5. Choice under Uncertainty
|
|
- Ἁλκυόνη Καζαντζής
- 5 χρόνια πριν
- Προβολές:
Transcript
1 5. Choice under Uncertainty Daisuke Oyama Microeconomics I May 23, 2018
2 Formulations von Neumann-Morgenstern (1944/1947) X: Set of prizes Π: Set of probability distributions on X : Preference relation on Π Representation: satisfies certain axioms if and only if there exists a function U : X R such that u(π) = x X U(x)π(x) represents. 1 / 22
3 Savage (1954) S: Set of states of nature X: Set of consequences/prizes An act is a function from S to X. A: Set of all acts (all functions from S to X) : Preference relation on A Representation: satisfies certain axioms if and only if there exist a probability distribution p on S and a function U : X R such that u(a) = s S U(a(s))p(s) represents. 2 / 22
4 Anscombe-Aumann (1963) S: Set of states of nature X: Set of consequences/prizes Π: Set of probability distributions on X H: Set of all functions from S to Π : Preference relation on H Representation: satisfies certain axioms if and only if there exist a probability distribution p on S and a function U : X R such that u(h) = [ ] U(x)h(s)(x) p(s) s S x X represents. 3 / 22
5 von Neumann-Morgenstern X = {x 1,..., x n }: Finite set of prizes (for simplicity) Π: Set of probability distributions on X, called lotteries δ x Π: Lottery that gives x X with probability 1 Compound lotteries: For π, ρ Π, the compound lottery aπ + (1 a)ρ is identified with the corresponding lottery. Example: X = {x, y, z, w} π = 0.3x + 0.1y + 0.6z + 0w ρ = 0.6x + 0y + 0z + 0.4w (1/3)π + (2/3)ρ is identified with the lottery 0.5x + (1/30)y + 0.2z + (4/15)w : Preference relation on Π 4 / 22
6 Axioms Order: is complete and transitive. Substitution (Independence): For all π, ρ, φ, φ Π and all a [0, 1], aπ+(1 a)φ aρ+(1 a)φ aπ+(1 a)φ aρ+(1 a)φ. Continuity: For all π, ρ, φ Π, π ρ = a (0, 1) : a (a, 1] : aπ + (1 a)φ ρ and π aρ + (1 a)φ. 5 / 22
7 Under Order, Substitution is equivalent to the following: Substitution : For all π, ρ, φ, φ Π and all a [0, 1], aπ+(1 a)φ aρ+(1 a)φ aπ+(1 a)φ aρ+(1 a)φ. Substitution implies the following: Indifference Substitution: For all π, ρ, φ, φ Π and all a [0, 1], aπ+(1 a)φ aρ+(1 a)φ aπ+(1 a)φ aρ+(1 a)φ. 6 / 22
8 Under Order, Continuity is equivalent to the following: Continuity : For all π, ρ, φ Π, {a [0, 1] aπ+(1 a)φ ρ} and {a [0, 1] ρ aπ+(1 a)φ} are open (relative to [0, 1]). Continuity : For all π, ρ, φ Π, {a [0, 1] aπ+(1 a)φ ρ} and {a [0, 1] ρ aπ+(1 a)φ} are closed (relative to [0, 1]). 7 / 22
9 Expected Utility Representation Proposition 5.3 For a preference relation on Π, the following statements are equivalent: 1. satisfies Order, Substitution, and Continuity. 2. There exists a function U : X R such that the utility function u: Π R defined by u(π) = x X U(x)π(x) represents. Moreover, U is unique up to positive affine transformations, i.e., x X V (x)π(x) represents if and only if there exist A > 0 and B such that V (x) = AU(x) + B for all x X. U is sometimes called a von Neumann-Morgenstern function. 8 / 22
