Supplement 1: Toolkit Functions
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On the Number of Cubic Orders of Bounded Discriminant Having
On the number of cubic orders of bounded discriminant having automorphism group C3, and related problems Manjul Bhargava and Ariel Shnidman August 8, 2013 Abstract For a binary quadratic form Q, we consider the action of SOQ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and of a coregular space whose underlying group is not semisimple. We show that the nondegenerate integer orbits of this representation are in natural bijection with orders in cubic fields having a fixed “lattice shape”. Moreover, this correspondence is discriminant-preserving: the value of the invariant polynomial of an element in this representation agrees with the discriminant of the corresponding cubic order. We use this interpretation of the integral orbits to solve three classical-style count- ing problems related to cubic orders and fields. First, we give an asymptotic formula for the number of cubic orders having bounded discriminant and nontrivial automor- phism group. More generally, we give an asymptotic formula for the number of cubic orders that have bounded discriminant and any given lattice shape (i.e., reduced trace form, up to scaling). Via a sieve, we also count cubic fields of bounded discriminant whose rings of integers have a given lattice shape. We find, in particular, that among cubic orders (resp. fields) having lattice shape of given discriminant D, the shape is equidistributed in the class group ClD of binary quadratic forms of discriminant D. As a by-product, we also obtain an asymptotic formula for the number of cubic fields of bounded discriminant having any given quadratic resolvent field. -
Introduction Into Quaternions for Spacecraft Attitude Representation
Introduction into quaternions for spacecraft attitude representation Dipl. -Ing. Karsten Groÿekatthöfer, Dr. -Ing. Zizung Yoon Technical University of Berlin Department of Astronautics and Aeronautics Berlin, Germany May 31, 2012 Abstract The purpose of this paper is to provide a straight-forward and practical introduction to quaternion operation and calculation for rigid-body attitude representation. Therefore the basic quaternion denition as well as transformation rules and conversion rules to or from other attitude representation parameters are summarized. The quaternion computation rules are supported by practical examples to make each step comprehensible. 1 Introduction Quaternions are widely used as attitude represenation parameter of rigid bodies such as space- crafts. This is due to the fact that quaternion inherently come along with some advantages such as no singularity and computationally less intense compared to other attitude parameters such as Euler angles or a direction cosine matrix. Mainly, quaternions are used to • Parameterize a spacecraft's attitude with respect to reference coordinate system, • Propagate the attitude from one moment to the next by integrating the spacecraft equa- tions of motion, • Perform a coordinate transformation: e.g. calculate a vector in body xed frame from a (by measurement) known vector in inertial frame. However, dierent references use several notations and rules to represent and handle attitude in terms of quaternions, which might be confusing for newcomers [5], [4]. Therefore this article gives a straight-forward and clearly notated introduction into the subject of quaternions for attitude representation. The attitude of a spacecraft is its rotational orientation in space relative to a dened reference coordinate system. -
Math 2250 HW #10 Solutions
Math 2250 Written HW #10 Solutions 1. Find the absolute maximum and minimum values of the function g(x) = e−x2 subject to the constraint −2 ≤ x ≤ 1. Answer: First, we check for critical points of g, which is differentiable everywhere. By the Chain Rule, 2 2 g0(x) = e−x · (−2x) = −2xe−x : Since e−x2 > 0 for all x, g0(x) = 0 when 2x = 0, meaning when x = 0. Hence, x = 0 is the only critical point of g. Now we just evaluate g at the critical point and the endpoints: 2 g(−2) = e−(−2) = e−4 = 1=e4 2 g(0) = e−0 = e0 = 1 2 g(2) = e−1 = e−1 = 1=e: Since 1 > 1=e > 1=e4, we see that the absolute maximum of g(x) on this interval is at (0; 1) and the absolute minimum is at (−2; 1=e4). 2. Find all local maxima and minima of the curve y = x2 ln x. Answer: Notice, first of all, that ln x is only defined for x > 0, so the function x2 ln x is likewise only defined for x > 0. This function is differentiable on its entire domain, so we differentiate in search of critical points. Using the Product Rule, 1 y0 = 2x ln x + x2 · = 2x ln x + x = x(2 ln x + 1): x Since we're only allowed to consider x > 0, we see that the derivative is zero only when 2 ln x + 1 = 0, meaning ln x = −1=2. Therefore, we have a critical point at x = e−1=2. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
