DOCSLIB.ORG

Chapter 16 the Tangent Space and the Notion of Smoothness

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 16 the Tangent Space and the Notion of Smoothness Chapter 16 The tangent space and the notion of smoothness We will always assume K algebraically closed. In this chapter we follow the approach of ˇ n Safareviˇc[S]. We define the tangent space TX,P at a point P of an affine variety X A as ⇢ the union of the lines passing through P and “ touching” X at P .Itresultstobeanaffine n subspace of A .Thenwewillfinda“local”characterizationofTX,P ,thistimeinterpreted as a vector space, the direction of T ,onlydependingonthelocalring :thiswill X,P OX,P allow to define the tangent space at a point of any quasi–projective variety. 16.1 Tangent space to an affine variety Assume first that X An is closed and P = O =(0,...,0). Let L be a line through P :if ⇢ A(a1,...,an)isanotherpointofL,thenageneralpointofL has coordinates (ta1,...,tan), t K.IfI(X)=(F ,...,F ), then the intersection X L is determined by the following 2 1 m \ system of equations in the indeterminate t: F (ta ,...,ta )= = F (ta ,...,ta )=0. 1 1 n ··· m 1 n The solutions of this system of equations are the roots of the greatest common divisor G(t)of the polynomials F1(ta1,...,tan),...,Fm(ta1,...,tan)inK[t], i.e. the generator of the ideal they generate. We may factorize G(t)asG(t)=cte(t ↵ )e1 ...(t ↵ )es ,wherec K, − 1 − s 2 ↵ ,...,↵ =0,e, e ,...,e are non-negative, and e>0ifandonlyifP X L.The 1 s 6 1 s 2 \ number e is by definition the intersection multiplicity at P of X and L. If G(t)is identically zero, then L X and the intersection multiplicity is, by definition, + . ⇢ 1 115 Note that the polynomial G(t)doesn’tdependonthechoiceofthegeneratorsF1,...,Fm of I(X), but only on the ideal I(X)andonL. Definition 16.1.1. The line L is tangent to the variety X at P if the intersection multiplicity of L and X at P is at least 2 (in particular, if L X). The tangent space to ⇢ X at P is the union of the lines that are tangent to X at P ;itisdenotedTP,X. n We will see now that TP,X is an affine subspace of A . Assume that P X: then the 2 polynomials Fi may be written in the form Fi = Li + Gi,whereLi is a homogeneous linear polynomial (possibly zero) and G contains only terms of degree 2. Then i ≥ Fi(ta1,...,tan)=tLi(a1,...,an)+Gi(ta1,...,tan), 2 where the last term is divisible by t .LetL be the line OA,withA =(a1,...,an). We note that the intersection multiplicity of X and L at P is the maximal power of t dividing the greatest common divisor, so L is tangent to X at P if and only if Li(a1,...,an)=0forall i =1,...,m. Therefore the point A belongs to TP,X if and only if L (a ,...,a )= = L (a ,...,a )=0. 1 1 n ··· m 1 n n This shows that TP,X is a linear subspace of A , whose equations are the linear components of the equations defining X. n n Example 16.1.2. (i) TO,An = A ,becauseI(A )=(0). (ii) If X is a hypersurface, with I(X)=(F ), we write as above F = L + G;then n TO,X = V (L): so TO,X is either a hyperplane if L =0,orthewholespaceA if L =0.For 6 3 2 2 2 instance, if X is the affine plane cuspidal cubic V (x y ) A , TO,X = A . − ⇢ Assume now that P X has coordinates (y ,...,y ). With an affine transformation 2 1 n we may translate P to the origin (0,...,0), taking as new coordinates functions on An x y ,...,x y .ThiscorrespondstoconsideringtheK-isomorphism K[x ,...,x ] 1 − 1 n − n 1 n ! K[x y ,...,x y ], which takes a polynomial F (x ,...,x )toitsTaylorexpansion 1 − 1 n − n 1 n G(x y ,...,x y )=F (y ,...,y )+d F + d(2)F + ..., 1 − 1 n − n 1 n P P (i) th where dP F denotes the i di↵erential of F at P :itisahomogeneouspolynomialofdegree i in the variables x y ,...,x y .Inparticularthelineartermis 1 − 1 n − n @F @F dP F = (P )(x1 y1)+ + (P )(xn yn). @x1 − ··· @xn − 116 n We get that, if I(X)=(F1,...,Fm), then TP,X is the affine subspace of A defined by the equations d F = = d F =0. P 1 ··· P m The affine space An,whichmayidentifiedwithKn,canbegivenanaturalstructureof