10 Just to recall: Order: is complete and transitive. Substitution: For all π, ρ, φ, φ Π and all a [0, 1], aπ+(1 a)φ aρ+(1 a)φ aπ+(1 a)φ aρ+(1 a)φ. Continuity: For all π, ρ, φ Π, π ρ = a (0, 1) : a (a, 1] : aπ + (1 a)φ ρ and π aρ + (1 a)φ. 9 / 22
11 Proof of 1 2 In the following, we assume that satisfies Order. Step 1 (Lemmas ) Strict Betweenness: For all π, ρ Π and all a (0, 1), π ρ = π aπ + (1 a)ρ ρ. Substitution + Continuity = Strict Betweenness Step 2 (Lemma ) Indifference Betweenness: For all π, ρ Π and all a [0, 1], π ρ = π aπ + (1 a)ρ ρ. Substitution + Continuity = Indifference Betweenness 10 / 22
12 Step 3 (Lemma 5.7) Mixture Monotonicity: For all π, ρ Π, if π ρ, then aπ + (1 a)ρ bπ + (1 b)ρ a b. Strict Betweenness = Mixture Monotonicity 11 / 22
13 Step 4 (Lemma 5.8) Indifference Independence: For all π, ρ, φ Π and all a [0, 1], π ρ = aπ + (1 a)φ aρ + (1 a)φ. Indifference Betweenness + Substitution = Indifference Independence Step 5 Strict Independence: For all π, ρ, φ Π and all a (0, 1), π ρ = aπ + (1 a)φ aρ + (1 a)φ. Strict Betweenness + Substitution = Strict Independence 12 / 22
14 Step 6 (Lemma 5.9) Mixture Intermediate Value: For all π, ρ, φ Π, π ρ φ = a [0, 1] : aπ + (1 a)φ ρ. Continuity = Mixture Intermediate Value Step 7 (Lemma 5.9) Unique Mixture Intermediate Value: For all π, ρ, φ Π, π φ, π ρ φ =! a [0, 1] : aπ + (1 a)φ ρ. Mixture Intermediate Value + Mixture Monotonicity = Unique Mixture Intermediate Value 13 / 22
15 Step 8 X: finite + Indifference Betweenness + Strict Betweenness = π, π Π : π π π for all π Π Notes: In the textbook, X is possibly an infinite set, and Π is the set of simple probability distributions (probability distributions with finite support). In the proof there, the existence of such π and π is assumed, whereas this assumption is not necessary; see Problem / 22
16 Step 9 Unique Mixture Intermediate Value + Mixture Monotonicity = Utility representation Proof Assume that π π. By Unique Mixture Intermediate Value, for each π Π, there is a unique a π [0, 1] such that π a π π + (1 a π )π. Define the function u: Π R by u(π) = a π for each π Π. Then u represents : for any π, ρ Π, π ρ u(π)π + (1 u(π))π u(ρ)π + (1 u(ρ))π u(π) u(ρ) by Mixture Monotonicity. 15 / 22
17 Step 10 Utility representation + Indifference Independence = Linear utility representation Proof Take any π, ρ Π and a [0, 1]. By the construction of u, we have π u(π)π + (1 u(π))π and ρ u(ρ)π + (1 u(ρ))π. By Indifference Independence, aπ + (1 a)ρ a[u(π)π + (1 u(π))π] + (1 a)ρ a[u(π)π + (1 u(π))π] + (1 a)[u(ρ)π + (1 u(ρ))π] = [au(π) + (1 a)u(ρ)]π + [1 (u(π) + (1 a)u(ρ))]π. Therefore, by the construction of u, we have u(aπ + (1 a)ρ) = au(π) + (1 a)u(ρ). 16 / 22
