Rudin, Chapter 4, Problem #3. The space L(T ) is separable since the trigonometric polynomials with complex coefficients whose real and imaginary parts are rational form a countable dense subset. (Denseness follows from Theorem 3.14 and Theorem 4.25; countability is clear since {e | n ∈ Z} is a countable basis of the trigonometric polynomials.) Meanwhile the space L(T ) is not separable. To see...

(b) We do several iterations of eliminating strictly dominated strategies. First, note that when v < 75, bidding 75 is strictly dominated: regardless of the other bidder’s strategy, you will win the auction at least half the time, giving strictly negative payoff, while bidding 0 ensures a payoff of 0. Second, assume your opponent never bids 75 unless his value is at least 75. Then a bid of 50 w...

In Part Four of Reasons and Persons Derek Parfit searches for “Theory X,” a satisfactory account of well-being. Theories of well-being cover the utilitarian part of ethics but don’t claim to cover everything. They say nothing, for example, about rights or justice. Most of the theories Parfit considers remain neutral on what well-being consists in; his ingenious problems concern form rather than...

So it suffices to consider the following two candidates for the shortest path from v and w. The first candidate follows a crossing edge to the opposite v′ of v, and then navigates around the cycle C2n to the target vertex w. The second candidate uses no crossing edges, and simply navigates around C2n. The first of these has length 1 + cv′w, and the second of these has length cvw; so to find a s...

This paper proposes a Trust-Region Based Augmented Method (TRALM) to solve a combined Environmental and Economic Power Dispatch (EEPD) problem. The EEPD problem is a multi-objective problem with competing and non-commensurable objectives. The TRALM produces a set of non-dominated Pareto optimal solutions for the problem. Fuzzy set theory is employed to extract a compromise non-dominated sol...

To every involutive non-degenerate set-theoretic solution [Formula: see text] of the Yang–Baxter equation on a finite set there is naturally associated solvable permutation group acting text]. We prove that primitive this type prime order Moreover, then so-called determined by cycle length This solves problem recently asked A. Ballester-Bolinches. The result opens new perspective possible appro...