Introduction Vectors in Function Spaces
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Relativistic Dynamics
Chapter 4 Relativistic dynamics We have seen in the previous lectures that our relativity postulates suggest that the most efficient (lazy but smart) approach to relativistic physics is in terms of 4-vectors, and that velocities never exceed c in magnitude. In this chapter we will see how this 4-vector approach works for dynamics, i.e., for the interplay between motion and forces. A particle subject to forces will undergo non-inertial motion. According to Newton, there is a simple (3-vector) relation between force and acceleration, f~ = m~a; (4.0.1) where acceleration is the second time derivative of position, d~v d2~x ~a = = : (4.0.2) dt dt2 There is just one problem with these relations | they are wrong! Newtonian dynamics is a good approximation when velocities are very small compared to c, but outside of this regime the relation (4.0.1) is simply incorrect. In particular, these relations are inconsistent with our relativity postu- lates. To see this, it is sufficient to note that Newton's equations (4.0.1) and (4.0.2) predict that a particle subject to a constant force (and initially at rest) will acquire a velocity which can become arbitrarily large, Z t ~ d~v 0 f ~v(t) = 0 dt = t ! 1 as t ! 1 . (4.0.3) 0 dt m This flatly contradicts the prediction of special relativity (and causality) that no signal can propagate faster than c. Our task is to understand how to formulate the dynamics of non-inertial particles in a manner which is consistent with our relativity postulates (and then verify that it matches observation, including in the non-relativistic regime). -
Interval Notation and Linear Inequalities
CHAPTER 1 Introductory Information and Review Section 1.7: Interval Notation and Linear Inequalities Linear Inequalities Linear Inequalities Rules for Solving Inequalities: 86 University of Houston Department of Mathematics SECTION 1.7 Interval Notation and Linear Inequalities Interval Notation: Example: Solution: MATH 1300 Fundamentals of Mathematics 87 CHAPTER 1 Introductory Information and Review Example: Solution: Example: 88 University of Houston Department of Mathematics SECTION 1.7 Interval Notation and Linear Inequalities Solution: Additional Example 1: Solution: MATH 1300 Fundamentals of Mathematics 89 CHAPTER 1 Introductory Information and Review Additional Example 2: Solution: 90 University of Houston Department of Mathematics SECTION 1.7 Interval Notation and Linear Inequalities Additional Example 3: Solution: Additional Example 4: Solution: MATH 1300 Fundamentals of Mathematics 91 CHAPTER 1 Introductory Information and Review Additional Example 5: Solution: Additional Example 6: Solution: 92 University of Houston Department of Mathematics SECTION 1.7 Interval Notation and Linear Inequalities Additional Example 7: Solution: MATH 1300 Fundamentals of Mathematics 93 Exercise Set 1.7: Interval Notation and Linear Inequalities For each of the following inequalities: Write each of the following inequalities in interval (a) Write the inequality algebraically. notation. (b) Graph the inequality on the real number line. (c) Write the inequality in interval notation. 23. 1. x is greater than 5. 2. x is less than 4. 24. 3. x is less than or equal to 3. 4. x is greater than or equal to 7. 25. 5. x is not equal to 2. 6. x is not equal to 5 . 26. 7. x is less than 1. 8. -
171 Composition Operator on the Space of Functions
Acta Math. Univ. Comenianae 171 Vol. LXXXI, 2 (2012), pp. 171{183 COMPOSITION OPERATOR ON THE SPACE OF FUNCTIONS TRIEBEL-LIZORKIN AND BOUNDED VARIATION TYPE M. MOUSSAI Abstract. For a Borel-measurable function f : R ! R satisfying f(0) = 0 and Z sup t−1 sup jf 0(x + h) − f 0(x)jp dx < +1; (0 < p < +1); t>0 R jh|≤t s n we study the composition operator Tf (g) := f◦g on Triebel-Lizorkin spaces Fp;q(R ) in the case 0 < s < 1 + (1=p). 1. Introduction and the main result The study of the composition operator Tf : g ! f ◦ g associated to a Borel- s n measurable function f : R ! R on Triebel-Lizorkin spaces Fp;q(R ), consists in finding a characterization of the functions f such that s n s n (1.1) Tf (Fp;q(R )) ⊆ Fp;q(R ): The investigation to establish (1.1) was improved by several works, for example the papers of Adams and Frazier [1,2 ], Brezis and Mironescu [6], Maz'ya and Shaposnikova [9], Runst and Sickel [12] and [10]. There were obtained some necessary conditions on f; from which we recall the following results. For s > 0, 1 < p < +1 and 1 ≤ q ≤ +1 n s n s n • if Tf takes L1(R ) \ Fp;q(R ) to Fp;q(R ), then f is locally Lipschitz con- tinuous. n s n • if Tf takes the Schwartz space S(R ) to Fp;q(R ), then f belongs locally to s Fp;q(R). The first assertion is proved in [3, Theorem 3.1]. -
Interval Computations: Introduction, Uses, and Resources
Interval Computations: Introduction, Uses, and Resources R. B. Kearfott Department of Mathematics University of Southwestern Louisiana U.S.L. Box 4-1010, Lafayette, LA 70504-1010 USA email: [email protected] Abstract Interval analysis is a broad field in which rigorous mathematics is as- sociated with with scientific computing. A number of researchers world- wide have produced a voluminous literature on the subject. This article introduces interval arithmetic and its interaction with established math- ematical theory. The article provides pointers to traditional literature collections, as well as electronic resources. Some successful scientific and engineering applications are listed. 1 What is Interval Arithmetic, and Why is it Considered? Interval arithmetic is an arithmetic defined on sets of intervals, rather than sets of real numbers. A form of interval arithmetic perhaps first appeared in 1924 and 1931 in [8, 104], then later in [98]. Modern development of interval arithmetic began with R. E. Moore’s dissertation [64]. Since then, thousands of research articles and numerous books have appeared on the subject. Periodic conferences, as well as special meetings, are held on the subject. There is an increasing amount of software support for interval computations, and more resources concerning interval computations are becoming available through the Internet. In this paper, boldface will denote intervals, lower case will denote scalar quantities, and upper case will denote vectors and matrices. Brackets “[ ]” will delimit intervals while parentheses “( )” will delimit vectors and matrices.· Un- derscores will denote lower bounds of· intervals and overscores will denote upper bounds of intervals. Corresponding lower case letters will denote components of vectors. -
Comparative Programming Languages
CSc 372 Comparative Programming Languages 10 : Haskell — Curried Functions Department of Computer Science University of Arizona [email protected] Copyright c 2013 Christian Collberg 1/22 Infix Functions Declaring Infix Functions Sometimes it is more natural to use an infix notation for a function application, rather than the normal prefix one: 5+6 (infix) (+) 5 6 (prefix) Haskell predeclares some infix operators in the standard prelude, such as those for arithmetic. For each operator we need to specify its precedence and associativity. The higher precedence of an operator, the stronger it binds (attracts) its arguments: hence: 3 + 5*4 ≡ 3 + (5*4) 3 + 5*4 6≡ (3 + 5) * 4 3/22 Declaring Infix Functions. The associativity of an operator describes how it binds when combined with operators of equal precedence. So, is 5-3+9 ≡ (5-3)+9 = 11 OR 5-3+9 ≡ 5-(3+9) = -7 The answer is that + and - associate to the left, i.e. parentheses are inserted from the left. Some operators are right associative: 5^3^2 ≡ 5^(3^2) Some operators have free (or no) associativity. Combining operators with free associativity is an error: 5==4<3 ⇒ ERROR 4/22 Declaring Infix Functions. The syntax for declaring operators: infixr prec oper -- right assoc. infixl prec oper -- left assoc. infix prec oper -- free assoc. From the standard prelude: infixl 7 * infix 7 /, ‘div‘, ‘rem‘, ‘mod‘ infix 4 ==, /=, <, <=, >=, > An infix function can be used in a prefix function application, by including it in parenthesis. Example: ? (+) 5 ((*) 6 4) 29 5/22 Multi-Argument Functions Multi-Argument Functions Haskell only supports one-argument functions. -
More on Vectors Math 122 Calculus III D Joyce, Fall 2012
More on Vectors Math 122 Calculus III D Joyce, Fall 2012 Unit vectors. A unit vector is a vector whose length is 1. If a unit vector u in the plane R2 is placed in standard position with its tail at the origin, then it's head will land on the unit circle x2 + y2 = 1. Every point on the unit circle (x; y) is of the form (cos θ; sin θ) where θ is the angle measured from the positive x-axis in the counterclockwise direction. u=(x;y)=(cos θ; sin θ) 7 '$θ q &% Thus, every unit vector in the plane is of the form u = (cos θ; sin θ). We can interpret unit vectors as being directions, and we can use them in place of angles since they carry the same information as an angle. In three dimensions, we also use unit vectors and they will still signify directions. Unit 3 vectors in R correspond to points on the sphere because if u = (u1; u2; u3) is a unit vector, 2 2 2 3 then u1 + u2 + u3 = 1. Each unit vector in R carries more information than just one angle since, if you want to name a point on a sphere, you need to give two angles, longitude and latitude. Now that we have unit vectors, we can treat every vector v as a length and a direction. The length of v is kvk, of course. And its direction is the unit vector u in the same direction which can be found by v u = : kvk The vector v can be reconstituted from its length and direction by multiplying v = kvk u. -
Lecture 3.Pdf
ENGR-1100 Introduction to Engineering Analysis Lecture 3 POSITION VECTORS & FORCE VECTORS Today’s Objectives: Students will be able to : a) Represent a position vector in Cartesian coordinate form, from given geometry. In-Class Activities: • Applications / b) Represent a force vector directed along Relevance a line. • Write Position Vectors • Write a Force Vector along a line 1 DOT PRODUCT Today’s Objective: Students will be able to use the vector dot product to: a) determine an angle between In-Class Activities: two vectors, and, •Applications / Relevance b) determine the projection of a vector • Dot product - Definition along a specified line. • Angle Determination • Determining the Projection APPLICATIONS This ship’s mooring line, connected to the bow, can be represented as a Cartesian vector. What are the forces in the mooring line and how do we find their directions? Why would we want to know these things? 2 APPLICATIONS (continued) This awning is held up by three chains. What are the forces in the chains and how do we find their directions? Why would we want to know these things? POSITION VECTOR A position vector is defined as a fixed vector that locates a point in space relative to another point. Consider two points, A and B, in 3-D space. Let their coordinates be (XA, YA, ZA) and (XB, YB, ZB), respectively. 3 POSITION VECTOR The position vector directed from A to B, rAB , is defined as rAB = {( XB –XA ) i + ( YB –YA ) j + ( ZB –ZA ) k }m Please note that B is the ending point and A is the starting point. -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
Multidisciplinary Design Project Engineering Dictionary Version 0.0.2
Multidisciplinary Design Project Engineering Dictionary Version 0.0.2 February 15, 2006 . DRAFT Cambridge-MIT Institute Multidisciplinary Design Project This Dictionary/Glossary of Engineering terms has been compiled to compliment the work developed as part of the Multi-disciplinary Design Project (MDP), which is a programme to develop teaching material and kits to aid the running of mechtronics projects in Universities and Schools. The project is being carried out with support from the Cambridge-MIT Institute undergraduate teaching programe. For more information about the project please visit the MDP website at http://www-mdp.eng.cam.ac.uk or contact Dr. Peter Long Prof. Alex Slocum Cambridge University Engineering Department Massachusetts Institute of Technology Trumpington Street, 77 Massachusetts Ave. Cambridge. Cambridge MA 02139-4307 CB2 1PZ. USA e-mail: [email protected] e-mail: [email protected] tel: +44 (0) 1223 332779 tel: +1 617 253 0012 For information about the CMI initiative please see Cambridge-MIT Institute website :- http://www.cambridge-mit.org CMI CMI, University of Cambridge Massachusetts Institute of Technology 10 Miller’s Yard, 77 Massachusetts Ave. Mill Lane, Cambridge MA 02139-4307 Cambridge. CB2 1RQ. USA tel: +44 (0) 1223 327207 tel. +1 617 253 7732 fax: +44 (0) 1223 765891 fax. +1 617 258 8539 . DRAFT 2 CMI-MDP Programme 1 Introduction This dictionary/glossary has not been developed as a definative work but as a useful reference book for engi- neering students to search when looking for the meaning of a word/phrase. It has been compiled from a number of existing glossaries together with a number of local additions. -
