if one expecs J o also be coninuous, as we should have here in our example, i is probably wise o consider working wih Proposiion 5.3 insead of working wih he definiion and Proposiion 5.1 when aemping o esablish ha a direcional derivaive is a Fréche derivaive. 8. The Second Direcional Variaion. Again consider f : X Y where X is a vecor space and Y is a opological vecor space. As we have shown his allows us sufficien srucure o define he direcional variaion. We now urn our aenion owards he ask of defining a second variaion. I is mahemaically saisfying o be able o define a noion of a second derivaive as he derivaive of he derivaive. If f is direcionally differeniable in X, hen we know ha f : X [X, Y ], where [X, Y ] denoes he vecor space of linear operaors from X ino Y. Since in his generaliy, [X, Y ] is no a opological vecor space we can no consider he direcional variaion of he direcional derivaive as he second direcional variaion. If we require he addiional srucure ha X and Y are normed linear spaces and f is Gâeaux differeniable, we have f : X L[X, Y ] where L[X, Y ] is he normed linear space of bounded linear operaors from X ino Y described in Appendix??. We can now consider he direcional variaion, direcional derivaive, and Gâeaux derivaive. However, requiring such srucure is excessively resricive for our vecor space applicaions. Hence we will ake he approach used successfully by workers in he early calculus of variaions. Moreover, for our purposes we do no need he generaliy of Y being a general opological vecor space and i suffices o choose Y = IR. Furhermore, because of (ii) of Proposiion 2.7 we will no consider he generaliy of onesided second variaions. Definiion 8.1. Consider f : X IR, where X is a real vecor space and x X. For η 1, η 2 X by he second direcional variaion of f a x in he direcions η 1 and η 2, we mean f (x)(η 1, η 2) = f (x + η 1)(η 2) f (x)(η 2) lim, (8.1) 0 whenever hese direcional derivaives and he limi exis. When he second direcional variaion of f a x is defined for all η 1, η 2 X we say ha f has a second direcional variaion a x. Moreover, if f (x)(η 1, η 2) is bilinear, i.e., linear in η 1 and η 2, we call f (x) he second direcional derivaive of f a x and say ha f is wice direcionally differeniable a x. We now exend our previous observaion o include he second direcional variaion. Proposiion 8.2. Consider a funcional f : X IR. Given x, η X, le Then, and φ() = f(x + η). φ (0) = f (x)(η) (8.2) φ (0) = f (x)(η, η) (8.3) 15
17 Moreover, given x, η 1, η 2 X, le ω() = f (x + η 1)(η 2). Then, ω (0) = f (x)(η 1, η 2) (8.4) Proof. The proof follows direcly from he definiions. Example 8.3. Given f(x) = x T A x, wih A IR n n and x, η IR n, find f (x)(η, η). From Example 2.11, we have φ () = x T Aη + η T Ax + 2η T Aη. Thus, and Moreover, leing we have φ () = 2 η T Aη, φ (0) = 2 η T Aη = f (x)(η, η). ω() = (x + η 1) T Aη 2 + η T 2 A(x + η 1), ω (0) = η T 1 Aη 2 + η T 2 Aη 1 = f (x)(η 1, η 2). In essenially all our applicaions we will be working wih he second direcional variaion only in he case where he direcions η 1, and η 2 are he same. The erminology second direcional variaion a x in he direcion η will be used o describe his siuaion. Some auhors only consider he second variaion when he direcions η 1 and η 2 are he same. They hen use (8.4) as he definiion of f (x)(η, η). While his is raher convenien and would acually suffice in mos of our applicaions, we are concerned ha i masks he fac ha he second derivaive a a poin is inherenly a funcion of wo independen variables wih he expecaion ha i be a symmeric bilinear form under reasonable condiions. This will become apparen when we sudy he second Fréche derivaive in laer secions. The