41. Shapes Of Constant Width The length of the boundary of shapes of constant width depends only on R.Honsberger,Ingenuity in mathematics, MAA, New math Library, 1970; H.Rademacher and O http://www.cut-the-knot.com/do_you_know/cwidth.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Shapes of constant width Yes there are shapes of constant width other than the circle No - you can't drill square holes. But saying this was not just an attention catcher. As the applet on the right illustrates, you can drill holes that are almost square - drilled holes whose border includes straight line segments! Now then let us define the subject of our discussion. First we need a notion of width. Let there be a bounded shape. Pick two parallel lines so that the shape lies between the two. Move each line towards the shape all the while keeping it parallel to its original direction. After both lines touched our figure, measure the distance between the two. This will be called the width of the shape in the direction of the two lines. A shape is of constant width if its (directional) width does not depend on the direction. This unique number is called the width of the figure. For the circle, the width and the diameter coincide. The curvilinear triangle above is built the following way. Start with an equilateral triangle. Draw three arcs with radius equal to the side of the triangle and each centered at one of the vertices. The figure is known as the Reuleaux triangle. Convince yourself that the construction indeed results in a figure of constant width. Starting with this we can create more. Rotating Reuleaux triangle covers most of the area of the enclosing square. For the width=1 the following formula is cited in

42. Roger Sayle - [PATCH] Perform Constant Folding Of Math Builtins PATCH Perform constant folding of math builtins. + Verify that builtin mathfunction constant folding doesn't + cause any problems for the compiler. http://gcc.gnu.org/ml/gcc-patches/2002-08/msg01494.html

This is the mail archive of the gcc-patches@gcc.gnu.org mailing list for the GCC project Index Nav: Date Index Subject Index Author Index Thread Index Message Nav: Date Prev Date Next Thread Prev Thread Next Other format: Raw text [PATCH] Perform constant folding of math builtins From To Date : Sun, 25 Aug 2002 20:17:55 -0600 (MDT) Subject : [PATCH] Perform constant folding of math builtins http://www.eyesopen.com/ Suite 1107, 3600 Cerrillos Road, Tel: (+1) 505-473-7385 Santa Fe, New Mexico, 87507. Fax: (+1) 505-473-0833 Follow-Ups Re: [PATCH] Perform constant folding of math builtins From: Gabriel Dos Reis Re: [PATCH] Perform constant folding of math builtins From: Kaveh R. Ghazi Re: [PATCH] Perform constant folding of math builtins From: Mark Mitchell Re: [PATCH] Perform constant folding of math builtins From: Richard Henderson Index Nav: Date Index Subject Index Author Index Thread Index Message Nav: Date Prev Date Next Thread Prev Thread Next

43. Roger Sayle - Re: [PATCH] Perform Constant Folding Of Math Builtins Re PATCH Perform constant folding of math builtins. From Roger SubjectRe PATCH Perform constant folding of math builtins. Hi Brad http://gcc.gnu.org/ml/gcc-patches/2002-08/msg01498.html

This is the mail archive of the gcc-patches@gcc.gnu.org mailing list for the GCC project Index Nav: Date Index Subject Index Author Index Thread Index Message Nav: Date Prev Date Next Thread Prev Thread Next Other format: Raw text Re: [PATCH] Perform constant folding of math builtins From To Cc Date : Sun, 25 Aug 2002 22:08:35 -0600 (MDT) Subject : Re: [PATCH] Perform constant folding of math builtins Follow-Ups Re: [PATCH] Perform constant folding of math builtins From: Gabriel Dos Reis Re: [PATCH] Perform constant folding of math builtins From: Brad Lucier Re: [PATCH] Perform constant folding of math builtins From: Brad Lucier Re: [PATCH] Perform constant folding of math builtins From: Brad Lucier References Re: [PATCH] Perform constant folding of math builtins From: Brad Lucier Index Nav: Date Index Subject Index Author Index Thread Index Message Nav: Date Prev Date Next Thread Prev Thread Next

44. Fine Structure Constant, Alpha Solution to a 20th century mystery. Feynman's conjecture of a relation between alpha, the fine structure Category Science Physics Quantum Mechanics Alternative...... Twisting String. David Thomson writes about the finestructure constant in relationto his Energy Pulse Model-. Tesla. Chekad Sarami's List of math Sites-. http://www.fine-structure-constant.org/

James G. Gilson, Fine Structure Constant The fine structure constant, a 20th century mystery The question of how to calculate the numerical value of the fine structure constant from theory was one of the most outstandingly difficult problems in mathematical physics for the greater part of the 20th century. There were many unsuccessful attempts by researchers including famous ones such as P. A. M. Dirac to find a formula for the fine structure constant. See John Baez's website page Open Questions in Physics and also the PhysLINK website. Substantial progress with this fundamental problem is outlined on this web page. fundamental quantum coupling constants are given by the very simple formula depending on two integer parameters n and n ,n ) = n The values of the quantum coupling constants are often denoted by the lower case Greek letter alpha alpha with no subscript. = 137 with n The latest CODATA Briefly, the hindsight obviousness of Feynman's conjecture is inherent in the simplest idea of coupling between two systems. Taking one of the objects as fixed in space then, if a second object moves in circles ,n i o i (n), the distance from its center to the center of a side, and a large radius r

45. ABE Florida: Math Lesson "Packages (B) - Constant Area And Constant Perimeter" Resource Guide. Benchmarks 39.1, 39.6, 39.7 Title Packages (B) constant Area and constant Perimeter. Contributor Jan Robinson http://abeflorida.org/math-lesson26.htm

Mathematics Level Lesson Plan for Standard This activity works well as the second in a series of activities by Jan Robinson. All activities, A through E, apply to Standard 39 and are in this Resource Guide. Benchmarks: 39.1, 39.6, 39.7 Title: Packages (B) - Constant Area and Constant Perimeter Contributor: Jan Robinson Instructional Specialist, School District 21, Illinois Presented at ASCD Classroom Leadership Conference Orlando, Florida July 1999. Contact Information: jrobinso@d21.k12.il.us Materials: 24 colored tiles per student or group; constant area chart and graph sheet constant perimeter chart and graph sheet Procedure: This activity has two components that allow for the discovery of patterns which suggest the formulas for area and perimeter. Students may work individually or in groups, but groups are recommended. Constant Area: First, students are to find all the rectangular arrays that can be made using 24 tiles. For this exercise, no tile may be stacked on top of another. Students then chart the area and perimeter of each rectangular array made. (The area will always be twenty four, but the perimeter will change.) After recording the data on the chart, students must graph their findings. Finally, students should write 3 observations about the data on either their chart or graph sheet. Constant Perimeter: Second, students are to hold the perimeter of the rectangular arrays to 24 units. This may be illustrated by talking in terms of tables and the number of people who could be seated at each table. Again, no tile may be stacked on another. Students should record the area and perimeter of their arrays on the chart. ( Here the perimeter will always be 24, but the area will change.) After recording the data on the chart, students must graph their findings. Finally, students should write 3 observations about their data either on their chart or on their graph sheet.

46. ABE Florida: Math Lesson "Constant Area Record Sheet" Handout Name_ Date.constant Area Record Sheet. Dimensions, Area, Perimeter. Bottom Edge, http://abeflorida.org/handouts/package-area.htm

Name_ Date Constant Area Record Sheet Dimensions Area Perimeter Bottom Edge Side Edge Number of Tables Number of People

47. Math & Science: A Constant? Title a constant? From eagerbob Date 12/17/2002 1202AM PST Status Waiting forAnswer Points 20. 3) The universal gravitational constant (G) becomes zero. http://www.experts-exchange.com/Miscellaneous/Math_Science/Q_20428388.html

Premium Search Search Help Search Disabled - You must register before you can search Login Register Forgot Password Reload This Question ... Miscellaneous Viewing a Question Title: a constant? From: eagerbob Date: 12/17/2002 12:02AM PST Status: Waiting for Answer Points: Plutonium occasionally splits spontanious. A hydrogen atom excists only for a few billion years. Every thing has its end. WHEN DOES GRAVITY STOP? (is it today, tomorrow or when?) Think about it...it could really happen! Member Name (can not be changed) Email (requires verification) Password Confirm Password Receive our Monthly Newsletter I agree to the Member Agreement Question History Comment from tmmacpha 12/17/2002 12:39AM PST Gravity stops when all matter ceases to exist. Guess what... this isnt going to happen until (if it does happen) the universe colapses in on itself. There is always gravity between any two objects with mass. As a matter of fact, there is a force of gravity between me and you, wherever you are. It is incredibly tiny (on the scale of zero) but it exists because we are two objects with mass, radius (sort of...) and some distance between us. So, until the earth isnt here anymore or is small enough that the moon's gravitational pull is stronger than the earth's, there will still be gravity holding you to earth.

48. Help-Site: Math::BigInt::Constant Computer Help MathBigIntconstant. Search. Documents. MathBigIntconstant Arbitrary sized constant integers 9 Dec 2002. If you can't find http://help-site.com/c.m/prog/lang/perl/cpan/06/math/math-bigint-constant/

[Main Index] -> [Programming] -> [Programming Languages] -> [Perl Programming] ... -> [Math] [Directory] [Forums] Math::BigInt::Constant Search options Learning Perl Other Perl Programming books to buy at Amazon. Documents Math::BigInt::Constant - Arbitrary sized constant integers [9 Dec 2002] If you can't find the help you are looking for on the main site you can now visit the new Help-Site Forums to ask for help. Save 10% on high-quality Crucial RAM. Order online at Crucial's factory-direct Web site. Crucial Technology, The Memory Experts. [New Links] [Add Url] [About]

49. Help-Site: Math::BigInt::Constant Computer Help mathBigIntconstant Arbitrary sized constant integers. AUTHORS. Telshttp//bloodgate.com in early 2001. (math-BigInt-constant-1.04.tar.gz) http://help-site.com/c.m/prog/lang/perl/cpan/06/math/math-bigint-constant/_d1280

[Main Index] -> [Programming] -> [Programming Languages] -> [Perl Programming] ... -> [Math] [Directory] [Forums] Math::BigInt::Constant Search options Learning Perl Other Perl Programming books to buy at Amazon. Math::BigInt::Constant - Arbitrary sized constant integers NAME SYNOPSIS DESCRIPTION EXAMPLES ... AUTHORS NAME Math::BigInt::Constant - Arbitrary sized constant integers SYNOPSIS use Math::BigInt::Constant; # The following would modify and thus are illegal: # The following do not modify their arguments and are ok: bgcd(@values); # greatest common divisor blcm(@values); # lowest common multiplicator DESCRIPTION This module let's you defined constant BigInt's on a per-object basis. The usual use Math::BigInt ':constant' will catch all integer constants in the script at compile time, but will not let you create constant values on the fly, nor work for strings and/or floating point constants like EXAMPLES these provide runtime checks and can be interpolated into strings: use Math::BigInt::Constant;

50. JavaScript: The Math Object values math.PI, Returns the constant pi. math.SQRT2, Returns thesquare root of 2. math.SQRT1_2, Returns the square root of 1/2. Below http://www.pageresource.com/jscript/jmath.htm

The Math Object Helpful methods for calculations JS Main Basics Advanced Complete List ... Advanced /The Math Object Browser Compatibility: NS 4+, IE 4+ The Math object of JavaScript allows you to perform certain calculations by using method functions of the Math object. Also, it provides a few constants such as pi. The easiest way to get a value is to define a variable and set its value to a property or function of the Math Object. If you want to use a property, which returns a constant value (such as pi), you would write something like this: var my_car=Math.property; You would replace the word "property" with the property you want to use. So, if you want to use the value of pi in a variable, you would use the PI property of the Math object. You would write: var my_pie=Math.PI; This returns the pi constant, a number with lots of decimals, or 3.14..... If you want to use a member function, which performs a calculation, you would write something like this: var my_house=Math.function(x); Most of the member functions have one or more parameters, which is what the "x" is for. You can replace x with a number or variable. You would replace the word "function" with the function you want to use. For instance, if you want the square root of a number, you can call the square root member function of the Math object with the number as the parameter: var my_money=Math.sqrt(2);

51. Math Help - Calculus - Antiderivatives And The Riemann Integral The constant C is required since we know derivatives destroy all constants;therefore the integral must create a constant. We don http://www.hyper-ad.com/tutoring/math/calculus/Antiderivatives and the Riemann I

Technical Tutoring Home Site Index Advanced Books Speed Arithmetic ... Gift Shop Antiderivatives and the Riemann Integral Introduction Derivatives and antiderivatives Integrable functions Indefinite integrals ... Recommended Books Introduction In a previous article , we introduced the Riemann integral and developed a limit process for calculating it. We mentioned that the integral represents the area under the curve f(x) to the x-axis starting with x = a and ending with x = b, and stated without any support at all that the integral was the inverse operation to the derivative. We now take up this idea in detail, and supply some more technical groundwork for the theory of the integral. return to top Derivatives and antiderivatives Elsewhere In practice, this definition can get difficult to use rather fast, so we derived an easier-to-use formula that works for powers of x only: Although good only for polynomial functions, this is a very useful formula. Using the formula is equivalent to the operations: multiply by the power of x and then subtract 1 from the exponent. Were we to write this process as an operator acting on the function, we might define the operator as: The operator acts on the nth-order polynomial function x n by multiplying by n the next smallest polynomial function x n-1 . We bring up operators (naively, to be sure) to point out that there can then be an "inverse" operator I

52. Gamma Constant Search, Gamma constant. (math Miscellaneous constants Gamma).gamma = = 0.5772156649 0153286061 = lim (n ) ( 1 + 1/2 + 1 http://www.math.com/tables/constants/gamma.htm

Home Teacher Parents Glossary ... Email this page to a friend Resources Cool Tools References Test Preparation Study Tips ... Wonders of Math Search Gamma Constant Math Miscellaneous Constants gamma = = lim ( 1 + 1/2 + 1/3 + 1/4 + ... + 1/n - ln(n) ) = 0.5772156649... e ^-x ln x dx (see Gamma Function FREE Software Click here to get your FREE Software Exclusive Math.com Offer Click here Contact us Partnership Link to us ... Legal Notices . Please read our Privacy Policy

53. Constant Rule Proof constant Rule. d. = c * f(x). FREE Software, 1, 0, 3, 7, 6, 9,Click here to get your FREE Software. Exclusive math.com Offer, » Clickhere. http://www.math.com/tables/derivatives/identities/constant.htm

Home Teacher Parents Glossary ... Email this page to a friend Resources Cool Tools References Test Preparation Study Tips ... Wonders of Math Search Proof: Constant Rule Math Calculus Derivatives Identities c f(x) = c f(x) Proof of c f(x) = c f(x) from the definition We can use the definition of the derivative: f(x) = lim f(x+d)-f(x) d Therefore, c f(x) can be written as such: c f(x) lim cf(x+d) - cf(x) d c lim f(x+d) - f(x) d c * f(x) FREE Software Click here to get your FREE Software Exclusive Math.com Offer Click here Contact us Partnership Link to us ... Legal Notices . Please read our Privacy Policy

54. Math Trek : Sound-Byte Math Music, Science News Online, Jan. 4, 2003 SoundByte math Music. Ivars Peterson. The decimal digits of themathematical constant pi, 3.14159265. . ., ring out an intricate http://www.sciencenews.org/20030104/mathtrek.asp

Home page. Math Trek Sound-Byte Math Music Food for Thought Mercurial Effects of Fish-Rich Diets Science Safari Wonderful World of Color TimeLine 70 Years Ago in Science News Week of Jan. 4, 2003; Vol. 163, No. 1 Sound-Byte Math Music Ivars Peterson The decimal digits of the mathematical constant pi, 3.14159265. . ., ring out an intricate melody that sounds vaguely medieval. Those of the constant e , 2.718281828. . ., progress at a relentless, suspenseful pace. Euler's prime-number-powered phi function bounces about with a semitropical rhythm. Lorenz's butterfly meanders through a ragged soundscape. Pascal's triangle echoes with an eerie beat. Lorenz's butterfly. Created by Swedish composer Daniel Cummerow, these mathematical sound bytes belong to a category known as algorithmic music. Each musical fragment is determined by a mathematical recipe—a formula that links digits with musical notes and their duration, as governed by the musical preferences of the composer. If you have browser software that can play MIDI (Musical Instrument Digital Interface) files, you can sample Cummerow's compositions at his Web page ( http://www.geocities.com/Vienna/9349/

55. The EZ Math Trix Series - TMW Media Group This program covers · Introduction. · Siri Time constant. · Number Fun. · MathGames. · Dice Trick. · Domino Trick. For students ages 8 years and over. http://www.tmwmedia.com/ezmathtrix.html

The EZ Math Trix Series Improve Your Math Skills in Minutes! It is no secret that people struggle with math. Even though it is a difficult subject for some, it can now be very exciting to master! By using the tricks and techniques provided in the video series and DVD, students will gain the ability to do their own calculations quickly and accurately. Results have shown, using these skills can dramatically improve student’s math skills. Siri Grandhe Siri Grandhe is the inspiration for this video and DVD series. Watch her perform the “Siri Time Constant” trick in which she adds and subtracts time units quickly and easily. Due to her unusual skills, by the age of 10, Siri had already appeared on several television shows such as Nickelodeon, The Tonight Show with Jay Leno and NBC News. She has also been featured in magazines including Children’s National Geographic and India Today. Addition and Subtraction: Vol. 1 Learn to improve your math skills using quick addition and subtraction tricks!

56. Colby Community College Mathematics Department Information And Math Tables, Form GAMMA constant. gamma = = 0.5772156649 0153286061 Summary A collectionof math related tables, facts, information and formulas http://www.colbycc.org/www/math/constants/gamma.htm

GAMMA Constant gamma = = lim ( 1 + 1/2 + 1/3 + 1/4 + ... + 1/n - ln(n) ) = 0.5772156649... e ^-x ln x dx (see Gamma Function Summary: A collection of math related tables, facts, information and formulas ... with the main title and index page located at: http://www.colbycc.org/www/math/math.htm . These pages are maintained by Colby Community College and the Colby Community College Mathematics Department. If you have any comments, suggestions or a page that you've constructed (that you believe would be valuable and appropriate to include on/in/at our math site), please contact our CCC Math Department Faculty Page Keywords: Mathematics, Mathematic, Mathematical, Math, Mathematically, Reference, Table, Tables, Formula, Formulas, Fact, Facts, Information, Data, Computation, Computational, Algebra, Geometry, Trigonometry, Trig, Analysis, Calculus, Calc, Addition, Subtraction, Multiplication, Division, Powers, Roots, Exponents, Identities, Chart, Graph, List, Java, On-line, Games, Puzzles, Listing, Equations, Equation, Graphs, Graph, Conversions, Convert, Number, Numerical, Factorial, Sequence, Series, Conic, Integral, Differential, Mensuration, Measure, Function, Statistics, Stat, Math Reference.

57. Math Unit III: More On The Derivative And Differential Equations Exact definition of derivation and calculating the relationship of derivatives of related functions.Category Science math Differential Equations......math Unit III More on the derivative and differential equations. rule is worth aspecial mention If one of the factors u or v or is a constant function then http://dept.physics.upenn.edu/courses/gladney/mathphys/subsection3_1_1.html

Back to Contents! Next: Force Revisited Up: CONSERVING EQUATIONS Previous: CONSERVING EQUATIONS Math Unit III: More on the derivative and differential equations In the last unit, we discussed how mathematicians and scientists deal with quantities that change in ways other than linearly. The key idea turns out to be the rate of change of the quantity. The lowbrow way to measure the rate of change is to compute the average rate of change over a small interval. The high-class way to talk about change is to try to compute the derivative , or instantaneous rate of change In real-world situations, when one deals with measured data, it is often the case that the data has been measured only for certain specific values of the independent variable. Then, one can only compute average rates of change between the data points. But in mathematics, when one deals with abstractly-defined functions, it is possible to compute derivatives. Often, mathematical models are developed using derivatives, predictions are made based upon these mathematical models, and then experimental results are compared to the predictions to see how well the models reflect reality. 1. Review and extensions

58. Class Crw.math.CMatrix java.lang.Object +crw.math.CMatrix add(CMatrix, Complex) Returnsaddition of a square matrix and a constant times the unity matrix. http://jas2.eng.buffalo.edu/semiconductor/apiWIE/crw.math.CMatrix.html

All Packages Class Hierarchy This Package Previous ... Index Class crw.math.CMatrix public class CMatrix extends Object Class for a matrix of complex numbers. It is useful mostly for a square matrix. Most methods are un-optimized straightforward routines. Version: Author: C. R. Wie CMatrix Creates a 1x1 complex matrix with a complex object initialized to a zero value. CMatrix (CMatrix) Creates a Cmatrix object initialized to the argument. CMatrix (Complex[][]) Creates a CMatrix object initialized to a complex matrix. CMatrix (int) Creates a square complex matrix with elements initialized to zero values. CMatrix (int, int) Creates a i x j complex matrix, elements initialized to complex numbers of 0. add (CMatrix) Adds the argument matrix to this one. add (CMatrix, CMatrix) Returns the addition of two matrices of equal dimensions, null if different dimensions. add (CMatrix, Complex) Returns addition of a square matrix and a constant times the unity matrix. add (Complex) Adds a constant to the diagonal elements of this matrix. add (Complex, CMatrix)

59. Conic Sections Dave's math Tables Conic Sections. distance to the origin is constant, sum of distancesto each focus is constant, distance to focus = distance to directrix, http://www.math2.org/math/algebra/conics.htm

Dave's Math Tables: Conic Sections Math Algebra Conics Circle Ellipse (h) Parabola (h) Hyperbola (h) Definition: A conic section is the intersection of a plane and a cone. Ellipse (v) Parabola (v) Hyperbola (v) By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or hyperbola; or in the special case when the plane touches the vertex: a point, line or 2 intersecting lines. Point Line Double Line The General Equation for a Conic Section: Ax + Bxy + Cy + Dx + Ey + F = The type of section can be found from the sign of: B If B - 4AC is... then the curve is a... ellipse, circle, point or no curve. parabola, 2 parallel lines, 1 line or no curve. hyperbola or 2 intersecting lines. The Conic Sections. For any of the below with a center (j, k) instead of (0, 0), replace each x term with (x-j) and each y term with (y-k). Circle Ellipse Parabola Hyperbola Equation (horiz. vertex): x + y = r x / a + y / b 4px = y x / a - y / b Equations of Asymptotes: Equation (vert. vertex): x + y = r y / a + x / b 4py = x y / a - x / b Equations of Asymptotes: Variables: r = circle radius a = major radius (= 1/2 length major axis) b = minor radius (= 1/2 length minor axis) c = distance center to focus p = distance from vertex to focus (or directrix) a = 1/2 length major axis b = 1/2 length minor axis c = distance center to focus Eccentricity: c/a c/a Relation to Focus: p = a - b = c p = p a + b = c Definition: is the locus of all points which meet the condition...

60. Gamma Constant Dave's math Tables Gamma constant. http://www.math2.org/math/constants/gamma.htm

Dave's Math Tables: Gamma Constant Math OddsEnds Constants Gamma gamma = = lim ( 1 + 1/2 + 1/3 + 1/4 + ... + 1/n - ln(n) ) = 0.5772156649... e ^-x ln x dx (see Gamma Function

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