K-vector space with origin P ,soinanaturalwayTP,X is a vector subspace (with origin n P ). The functions x y ,...,x y form a basis of the dual space (K )⇤ and their 1 − 1 n − n restrictions generate TP,X⇤ . Note moreover that dim TP,X =dimTP,X⇤ = k if and only if n k is the maximal number of polynomials linearly independent among d F ,...,d F . − P 1 P m If dP F1,...,dP Fn k are these polynomials, then they form a basis of the orthogonal TP,X? of − n the vector space TP,X in (K )⇤,becausetheyvanishonTP,X. 16.2 Zariski tangent space Let us define now the di↵erential of a regular function. Let f (X)bearegularfunction 2O on X.Wewanttodefinethedi↵erentialoff at P .SinceX is closed in An, f is induced by a polynomial F K[x ,...,x ]aswellasbyallpolynomialsoftheformF + G with 2 1 n G I(X). Fix P X:thend (F + G)=d F + d G so the di↵erentials of two polynomials 2 2 P P P inducing the same function f on X di↵er by the term d G with G I(X). By definition, P 2 dP G is zero along TP,X,sowemaydefinedpf as a regular function on TP,X,thedi↵erential of f at P :itisthefunctiononTP,X induced by dP F .SincedP F is a linear combination of x y ,...,x y , d f can also be seen as an element of T ⇤ . 1 − 1 n − n p P,X There is a natural map d : (X) T ⇤ ,whichsendsf to d f.Becauseoftherules p O ! P,X p of derivation, it is clear that dP (f + g)=dP f + dP g and dP (fg)=f(P )dP g + g(P )dP f.In particular, if c K, d (cf)=cd f.Sod is a linear map of K-vector spaces. We denote 2 p P p again by dP the restriction of dP to IX (P ), the maximal ideal of the regular functions on X which are zero at P .Sinceclearlyf = f(P )+(f f(P )) then d f = d (f f(P )), so this − P P − restriction doesn’t modify the image of the map. 2 Proposition 16.2.1. The map dP : IX (P ) TP,X⇤ is surjective and its kernel is IX (P ) . 2 ! Therefore T ⇤ I (P )/I (P ) as K-vector spaces. P,X ' X X n Proof. Let ' T ⇤ be a linear form on T . ' is the restriction of a linear form on K : 2 P,X P,X λ (x y )+...+ λ (x y ), with λ ,...,λ K.LetG be the polynomial of degree 1 1 1 − 1 n n − n 1 n 2 λ (x y )+...+ λ (x y ): the function g induced by G on X is zero at P and coincides 1 1 − 1 n n − n with its own di↵erential, so ' = dP g and dP is surjective. 117 Let now g IX (P )suchthatdpg =0,g induced by a polynomial G. Note that dP G may 2 n be interpreted as a linear form on K which vanishes on TP,X,henceasanelementofTP,X? . So dP G = c1dpF1 + ...+ cmdpFm (c1,...,cm suitable elements of K). Let us consider the polynomial G c F ... c F :sinceitsdi↵erentialatP is zero, it doesn’t have any term − 1 1 − − m m of degree 0 or 1 in x y ,...,x y ,soitbelongstoI(P )2.SinceG c F ... c F 1 − 1 n − n − 1 1 − − m m defines the function g on X,weconcludethatg I (P )2. 2 X ⇤ 2 Corollary 16.2.2. The tangent space TP,X is isomorphic to (IX (P )/IX (P ) )⇤ as an abstract K-vector space. Corollary 16.2.3. Let ' : X Y be an isomorphism of affine varieties and P X, ! 2 Q = '(P ). Then the tangent spaces TP,X and TQ,Y are isomorphic. Proof. ' induces the comorphism '⇤ : (Y ) (X), which results to be an isomorphism O2 !O2 such that '⇤IY (Q)=IX (P )and'⇤IY (Q) = IX (P ) .Sothereisaninducedhomomorphism I (Q)/I (Q)2 I (P )/I (P )2. Y Y ! X X which is an isomorphism of K-vector spaces. By dualizing we get the claim. ⇤ The above map from TP,X to TQ,Y is called the di↵erential of ' at P and is denoted by dP '. Now we would like to find a “more local” characterization of TP,X.Tothisendweconsider the local ring of P in X: .Werecallthenaturalmap (X) = (X) ,the OP,X O !OP,X O IX (P ) last one being the localization. It is natural to extend the map d : (X) T ⇤ to P O ! P,X OP,X setting f g(P )d f f(P )d g d = P − P . P g g(P )2 ⇣ ⌘ As in the proof of Proposition 16.2.1 one proves that the map dP : P,X TP,X⇤ induces 2 O ! an isomorphism P,X/ P,X TP,X⇤ ,where P,X is the maximal ideal of P,X.Soby M M ! 2 M O duality we have: T ( / )⇤.ThisprovesthatthetangentspaceT is a local P,X ' MP,X MP,X P,X invariant of P in X.