18 Step 11 Linear utility representation = Expected utility representation Proof Define U : X R by U(x) = u(δ x ) for x X. For each π Π, which is written as x X π(x)δ x, by the linearity of u we have u(π) = x X π(x)u(δ x ) = x X π(x)u(x). 17 / 22
19 Step 12 Linear utility representation = Unique linear utility representation up to positive affine transformation i.e., if π ρ u(π) u(ρ) and π ρ v(π) v(ρ), and u and v are linear, then v( ) = Au( ) + B for some A > 0 and B. 18 / 22
20 Proof Assume that π π. Take any π Π, and define a π = u(π) u(π) [0, 1]. u(π) u(π) Since u is linear, u(a π π + (1 a π )π) = a π u(π) + (1 a π )u(π) = u(π). Since u is a representation of, this implies that a π π + (1 a π )π π. Since v is a representation of, this implies that v(a π π + (1 a π )π) = v(π), where v(a π π + (1 a π )π) = a π v(π) + (1 a π )v(π) since v is linear. Hence v(π) = Au(π) + B where A = v(π) v(π) u(π) u(π) > 0, which are independent of π. B = Au(π) + v(π), 19 / 22
21 Step 13 Unique linear utility representation up to positive affine transformation = Unique expected utility representation up to positive affine transformation Proof i.e., if π ρ x X U(x)π(x) x X U(x)ρ(x) and π ρ x X V (x)π(x) x X V (x)ρ(x), then V ( ) = AU( ) + B for some A > 0 and B. Let u(π) = x X U(x)π(x) and v(π) = x X V (x)π(x). By v( ) = Au( ) + B, A > 0, for all x X we have V (x) = v(δ x ) = Au(δ x ) + B = AU(x) + B. 20 / 22
22 Implications of Substitution + Continuity Assume that X = {x 1, x 2, x 3 } and δ x1 δ x2 δ x3. Indifference curves are straight lines: if π ρ, then aπ + (1 a)ρ ρ for all a [0, 1]. (Indifference Betweenness) Indifference curves are parallel straight lines: if π ρ, then aπ + (1 a)δ x1 aρ + (1 a)δ x1 and aπ + (1 a)δ x3 aρ + (1 a)δ x3 for all a [0, 1]. (Indifference Independence) 21 / 22
23 Critiques Allais Paradox Ellsberg Paradox Chapter / 22
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραORDINAL ARITHMETIC JULIAN J. SCHLÖDER
ORDINAL ARITHMETIC JULIAN J. SCHLÖDER Abstract. We define ordinal arithmetic and show laws of Left- Monotonicity, Associativity, Distributivity, some minor related properties and the Cantor Normal Form.
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότεραLecture 13 - Root Space Decomposition II
Lecture 13 - Root Space Decomposition II October 18, 2012 1 Review First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra h (which we are calling a CSA, or Cartan Subalgebra).
Διαβάστε περισσότεραFractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραSequent Calculi for the Modal µ-calculus over S5. Luca Alberucci, University of Berne. Logic Colloquium Berne, July 4th 2008
Sequent Calculi for the Modal µ-calculus over S5 Luca Alberucci, University of Berne Logic Colloquium Berne, July 4th 2008 Introduction Koz: Axiomatisation for the modal µ-calculus over K Axioms: All classical
Διαβάστε περισσότεραMath 446 Homework 3 Solutions. (1). (i): Reverse triangle inequality for metrics: Let (X, d) be a metric space and let x, y, z X.
Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequalit for metrics: Let (X, d) be a metric space and let x,, z X. Prove that d(x, z) d(, z) d(x, ). (ii): Reverse triangle inequalit for norms:
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραMINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS
MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS FUMIE NAKAOKA AND NOBUYUKI ODA Received 20 December 2005; Revised 28 May 2006; Accepted 6 August 2006 Some properties of minimal closed sets and maximal closed
Διαβάστε περισσότεραBounding Nonsplitting Enumeration Degrees
Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007 Goal: Introduce a form of Σ 0 2-permitting for the enumeration degrees. Till now,
Διαβάστε περισσότεραGenerating Set of the Complete Semigroups of Binary Relations
Applied Mathematics 06 7 98-07 Published Online January 06 in SciRes http://wwwscirporg/journal/am http://dxdoiorg/036/am067009 Generating Set of the Complete Semigroups of Binary Relations Yasha iasamidze
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραChapter 3: Ordinal Numbers
Chapter 3: Ordinal Numbers There are two kinds of number.. Ordinal numbers (0th), st, 2nd, 3rd, 4th, 5th,..., ω, ω +,... ω2, ω2+,... ω 2... answers to the question What position is... in a sequence? What
Διαβάστε περισσότεραSOME PROPERTIES OF FUZZY REAL NUMBERS
Sahand Communications in Mathematical Analysis (SCMA) Vol. 3 No. 1 (2016), 21-27 http://scma.maragheh.ac.ir SOME PROPERTIES OF FUZZY REAL NUMBERS BAYAZ DARABY 1 AND JAVAD JAFARI 2 Abstract. In the mathematical
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότερα12. Radon-Nikodym Theorem
Tutorial 12: Radon-Nikodym Theorem 1 12. Radon-Nikodym Theorem In the following, (Ω, F) is an arbitrary measurable space. Definition 96 Let μ and ν be two (possibly complex) measures on (Ω, F). We say
Διαβάστε περισσότεραTridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Διαβάστε περισσότεραCongruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Διαβάστε περισσότεραNowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραLecture 15 - Root System Axiomatics
Lecture 15 - Root System Axiomatics Nov 1, 01 In this lecture we examine root systems from an axiomatic point of view. 1 Reflections If v R n, then it determines a hyperplane, denoted P v, through the
Διαβάστε περισσότεραLecture 21: Properties and robustness of LSE
Lecture 21: Properties and robustness of LSE BLUE: Robustness of LSE against normality We now study properties of l τ β and σ 2 under assumption A2, i.e., without the normality assumption on ε. From Theorem
Διαβάστε περισσότεραF A S C I C U L I M A T H E M A T I C I
F A S C I C U L I M A T H E M A T I C I Nr 46 2011 C. Carpintero, N. Rajesh and E. Rosas ON A CLASS OF (γ, γ )-PREOPEN SETS IN A TOPOLOGICAL SPACE Abstract. In this paper we have introduced the concept
Διαβάστε περισσότεραω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω
0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραChap. 6 Pushdown Automata
Chap. 6 Pushdown Automata 6.1 Definition of Pushdown Automata Example 6.1 L = {wcw R w (0+1) * } P c 0P0 1P1 1. Start at state q 0, push input symbol onto stack, and stay in q 0. 2. If input symbol is
Διαβάστε περισσότερα1. Introduction and Preliminaries.
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.yu/filomat Filomat 22:1 (2008), 97 106 ON δ SETS IN γ SPACES V. Renuka Devi and D. Sivaraj Abstract We
Διαβάστε περισσότεραTHE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS. Daniel A. Romano
235 Kragujevac J. Math. 30 (2007) 235 242. THE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS Daniel A. Romano Department of Mathematics and Informatics, Banja Luka University, Mladena Stojanovića
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραHomomorphism in Intuitionistic Fuzzy Automata
International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic
Διαβάστε περισσότεραDepth versus Rigidity in the Design of International Trade Agreements. Leslie Johns
Depth versus Rigidity in the Design of International Trade Agreements Leslie Johns Supplemental Appendix September 3, 202 Alternative Punishment Mechanisms The one-period utility functions of the home
Διαβάστε περισσότεραΑπόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότεραA Note on Intuitionistic Fuzzy. Equivalence Relation