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 -
Concept of a Dyad and Dyadic: Consider Two Vectors a and B Dyad: It Consists of a Pair of Vectors a B for Two Vectors a a N D B
1/11/2010 CHAPTER 1 Introductory Concepts • Elements of Vector Analysis • Newton’s Laws • Units • The basis of Newtonian Mechanics • D’Alembert’s Principle 1 Science of Mechanics: It is concerned with the motion of material bodies. • Bodies have different scales: Microscropic, macroscopic and astronomic scales. In mechanics - mostly macroscopic bodies are considered. • Speed of motion - serves as another important variable - small and high (approaching speed of light). 2 1 1/11/2010 • In Newtonian mechanics - study motion of bodies much bigger than particles at atomic scale, and moving at relative motions (speeds) much smaller than the speed of light. • Two general approaches: – Vectorial dynamics: uses Newton’s laws to write the equations of motion of a system, motion is described in physical coordinates and their derivatives; – Analytical dynamics: uses energy like quantities to define the equations of motion, uses the generalized coordinates to describe motion. 3 1.1 Vector Analysis: • Scalars, vectors, tensors: – Scalar: It is a quantity expressible by a single real number. Examples include: mass, time, temperature, energy, etc. – Vector: It is a quantity which needs both direction and magnitude for complete specification. – Actually (mathematically), it must also have certain transformation properties. 4 2 1/11/2010 These properties are: vector magnitude remains unchanged under rotation of axes. ex: force, moment of a force, velocity, acceleration, etc. – geometrically, vectors are shown or depicted as directed line segments of proper magnitude and direction. 5 e (unit vector) A A = A e – if we use a coordinate system, we define a basis set (iˆ , ˆj , k ˆ ): we can write A = Axi + Ay j + Azk Z or, we can also use the A three components and Y define X T {A}={Ax,Ay,Az} 6 3 1/11/2010 – The three components Ax , Ay , Az can be used as 3-dimensional vector elements to specify the vector. -
Two Worked out Examples of Rotations Using Quaternions
TWO WORKED OUT EXAMPLES OF ROTATIONS USING QUATERNIONS This note is an attachment to the article \Rotations and Quaternions" which in turn is a companion to the video of the talk by the same title. Example 1. Determine the image of the point (1; −1; 2) under the rotation by an angle of 60◦ about an axis in the yz-plane that is inclined at an angle of 60◦ to the positive y-axis. p ◦ ◦ 1 3 Solution: The unit vector u in the direction of the axis of rotation is cos 60 j + sin 60 k = 2 j + 2 k. The quaternion (or vector) corresponding to the point p = (1; −1; 2) is of course p = i − j + 2k. To find −1 θ θ the image of p under the rotation, we calculate qpq where q is the quaternion cos 2 + sin 2 u and θ the angle of rotation (60◦ in this case). The resulting quaternion|if we did the calculation right|would have no constant term and therefore we can interpret it as a vector. That vector gives us the answer. p p p p p We have q = 3 + 1 u = 3 + 1 j + 3 k = 1 (2 3 + j + 3k). Since q is by construction a unit quaternion, 2 2 2 4 p4 4 p −1 1 its inverse is its conjugate: q = 4 (2 3 − j − 3k). Now, computing qp in the routine way, we get 1 p p p p qp = ((1 − 2 3) + (2 + 3 3)i − 3j + (4 3 − 1)k) 4 and then another long but routine computation gives 1 p p p qpq−1 = ((10 + 4 3)i + (1 + 2 3)j + (14 − 3 3)k) 8 The point corresponding to the vector on the right hand side in the above equation is the image of (1; −1; 2) under the given rotation.