Real-Time Smoothness Measurements for Portland Cement Concrete Pavements

Total Page:16

File Type:pdf, Size:1020Kb

Real-Time Smoothness Measurements for Portland Cement Concrete Pavements “Moving Advancements into Practice” MAP Brief August 2016 Best practices and promising technologies that can be used now to enhance concrete paving Real-Time Smoothness Measurements for Portland Cement Concrete Pavements www.cproadmap.org August 2016 ROAD MAP TRACK 8 Introduction laptop (figure 2). The Gomaco unit uses Pavement smoothness is one of the most sonic sensors and a dedicated computer PROJECT TITLE important factors affecting user (driver) (figure 3). Both are configured similarly Real-Time Smoothness satisfaction. As far back as the AASHO with sensors mounted to the back of the Measurements for Portland Road Test it has been recognized that road paver to measure the pavement profile Cement Concrete Pavements users judge the quality of a road primarily and send it to the data collection hardware and software for processing and display in TECHNICAL WRITER based on its ride quality. Gary Fick real-time (figure 1). However, initial smoothness of a portland cement concrete pavement (PCCP) also The primary difference between the EDITOR has a direct impact on the life of the pave- systems is the sensor technology used, the Sabrina Shields-Cook ment. According to the findings of Perera, GSI uses acoustic (ultrasonic) sensors and et al. (2005), “… pavements that are built the RTP uses lasers. When mounted to the SPONSORS back of the paver, both systems capture Federal Highway Administration smoother will provide a longer service National Concrete Consortium life before reaching a terminal roughness profile data by measuring the height of value, compared to pavements having a the sensor relative to the fresh pavement MORE INFORMATION lower initial smoothness level.” directly behind the paver (typically 6 in. to Gary Fick 12 in. behind the pan or trailing pan). trinity construction management Therefore, from both a user and a life- services, inc. cycle cost perspective, it is desirable to Both systems use a combination of [email protected] construct smooth PCCP. The use of real- height, slope, and distance data that is time smoothness equipment can assist continuously fed to the software where the contractor in improving the initial it is converted to a real-time profile and smoothness of PCCP. smoothness statistics (IRI, PI, must grinds, and localized roughness). Distance data is collected using a calibrated bicycle wheel, Real-Time Smoothness (RTS) a wheel mounted to a paver track, or an The Long-Term Plan for Concrete internal encoder Pavement Research and Systems Technology (CP Road Map) is a national research plan developed There are currently two systems com- Each of these systems underwent a thor- and jointly implemented by the mercially available for measuring PCCP ough independent evaluation as part of concrete pavement stakeholder community. Publications and smoothness in real-time: Ames Real-Time the SHRP2 R06E project Real-Time Smooth- other support services are Profiler (RTP), and Gomaco Smoothness ness Measurements on Portland Cement provided by the Operations Support Group and funded by the Indicator (GSI). The Ames unit is a laser- Concrete Pavements During Construction Federal Highway Administration. based sensor combined with a ruggedized (Rasumssen et al. 2013). Findings from Moving Advancements into Practice (MAP) Briefs describe innovative research and promising technologies that can be used now to enhance concrete paving practices. The August 2016 MAP Brief provides information relevant to Track 8 of the CP Road Map: Concrete Pavement Construction, Reconstruction, and Overlays. This MAP Brief is available at www.cproadmap. Figure 1. Diagram of a real-time smoothness system org/publications/ MAPbriefAugust2016.pdf. CP Road MAP Brief August 2016 Figure 2. Ames RTP system installed on a paver (from the left: RTP sensors mounted at the back of the paver, computer showing real-time smoothness information, and bicycle wheel collecting distance data) Figure 3. Gomaco GSI system installed on a paver (from the left: GSI sensors mounted at the back of the paver, computer showing real-time smoothness information, and wheel mounted to the paver track collecting distance data) this project concluded that both systems demonstrated ty assurance (acceptance level) smoothness measurements, their value as a quality control (QC) tool for the contractor the pavement must have adequate strength and all saw- in assessing initial pavement smoothness and providing ing must be completed before they can be operated on the real-time feedback for process adjustments. pavement surface, resulting in a minimum 12- to 24-hour delay in the feedback on smoothness numbers. Real-time Based on these findings, the Federal Highway Admin- smoothness systems provide the same profile information istration (FHWA) nominated