| № |
Condition |
free/or 0.5$ |
| m83092 | The following facts about bases and dimension are often used in practice. |
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| m83093 | The following interactions between these orderings are often used in practice. |
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| m83099 | The function f: X → Y has an inverse function f -1: Y → X if and only if f is one-to-one and onto. |
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| m83103 | The functional ||x|| = √xTx is a norm on X. |
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| m83105 | The general power function f (x) = xa is strictly increasing on ℜ+ for all a > 0 and strictly decreasing for a < 0. |
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| m83114 | The inner product is a continuous bilinear functional. |
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| m83115 | The interior of a set S comprises the set minus its boundary, that is, int S = S(S) |
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| m83117 | The intersection of any collection of convex sets is convex. |
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| m83119 | The inverse of a (nonsingular) linear function is linear. |
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| m83127 | The linear hull of a set of vectors S is the smallest subspace of X containing S. |
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| m83147 | The operations ∨ and ∧ have the following consistency properties. For every x; y in a lattice (X; ≿),
1. x ∨ y ≿ x ≿ x ∧ y
2. x ≿ y =⇒ x ∨ y = x and x ∧ y = y
3. x ∨ x ∧ y = x = x ∧(x ∨ y) |
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| m83158 | The power function (example 2.2)
f(x) = xn, n = 1, 2, . . .
is differentiable with derivative
Df[x] = f′[x] = nxn-1 |
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| m83159 | The preceding example is more familiar where the firm produces a single output and we distinguish inputs and outputs. Assume that a competitive firm produces a single output y from n inputs x = (x1, x2, . . . , xn) according to the production function y = f(x) so as to maximize profit
Assume that there is a unique optimum for every p and w. Show that the input demand x*I (w, p) and supply y*(w, p) functions have the following properties: |
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| m83161 | The preference relation ≿ is convex if and only if the upper preference sets ≿(y) are convex for every y. |
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| m83164 | The product of two lattices is a lattice.
The previous exercise implies that the product of n lattices is a lattice, with ∨ and ∨ defined componentwise, that is,
x ∨ y = (x1 ∨ y1, x24y2; . . . ; xn ∨ yn)
x ∧ y = x1 ∧ y1, x2 ∧ y2, . . . , xn ∧ yn) |
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| m83178 | The sample space for tossing a single die is {1; 2; 3; 4; 5; 6}. Assuming that the die is fair, so that all outcomes are equally likely, what is the probability of the event E that the result is even (See exercise 1.4)? |
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| m83182 | The set of balanced games forms a convex cone in the set of all TP-coalitional games GN. |
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| m83183 | The set of imputations of an essential TP-coalitional game (N, w)is an
(n - 1)-dimensional simplex in ℜ |
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| m83184 | The set of solutions to a homogeneous system of linear inequalities Ax ≤ 0 is a convex cone. |
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| m83185 | The set of solutions to a nonhomogeneous system of linear equations Ax = c is an affine set.
The converse is also true. |
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