New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
mathematics
numerical analysis

Foundations of Mathematical Economics 1st edition Michael Carter - Solutions

Let y be a boundary point of a nonempty, convex set S in a Euclidean space X. There exists a supporting hyperplane at y; that is, there exists a continuous linear functional f ∈ X* such that f (y) ≤ f (x) for every x ∈ S
Generalize exercise 3.182 to dispense with the assumption that S is closed. That is, let S be a nonempty, convex set in a Euclidean space X and y ∉ S. There exists a continuous linear functional f ∈ X* such that f (y) ≤ f (x). for every x ∈ S
Let S be an open convex subset of a linear space X and f ∈ X' a nonzero linear functional on X. Then f (S) is an open interval in ℜ.
Prove theorem 3.2 (assuming that X is Euclidean). [Hint: Apply exercise 3.184 to separate 0 from S = B + (- A) Then use exercise 3.185.]
Let Hf (c) be a bounding hyperplane of a cone C in a normed linear space X, that is, f (x) ≤ c for every x ∈ C. Then f (x) ≥ 0 for every x ∈ C
A linear function f: X → Y is nonsingular if and only if kernel f = {0} and f(X) = Y.
Let Hf (c) be a bounding hyperplane of a subspace Z of a normed linear space X, that is, f (x) ≤ c for every x ∈ Z. Then Z is contained in the kernel of f, that is, f (x) = 0 for every x ∈ Z
Prove corollary 3.2.3. Separation theorems are so pervasive in mathematical economics that it is necessary to have a range of variations in the armory. In the following sections we develop some refinements of the basic separating hyperplane theorem that are useful in applications.
Let A and B be nonempty, convex subsets in a normed linear space X with int A ‰ ˆ… and int A‹‚B ˆ…. Then A and B can be properly separated.Frequently stronger forms of separation are required. Two sets A and B are strictly separated by a hyperplane Hf. if A
Prove proposition 3.14 for a finite-dimensional space X. [Hint: Apply exercise 3.182 to the set B - A. Compactness of A is necessary to ensure that B - A is closed.] The following exercise shows that compactness of A is essential in proposition 3.14.
1. Let A and B be nonempty, disjoint, convex subsets in a normed linear space X. A and B can be strongly separated if and only if there exists a convex neighborhood of U of 0 such that (A + U) ⋂B = ∅ 2. Prove proposition 3.14.
Let A and B be convex subsets in a finite-dimensional normed linear space X. A and B can be strongly separated if and only if P(A, B) = inf{||x - y|| : x ∈ A, y ∈ B} > 0 Combining proposition 3.14 with corollary 3.2.3 gives the following important result, a geometric form of the Hahn-Banach
Let M be a nonempty, closed, subspace of a linear space X and y ∉ M. Then there exists a continuous linear functional f ∈ X* such that f (y) > 0 and f (x) = 0 for every x ∈ As an application of the previous result, we use it in the following exercise to provide an alternative derivation of
Let g1; g2; . . . ; gm be linear functional on a linear space X, and letS ={x ˆˆ X : gj(x) = 0, j = 1,2,..., m}=Suppose that f ^ 0 is another linear functional such that such that f (x)= 0 for every x e S. Show that Figure 3.20 The Fredholm alternative via separation 1. The set Z . {f} x,
Show that the set L(X, Y) of all linear functions X → Y is a linear space.
The inverse of a (nonsingular) linear function is linear.
Show the converse; that is, if f (x) =then f .(x). 0 for every x ˆˆ S where S . {x ˆˆ X : gj(x) = 0 = j . 1; 2 . . .m}.
Let g1, g, . . . , gm be linear functionals on a linear space X. For fixed numbers cj, the systems of equationsgj(x) = cj, j = 1, 2, . . . ,mis consistent if and only iffor every set of numbers Î»1, Î»2, . . . , Î»m.
