Composition of Functions

Int. J. Med. Sci. 2007, 4 45International Journal of Medical Sciences ISSN 1449-1907 www.medsci.org 2007 4(1):45-52 © Ivyspring International Publisher. All rights reserved Research Paper A Dietary Supplement Containing Standardized Phaseolus vulgaris Extract Influences Body Composition of Overweight Men and Women Leonardo Celleno 1, Maria Vittoria Tolaini 1, Alessandra D’Amore 1, Nicholas V. Perricone 2, Harry G. Preuss 3 1. Cosmetic Research Center, dell’Università Cattolica di Roma, Rome, Italy 2. NV Perricone, MD, Ltd, Meriden, CT 06450, USA 3. Georgetown University Medical Center, Department of Physiology, Washington, DC 20057, USA Correspondence to: Harry G. Preuss MD, Professor of Physiology, Medicine, & Pathology, Basic Science Building, Room 231B, Georgetown University Medical Center, Washington, DC 20057. preusshg@georgetown.edu Received: 2006.12.13; Accepted: 2007.01.23; Published: 2007.01.24 Background: More than one billion human adults worldwide are overweight and, therefore, are at higher risk of developing cardiovascular diseases, diabetes, and a variety of other chronic perturbations. Many believe that use of natural dietary supplements could aid in the struggle against obesity. So-called "starch blockers" are listed among natural weight loss supplements. Theoretically, they may promote weight loss by interfering with the breakdown of complex carbohydrates thereby reducing, or at least slowing, the digestive availability of carbo-hydrate-derived calories and/or by providing resistant starches to the lower gastrointestinal tract. Aims: The present research study examines a dietary supplement containing 445 mg of Phaseolus vulgaris extract derived from the white kidney bean, previously shown to inhibit the activity of the digestive enzyme alpha amylase, on body composition of overweight human subjects. Methods: A randomized, double-blinded, placebo-controlled study was conducted on 60 pre-selected, slightly overweight volunteers, whose weight had been essentially stable for at least six months. The volunteers were divided into two groups, homogeneous for age, gender, and body weight. The test product containing Phaseolus vulgaris extract and the placebo were taken one tablet per day for 30 consecutive days before a main meal rich in carbohydrates. Each subject’s body weight, fat and non-fat mass, skin fold thickness, and waist/hip/thigh cir-cumferences were measured. Results: After 30 days, subjects receiving Phaseolus vulgaris extract with a carbohydrate-rich, 2000- to 2200-calorie diet had significantly (p<0.001) greater reduction of body weight, BMI, fat mass, adipose tissue thickness, and waist,/hip/ thigh circumferences while maintaining lean body mass compared to subjects receiving placebo. Conclusion: The results indicate that Phaseolus vulgaris extract produces significant decrements in body weight and suggest decrements in fat mass in the face of maintained lean body mass. Key words: starch blockers, weight loss, obesity, amylase inhibitors, bean extract 1. Introduction Excess accumulation of body fat (over-weight/obesity), a chronic disequilibrium between food consumption and energy expenditure, is becom-ing noticeably more prevalent [1-4]. This is unfortu-nate for more reasons than just poor physical appear-ance, because the overweight/obesity states increase the risk of hypertension, type II diabetes, arthritis, elevated Composition of Functions Composition of Functions By: OpenStaxCollege Suppose we want to calculate how much it costs to heat a house on a particular day of the year The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day Using descriptive variables, we can notate these two functions The function C(T) gives the cost C of heating a house for a given average daily temperature in T degrees Celsius The function T(d) gives the average daily temperature on day d of the year For any given day, Cost = C(T(d)) means that the cost depends on the temperature, which in turns depends on the day of the year Thus, we can evaluate the cost function at the temperature T(d) For example, we could evaluate T(5) to determine the average daily temperature on the 5th day of the year Then, we could evaluate the cost function at that temperature We would write C(T(5)) By combining these two relationships into one function, we have performed function