International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραSOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max
Διαβάστε περισσότεραMean-Variance Analysis
Mean-Variance Analysis Jan Schneider McCombs School of Business University of Texas at Austin Jan Schneider Mean-Variance Analysis Beta Representation of the Risk Premium risk premium E t [Rt t+τ ] R1
Διαβάστε περισσότερα6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Διαβάστε περισσότεραAffine Weyl Groups. Gabriele Nebe. Summerschool GRK 1632, September Lehrstuhl D für Mathematik
Affine Weyl Groups Gabriele Nebe Lehrstuhl D für Mathematik Summerschool GRK 1632, September 2015 Crystallographic root systems. Definition A crystallographic root system Φ is a finite set of non zero
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραOperation Approaches on α-γ-open Sets in Topological Spaces
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 10, 491-498 Operation Approaches on α-γ-open Sets in Topological Spaces N. Kalaivani Department of Mathematics VelTech HighTec Dr.Rangarajan Dr.Sakunthala
Διαβάστε περισσότεραOverview. Transition Semantics. Configurations and the transition relation. Executions and computation
Overview Transition Semantics Configurations and the transition relation Executions and computation Inference rules for small-step structural operational semantics for the simple imperative language Transition
Διαβάστε περισσότεραA Two-Sided Laplace Inversion Algorithm with Computable Error Bounds and Its Applications in Financial Engineering
Electronic Companion A Two-Sie Laplace Inversion Algorithm with Computable Error Bouns an Its Applications in Financial Engineering Ning Cai, S. G. Kou, Zongjian Liu HKUST an Columbia University Appenix
Διαβάστε περισσότεραThe semiclassical Garding inequality
The semiclassical Garding inequality We give a proof of the semiclassical Garding inequality (Theorem 4.1 using as the only black box the Calderon-Vaillancourt Theorem. 1 Anti-Wick quantization For (q,
Διαβάστε περισσότεραParametrized Surfaces
Parametrized Surfaces Recall from our unit on vector-valued functions at the beginning of the semester that an R 3 -valued function c(t) in one parameter is a mapping of the form c : I R 3 where I is some
Διαβάστε περισσότεραDIRECT PRODUCT AND WREATH PRODUCT OF TRANSFORMATION SEMIGROUPS
GANIT J. Bangladesh Math. oc. IN 606-694) 0) -7 DIRECT PRODUCT AND WREATH PRODUCT OF TRANFORMATION EMIGROUP ubrata Majumdar, * Kalyan Kumar Dey and Mohd. Altab Hossain Department of Mathematics University
Διαβάστε περισσότεραSCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Διαβάστε περισσότεραLecture 34 Bootstrap confidence intervals
Lecture 34 Bootstrap confidence intervals Confidence Intervals θ: an unknown parameter of interest We want to find limits θ and θ such that Gt = P nˆθ θ t If G 1 1 α is known, then P θ θ = P θ θ = 1 α
Διαβάστε περισσότεραSolution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Διαβάστε περισσότεραLecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
Διαβάστε περισσότερα3.2 Simple Roots and Weyl Group
44 CHAPTER 3. ROOT SYSTEMS 3.2 Simple Roots and Weyl Group In this section, we fix a root system Φ of rank l in a euclidean space E, with Weyl group W. 3.2.1 Bases and Weyl chambers Def. A subset of Φ
Διαβάστε περισσότερα2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
Διαβάστε περισσότεραSome new generalized topologies via hereditary classes. Key Words:hereditary generalized topological space, A κ(h,µ)-sets, κµ -topology.
Bol. Soc. Paran. Mat. (3s.) v. 30 2 (2012): 71 77. c SPM ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v30i2.13793 Some new generalized topologies via hereditary
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραk A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Διαβάστε περισσότεραAbout these lecture notes. Simply Typed λ-calculus. Types
About these lecture notes Simply Typed λ-calculus Akim Demaille akim@lrde.epita.fr EPITA École Pour l Informatique et les Techniques Avancées Many of these slides are largely inspired from Andrew D. Ker
Διαβάστε περισσότεραThe Pohozaev identity for the fractional Laplacian
The Pohozaev identity for the fractional Laplacian Xavier Ros-Oton Departament Matemàtica Aplicada I, Universitat Politècnica de Catalunya (joint work with Joaquim Serra) Xavier Ros-Oton (UPC) The Pohozaev