definiion (7.10) reains he flavor of wo independen argumens. The second direcional variaion is usually linear in each of η 1 and η 2, and i is usually symmeric, in he sense ha f (x)(η 1, η 2) = f (x)(η 2, η 1). This is he case in Example 8.3 above even when A is no symmeric. However, his is no always he case. Example 8.4. Consider he following example. Define f : IR 2 IR 3 by f(0) = 0 and by f(x) = x1x2(x2 1 x 2 2) x x2 2 for x 0. Clearly, f (0)(η) = 0 η IR 2. Now, for x 0, he parial derivaives of f are clearly coninuous. Hence f is Fréche differeniable for x 0 by Proposiion 5.3. I follows ha we can wrie A direc calculaion of he wo parial derivaives allows us o show ha f (x)(η) = f(x), η. (8.5) f (x)(η) = f(x), η = f(x), η = f (x)(η). (8.6) Using (8.6) we have ha» f f (u)(v) f (0)(v) (0)(u, v) = lim = f (u)(v). 0 Hence, we can wrie f (0)(u, v) = f(u), v. (8.7) Again, appealing o he calculaed parial derivaives we see ha f (0) as given by (8.7) is no linear in u and is no symmeric. We conclude ha f (0) as a second direcional variaion is neiher symmeric nor a second direcional derivaive (bilinear). However, as we shall demonsrae in 10 he second Fréche derivaive a a poin is always a symmeric bilinear form. 16
18 In he following secion of his appendix we will define he second Fréche derivaive. A cerain amoun of saisfacion will be gained from he fac ha, unlike he second direcional variaion (derivaive), i is defined as he derivaive of he derivaive. However, in order o make our presenaion complee, we firs define he second Gâeaux derivaive. I oo will be defined as he derivaive of he derivaive, and perhaps represens he mos general siuaion in which his can be done. We hen prove a proposiion relaing he second Gâeaux derivaive o Fréche differeniabiliy. Afer several examples concerning Gâeaux differeniabiliy in IR n, we close he secion wih a proposiion giving several basic resuls in IR n. Definiion 8.5. Le X and Y be normed linear spaces. Consider f : X Y. Suppose ha f is Gâeaux differeniable in an open se D X. Then by he second Gâeaux derivaive of f we mean f : D X L[X, L[X, Y ]], he Gâeaux derivaive of he Gâeaux derivaive f : D X L[X, Y ]. As we show in deail in he nex secion, L[X, L[X, Y ]] is isomerically isomorphic o [X 2, Y ] he bounded bilinear forms defined on X ha map ino Y. Hence he second Gâeaux derivaive can be viewed naurally as a bounded bilinear form. The following proposiion is essenially Proposiion 5.2 resaed o accommodae he second Gâeaux derivaive. Proposiion 8.6. Le X and Y be normed linear spaces. Suppose ha f : X Y has a firs and second Gâeaux derivaive in an open se D X and f is coninuous a x D. Then boh f (x) and f (x) are Fréche derivaives. Proof. By coninuiy f (x) is a Fréche derivaive (Proposiion 5.1). Hence f is coninuous a x (Proposiion 5.2) and is herefore he Fréche derivaive a x (Proposiion 5.1). Some concern immediaely arises. According o our definiion given in he nex secion, he second Fréche derivaive is he Fréche derivaive of he Fréche derivaive. So i seems as if we have idenified a slighly more general siuaion here where he second Fréche derivaive could be defined as he Fréche derivaive of he Gâeaux derivaive f : X X. However, his siuaion is no more general. For if f is he Fréche derivaive of f, hen by Proposiion 4.4 f is coninuous and, in urn by Proposiion 5.2 f is a Fréche derivaive. So, a second Gâeaux derivaive which is a Fréche derivaive is a second Fréche derivaive according o he definiion ha will be given. Recall ha if f : IR n IR, hen he noions of direcional derivaive and Gâeaux derivaive coincide since in heis seing all linear operaors are bounded, see??. Consider f : IR n IR. Suppose ha f is Gâeaux differeniable. Then we showed in he Example 2.15 ha he gradien vecor is he represener of he Gâeaux derivaive, i.e., for x, η IR n f (x)(η) = f(x), η. Now suppose ha f is wice Gâeaux differeniable. The firs hing ha we observe, via Example 2.16, is ha he Hessian marix 2 f(x) is he Jacobian marix of he gradien vecor; specifically 2 1 1f(x)... 