composition, which is the focus of this section Combining Functions Using Algebraic Operations Function composition is only one way to combine existing functions Another way is to carry out the usual algebraic operations on functions, such as addition, subtraction, multiplication and division We this by performing the operations with the function outputs, defining the result as the output of our new function 1/29 Composition of Functions Suppose we need to add two columns of numbers that represent a husband and wife’s separate annual incomes over a period of years, with the result being their total household income We want to this for every year, adding only that year’s incomes and then collecting all the data in a new column If w(y) is the wife’s income and h(y) is the husband’s income in year y, and we want T to represent the total income, then we can define a new function T(y) = h(y) + w(y) If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write T=h+w Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which have to be numbers so that the usual operations of algebra can apply to them, and which also must have the same units or no units when we add and subtract) In this way, we can think of adding, subtracting, multiplying, and dividing functions For two functions f(x) and g(x) with real number outputs, we define new functions f f + g, f − g, fg, and g by the relations (f + g)(x) = f(x) + g(x) (f − g)(x) = f(x) − g(x) (fg)(x) = f(x)g(x) f(x) ( gf )(x) = g(x) Performing Algebraic Operations on Functions Find and simplify the functions (g − f)(x) and ( gf )(x), given f(x) = x − and g(x) = x2 − Are they the same function? Begin by writing the general form, and then substitute the given functions 2/29 Composition of Functions (g − f)(x) = g(x) − f(x) (g − f)(x) = x2 − − (x − 1) = x2 − x = x(x − 1) ( gf )(x) = g(x) f(x) ( gf )(x) = xx −− 11 = (x + 1)(x − 1) x−1 where x ≠ =x+1 No, the functions are not the same (g) Note: For f (x), the condition x ≠ is necessary because when x = 1, the denominator is equal to 0, which makes the function undefined Try It Find and simplify the functions (fg)(x) and (f − g)(x) f (x ) = x − and g(x) = x2 − Are they the same function? (fg)(x) = f(x)g(x) = (x − 1)(x2 − 1) = x3 − x2 − x + (f − g)(x) = f(x) − g(x) = (x − 1) − (x2 − 1) = x − x2 No, the functions are not the same Create a Function by Composition of Functions Performing algebraic operations on functions combines them into a new function, but we can also create functions by composing functions When we wanted to compute a heating cost from a day of the year, we created a new function that takes a day as input and yields a cost as output The process of combining functions so that the output of 3/29 Composition of Functions one function becomes the input of another is known as a composition of functions The resulting function is known as a composite function We represent this combination by the following notation: (f ∘ g)(x) = f(g(x)) We read the left-hand side as“f composed with g at x ,” and the right-hand side as“f of g of x.”The two sides of the equation have the same mathematical meaning and are equal The open circle symbol ∘ is called the composition operator We use this operator mainly when we wish to emphasize the relationship between the functions themselves without referring to any particular input value Composition is a binary operation that takes two functions and forms a new function, much as addition or multiplication takes two numbers and gives a new number However, it is important not to confuse function composition with ... Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). Chapter 4. Integration of Functions 4.0 Introduction Numerical integration,which is also called quadrature, has a history extending back to the invention of calculus and before. The fact that integrals of elementary functions could not, in general, be computed analytically, while derivatives could be, served to give the field a certain panache, and to set it a cut above the arithmetic drudgery of numerical analysis during the whole of the 18th and 19th centuries. With the invention of automatic computing, quadrature became just one numer- ical task among many, and not a very interesting one at that. Automatic computing, even the most primitivesort involvingdesk calculators and roomsfull of “computers” (that were, until