Load more

Recommended publications
  • High Energy and Smoothness Asymptotic Expansion of the Scattering Amplitude
    HIGH ENERGY AND SMOOTHNESS ASYMPTOTIC EXPANSION OF THE SCATTERING AMPLITUDE D.Yafaev Department of Mathematics, University Rennes-1, Campus Beaulieu, 35042, Rennes, France (e-mail : [email protected]) Abstract We find an explicit expression for the kernel of the scattering matrix for the Schr¨odinger operator containing at high energies all terms of power order. It turns out that the same expression gives a complete description of the diagonal singular- ities of the kernel in the angular variables. The formula obtained is in some sense universal since it applies both to short- and long-range electric as well as magnetic potentials. 1. INTRODUCTION d−1 d−1 1. High energy asymptotics of the scattering matrix S(λ): L2(S ) → L2(S ) for the d Schr¨odinger operator H = −∆+V in the space H = L2(R ), d ≥ 2, with a real short-range potential (bounded and satisfying the condition V (x) = O(|x|−ρ), ρ > 1, as |x| → ∞) is given by the Born approximation. To describe it, let us introduce the operator Γ0(λ), −1/2 (d−2)/2 ˆ 1/2 d−1 (Γ0(λ)f)(ω) = 2 k f(kω), k = λ ∈ R+ = (0, ∞), ω ∈ S , (1.1) of the restriction (up to the numerical factor) of the Fourier transform fˆ of a function f to −1 −1 the sphere of radius k. Set R0(z) = (−∆ − z) , R(z) = (H − z) . By the Sobolev trace −r d−1 theorem and the limiting absorption principle the operators Γ0(λ)hxi : H → L2(S ) and hxi−rR(λ + i0)hxi−r : H → H are correctly defined as bounded operators for any r > 1/2 and their norms are estimated by λ−1/4 and λ−1/2, respectively.
    [Show full text]
  • Bertini's Theorem on Generic Smoothness
    U.F.R. Mathematiques´ et Informatique Universite´ Bordeaux 1 351, Cours de la Liberation´ Master Thesis in Mathematics BERTINI1S THEOREM ON GENERIC SMOOTHNESS Academic year 2011/2012 Supervisor: Candidate: Prof.Qing Liu Andrea Ricolfi ii Introduction Bertini was an Italian mathematician, who lived and worked in the second half of the nineteenth century. The present disser- tation concerns his most celebrated theorem, which appeared for the first time in 1882 in the paper [5], and whose proof can also be found in Introduzione alla Geometria Proiettiva degli Iperspazi (E. Bertini, 1907, or 1923 for the latest edition). The present introduction aims to informally introduce Bertini’s Theorem on generic smoothness, with special attention to its re- cent improvements and its relationships with other kind of re- sults. Just to set the following discussion in an historical perspec- tive, recall that at Bertini’s time the situation was more or less the following: ¥ there were no schemes, ¥ almost all varieties were defined over the complex numbers, ¥ all varieties were embedded in some projective space, that is, they were not intrinsic. On the contrary, this dissertation will cope with Bertini’s the- orem by exploiting the powerful tools of modern algebraic ge- ometry, by working with schemes defined over any field (mostly, but not necessarily, algebraically closed). In addition, our vari- eties will be thought of as abstract varieties (at least when over a field of characteristic zero). This fact does not mean that we are neglecting Bertini’s original work, containing already all the rele- vant ideas: the proof we shall present in this exposition, over the complex numbers, is quite close to the one he gave.