Let K be a closed convex cone in a finite-dimensional linear space X and M a subspace with K ∩ M = {0}. Then there exists a linear functional f ∈ X* such that f(x) > 0 for every x ∈ K\{0} and f(x) = 0 for every x ∈ M
Suppose that f0 is a linear functional defined on a subspace Z of X such thatf0(x) ‰¤ g(x)† for every x ˆˆ Zwhere g ˆˆ X* is convex. Show that1. The setsand are convex subsets of the linear space Y = X Ã— „œ (figure 3.21) 2. int A
Let f0 be a bounded linear functional on a subspace Z of a normed linear space X. Then f0 can be extended to a linear functional on the whole space X without increasing its norm, that is,
Let x0 be an element of a normed linear space X. There exists a linear functional f ∈ X* such that ||f|| = 1 and f(x0) = ||x0||.
Let x1, x2 be distinct points in a normed linear space X. There exists a continuous linear functional f ˆˆ X* that evaluates them differently, that is, such that
Let S be a nonempty compact convex subset of a normed linear space X. 1. Let
Let S be a compact convex set. Every supporting hyperplane to S contains an extreme point of S.
Every compact, convex set is the closed convex hull of its extreme points.
If f, g are nonsingular linear functions, then so is their composition g ∘ f with (g ∘ f)-1 = f-1 ∘ g-1 The following converse of 3.14 is the linear version of the important implicit function theorem (theorem 4.5).
Let {S1, S2, . . . , Sn} be a collection of nonempty compact subsets of an m-dimensional linear space, and let x ˆˆ convWe consider the Cartesian product of the convex hulls of Si, namely Every point in P is an n-tuple (x1, x2, . . . , xn) where each xi belongs to the corresponding conv
Let S1 = {0, l}, S2 = {2, 3}, and x = 2.5. Illustrate the sets conv Si, P, and P(x).
Extend exercise 3.210 to allow for noncompact Si, thus proving the Shapley-Folkman theorem.Exercise 3.210Let {S1, S2, . . . , Sn} be a collection of nonempty compact subsets of an m-dimensional linear space, and let x ˆˆ convWe consider the Cartesian product of the convex hulls of Si,
Prove proposition 3.17.Proposition 3.17 (Minkowski's theorem)A closed, convex set in a normed linear space is the intersection of the closed halfspaces that contain it.Minkowski's theorem is illustrated in figure 3.22.
Show that 1. V*(y) is a closed convex set containing V(y) 2. If V(y) is convex and monotonic, then V*(y) = V(y)
Assume that the cost function for a convex, monotonic technology V(y) is linear in y, that is, c(w, y) = (w)y. Then the technology exhibits constant returns to scale.
Every polyhedral set is closed and convex.
Let A be an m × n matrix. The set of solutions to the system of linear inequalities Ax ≤ c is a polyhedron in ℜn.
Show that a polytope can be defined alternatively as • The convex hull of a finite set of points • A nonempty compact polyhedral set That is, show the equivalence of these two definitions.
If f: X → Y and h: X → Z are linear functions with kernel f ⊆ kernel h, then there exists a linear function g: f X → Z such that h = g ∘ f.
Let S be a nonempty set in a normed linear space X. Show 1. S* is a closed convex cone in X* 2. S** is a closed convex cone in X 3. S ⊆ S**
Let S1, S2 be nonempty sets in a normed linear space.Whatever the nature of the set S, S* and S** are always cones (in X* and X respectively) and S Š† S**. Under what conditions is S identical to its bipolar S** (rather than a proper subset)? Clearly, a necessary condition is that S is a
Let S be a nonempty convex cone in a normed linear space X. Then S = S** if and only if S is closed.
Use proposition 3.14 directly to prove the Farkas lemma when X is a Hilbert space.
Let A be an m × n matrix. A necessary and sufficient condition for the system of linear equations AT y = c to have a nonnegative solution y ∈ ℜm+ is that cTx ≤ 0 for every x ∈ ℜn satisfying Ax ≤ 0.