this technology for SHRP2 as the profilograph and inertial profiler, but in real-time implementation funding. In cooperation with FHWA, The during paving. It should be noted that these systems are National Concrete Pavement Technology Center (CP Tech not intended for, and should not be used for, acceptance Center) has been involved in furthering the implementa- measurements. (See Real-Time IRI vs. Hardened IRI page 4 tion of real-time smoothness by exposing contractors to for further details). the technology through equipment loans and workshops designed to help contractors realize the benefits of this However, having this information in real-time allows the tool for improving the initial smoothness of PCCP. contractor to make process adjustments sooner and allows for corrections to be made during finishing, providing the contractor the opportunity to construct smoother pave- Benefits of Using Real-Time ments. Smoothness Systems The primary process adjustments that can be validated by When properly implemented into the contractor’s paving use of these systems include but are not limited to: operation, real-time smoothness systems provide valu- able feedback that allows the contractor to adjust their • Tuning the paver—there are numerous adjustments and processes to improve the initial smoothness characteristics operational characteristics of slipform pavers that impact (overall smoothness and localized roughness) of the new pavement smoothness (see the following section for PCCP. more detail). Mixture adjustments aimed at improving the overall work- While profilographs and lightweight inertial profilers ability and/or edge stability. have traditionally been used for quality control and quali- In addition, localized roughness caused by major profile 2 CP Road MAP Brief August 2016 events (e.g., loss of vertical control, paver stops, etc.) result- iv. Total mass of cementitious materials. ing in dips and bumps that need to be corrected by hand v. Ratio of supplementary cementitious materials finishing can be identified using real-time smoothness to portland cement. systems. The effectiveness of the correction made by hand finishing is a matter of workmanship, there is no real-time 4. Identify repeating profile features using the power verification for this process. spectral density analysis (PSD) in ProVAL and use the real-time system to mitigate the roughness from these The power of these systems to improve the initial smooth- features. The PSD function of road profiles is a statisti- ness of PCC pavements lies in the timely use of profile infor- cal representation of the importance of various wave- mation. No improvement to smoothness occurs by installing lengths. It provides valuable information regarding a system on a paver; improvements are only possible when what repeating wavelengths are contributing to pave- the crew members embrace the technology and act on the ment roughness. feedback provided in real-time. a. Doweled joints. b. Dumping/Spreading loads. Using Real-Time Smoothness Systems c. CRCP bar supports. Through the experience gained from the equipment loans, the CP Tech Center team has developed recommendations Lessons Learned from Real-Time for contractors who are interested in using these systems. Smoothness Equipment Loans This four step implementation process includes: Six real-time smoothness equipment loans have been 1. Establish a baseline—monitor the process. completed by the CP Tech Center as part of the SHRP2 a. Install a real-time smoothness system. Technology Implementation program. A sampling of the b. Monitor results for 1 to 2 days. lessons learned from utilization of the real-time smooth- c. Keep processes static, but make ordinary adjust ness systems on these projects in Idaho, Iowa, Nebraska, ments (mixture, vibrators, paving speed, head, etc.). Michigan, Pennsylvania, and Texas shows the potential d. Observe typical responses to the ordinary adjustments benefit of utilizing real-time profile feedback. and make notes or add event markers in the RTS. Pennsylvania 2. Eliminate large events—actively utilize the real-time sys- The Gomaco GSI was utilized in this real-time smooth- tem to reduce the impact of major profile features. ness equipment loan in October/November of 2015 on the a. Stringline/stringless interference. northbound lanes of I-81 near Pine Grove, PA (figure 4). b. Paver stops. Paving was 24 ft. wide and consisted of two typical sec- c. Padline issues. tions: 8 in. thick JPCP unbonded overlay and 13 in. thick d. Other mixture or process impacts. JPCP reconstruction sections. 3. Fine-tune the paving process—utilize the real-time feedback when making intentional adjustments to the processes. a. Paver adjustments. i. Maintaining/adjusting concrete head in the grout box. ii.
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]
  • 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.
    [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]