    [Show full text]
  • Jia 84 (1958) 0125-0165
    INSTITUTE OF ACTUARIES A MEASURE OF SMOOTHNESS AND SOME REMARKS ON A NEW PRINCIPLE OF GRADUATION BY M. T. L. BIZLEY, F.I.A., F.S.S., F.I.S. [Submitted to the Institute, 27 January 19581 INTRODUCTION THE achievement of smoothness is one of the main purposes of graduation. Smoothness, however, has never been defined except in terms of concepts which themselves defy definition, and there is no accepted way of measuring it. This paper presents an attempt to supply a definition by constructing a quantitative measure of smoothness, and suggests that such a measure may enable us in the future to graduate without prejudicing the process by an arbitrary choice of the form of the relationship between the variables. 2. The customary method of testing smoothness in the case of a series or of a function, by examining successive orders of differences (called hereafter the classical method) is generally recognized as unsatisfactory. Barnett (J.1.A. 77, 18-19) has summarized its shortcomings by pointing out that if we require successive differences to become small, the function must approximate to a polynomial, while if we require that they are to be smooth instead of small we have to judge their smoothness by that of their own differences, and so on ad infinitum. Barnett’s own definition, which he recognizes as being rather vague, is as follows : a series is smooth if it displays a tendency to follow a course similar to that of a simple mathematical function. Although Barnett indicates broadly how the word ‘simple’ is to be interpreted, there is no way of judging decisively between two functions to ascertain which is the simpler, and hence no quantitative or qualitative measure of smoothness emerges; there is a further vagueness inherent in the term ‘similar to’ which would prevent the definition from being satisfactory even if we could decide whether any given function were simple or not.
    [Show full text]
  • Lecture 2 — February 25Th 2.1 Smooth Optimization
    Statistical machine learning and convex optimization 2016 Lecture 2 — February 25th Lecturer: Francis Bach Scribe: Guillaume Maillard, Nicolas Brosse This lecture deals with classical methods for convex optimization. For a convex function, a local minimum is a global minimum and the uniqueness is assured in the case of strict convexity. In the sequel, g is a convex function on Rd. The aim is to find one (or the) minimum θ Rd and the value of the function at this minimum g(θ ). Some key additional ∗ ∗ properties will∈ be assumed for g : Lipschitz continuity • d θ R , θ 6 D g′(θ) 6 B. ∀ ∈ k k2 ⇒k k2 Smoothness • d 2 (θ , θ ) R , g′(θ ) g′(θ ) 6 L θ θ . ∀ 1 2 ∈ k 1 − 2 k2 k 1 − 2k2 Strong convexity • d 2 µ 2 (θ , θ ) R ,g(θ ) > g(θ )+ g′(θ )⊤(θ θ )+ θ θ . ∀ 1 2 ∈ 1 2 2 1 − 2 2 k 1 − 2k2 We refer to Lecture 1 of this course for additional information on these properties. We point out 2 key references: [1], [2]. 2.1 Smooth optimization If g : Rd R is a convex L-smooth function, we remind that for all θ, η Rd: → ∈ g′(θ) g′(η) 6 L θ η . k − k k − k Besides, if g is twice differentiable, it is equivalent to: 0 4 g′′(θ) 4 LI. Proposition 2.1 (Properties of smooth convex functions) Let g : Rd R be a con- → vex L-smooth function. Then, we have the following inequalities: L 2 1.