the 1950s, people rather than machines), opened to feasibility the much richer field of numerical integration of differential equations. Quadrature is merely the simplest special case: The evaluation of the integral I =  b a f(x)dx (4.0.1) is precisely equivalent to solving for the value I ≡ y(b) the differential equation dy dx = f(x)(4.0.2) with the boundary condition y(a)=0 (4.0.3) Chapter 16 of this book deals with the numerical integration of differential equations. In that chapter, much emphasis is given to the concept of “variable” or “adaptive” choices of stepsize. We will not, therefore, develop that material here. If the function that you propose to integrate is sharply concentrated in one or more peaks, or if its shape is not readily characterized by a single length-scale, then it is likely that you should cast the problem in the form of (4.0.2)–(4.0.3) and use the methods of Chapter 16. The quadrature methods in this chapter are based, in one way or another, on the obvious device of adding up the value of the integrand at a sequence of abscissas within the range of integration. The game is to obtain the integral as accurately as possible with the smallest number of function evaluations of the integrand. Just as in the case of interpolation (Chapter 3), one has the freedom to choose methods 129 130 Chapter 4. Integration of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). of various orders, with higher order sometimes, but not always, giving higher accuracy. “Romberg integration,” which is discussed in §4.3, is a general formalism for making use of integration methods of a variety of different orders, and we recommend it highly. Apart from the methods of this chapter and of Chapter 16, there are yet other 130 Chapter 4. Integration of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). of various orders, with higher order sometimes, but not always, giving higher accuracy. “Romberg integration,” which is discussed in §4.3, is a general formalism for making use of integration methods of a variety of different orders, and we recommend it highly. Apart from the methods of this chapter and of Chapter 16, there are yet other methods for obtaining integrals. One important class is based on function approximation. We discuss explicitly the integration of functions by Chebyshev approximation (“Clenshaw-Curtis” quadrature) in §5.9. Although not explicitly discussed here, you ought to be able to figure out how to do cubic spline quadrature using the output of the routine spline in §3.3. (Hint: Integrate equation 3.3.3 over x analytically. See [1] .) Some integrals related to Fourier transforms can be calculated using the fast Fourier transform (FFT) algorithm. This is discussed in §13.9. Multidimensional integrals are another whole multidimensional bag of worms. Section 4.6 is an introductory discussion in this chapter; the important technique of Monte-Carlo integration is treated in Chapter 7. CITED REFERENCES AND FURTHER READING: Carnahan, B., Luther, H.A., and Wilkes, J.O. 1969, Applied Numerical Methods (New York: Wiley), Chapter 2. Isaacson, E., and Keller, H.B. 1966, Analysis of Numerical Methods (New York: Wiley), Chapter 7. Acton, F.S. 1970, Numerical Methods That Work ; 1990, corrected edition (Washington: Mathe- matical Association of America), Chapter 4. Stoer, J., and Bulirsch, R. 1980, Introduction to Numerical Analysis (New York: Springer-Verlag), Chapter 3. Ralston, A., and Rabinowitz, P. 1978, A First Course in Numerical Analysis , 2nd ed. (New York: McGraw-Hill), Chapter 4. Dahlquist, G., and Bjorck, A. 1974, Numerical Methods (Englewood Cliffs, NJ: Prentice-Hall), § 7.4. Kahaner, D., Moler, C., and Nash, S. 1989, Numerical Methods and Software (Englewood Cliffs, NJ: Prentice Hall), Chapter 5. Forsythe, G.E., Malcolm, M.A., and Moler, C.B. 1977, Computer Methods for Mathematical Computations (Englewood Cliffs, NJ: Prentice-Hall), § 5.2, p. 89. [1] Davis, P., and Rabinowitz, P. 1984, Methods of Numerical Integration , 2nd ed. (Orlando, FL: Academic Press). 4.1 Classical Formulas for Equally Spaced Abscissas Where would any book on numerical analysis be without Mr. Simpson and his “rule”? The classical formulas for integrating a function whose value is known at equally spaced steps have a certain elegance about them, and they are redolent with historical association. Through them, the modern numerical analyst communes with the spirits of his or her predecessors back across the centuries, as far as the time of Newton, if not farther. Alas, times do change; with the exception of two of the most modest formulas (“extended trapezoidal rule,” equation 4.1.11, and “extended 4.1 Classical Formulas for Equally Spaced Abscissas 131 Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is Description of Functions Description of Functions 11.2003 Edition sinumerik SINUMERIK 802D SINUMERIK 802D base line Preface, Table of Contents EMERGENCY STOP (N2) 1 Axis Monitoring (A3) 2 Velocities, Setpoint/Actual-Value Systems, Closed-Loop Control (G2) 3 Acceleration (B2) 4 Spindle (S1) 5 Rotary Axes (R2) 6 Transverse Axes (P1) 7 Reference Point Approach (R1) 8 Manual Traversing and Hand- wheel Traversing (H1) 9 Operating Modes, Program Mode (K1) 10 Feed (V1) 11 Continous Path Mode, Exact Stop and LookAhead (B1) 12 Output of Auxiliary Functions to the PLC (H2) 13 Tool Compensation (W1) 14 Measuring (M5) 15 Compensation (K3) 16 Traversing to Fixed Stop (F1) 17 Kinematic Transformations (M1) 18 Various Interface Signals (A2) 19 PLC User Interface 20 Various Machine Data 21 Glossary, Index 11.03 Edition SINUMERIK 802D Description of Functions SINUMERIK  Documentation Printing history Brief details of this edition and previous edition are listed below. The status of each edition is shown by the code in the “Remarks” column. Status code in the “Remarks” column: A New documentation . . . . B Unrevised reprint with new Order No . . . . C Revised edition with new status. . . . . . If actual changes have been made on the page since the last edition, this is indicated by a new edition coding in the header on that page. Edition Order No. Remarks 11.00 6FC5 697-2AA10-0BP0 A 10.02 6FC5 697-2AA10-0BP1 C 11.03 6FC5 697-2AA10-0BP2 C This Manual is included on the documentation on CD ROM (DOCONCD) E Trademarks SIMATICr, SIMATIC HMIr, SIMATIC NETr, SIROTECr, SINUMERIKr and SIMODRIVEr are registered trademarks of Siemens. Third parties using for their own purpose any other names in this document which refer to trademarks might infringe upon the rights of trademark owners. This publication was produced with Interleaf V 7. The reproduction, transmission or use of this document or its contents is not permitted without express written authority.Offenders will be made liable for damages. All rights, including rights created by patent grant or registration of utility model or design, are reserved.  Siemens AG 2003. All rights reserved. Other functions not described in this documentation might be executable in the control. This does not, however, represent an obligation to supply such functions with a new control or when servicing. We have checked that the contents of this document correspond to the hardware and software described. Nonetheless, differences might exist and therefore we cannot guarantee that they are completely identical. The information contained in this document is, however, reviewed regularly and any necessary changes will be included in the next edition. We welcome suggestions for improvement. Subject to change without prior notice. Siemens -AktiengesellschaftOrder No.: 6FC5 697-2AA10 - 0BP2 Printed in the Federal Republic of Germany 3ls v SINUMERIK 802D, 802D base line 6FC5 697-2AA10-0BP2 (11.03) (DF) Preface Notes for the reader The descriptions of functions are only valid for or up to the specified software release. When new software releases are issued, the relevant descriptions of functions must be requested. Old descriptions of functions can only partially be used for new software releases. Note Other functions not described in this documentation might be executable in the control. This does not, however, represent an obligation to supply such functions with a new control or when servicing. Technical notes Notations The following notations and abbreviations are used in this Documentation: S PLC interface signals -> IS ”Signal name” (signal data) Example: IS ”Feed override“ (VB380x 0000) The variable byte is in the range “at axis“, “x” stands for the axis: 0 axis 1 1 aixs 2 n axis n+1. S Machine data -> Composition of transpositions and equality of ribbon Schur Q-functions Farzin Barekat Department of Mathematics University of British Columbia Vancouver, BC V6T 1Z2, Canada farzin barekat@yahoo.com Stephanie van Willigenburg Department of Mathematics University of British Columbia Vancouver, BC V6T 1Z2, Canada steph@math.ubc.ca Submitted: Apr 1, 2009; Accepted: Aug 24, 2009; Published: Aug 31, 2009 Mathematics Subject Classification: Primary 05A19, 05E10; Secondary 05A17, 05E05 Keywords: compositions, Eulerian posets, ribbons, Schur Q-functions, tableaux Abstract We introduce a new operation on skew diagrams called composition of trans- positions, and use it and a Jacobi-Trudi style formula to derive equalities on skew Schur Q-functions whose indexing shifted skew diagram is an ordinary skew dia- gram. When this skew diagram is a ribbon, we conjecture necessary and sufficient conditions for equality of ribbon Schur Q-functions. Moreover, we determine all relations between ribbon Schur Q-functions; show they supply a Z-basis for skew Schur Q-functions; assert their irreducibility; and show that the non-commutative analogue of ribbon Schur Q-functions is the flag h-vector of Eulerian posets. Contents 1 Introduction 2 2 Diagrams 3 2.1 Operations on diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Preliminary properties of • . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Skew Schur Q-functions 6 3.1 Symmetric functions and θ . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 New bases and relations in Ω . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Equivalence of relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Equality of ordinary skew Schur Q-functions 15 5 Ribbon Schur Q-functions 22 5.1 Equality of ribbon Schur Q-functions . . . . . . . . . . . . . . . . . . . . . 23 the electronic journal of combinatorics 16 (2009), #R110 1 1 Introduction In the algebra of symmetric functions there is interest in determining when two skew Schur functions are equal [4, 7, 11, 12, 17]. The equalities are described in terms of equivalence relations on skew diagrams. It is consequently natura l to investigate whether new equivalence relations on skew diagrams arise when we restrict our attention to the subalgebra of skew Schur Q-functions. This is a particularly interesting subalgebra to study since the combinatorics of skew Schur Q-functions also arises in the representation theory of the twisted symmetric group [1, 13, 15], and the theory o f enriched P -partitions [16], and hence skew Schur Q-function equality would impact these areas. The study of skew Schur Q-function equality was begun in [8], where a series of technical conditions classified when a skew Schur Q-function is equal to a Schur Q-f unction. In this paper we extend this study to the equality of ribbon Schur Q-functions. O ur motivation for focussing on this family is because the study of ribbon Schur function equality is funda- mental to the general study of skew Schur function equality, as evidenced by [4, 11, 12]. Our method of proof is to study a slightly more general family of skew Schur Q-functions, and then restrict our attention to ribbon Schur Q-functions. Since the combinatorics of skew Schur Q-functions is more technical than that o f skew Schur functions, we provide detailed proofs to highlight the subtleties needed to be considered for the general study of equality of skew Schur Q-functions. The rest of this paper is structured as follows. In the next section we review operations on skew diag r ams, introduce the skew diagram operation composition of transpositions and derive some basic properties for it, including associativity in Proposition 2 .5 . In Section 3 we recall Ω, the algebra of Schur Q-functions, discover new bases for this algebra in Proposition 3.6 and Corollary 3.7. We see the prominence of ribbon Schur Q-functions in the latter, which states Result. The set of all ribbon Schur ... General Note Composition of Functions When the output of one function is used as the input of another, we call the entire operation a composition of functions For any input x and functions f and... process of combining functions so that the output of 3/29 Composition of Functions one function becomes the input of another is known as a composition of functions The resulting function is known as... domain of f consists of all real numbers except So we need to exclude from the domain of g(x) that value of x for which g(x) = 13/29 Composition of Functions =1 3x − = 3x − = 3x x=2 So the domain of