Let S be convex set in ℜn which contains no interior points of the nonnegative orthant ℜn+. Then there exists a hyperplane with nonnegative normal p ≩ 0 (that is p ≠ 0, pi ≥ 0) such that pTx ≤ 0 for every x ∈ S.
Suppose that the production possibility set Y is convex. Then every efficient production plan y ∈ Y is profit maximizing for some nonzero price system p ≥ 0.
Let S be convex set in ℜn that contains no interior points of the non positive orthant ℜn-. Then there exists a hyperplane with nonnegative normal p ≩ 0 such that pTx ≥ 0 for every x ∈ S
Suppose that x* = (x*1, x*2,..., x*n) is a Pareto efficient allocation in an exchange economy with l commodities and n consumers (example 1.117). Assume that€¢ Individual preferences are convex, continuous and strongly monotonic.€¢ x*Show that 1. The set is the set of all
Suppose that x* = (x*1, x*2,..., x*n) is a Pareto-efficient allocation in an exchange economy (example 1.117) in which each of the n consumers has an endowment wi ˆˆ „œl+ of the l commodities. Assume that€¢ Individual preferences ‰¿i are convex, continuous
Suppose that f: X → Y is a linear function and B ⊆ X is a basis for X. Then f (B) spans f (X).
Let K be a closed convex cone that intersects the nonnegative orthant at 0, that is, K ∩ ℜn+ = {0}. Then there exists a hyperplane with positive normal p > 0 (i.e., pi > 0 for every i) such that pTx ≤ 0 for every x ∈ K.
There is no arbitrage if and only if there exist state prices.
Let A be an m × n matrix and c be a nonzero vector in ℜn. Then exactly one of the following systems has a nonnegative solution: Either I Ax ≤ 0 and cx > 0 for some x ∈ Rn+ or II ATy ≥ c for some y ∈ Rm+. Still other variants can be derived by creative reformulations of the alternative
Let A be an m × n matrix and c be a nonzero vector in ℜn. Then exactly one of the following systems has a solution: Either I Ax = 0 and cTx > 0 for some x ∈ ℜn+ or II ATy ≥ c for some y ∈ ℜm.
Let g1, g2, ..., gm be linear functionals on „œn. For fixed numbers c1, c2,..., cm, the system of inequalitiesgj(x) ‰¥ cj, j = 1, 2,..., m (31)is consistent for some x ˆˆ „œn if and only ifimplies that for every set of nonnegative numbers Î»1,
Let A be an m × n matrix and c be a nonzero vector in ℜn. Then exactly one of the following systems has a solution: Either I Ax = 0 and cx > 0 or II ATy = c
Derive the Fredholm alternative (exercise 3.237) directly from exercise 3.199.
Let A be an m × n matrix. Then exactly one of the following systems as a solution: Either I Ax > 0 for some x ∈ ℜn or II ATy = 0, y ≩ 0 for some y ∈ ℜm
Suppose that f: X → Y is a linear function. If X is finite-dimensional, then rank f + nullity f = dim X
Derive Gordan's theorem from the Farkas alternative. [Apply the Farkas alternative to the matrix B = (A, 1), the matrix B augmented with a column of ones.]
Let S be a subspace in ℜn and S⊥ its orthogonal complement. Either I S contains a positive vector y > 0 or II S⊥ contains a nonnegative vector y ≩ 0
Let A be an m × n matrix. Then exactly one of the following systems has a solution: Either I Ax ≩ 0 for some x ∈ ℜn or II ATy = 0, y > 0 for some y ∈ ℜm++
Deduce Stiemke's theorem directly from exercise 3.230.
Let A be an m × n matrix. Then exactly one of the following systems has a nonnegative solution: Either I Ax > 0, x > 0 for some x ∈ ℜn or II ATy ≤ 0, y ≩ 0 for some y ∈ ℜm.