    [Show full text]
  • Convergence Theorems for Gradient Descent
    Convergence Theorems for Gradient Descent Robert M. Gower. October 5, 2018 Abstract Here you will find a growing collection of proofs of the convergence of gradient and stochastic gradient descent type method on convex, strongly convex and/or smooth functions. Important disclaimer: Theses notes do not compare to a good book or well prepared lecture notes. You should only read these notes if you have sat through my lecture on the subject and would like to see detailed notes based on my lecture as a reminder. Under any other circumstances, I highly recommend reading instead the first few chapters of the books [3] and [1]. 1 Assumptions and Lemmas 1.1 Convexity We say that f is convex if d f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y); 8x; y 2 R ; t 2 [0; 1]: (1) If f is differentiable and convex then every tangent line to the graph of f lower bounds the function values, that is d f(y) ≥ f(x) + hrf(x); y − xi ; 8x; y 2 R : (2) We can deduce (2) from (1) by dividing by t and re-arranging f(y + t(x − y)) − f(y) ≤ f(x) − f(y): t Now taking the limit t ! 0 gives hrf(y); x − yi ≤ f(x) − f(y): If f is twice differentiable, then taking a directional derivative in the v direction on the point x in (2) gives 2 2 d 0 ≥ hrf(x); vi + r f(x)v; y − x − hrf(x); vi = r f(x)v; y − x ; 8x; y; v 2 R : (3) Setting y = x − v then gives 2 d 0 ≤ r f(x)v; v ; 8x; v 2 R : (4) 1 2 d The above is equivalent to saying the r f(x) 0 is positive semi-definite for every x 2 R : An analogous property to (2) holds even when the function is not differentiable.
    [Show full text]
  • Online Methods in Machine Learning: Recitation 1
    Online Methods in Machine Learning: Recitation 1. As seen in class, many offline machine learning problems can be written as: n 1 X min `(w, (xt, yt)) + λR(w), (1) w∈C n t=1 where C is a subset of Rd, R is a penalty function, and ` is the loss function that measures how well our predictor, w, explains the relationship between x and y. Example: Support Vector λ 2 w Machines with R(w) = ||w|| and `(w, (xt, yt)) = max(0, 1 − ythw, xti) where h , xti is 2 2 ||w||2 the signed distance to the hyperplane {x | hw, xi = 0} used to classify the data. A couple of remarks: 1. In the context of classification, ` is typically a convex surrogate loss function in the sense that (i) w → `(w, (x, y)) is convex for any example (x, y) and (ii) `(w, (x, y)) is larger than the true prediction error given by 1yhw,xi≤0 for any example (x, y). In general, we introduce this surrogate loss function because it would be difficult to solve (1) computationally. On the other hand, convexity enables the development of general- purpose methods (e.g. Gradient Descent) that are fast, easy to implement, and simple to analyze. Moreover, the techniques and the proofs generalize very well to the online setting, where the data (xt, yt) arrive sequentially and we have to make predictions online. This theme will be explored later in the class. 2. If ` is a surrogate loss for a classification problem and we solve (1) (e.g with Gradient Descent), we cannot guarantee that our predictor will perform well with respect to our original classification problem n X min 1yhw,xi≤0 + λR(w), (2) w∈C t=1 hence it is crucial to choose a surrogate loss function that reflects accurately the classifi- cation problem at hand.
    [Show full text]
  • Be a Smooth Manifold, and F : M → R a Function
    LECTURE 3: SMOOTH FUNCTIONS 1. Smooth Functions Definition 1.1. Let (M; A) be a smooth manifold, and f : M ! R a function. (1) We say f is smooth at p 2 M if there exists a chart f'α;Uα;Vαg 2 A with −1 p 2 Uα, such that the function f ◦ 'α : Vα ! R is smooth at 'α(p). (2) We say f is a smooth function on M if it is smooth at every x 2 M. −1 Remark. Suppose f ◦ 'α is smooth at '(p). Let f'β;Uβ;Vβg be another chart in A with p 2 Uβ. Then by the compatibility of charts, the function −1 −1 −1 f ◦ 'β = (f ◦ 'α ) ◦ ('α ◦ 'β ) must be smooth at 'β(p). So the smoothness of a function is independent of the choice of charts in the given atlas. Remark. According to the chain rule, it is easy to see that if f : M ! R is smooth at p 2 M, and h : R ! R is smooth at f(p), then h ◦ f is smooth at p. We will denote the set of all smooth functions on M by C1(M). Note that this is a (commutative) algebra, i.e. it is a vector space equipped with a (commutative) bilinear \multiplication operation": If f; g are smooth, so are af + bg and fg; moreover, the multiplication is commutative, associative and satisfies the usual distributive laws. 1 n+1 i n Example. Each coordinate function fi(x ; ··· ; x ) = x is a smooth function on S , since ( 2yi −1 1 n 1+jyj2 ; 1 ≤ i ≤ n fi ◦ '± (y ; ··· ; y ) = 1−|yj2 ± 1+jyj2 ; i = n + 1 are smooth functions on Rn.