Διαβάστε περισσότεραCHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραA General Note on δ-quasi Monotone and Increasing Sequence
International Mathematical Forum, 4, 2009, no. 3, 143-149 A General Note on δ-quasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in
Διαβάστε περισσότεραCovariance and Pseudo-Covariance of Complex Uncertain Variables
Covariance and Pseudo-Covariance of Complex Uncertain Variables Rong Gao 1, Hamed Ahmadzade 2, Mojtaba Esfahani 3 1. School of Economics and Management, Hebei University of Technology, Tianjin 341, China
Διαβάστε περισσότεραCyclic or elementary abelian Covers of K 4
Cyclic or elementary abelian Covers of K 4 Yan-Quan Feng Mathematics, Beijing Jiaotong University Beijing 100044, P.R. China Summer School, Rogla, Slovenian 2011-06 Outline 1 Question 2 Main results 3
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραPractice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Διαβάστε περισσότεραIterated trilinear fourier integrals with arbitrary symbols
Cornell University ICM 04, Satellite Conference in Harmonic Analysis, Chosun University, Gwangju, Korea August 6, 04 Motivation the Coifman-Meyer theorem with classical paraproduct(979) B(f, f )(x) :=
Διαβάστε περισσότεραTakeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, RIMS
Takeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, 2017 @ RIMS Contents Introduction Generalized Karcher equation Ando-Hiai inequalities Problem Introduction PP: The set of all positive definite operators
Διαβάστε περισσότεραde Rham Theorem May 10, 2016
de Rham Theorem May 10, 2016 Stokes formula and the integration morphism: Let M = σ Σ σ be a smooth triangulated manifold. Fact: Stokes formula σ ω = σ dω holds, e.g. for simplices. It can be used to define
Διαβάστε περισσότεραNew bounds for spherical two-distance sets and equiangular lines
New bounds for spherical two-distance sets and equiangular lines Michigan State University Oct 8-31, 016 Anhui University Definition If X = {x 1, x,, x N } S n 1 (unit sphere in R n ) and x i, x j = a
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραSOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr (T t N n) Pr (max (X 1,..., X N ) t N n) Pr (max
Διαβάστε περισσότεραAn Inventory of Continuous Distributions
Appendi A An Inventory of Continuous Distributions A.1 Introduction The incomplete gamma function is given by Also, define Γ(α; ) = 1 with = G(α; ) = Z 0 Z 0 Z t α 1 e t dt, α > 0, >0 t α 1 e t dt, α >
Διαβάστε περισσότεραMath 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
Διαβάστε περισσότεραSTAT200C: Hypothesis Testing
STAT200C: Hypothesis Testing Zhaoxia Yu Spring 2017 Some Definitions A hypothesis is a statement about a population parameter. The two complementary hypotheses in a hypothesis testing are the null hypothesis
Διαβάστε περισσότεραStatistics 104: Quantitative Methods for Economics Formula and Theorem Review
Harvard College Statistics 104: Quantitative Methods for Economics Formula and Theorem Review Tommy MacWilliam, 13 tmacwilliam@college.harvard.edu March 10, 2011 Contents 1 Introduction to Data 5 1.1 Sample
Διαβάστε περισσότεραECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Διαβάστε περισσότεραResearch Article Existence of Positive Solutions for m-point Boundary Value Problems on Time Scales
Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 29, Article ID 189768, 12 pages doi:1.1155/29/189768 Research Article Existence of Positive Solutions for m-point Boundary
Διαβάστε περισσότεραGAUGES OF BAIRE CLASS ONE FUNCTIONS
GAUGES OF BAIRE CLASS ONE FUNCTIONS ZULIJANTO ATOK, WEE-KEE TANG, AND DONGSHENG ZHAO Abstract. Let K be a compact metric space and f : K R be a bounded Baire class one function. We proved that for any
Διαβάστε περισσότεραSection 9.2 Polar Equations and Graphs
180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραAbstract Storage Devices
Abstract Storage Devices Robert König Ueli Maurer Stefano Tessaro SOFSEM 2009 January 27, 2009 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD
Διαβάστε περισσότεραChapter 2. Ordinals, well-founded relations.
Chapter 2. Ordinals, well-founded relations. 2.1. Well-founded Relations. We start with some definitions and rapidly reach the notion of a well-ordered set. Definition. For any X and any binary relation
Διαβάστε περισσότερα