How to Enter Answers in Webwork
Introduction to WeBWorK 1 How to Enter Answers in WeBWorK Addition + a+b gives ab Subtraction - a-b gives ab Multiplication * a*b gives ab Multiplication may also be indicated by a space or juxtaposition, such as 2x, 2 x, 2*x, or 2(x+y). Division / a a/b gives b Exponents ^ or ** a^b gives ab as does a**b Parentheses, brackets, etc (...), [...], {...} Syntax for entering expressions Be careful entering expressions just as you would be careful entering expressions in a calculator. Sometimes using the * symbol to indicate multiplication makes things easier to read. For example (1+2)*(3+4) and (1+2)(3+4) are both valid. So are 3*4 and 3 4 (3 space 4, not 34) but using an explicit multiplication symbol makes things clearer. Use parentheses (), brackets [], and curly braces {} to make your meaning clear. Do not enter 2/4+5 (which is 5 ½ ) when you really want 2/(4+5) (which is 2/9). Do not enter 2/3*4 (which is 8/3) when you really want 2/(3*4) (which is 2/12). Entering big quotients with square brackets, e.g. [1+2+3+4]/[5+6+7+8], is a good practice. Be careful when entering functions. It is always good practice to use parentheses when entering functions. Write sin(t) instead of sint or sin t. WeBWorK has been programmed to accept sin t or even sint to mean sin(t). But sin 2t is really sin(2)t, i.e. (sin(2))*t. Be careful. Be careful entering powers of trigonometric, and other, functions. -
Calculus Formulas and Theorems
Formulas and Theorems for Reference I. Tbigonometric Formulas l. sin2d+c,cis2d:1 sec2d l*cot20:<:sc:20 +.I sin(-d) : -sitt0 t,rs(-//) = t r1sl/ : -tallH 7. sin(A* B) :sitrAcosB*silBcosA 8. : siri A cos B - siu B <:os,;l 9. cos(A+ B) - cos,4cos B - siuA siriB 10. cos(A- B) : cosA cosB + silrA sirrB 11. 2 sirrd t:osd 12. <'os20- coS2(i - siu20 : 2<'os2o - I - 1 - 2sin20 I 13. tan d : <.rft0 (:ost/ I 14. <:ol0 : sirrd tattH 1 15. (:OS I/ 1 16. cscd - ri" 6i /F tl r(. cos[I ^ -el : sitt d \l 18. -01 : COSA 215 216 Formulas and Theorems II. Differentiation Formulas !(r") - trr:"-1 Q,:I' ]tra-fg'+gf' gJ'-,f g' - * (i) ,l' ,I - (tt(.r))9'(.,') ,i;.[tyt.rt) l'' d, \ (sttt rrJ .* ('oqI' .7, tJ, \ . ./ stll lr dr. l('os J { 1a,,,t,:r) - .,' o.t "11'2 1(<,ot.r') - (,.(,2.r' Q:T rl , (sc'c:.r'J: sPl'.r tall 11 ,7, d, - (<:s<t.r,; - (ls(].]'(rot;.r fr("'),t -.'' ,1 - fr(u") o,'ltrc ,l ,, 1 ' tlll ri - (l.t' .f d,^ --: I -iAl'CSllLl'l t!.r' J1 - rz 1(Arcsi' r) : oT Il12 Formulas and Theorems 2I7 III. Integration Formulas 1. ,f "or:artC 2. [\0,-trrlrl *(' .t "r 3. [,' ,t.,: r^x| (' ,I 4. In' a,,: lL , ,' .l 111Q 5. In., a.r: .rhr.r' .r r (' ,l f 6. sirr.r d.r' - ( os.r'-t C ./ 7. /.,,.r' dr : sitr.i'| (' .t 8. tl:r:hr sec,rl+ C or ln Jccrsrl+ C ,f'r^rr f 9. -
As1.4: Polynomial Long Division
AS1.4: POLYNOMIAL LONG DIVISION One polynomial may be divided by another of lower degree by long division (similar to arithmetic long division). Example (x32+ 3 xx ++ 9) (x + 2) x + 2 x3 + 3x2 + x + 9 1. Write the question in long division form. 3 2. Begin with the x term. 2 x3 divided by x equals x2. x 2 Place x above the division bracket x+2 x32 + 3 xx ++ 9 as shown. subtract xx32+ 2 2 3. Multiply x + 2 by x . x2 xx2+=+22 x 32 x ( ) . Place xx32+ 2 below xx32+ 3 then subtract. x322+−+32 x x xx = 2 ( ) 4. “Bring down” the next term (x) as 2 x + x indicated by the arrow. +32 +++ Repeat this process with the result until xxx23x 9 there are no terms remaining. x32+↓2x 2 2 5. x divided by x equals x. x + x Multiply xx( +=+22) x2 x . xx2 +−1 ( xx22+−) ( x +2 x) =− x 32 x+23 x + xx ++ 9 6. Bring down the 9. xx32+2 ↓↓ 2 7. − x divided by x equals − 1. xx+↓ Multiply xx2 +↓2 −12( xx +) =−− 2 . −+x 9 (−+xx9) −−−( 2) = 11 −−x 2 The remainder is 11 +11 (x32+ 3 xx ++ 9) equals xx2 +−1, with remainder 11. (x + 2) AS 1.4 – Polynomial Long Division Page 1 of 4 June 2012 If the polynomial has an ‘xn ‘ term missing, add the term with a coefficient of zero. Example (2xx3 − 31 +÷) ( x − 1) Rewrite (2xx3 −+ 31) as (2xxx32+ 0 −+ 31) Divide using the method from the previous example. 