    [Show full text]
  • Notes on Smooth Manifolds
    UNIVERSITY OF SÃO PAULO Notes on Smooth Manifolds CLAUDIO GORODSKI December ii TO MY SON DAVID iv Foreword The concept of smooth manifold is ubiquitous in Mathematics. Indeed smooth manifolds appear as Riemannian manifolds in differential geom- etry, space-times in general relativity, phase spaces and energy levels in mechanics, domains of definition of ODE’s in dynamical systems, Riemann surfaces in the theory of complex analytic functions, Lie groups in algebra and geometry..., to name a few instances. The notion took some time to evolve until it reached its present form in H. Whitney’s celebrated Annals of Mathematics paper in 1936 [Whi36]. Whitney’s paper in fact represents a culmination of diverse historical de- velopments which took place separately, each in a different domain, all striving to make the passage from the local to the global. From the modern point of view, the initial goal of introducing smooth manifolds is to generalize the methods and results of differential and in- tegral calculus, in special, the inverse and implicit function theorems, the theorem on existence, uniqueness and regularity of ODE’s and Stokes’ the- orem. As usual in Mathematics, once introduced such objects start to atract interest on their own and new structure is uncovered. The subject of dif- ferential topology studies smooth manifolds per se. Many important results about the topology of smooth manifolds were obtained in the 1950’s and 1960’s in the high dimensional range. For instance, there exist topological manifolds admitting several non-diffeomorphic smooth structures (Milnor, 1956, in the case of S7), and there exist topological manifolds admitting no smooth strucuture at all (Kervaire, 1961).
    [Show full text]
  • PN 470 - 4/20/2007 – Thin Lift Asphalt Surface Smoothness Requirements
    PN 470 - 4/20/2007 – Thin Lift Asphalt Surface Smoothness Requirements Add the following requirements to 401.19: Determine the Profile Index and International Roughness Index (IRI) for each 0.10-mile (0.16- km) section of each asphalt pavement lane. Use equipment and computer printouts conforming to Supplement 1058 and that can provide both Profile Index (PI) and International Roughness Index (IRI) measurements. Notify the Engineer before beginning profile testing. Remove any dirt and debris from the pavement surface and then test the pavement surface in both wheel paths of the lane. The wheel paths are parallel to the centerline of the pavement and approximately 3.0 feet (1.0 m) measured transversely, from both lane edges. Provide all traffic control and survey stationing needed to perform the smoothness measurements. Develop the PI trace in accordance with California Test 526, 1978, using a 0.2-inch blanking band and the IRI in accordance with ASTM E1926 for each pavement section. Provide the Engineer with the two (2) copies of traces and calculations for determining pay. Include electronic copies of all longitudinal pavement profiles in the University of Michigan Transportation Research Institute’s Engineering Research Division [ERD] format. The Engineer will send a copy of all traces, electronic ERD files, evaluation results, and the project identification to the Pavement Management Section of the Office of Pavement Engineering. The Department will determine the unit price adjustment of the completed asphalt concrete pavement as follows: Pay Schedule Profile Index Price Adjust. Mean International (inches per mile Percent Roughness Index per 0.10 mile (1) (inches per mile) section) (IRI per 0.10 mile section) 1 or less 105 45 or less Over 1 to 2 103 Over 45 to 50 Over 2 to 3 102 Over 50 to 55 Over 3 to 4 101 Over 55 to 60 Over 4 100 Over 60 (1) The Department will determine the price adjustment percent using the IRI of the lane.