3 n 1f(x) J f(x) = (8.8) 1 nf(x)... n n(x) Nex, observe ha f f (x + η 1)(η 2) f (x)(η 2) (x)(η 1, η 2) = lim 0 f(x + η 1), η 2 f(x), η 2 = lim 0 f(x + η 1) f(x) = lim, η 0 2 = 2 f(x)η 1, η 2 = η T 2 2 f(x)η 1. Following in his direcion consider wice Gâeaux differeniable f : IR n IR m. Turning o he componen funcions we wrie f(x) = (f 1(x)...., f m(x)) T. Then, and f (x)(η) = Jf(x)η = ( f 1(x), η,..., f m(x), η ) T, f (x)(η 1, η 2) = ( 2 f 1(x)η 1, η 2,..., 2 f m(x)η 1, η 2 ) T = (η T 1 2 f 1(x)η 2,..., η T 1 2 f m(x)η 2) T. (8.9) This discussion leads naurally o he following proposiion. Proposiion 8.7. Consider f : D IR n IR m where D is an open se in IR n. Suppose ha f is wice Gâeaux differeniable in D. Then for x D (i) f (x) is symmeric if and only if he Hessian marices of he componen funcions a x, 2 f 1(x),..., 2 f m(x), are symmeric. 17
19 (ii) f is coninuous a x if and only if all secondorder parial derivaives of he componen funcions f 1,..., f m are coninuous a x. (iii) If f is coninuous a x, hen f (x) and f (x) are Fréche derivaives and f (x) is symmeric. Proof. The proof of (i) follows direcly from (8.9) since secondorder parial derivaives are firsorder parial derivaives of he firsorder parial derivaives. Par (ii) follows from Corollary 5.6. Par (iii) follows Proposiion 5.3, Proposiion 4.6, Proposiion 5.3 again, and he ye o be proved Proposiion 9.4, all in ha order. Hence he Hessian marix is he represener, via he inner produc, of he bilinear form f (x). So, he gradien vecor is he represener of he firs derivaive and he Hessian marix is he represener of he second derivaive. This is saisfying. A his poin we pause o collec hese observaions in proposiion form, so hey can be easily referenced. Proposiion 8.8. Consider f : IR n IR. (i) If f is direcionally differeniable a x, hen f (x) = f(x), η where f(x) is he gradien vecor of f a x defined in (1.3). η IR n (ii) If f is direcionally differeniable in a neighborhood of x and wice direcionally differeniable a x, hen f (x)(η 1, η 2) = 2 f(x)η 1, η 2 where 2 f(x) is he Hessian marix of f a x, described in (8.8). η 1, η 2 X Remark 8.9. I is worh noing ha he Hessian marix can be viewed as he Jacobian marix of he gradien vecor funcion. Moreover, in his conex our direcional derivaives are acually Gâeaux derivaives, since we are working in IR n. Finally, in his generaliy we can no guaranee ha he Hessian marix is symmeric. This would follow if f (x) was acually a Fréche derivaive, see Proposiion The following proposiion plays a srong role in Chaper?? where we develop our fundamenal principles for secondorder necessiy and sufficiency. Our presenaion follows ha of Orega and Rheinbold []. Proposiion Assume ha F : D X Y, where X and Y are normed linear spaces and D is an open se in X, is Gâeaux differeniable in D and has a second Gâeaux derivaive a x D. Then, 1 (i) lim 0 ˆF (x + h) F (x) F (x)(h) 1 F (x)(h, h) = for an h X. Moreover, if f (x) is a Fréche derivaive, hen 1 (ii) lim h 0 h ˆF (x + h) F (x) F (x)(h) 1 F (x)(h, h). 2 2 Proof. For given h X and sufficiency small we have ha x + h D and is welldefined. Clearly, G() = F (x + h) F (x) F (x)(h) 1 2 F (x)(h, h) G () = F (x + h)(h) F (x)(h) F (x)(h, h). From he definiion of f (x)(h, h) we have ha given ε > 0 δ > 0 so ha when < δ. Now, from (iii) of Proposiion 2.12 G () ε G() = G() G(0) sup G (θ) ε 2 0 θ 1 whenever < δ. Since ε > 0 was arbirary we have esablished (i). We herefore urn our aenion o (ii) by leing R(h) = F (x + h) F (x) F (x)(h) 1 2 F (x)(h, h). Noice ha R is welldefined for h sufficienly small and is Gâeaux differeniable in a neighborhood of h = 0, since F is defined in a neighborhood of x. I is acually Fréche differeniable in his neighborhood, bu we only need Gâeaux differeniabiliy. Since F (x) is a Fréche derivaive, by definiion, given ε > 0 δ > 0 so ha provided h < δ. As before, (iii) of Proposiion 2.12 gives R (h) = F (x + h) F (x) F (x)(h, ) ε h R(h) = R(h) R(0) sup R (θh) h ε h 2 0 θ 1 provided h < δ. Since ε was arbirary we have esablished (ii) and he proposiion. 18
20 9. Higher Order Fréche Derivaives. Le X and Y be normed linear spaces. Suppose f : X Y is Fréche differeniable in D an open subse of X. Le L 1[X, Y ] denoe L[X, Y ] he normed linear space of bounded linear operaors from X ino Y. Recall ha f : D X L 1[X, Y ]. Consider f, he Fréche derivaive of he Fréche derivaive of f. Clearly f : D X L 1 [X, L 1[X, Y ]]. (9.1) In general le L n[x, Y ] denoe L 1[X, L n 1[X, Y ]], n = 2, 3,. Then f (n), n = 2, 3,..., he nh Fréche derivaive of f is by definiion he Fréche derivaive of f (n 1), he (n 1)s Fréche derivaive of f. Clearly f (n) : D X L n[x, Y ]. I is no immediaely obvious how o inerpre he elemens of L n[x, Y ]. The following inerpreaion is very helpful. Recall ha he Caresian produc of wo ses U and V is by definiion U V = {(u, v) : u U, v V }. Also by U 1 we mean U and by U n we mean U U n 1, n = 2, 3,. Clearly U n is a vecor space in he obvious manner whenever U is a vecor space. An operaor K : X n Y is said o be an nlinear operaor from X ino Y if i is linear in each of he n variables, i.e., for real α and β K(x 1,, αx i + βx i,, x n) = αk(x 1,, x i,, x n) + βk(x 1,, x i,, x n), i = 1,, n. The nlinear operaor K is said o be bounded if here exiss M > 0 such ha K(x 1,, x n) M x 1 x 2 x n, for all (x 1,, x n) X n. (9.2) The vecor space of all bounded nlinear operaors from X ino Y becomes a normed linear space if we define K o be he infimum of all M saisfying (9.2). This normed linear space we denoe by [X n, Y ]. Clearly by a 1linear operaor we mean a linear operaor. Also a 2linear operaor is usually called a bilinear operaor or form. The following proposiion shows ha he spaces L n[x, Y ] and [X n, Y ] are essenially he same excep for noaion. Proposiion 9.1. The normed linear spaces L n[x, Y ] and [X n, Y ] are isomerically isomorphic. Proof. These wo spaces are isomorphic if here exiss T n : L n[x, Y ] [X n, Y ] which is linear, oneone, and ono. The isomorphism is an isomery if i is also norm preserving. Clearly Tn 1 will have he same properies. Since L 1[X, Y ] = [X 1, Y ], le T 1 be he ideniy operaor. Assume we have consruced T n 1 wih he desired properies. Define T n as follows. For W L n[x, Y ] le T n(w ) be he nlinear operaor from X ino Y defined by T n(w )(x 1,, x n) = T n 1(W (x 1))(x 2,, x n) for (x 1,, x n) X n. Clearly T n(w ) W ; hence T n(w ) [X n, Y ]. Also T n is linear. If U [X n, Y ], hen for each x X le W (x) = T 1 n 1 (U(x,,, )). I follows ha W : X Ln 1[X, Y ] is linear and W U ; hence W L n[x, Y ] and T n(w ) = U.This shows ha T n is ono and norm preserving. Clearly a linear norm preserving operaor mus be oneone. This proves he proposiion. Remark 9.2. The Proposiion impacs he Fréche derivaive in he following manner. The nh Fréche derivaive of f : X Y a a poin can be viewed as a bounded nlinear operaor from X n ino Y. I is quie saisfying, ha wihou asking for coninuiy of he nh Fréche derivaive, f n : X [X n, Y ] we acually have symmery as is described in he following proposiion. Proposiion 9.3. If f : X Y is n imes Fréche differeniable in an open se D X, hen he nlinear operaor f n (x) is symmeric for each x D. A proof of his symmery proposiion can be found in Chaper 8 of Dieudonné []. There he resul is proved by inducion on n. The inducion process is iniiaed by firs esablishing he resul for n = 2. We will include ha proof because i is insrucive, ingenious, and elegan. However, he version of he proof presened in Dieudonné is difficul o follow. Hence, we presen Orega and Rheinbold s [] adapaion of ha proof. Proposiion 9.4. Suppose ha f : X Y is Fréche differeniable in an open se D X and has a second Fréche derivaive a x D. Then he bilinear operaor f (x) is symmeric. Proof. The second Fréche derivaive is by definiion he Fréche derivaive of he Fréche derivaive. Hence applying (4.2) for second Fréche derivaives we see ha for given ɛ > 0 δ > 0 so ha y D and f (y) f (x) f (x)(y x, ) ɛ x y (9.3) whenever x y < δ. Consider u, v B(0, δ ). The funcion G : [0, 1] Y defined by 2 G() = f(x + u + v) f(x + u) 19
The Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationnonlocal conditions.
ISSN 17493889 prin, 17493897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.39 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More information3 RungeKutta Methods
3 RungeKua Mehods In conras o he mulisep mehods of he previous secion, RungeKua mehods are singlesep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationA NOTE ON THE ALMOST EVERYWHERE CONVERGENCE OF ALTERNATING SEQUENCES WITH DUNFORD SCHWARTZ OPERATORS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. I A OTE O THE ALMOST EVERYWHERE COVERGECE OF ALTERATIG SEQUECES WITH DUFORD SCHWARTZ OPERATORS BY RYOTARO S A T O (OKAYAMA) 1. Inroducion.
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationPATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM
PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu
More informationLectures # 5 and 6: The Prime Number Theorem.
Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζfuncion o skech an argumen which would give an acual formula for π( and sugges
More informationINDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES
Journal of Applied Analysis 1, 1 (1995), pp. 39 45 INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES A. LUCZAK Absrac. We find exponens of independen marginals
More informationForecasting and Information Sharing in Supply Chains Under QuasiARMA Demand
Forecasing and Informaion Sharing in Supply Chains Under QuasiARMA Demand Avi Giloni, Clifford Hurvich, Sridhar Seshadri July 9, 2009 Absrac In his paper, we revisi he problem of demand propagaion in
More informationKeldysh Formalism: Nonequilibrium Green s Function
Keldysh Formalism: Nonequilibrium Green s Funcion Jinshan Wu Deparmen of Physics & Asronomy, Universiy of Briish Columbia, Vancouver, B.C. Canada, V6T 1Z1 (Daed: November 28, 2005) A review of Nonequilibrium
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationMeasuring macroeconomic volatility Applications to export revenue data, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More informationModule 3 Design for Strength. Version 2 ME, IIT Kharagpur
Module 3 Design for Srengh Lesson 2 Sress Concenraion Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More information1 HALFLIFE EQUATIONS
R.L. Hanna Page HALFLIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of halflives, and / log / o calculae he age (# ears): age (halflife)
More informationEntropy: From the Boltzmann equation to the Maxwell Boltzmann distribution
Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationImproper Integrals. Dr. Philippe B. laval Kennesaw State University. September 19, 2005. f (x) dx over a finite interval [a, b].
Improper Inegrls Dr. Philippe B. lvl Kennesw Se Universiy Sepember 9, 25 Absrc Noes on improper inegrls. Improper Inegrls. Inroducion In Clculus II, sudens defined he inegrl f (x) over finie inervl [,
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More information5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.
5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationThe Heisenberg group and Pansu s Theorem
The Heisenberg group and Pansu s Theorem July 31, 2009 Absrac The goal of hese noes is o inroduce he reader o he Heisenberg group wih is Carno Carahéodory meric and o Pansu s differeniaion heorem. As
More informationNiche Market or Mass Market?
Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.
More informationABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION
QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS BY MOGENS STEFFENSEN ABSTRACT Quadraic opimizaion is he classical approach o opimal conrol of pension funds. Usually he paymen sream is approximaed
More informationUSE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES
USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were
More informationI. Basic Concepts (Ch. 14)
(Ch. 14) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationWHAT ARE OPTION CONTRACTS?
WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationAppendix D Flexibility Factor/Margin of Choice Desktop Research
Appendix D Flexibiliy Facor/Margin of Choice Deskop Research Cheshire Eas Council Cheshire Eas Employmen Land Review Conens D1 Flexibiliy Facor/Margin of Choice Deskop Research 2 Final Ocober 2012 \\GLOBAL.ARUP.COM\EUROPE\MANCHESTER\JOBS\200000\22348900\4
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More informationMotion Along a Straight Line
Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 57 85 Barcelona Absrac We see ha he price of an european call opion
More informationDynamic Information. Albina Danilova Department of Mathematical Sciences Carnegie Mellon University. September 16, 2008. Abstract
Sock Marke Insider Trading in Coninuous Time wih Imperfec Dynamic Informaion Albina Danilova Deparmen of Mahemaical Sciences Carnegie Mellon Universiy Sepember 6, 28 Absrac This paper sudies he equilibrium
More informationDistributing Human Resources among Software Development Projects 1
Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources
More informationOn Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations
On Galerkin Approximaions for he Zakai Equaion wih Diffusive and Poin Process Observaions An der Fakulä für Mahemaik und Informaik der Universiä Leipzig angenommene DISSERTATION zur Erlangung des akademischen
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationMultiprocessor SystemsonChips
Par of: Muliprocessor SysemsonChips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationSEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION. Chavchavadze Ave. 17 a, Tbilisi, Georgia, Email: toronj333@yahoo.
SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION N. LAZRIEVA, 2, T. SHARIA 3, 2 AND T. TORONJADZE Georgian American Universiy, Business School, 3, Alleyway II, Chavchavadze Ave.
More information1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,
Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationAP Calculus AB 2010 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College
More informationDifferential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.
Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given
More informationDifferential Equations in Finance and Life Insurance
Differenial Equaions in Finance and Life Insurance Mogens Seffensen 1 Inroducion The mahemaics of finance and he mahemaics of life insurance were always inersecing. Life insurance conracs specify an exchange
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationDifferential Equations and Linear Superposition
Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y
More informationT ϕ t ds t + ψ t db t,
16 PRICING II: MARTINGALE PRICING 2. Lecure II: Pricing European Derivaives 2.1. The fundamenal pricing formula for European derivaives. We coninue working wihin he Black and Scholes model inroduced in
More informationThe Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas
The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he
More informationChapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m
Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m
More informationTwo Compartment Body Model and V d Terms by Jeff Stark
Two Comparmen Body Model and V d Terms by Jeff Sark In a onecomparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics  By his, we mean ha eliminaion is firs order and ha pharmacokineic
More informationSecond Order Linear Differential Equations
Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous
More informationHOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES?
HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Absrac. We compare he opion pricing formulas of Louis Bachelier and BlackMeronScholes
More informationSTABILITY OF LOAD BALANCING ALGORITHMS IN DYNAMIC ADVERSARIAL SYSTEMS
STABILITY OF LOAD BALANCING ALGORITHMS IN DYNAMIC ADVERSARIAL SYSTEMS ELLIOT ANSHELEVICH, DAVID KEMPE, AND JON KLEINBERG Absrac. In he dynamic load balancing problem, we seek o keep he job load roughly
More informationII.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal
Quarerly Repor on he Euro Area 3/202 II.. Deb reducion and fiscal mulipliers The deerioraion of public finances in he firs years of he crisis has led mos Member Saes o adop sizeable consolidaion packages.
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationChapter 2: Principles of steadystate converter analysis
Chaper 2 Principles of SeadySae Converer Analysis 2.1. Inroducion 2.2. Inducor volsecond balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer
More informationInventory Planning with Forecast Updates: Approximate Solutions and Cost Error Bounds
OPERATIONS RESEARCH Vol. 54, No. 6, November December 2006, pp. 1079 1097 issn 0030364X eissn 15265463 06 5406 1079 informs doi 10.1287/opre.1060.0338 2006 INFORMS Invenory Planning wih Forecas Updaes:
More informationTail Distortion Risk and Its Asymptotic Analysis
Tail Disorion Risk and Is Asympoic Analysis Li Zhu Haijun Li May 2 Revision: March 22 Absrac A disorion risk measure used in finance and insurance is defined as he expeced value of poenial loss under a
More informationEfficient Risk Sharing with Limited Commitment and Hidden Storage
Efficien Risk Sharing wih Limied Commimen and Hidden Sorage Árpád Ábrahám and Sarola Laczó March 30, 2012 Absrac We exend he model of risk sharing wih limied commimen e.g. Kocherlakoa, 1996) by inroducing
More informationMarkov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension
Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationChapter 4: Exponential and Logarithmic Functions
Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion
More informationSPECULATIVE DYNAMICS IN THE TERM STRUCTURE OF INTEREST RATES. Abstract
SPECULATIVE DYNAMICS IN THE TERM STRUCTURE OF INTEREST RATES KRISTOFFER P. NIMARK Absrac When long mauriy bonds are raded frequenly and raional raders have nonnesed informaion ses, speculaive behavior
More informationPart 1: White Noise and Moving Average Models
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
More information4 Convolution. Recommended Problems. x2[n] 1 2[n]
4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discreeime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.11.
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationadaptive control; stochastic systems; certainty equivalence principle; longterm
COMMUICATIOS I IFORMATIO AD SYSTEMS c 2006 Inernaional Press Vol. 6, o. 4, pp. 299320, 2006 003 ADAPTIVE COTROL OF LIEAR TIME IVARIAT SYSTEMS: THE BET O THE BEST PRICIPLE S. BITTATI AD M. C. CAMPI Absrac.
More informationVerification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing
MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp. 62 635 issn 364765X eissn 526547 364 62 hp://dx.doi.org/.287/moor..57 2 INFORMS Verificaion Theorems for Models of Opimal Consumpion
More informationSTABILITY OF LOAD BALANCING ALGORITHMS IN DYNAMIC ADVERSARIAL SYSTEMS
SIAM J. COMPUT. Vol. 37, No. 5, pp. 1656 1673 c 2008 Sociey for Indusrial and Applied Mahemaics STABILITY OF LOAD BALANCING ALGORITHMS IN DYNAMIC ADVERSARIAL SYSTEMS ELLIOT ANSHELEVICH, DAVID KEMPE, AND
More informationThe Grantor Retained Annuity Trust (GRAT)
WEALTH ADVISORY Esae Planning Sraegies for closelyheld, family businesses The Granor Reained Annuiy Trus (GRAT) An efficien wealh ransfer sraegy, paricularly in a low ineres rae environmen Family business
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buyside of a forward/fuures
More informationResearch on Inventory Sharing and Pricing Strategy of Multichannel Retailer with Channel Preference in Internet Environment
Vol. 7, No. 6 (04), pp. 365374 hp://dx.doi.org/0.457/ijhi.04.7.6.3 Research on Invenory Sharing and Pricing Sraegy of Mulichannel Reailer wih Channel Preference in Inerne Environmen Hanzong Li College
More information