Let A be an m × n matrix. Then exactly one of the following systems has a nonnegative solution: Either I ATx > 0, x ≩ 0 for some x ∈ ℜm or II Ay ≤ 0, y ≩ 0 for some y ∈ ℜn.
Assume the system Ax ≩ 0 has a solution, while Ax > 0 has no solution. Then A can be decomposed into two consistent subsystems Bx > 0 and Cx = 0 such that Cx ≩ 0 has no solution.
Let A be an m × n matrix. The dual pair Ax ≥ 0 and ATy = 0; y ≥ 0
Let A be an m × n matrix. The dual pair Ax ≥ 0; x ≥ 0 and ATy ≤ 0; y ≥ 0
Show that Gordan's and Stiemke's theorems are special cases of Tucker's theorem.
Derive the Farkas lemma from Tucker's theorem.
Let A, B, and C be matrices of order m1 × n, m2 × n and m3 × n respectively with A nonvacuous. Then either
Let S be a nonempty convex set in „œn with 0 ˆ‰ S.1. For every point a ˆˆ S, define the polar set S*a = {x ˆˆ „œn : ||x|| = 1; xTa ‰¥ 0}. S*a is a nonempty closed set.2. For any finite set of points {a1, a2, ..., am} in S, let A be the m
In any finite two person zero-sum game, at least one of the players has a mixed strategy that guarantees she cannot lose. That is, either v1(p) ≥ 0 or v2(q) ≤ 0.
Prove proposition 3.20 by showing 1. For every c ∈ ℜ, either v1 ≥ c or v2 ≤ c 2. v1 = v2
Let v denote the value of a two person zero-sum game. Show that 1. Player 1 has an optimal strategy p* that achieves the value v. Similarly player 2 has an optimal strategy q*. 2. The strategy pair (p*, q*) constitutes a Nash equilibrium of the game.
In the context of the previous example, prove that v1(p) Previous exampleConsider a zero-sum game in which player 1 has two strategies {s1, s2}, while player 2 has 5 strategies {t1, t2, t3, t4, t5}. The payoffs to player 1 are
Let A be a m Ã— n matrix which represents (exercise 3.253) the payoff function of a two-person zero-sum game in which player 1 has m pure strategies and player 2 has n strategies. Let Z be the convex hull of the columns of A, that is, Z = {z = Aq : q ˆˆ Î”n-1}.
Prove the minimax theorem by extending the previous exercise to an arbitrary two-person zero-sum game with v2 = c ≠ 0. Previous exercise Let A be a m × n matrix which represents (exercise 3.253) the payoff function of a two-person zero-sum game in which player 1 has m pure strategies and player
Assuming that X = ℜn and Y = ℜm, prove proposition 3.1 for the standard basis (example 1.79).
Show that the set of optimal strategies for each player is convex.
Let f be a bilinear functional on the product of two simplices, that is f: Î”m Ã— Î”n †’ R. Then
A classic example in coalitional game theory is the three-person majority game, in which the allocation of $1 among three persons is decided by majority vote. Let N = {1, 2, 3}. The characteristic function is w({i}) = 0, i = 1, 2, 3 w({i, j}) = 1, i, j ∈ N, i ≠ j w(N) = 1 1. Show that the
Show that cohesivity is necessary, but not sufficient, for the existence of a core.A partition {S1, S2,..., SK} is a family of coalitions in which each player I belongs to one and only one of the coalitions. A balanced family of coalitions B is a set of coalitions {S1, S2,..., Sk} together with a
List three other balanced families of coalitions for the three-player game.
Find a nontrivial (i.e., not a partition) balanced family of coalitions for the four-player game with N = {1, 2, 3, 4}.
A collection B of coalitions is balanced if and only if there exists positive weights {Î»S : S ˆˆ B} such thatwhere eS is the characteristic vector of the coalition S (example 3.19) and gS: X †’ „œ represents the share of coalition S at the allocation x
Why can we normalize so that μ = 1 in the previous proof?