    [Show full text]
  • Integration on Manifolds
    CHAPTER 4 Integration on Manifolds §1 Introduction Until now we have studied manifolds from the point of view of differential calculus. The last chapter introduces to our study the methods of integraI calculus. As the new tools are developed in the next sections, the reader may be somewhat puzzied about their reievancy to the earlierd material. Hopefully, the puzziement will be resolved by the chapter's en , but a pre­ liminary ex ampIe may be heIpful. Let p(z) = zm + a1zm-1 + + a ... m n, be a complex polynomial on p.a smooth compact region in the pIane whose boundary contains no zero of In Sectionp 3 of the previous chapter we showed that the number of zeros of inside n, counting multiplicities, equals the degree of the map argument principle, A famous theorem in complex variabie theory, called the asserts that this may also be calculated as an integraI, f d(arg p). òO 151 152 CHAPTER 4 TNTEGRATION ON MANIFOLDS You needn't understand the precise meaning of the expression in order to appreciate our point. What is important is that the number of zeros can be computed both by an intersection number and by an integraI formula. r Theo ems like the argument principal abound in differential topology. We shall show that their occurrence is not arbitrary orStokes fo rt uitoustheorem. but is a geometrie consequence of a generaI theorem known as Low­ dimensionaI versions of Stokes theorem are probably familiar to you from calculus : l. Second Fundamental Theorem of Calculus: f: f'(x) dx = I(b) -I(a).
    [Show full text]
  • The Smoothing Effect of Integration in R and the ANOVA Decomposition
    Wegelerstraße 6 • 53115 Bonn • Germany phone +49 228 73-3427 • fax +49 228 73-7527 www.ins.uni-bonn.de Michael Griebel, Frances Y. Kuo, and Ian H. Sloan The Smoothing Effect of integration in Rd and the ANOVA Decomposition INS Preprint No. 1007 December 2010 The smoothing effect of integration in Rd and the ANOVA decomposition Michael Griebel,∗ Frances Y. Kuo,y and Ian H. Sloanz May 2011 Abstract This paper studies the ANOVA decomposition of a d-variate function f defined on the whole of Rd, where f is the maximum of a smooth function and zero (or f could be the absolute value of a smooth function). Our study is motivated by option pricing problems. We show that under suitable conditions all terms of the ANOVA decomposition, except the one of highest order, can have unlimited smoothness. In particular, this is the case for arithmetic Asian options with both the standard and Brownian bridge constructions of the Brownian motion. 2010 Mathematics Subject Classification: Primary 41A63, 41A99. Secondary 65D30. Keywords: ANOVA decomposition, smoothing, option pricing. 1 Introduction In this paper we study the ANOVA decomposition of d-variate real-valued functions f defined on the whole of Rd, where f fails to be smooth because it is the maximum of a smooth function and zero. That is, we consider d f(x) = ϕ(x)+ := max(ϕ(x); 0); x 2 R ; (1) with ϕ a smooth function on Rd. The conclusions will apply equally to the absolute value of ϕ, since jϕ(x)j = ϕ(x)+ + (−ϕ(x))+: Our study is motivated by option pricing problems, which take the form of (1) because a financial option is considered to be worthless once its value drops below a specified `strike price'.
    [Show full text]
  • Smoothness, Low-Noise and Fast Rates
    Smoothness, Low-Noise and Fast Rates Nathan Srebro Karthik Sridharan Ambuj Tewari Toyota Technological Institute at Chicago 6045 S Kenwood Ave., Chicago, IL 60637 October 29, 2010 Abstract p ~ 2 ∗ We establish an excess risk bound of O HRn + HL Rn for ERM with an H-smooth loss function ∗ and a hypothesis class with Rademacher complexity Rn, where L is the best risk achievable by the p hypothesis class. For typical hypothesis classes where Rn = R=n, this translates to a learning rate of O~ (RH=n) in the separable (L∗ = 0) case and O~ RH=n + pL∗RH=n more generally. We also provide similar guarantees for online and stochastic convex optimization of a smooth non-negative objective. 1 Introduction Consider empirical risk minimization for a hypothesis class H = fh : X! Rg w.r.t. some non-negative loss function φ(t; y). That is, we would like to learn a predictor h with small risk L (h) = E [φ(h(X);Y )] by minimizing the empirical risk n 1 X L^(h) = φ(h(x ); y ) n i i i=1 of an i.i.d. sample (x1; y1);:::; (xn; yn). Statistical guarantees on the excess risk are well understood for parametric (i.e. finite dimensional) hypothesis classes. More formally, these are hypothesis classes with finite VC-subgraph dimension [24] (aka pseudo- dimension). For such classes learning guarantees can be obtained for any bounded loss function (i.e. s.t. jφj ≤ b < 1) and the relevant measure of complexity is the VC-subgraph dimension. Alternatively, even for some non-parametric hypothesis classes (i.e.
    [Show full text]