2 22xx32÷= x 2xx+− 21 2 32 −= − 2xx( 12) x 2 x 32 x−12031 xxx + −+ 20xx32−=−−(22xx32) 2x2 2xx32− 2 ↓↓ 2 22xx÷= x 2xx2 −↓ 3 2xx( −= 12) x2 − 2 x 2 2 2 2xx−↓ 2 23xx−=−−22xx− x ( ) −+x 1 −÷xx =−1 −+x 1 −11( xx −) =−+ 1 0 −+xx1− ( + 10) = Remainder is 0 (2xx32− 31 +÷) ( x −= 1) ( 2 xx + 21 −) with remainder = 0 ∴(2xx32 − 3 += 1) ( x − 12)( xx + 2 − 1) See Exercise 1. -
Section 1.4 Absolute Value
Section 1.4 Absolute Value Geometric Definition of Absolute Value As we study the number line, we observe a very useful property called symmetry. The numbers are symmetrical with respect to the origin. That is, if we go four units to the right of 0, we come to the number 4. If we go four units to the lest of 0 we come to the opposite of 4 which is −4 The absolute value of a number a, denoted |a|, is the distance from a to 0 on the number line. Absolute value speaks to the question of "how far," and not "which way." The phrase how far implies length, and length is always a nonnegative (zero or positive) quantity. Thus, the absolute value of a number is a nonnegative number. This is shown in the following examples: Example 1 1 Algebraic Definition of Absolute Value The absolute value of a number a is aaif 0 | a | aaif 0 The algebraic definition takes into account the fact that the number a could be either positive or zero (≥0) or negative (<0) 1) If the number a is positive or zero (≥ 0), the first part of the definition applies. The first part of the definition tells us that if the number enclosed in the absolute bars is a nonnegative number, the absolute value of the number is the number itself. 2) If the number a is negative (< 0), the second part of the definition applies. The second part of the definition tells us that if the number enclosed within the absolute value bars is a negative number, the absolute value of the number is the opposite of the number. -
A Tutorial on Euler Angles and Quaternions
A Tutorial on Euler Angles and Quaternions Moti Ben-Ari Department of Science Teaching Weizmann Institute of Science http://www.weizmann.ac.il/sci-tea/benari/ Version 2.0.1 c 2014–17 by Moti Ben-Ari. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/ by-sa/3.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA. Chapter 1 Introduction You sitting in an airplane at night, watching a movie displayed on the screen attached to the seat in front of you. The airplane gently banks to the left. You may feel the slight acceleration, but you won’t see any change in the position of the movie screen. Both you and the screen are in the same frame of reference, so unless you stand up or make another move, the position and orientation of the screen relative to your position and orientation won’t change. The same is not true with respect to your position and orientation relative to the frame of reference of the earth. The airplane is moving forward at a very high speed and the bank changes your orientation with respect to the earth. The transformation of coordinates (position and orientation) from one frame of reference is a fundamental operation in several areas: flight control of aircraft and rockets, move- ment of manipulators in robotics, and computer graphics. This tutorial introduces the mathematics of rotations using two formalisms: (1) Euler angles are the angles of rotation of a three-dimensional coordinate frame. -
Calculus Online Textbook Chapter 2
Contents CHAPTER 1 Introduction to Calculus 1.1 Velocity and Distance 1.2 Calculus Without Limits 1.3 The Velocity at an Instant 1.4 Circular Motion 1.5 A Review of Trigonometry 1.6 A Thousand Points of Light 1.7 Computing in Calculus CHAPTER 2 Derivatives The Derivative of a Function Powers and Polynomials The Slope and the Tangent Line Derivative of the Sine and Cosine The Product and Quotient and Power Rules Limits Continuous Functions CHAPTER 3 Applications of the Derivative 3.1 Linear Approximation 3.2 Maximum and Minimum Problems 3.3 Second Derivatives: Minimum vs. Maximum 3.4 Graphs 3.5 Ellipses, Parabolas, and Hyperbolas 3.6 Iterations x, + ,= F(x,) 3.7 Newton's Method and Chaos 3.8 The Mean Value Theorem and l'H8pital's Rule CHAPTER 2 Derivatives 2.1 The Derivative of a Function This chapter begins with the definition of the derivative. Two examples were in Chapter 1. When the distance is t2, the velocity is 2t. When f(t) = sin t we found v(t)= cos t. The velocity is now called the derivative off (t). As we move to a more formal definition and new examples, we use new symbols f' and dfldt for the derivative. 2A At time t, the derivativef'(t)or df /dt or v(t) is fCt -t At) -f (0 f'(t)= lim (1) At+O At The ratio on the right is the average velocity over a short time At. The derivative, on the left side, is its limit as the step At (delta t) approaches zero. -
Doing Physics with Quaternions
Doing Physics with Quaternions Douglas B. Sweetser ©2005 doug <[email protected]> All righs reserved. 1 INDEX Introduction 2 What are Quaternions? 3 Unifying Two Views of Events 4 A Brief History of Quaternions Mathematics 6 Multiplying Quaternions the Easy Way 7 Scalars, Vectors, Tensors and All That 11 Inner and Outer Products of Quaternions 13 Quaternion Analysis 23 Topological Properties of Quaternions 28 Quaternion Algebra Tool Set Classical Mechanics 32 Newton’s Second Law 35 Oscillators and Waves 37 Four Tests of for a Conservative Force Special Relativity 40 Rotations and Dilations Create the Lorentz Group 43 An Alternative Algebra for Lorentz Boosts Electromagnetism 48 Classical Electrodynamics 51 Electromagnetic Field Gauges 53 The Maxwell Equations in the Light Gauge: QED? 56 The Lorentz Force 58 The Stress Tensor of the Electromagnetic Field Quantum Mechanics 62 A Complete Inner Product Space with Dirac’s Bracket Notation 67 Multiplying quaternions in Polar Coordinate Form 69 Commutators and the Uncertainty Principle 74 Unifying the Representations of Integral and Half−Integral Spin 79 Deriving A Quaternion Analog to the Schrödinger Equation 83 Introduction to Relativistic Quantum Mechanics 86 Time Reversal Transformations for Intervals Gravity 89 Unified Field Theory by Analogy 101 Einstein’s vision I: Classical unified field equations for gravity and electromagnetism using Riemannian quaternions 115 Einstein’s vision II: A unified force equation with constant velocity profile solutions 123 Strings and Quantum Gravity 127 Answering Prima Facie Questions in Quantum Gravity Using Quaternions 134 Length in Curved Spacetime 136 A New Idea for Metrics 138 The Gravitational Redshift 140 A Brief Summary of Important Laws in Physics Written as Quaternions 155 Conclusions 2 What Are Quaternions? Quaternions are numbers like the real numbers: they can be added, subtracted, multiplied, and divided. -
Low-Degree Polynomial Roots
Low-Degree Polynomial Roots David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Created: July 15, 1999 Last Modified: September 10, 2019 Contents 1 Introduction 3 2 Discriminants 3 3 Preprocessing the Polynomials5 4 Quadratic Polynomials 6 4.1 A Floating-Point Implementation..................................6 4.2 A Mixed-Type Implementation...................................7 5 Cubic Polynomials 8 5.1 Real Roots of Multiplicity Larger Than One............................8 5.2 One Simple Real Root........................................9 5.3 Three Simple Real Roots......................................9 5.4 A Mixed-Type Implementation................................... 10 6 Quartic Polynomials 12 6.1 Processing the Root Zero...................................... 14 6.2 The Biquadratic Case........................................ 14 6.3 Multiplicity Vector (3; 1; 0; 0).................................... 15 6.4 Multiplicity Vector (2; 2; 0; 0).................................... 15 6.5 Multiplicity Vector (2; 1; 1; 0).................................... 15 6.6 Multiplicity Vector (1; 1; 1; 1).................................... 16 1 6.7 A Mixed-Type Implementation................................... 17 2 1 Introduction Consider a polynomial of degree d of the form d X i p(y) = piy (1) i=0 where the pi are real numbers and where pd 6= 0. A root of the polynomial is a number r, real or non-real (complex-valued with nonzero imaginary part) such that p(r) = 0. The polynomial can be factored as p(y) = (y − r)mf(y), where m is a positive integer and f(r) 6= 0.