The set of balanced games forms a convex cone in the set of all TP-coalitional games GN.
To construct a suitable space for the application of a separation argument, consider the set of pointswhere eS is characteristic vector of the coalition S (example 3.19) and w(S) is its worth. Let A be the conic hull of A0, that is, Let B be the interval Clearly, A and B are convex and nonempty. We
A TP-coalitional game (N, w) is called 0-1 normalized ifw({i}) = 0 for every i ˆˆ N and w(N) = 1Show that1. To every essential game (N, w) there is a corresponding 0-1 normalized game (N, w0)2. Core (N, w) = Î± core (N, w0) + w, where w = (w1, w2,..., wn), wi = w({i}),3.
Let (N, w) be a 0-1 normalized TP-coalitional game, and let A be the set of all nontrivial coalitions A = {S Š† N : w(S) > 0}. Consider the following two-player zero-sum game. Player 1 chooses a player i ˆˆ N and player 2 chooses a coalition S ˆˆ A. The payoff (from 2
Let (N, w) be a 0-1 normalized TP-coalitional game, and let G be the corresponding two-person zero-sum game described in the previous exercise with value Î´. Show that Î´ ‰¥ 1 if (N, w) is balanced.Previous exerciseLet (N, w) be a 0-1 normalized TP-coalitional game,
Core(N, w) ‰ ˆ… ‡’ (N, w) balanced.Combining the three previous exercises establishes the cycle of equivalences(N, w) balanced ‡’ Î´ ‰¥ 1 ‡’ (core(N, w) ‰ ˆ…) ‡’ (N, w) balanced and establishes
Describe the matrix representing the Shapley value. Specify the matrix for three-player games (|N|) = 3). If f: X → Y is a nonsingular linear function with matrix representation A, then the representation of the inverse function f -1 with respect to the same bases is called the matrix inverse of
A linear function f: X → Y is continuous if and only if it is continuous at 0. A linear function f: X → Y is bounded if there exists a constant M such that || f(x) || ≤ M ||x|| for every x ∊ X Note that boundedness does not imply that the range f (X) is bounded but rather that f (S) is
Show that f in example 3.1 is linear. The function f : ℜ2 → ℜ2 defined by F( x1, x2) = (x1 cos θ - x2 sin θ ,x1 sin θ + x2 cos θ), 0 ≤ 0 2π rotates any vector in the plane counter clock wise through the angle θ (figure 2.3). It is easily verified that f is linear. Linearity implies
A linear function f: X → Y is continuous if and only if it is bounded. Fortunately every linear function on a finite-dimensional space is bounded and therefore continuous.
A linear function f: X †’ Y is bounded if X has finite dimension.Rewriting (4), we have that a linear function f is bounded if there exists a constant MFor every x ˆŠ X The smallest constant M satisfying (5) is called the norm of f. It is given by Clearly,
If f is a bounded linear function, an equivalent definition of the least upper bound is
The space BL(X, Y) of all bounded linear functions from X to Y is a normed linear space, with normIt is a Banach space (complete normed linear space) if Y is complete. The following proposition is an important result regarding bounded linear functions.
Prove proposition 3.2 assuming that X is finite-dimensional as follows: Let B be the unit ball in X, that is, B = {x: ||x|| < 1} The boundary of B is the unit sphere S = {x: ||x|| < 1}. Show that 1. f (S) is a compact subset of Y which does not contain 0Y. 2. There exists an open ball T ⊆ (f
Let X be a finite-dimensional normed linear space, and {x1; x2,..., xn} any basis for X. The function f: „œn †’ X defined byis a linear homeomorphism (remark 2.12). That is, €¢ f is linear €¢ f is one-to-one and onto €¢ f and f-1 are continuous

Showing 2900 - 3000 of 3402

First
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved