Maxima by Example: Ch.4: Solving Equations ∗
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Plotting Direction Fields in Matlab and Maxima – a Short Tutorial
Plotting direction fields in Matlab and Maxima – a short tutorial Luis Carvalho Introduction A first order differential equation can be expressed as dx x0(t) = = f(t; x) (1) dt where t is the independent variable and x is the dependent variable (on t). Note that the equation above is in normal form. The difficulty in solving equation (1) depends clearly on f(t; x). One way to gain geometric insight on how the solution might look like is to observe that any solution to (1) should be such that the slope at any point P (t0; x0) is f(t0; x0), since this is the value of the derivative at P . With this motivation in mind, if you select enough points and plot the slopes in each of these points, you will obtain a direction field for ODE (1). However, it is not easy to plot such a direction field for any function f(t; x), and so we must resort to some computational help. There are many computational packages to help us in this task. This is a short tutorial on how to plot direction fields for first order ODE’s in Matlab and Maxima. Matlab Matlab is a powerful “computing environment that combines numeric computation, advanced graphics and visualization” 1. A nice package for plotting direction field in Matlab (although resourceful, Matlab does not provide such facility natively) can be found at http://math.rice.edu/»dfield, contributed by John C. Polking from the Dept. of Mathematics at Rice University. To install this add-on, simply browse to the files for your version of Matlab and download them. -
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. -
The Diamond Method of Factoring a Quadratic Equation
The Diamond Method of Factoring a Quadratic Equation Important: Remember that the first step in any factoring is to look at each term and factor out the greatest common factor. For example: 3x2 + 6x + 12 = 3(x2 + 2x + 4) AND 5x2 + 10x = 5x(x + 2) If the leading coefficient is negative, always factor out the negative. For example: -2x2 - x + 1 = -1(2x2 + x - 1) = -(2x2 + x - 1) Using the Diamond Method: Example 1 2 Factor 2x + 11x + 15 using the Diamond Method. +30 Step 1: Multiply the coefficient of the x2 term (+2) and the constant (+15) and place this product (+30) in the top quarter of a large “X.” Step 2: Place the coefficient of the middle term in the bottom quarter of the +11 “X.” (+11) Step 3: List all factors of the number in the top quarter of the “X.” +30 (+1)(+30) (-1)(-30) (+2)(+15) (-2)(-15) (+3)(+10) (-3)(-10) (+5)(+6) (-5)(-6) +30 Step 4: Identify the two factors whose sum gives the number in the bottom quarter of the “x.” (5 ∙ 6 = 30 and 5 + 6 = 11) and place these factors in +5 +6 the left and right quarters of the “X” (order is not important). +11 Step 5: Break the middle term of the original trinomial into the sum of two terms formed using the right and left quarters of the “X.” That is, write the first term of the original equation, 2x2 , then write 11x as + 5x + 6x (the num bers from the “X”), and finally write the last term of the original equation, +15 , to get the following 4-term polynomial: 2x2 + 11x + 15 = 2x2 + 5x + 6x + 15 Step 6: Factor by Grouping: Group the first two terms together and the last two terms together. -
Solving Cubic Polynomials
Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial monic. a 2. Then, given x2 + a x + a , substitute x = y − 1 to obtain an equation without the linear term. 1 0 2 (This is the \depressed" equation.) 3. Solve then for y as a square root. (Remember to use both signs of the square root.) a 4. Once this is done, recover x using the fact that x = y − 1 . 2 For example, let's solve 2x2 + 7x − 15 = 0: First, we divide both sides by 2 to create an equation with leading term equal to one: 7 15 x2 + x − = 0: 2 2 a 7 Then replace x by x = y − 1 = y − to obtain: 2 4 169 y2 = 16 Solve for y: 13 13 y = or − 4 4 Then, solving back for x, we have 3 x = or − 5: 2 This method is equivalent to \completing the square" and is the steps taken in developing the much- memorized quadratic formula. For example, if the original equation is our \high school quadratic" ax2 + bx + c = 0 then the first step creates the equation b c x2 + x + = 0: a a b We then write x = y − and obtain, after simplifying, 2a b2 − 4ac y2 − = 0 4a2 so that p b2 − 4ac y = ± 2a and so p b b2 − 4ac x = − ± : 2a 2a 1 The solutions to this quadratic depend heavily on the value of b2 − 4ac. -
CAS (Computer Algebra System) Mathematica
CAS (Computer Algebra System) Mathematica- UML students can download a copy for free as part of the UML site license; see the course website for details From: Wikipedia 2/9/2014 A computer algebra system (CAS) is a software program that allows [one] to compute with mathematical expressions in a way which is similar to the traditional handwritten computations of the mathematicians and other scientists. The main ones are Axiom, Magma, Maple, Mathematica and Sage (the latter includes several computer algebras systems, such as Macsyma and SymPy). Computer algebra systems began to appear in the 1960s, and evolved out of two quite different sources—the requirements of theoretical physicists and research into artificial intelligence. A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics Martin Veltman, who designed a program for symbolic mathematics, especially High Energy Physics, called Schoonschip (Dutch for "clean ship") in 1963. Using LISP as the programming basis, Carl Engelman created MATHLAB in 1964 at MITRE within an artificial intelligence research environment. Later MATHLAB was made available to users on PDP-6 and PDP-10 Systems running TOPS-10 or TENEX in universities. Today it can still be used on SIMH-Emulations of the PDP-10. MATHLAB ("mathematical laboratory") should not be confused with MATLAB ("matrix laboratory") which is a system for numerical computation built 15 years later at the University of New Mexico, accidentally named rather similarly. The first popular computer algebra systems were muMATH, Reduce, Derive (based on muMATH), and Macsyma; a popular copyleft version of Macsyma called Maxima is actively being maintained. -
Algorithmic Factorization of Polynomials Over Number Fields
Rose-Hulman Institute of Technology Rose-Hulman Scholar Mathematical Sciences Technical Reports (MSTR) Mathematics 5-18-2017 Algorithmic Factorization of Polynomials over Number Fields Christian Schulz Rose-Hulman Institute of Technology Follow this and additional works at: https://scholar.rose-hulman.edu/math_mstr Part of the Number Theory Commons, and the Theory and Algorithms Commons Recommended Citation Schulz, Christian, "Algorithmic Factorization of Polynomials over Number Fields" (2017). Mathematical Sciences Technical Reports (MSTR). 163. https://scholar.rose-hulman.edu/math_mstr/163 This Dissertation is brought to you for free and open access by the Mathematics at Rose-Hulman Scholar. It has been accepted for inclusion in Mathematical Sciences Technical Reports (MSTR) by an authorized administrator of Rose-Hulman Scholar. For more information, please contact [email protected]. Algorithmic Factorization of Polynomials over Number Fields Christian Schulz May 18, 2017 Abstract The problem of exact polynomial factorization, in other words expressing a poly- nomial as a product of irreducible polynomials over some field, has applications in algebraic number theory. Although some algorithms for factorization over algebraic number fields are known, few are taught such general algorithms, as their use is mainly as part of the code of various computer algebra systems. This thesis provides a summary of one such algorithm, which the author has also fully implemented at https://github.com/Whirligig231/number-field-factorization, along with an analysis of the runtime of this algorithm. Let k be the product of the degrees of the adjoined elements used to form the algebraic number field in question, let s be the sum of the squares of these degrees, and let d be the degree of the polynomial to be factored; then the runtime of this algorithm is found to be O(d4sk2 + 2dd3). -
Flow Routing Techniques for Environmental Modeling
EPA/600/B-18/256 | August 2018 | www.epa.gov/research Flow Routing Techniques for Environmental Modeling photo 0 EPA/600/B-18/256 August 2018 Flow Routing Techniques for Environmental Modeling by Jan Sitterson1 , Chris Knightes2 , Brian Avant3 1 Oak Ridge Associated Universities 2 U.S. Environmental Protection Agency, Office of Research and Development 3 Oak Ridge Institute for Science and Education Chris Knightes - Project Officer Office of Research and Development National Exposure Research Laboratory Athens, GA, 30605 1 Notice The U.S. Environmental Protection Agency (EPA) through its Office of Research and Development funded and managed the research described here. The research described herein was conducted at the Computational Exposure Division of the U.S. Environmental Protection Agency National Exposure Research Laboratory in Athens, GA. Any mention of trade names, products, or services does not imply an endorsement by the U.S. Government or the U.S. Environmental Protection Agency. The EPA does not endorse any commercial products, services, or enterprises. This document has been reviewed by the U.S. Environmental Protection Agency, Office of Research and Development, and approved for publication. 2 Abstract This report describes a few ways to simulate the movement of water through a network of streams. Flow routing connects excess water from precipitation and runoff to the stream to other surface water as part of the hydrological cycle. Simulating flow helps elucidate the transportation of nutrients through a stream system, predict flood events, inform decision makers, and regulate water quality and quantity issues. Three flow routing techniques are presented and discussed in this paper. -
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. -
Some Properties of the Discriminant Matrices of a Linear Associative Algebra*
570 R. F. RINEHART [August, SOME PROPERTIES OF THE DISCRIMINANT MATRICES OF A LINEAR ASSOCIATIVE ALGEBRA* BY R. F. RINEHART 1. Introduction. Let A be a linear associative algebra over an algebraic field. Let d, e2, • • • , en be a basis for A and let £»•/*., (hjik = l,2, • • • , n), be the constants of multiplication corre sponding to this basis. The first and second discriminant mat rices of A, relative to this basis, are defined by Ti(A) = \\h(eres[ CrsiCij i, j=l T2(A) = \\h{eres / ,J CrsiC j II i,j=l where ti(eres) and fa{erea) are the first and second traces, respec tively, of eres. The first forms in terms of the constants of multi plication arise from the isomorphism between the first and sec ond matrices of the elements of A and the elements themselves. The second forms result from direct calculation of the traces of R(er)R(es) and S(er)S(es), R{ei) and S(ei) denoting, respectively, the first and second matrices of ei. The last forms of the dis criminant matrices show that each is symmetric. E. Noetherf and C. C. MacDuffeeJ discovered some of the interesting properties of these matrices, and shed new light on the particular case of the discriminant matrix of an algebraic equation. It is the purpose of this paper to develop additional properties of these matrices, and to interpret them in some fa miliar instances. Let A be subjected to a transformation of basis, of matrix M, 7 J rH%j€j 1,2, *0). -
Integration and Differential Equations
R.S. Johnson Integration and differential equations Download free eBooks at bookboon.com 2 Integration and differential equations © 2012 R.S. Johnson & Ventus Publishing ApS ISBN 978-87-7681-970-5 Download free eBooks at bookboon.com 3 Integration and differential equations Contents Contents Preface to these two texts 8 Part I An introduction to the standard methods of elementary integration 9 List of Integrals 10 Preface 11 1 Introduction and Background 12 1.1 The Riemann integral 12 1.2 The fundamental theorem of calculus 17 1.4 Theorems on integration 19 1.5 Standard integrals 23 1.6 Integration by parts 26 1.7 Improper integrals 32 1.8 Non-uniqueness of representation 35 Exercises 1 36 2 The integration of rational functions 38 2.1 Improper fractions 38 2.2 Linear denominator, Q(x) 39 The next step for top-performing graduates Masters in Management Designed for high-achieving graduates across all disciplines, London Business School’s Masters in Management provides specific and tangible foundations for a successful career in business. This 12-month, full-time programme is a business qualification with impact. In 2010, our MiM employment rate was 95% within 3 months of graduation*; the majority of graduates choosing to work in consulting or financial services. As well as a renowned qualification from a world-class business school, you also gain access to the School’s network of more than 34,000 global alumni – a community that offers support and opportunities throughout your career. For more information visit www.london.edu/mm, email [email protected] or give us a call on +44 (0)20 7000 7573. -
Algebra II Level 1
Algebra II 5 credits – Level I Grades: 10-11, Level: I Prerequisite: Minimum grade of 70 in Geometry Level I (or a minimum grade of 90 in Geometry Topics) Units of study include: equations and inequalities, linear equations and functions, systems of linear equations and inequalities, matrices and determinants, quadratic functions, polynomials and polynomial functions, powers, roots, and radicals, rational equations and functions, sequence and series, and probability and statistics. PROFICIENCIES INEQUALITIES -Solve inequalities -Solve combined inequalities -Use inequality models to solve problems -Solve absolute values in open sentences -Solve absolute value sentences graphically LINEAR EQUATION FUNCTIONS -Solve open sentences in two variables -Graph linear equations in two variables -Find the slope of the line -Write the equation of a line -Solve systems of linear equations in two variables -Apply systems of equations to solve real word problems -Solve linear inequalities in two variables -Find values of functions and graphs -Find equations of linear functions and apply properties of linear functions -Graph relations and determine when relations are functions PRODUCTS AND FACTORS OF POLYNOMIALS -Simplify, add and subtract polynomials -Use laws of exponents to multiply a polynomial by a monomial -Calculate the product of two or more polynomials -Demonstrate factoring -Solve polynomial equations and inequalities -Apply polynomial equations to solve real world problems RATIONAL EQUATIONS -Simplify quotients using laws of exponents -Simplify -
Math: Honors Geometry UNIT/Weeks (Not Timeline/Topics Essential Questions Consecutive)
Math: Honors Geometry UNIT/Weeks (not Timeline/Topics Essential Questions consecutive) Reasoning and Proof How can you make a Patterns and Inductive Reasoning conjecture and prove that it is Conditional Statements 2 true? Biconditionals Deductive Reasoning Reasoning in Algebra and Geometry Proving Angles Congruent Congruent Triangles How do you identify Congruent Figures corresponding parts of Triangle Congruence by SSS and SAS congruent triangles? Triangle Congruence by ASA and AAS How do you show that two 3 Using Corresponding Parts of Congruent triangles are congruent? Triangles How can you tell whether a Isosceles and Equilateral Triangles triangle is isosceles or Congruence in Right Triangles equilateral? Congruence in Overlapping Triangles Relationships Within Triangles Mid segments of Triangles How do you use coordinate Perpendicular and Angle Bisectors geometry to find relationships Bisectors in Triangles within triangles? 3 Medians and Altitudes How do you solve problems Indirect Proof that involve measurements of Inequalities in One Triangle triangles? Inequalities in Two Triangles How do you write indirect proofs? Polygons and Quadrilaterals How can you find the sum of The Polygon Angle-Sum Theorems the measures of polygon Properties of Parallelograms angles? Proving that a Quadrilateral is a Parallelogram How can you classify 3.2 Properties of Rhombuses, Rectangles and quadrilaterals? Squares How can you use coordinate Conditions for Rhombuses, Rectangles and geometry to prove general Squares