3 The main hierarchy 2 1 Contextual categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Π-contextual categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Impredicative Π-universe structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Predicative Π-universe structures. . . . . . . . . . . . . . . . . . . . . . . . . ...

Note that ‖f‖Lp(X) may take the value ∞. Definition 1.2. The space L(X) is the set L(X) = {f : X → R | ‖f‖Lp(X) <∞} . The space L(X) satisfies the following vector space properties: (1) For each α ∈ R, if f ∈ L(X) then αf ∈ L(X); (2) If f, g ∈ L(X), then |f + g| ≤ 2p−1(|f | + |g|) , so that f + g ∈ L(X). (3) The triangle inequality is valid if p ≥ 1. The most interesting cases are p = 1, 2,∞, w...

Then, for example, taking (r, s) = (1, 3) gives us (x, y, z) = (3, 4, 5). (2) Geometric solution: Solve X2 + Y 2 = 1 in rationals, where X = xz and Y = y z . Therefore, we are looking for rational points on the circle. Finding real points is easy, just take X = sin θ, Y = cos θ, however this doesn’t help us very much. Instead, fix the point at (−1, 0) and look at lines from that point that inte...

Class Meeting # 24: Transport Equations and Burger’s Equation In these notes, we introduce a class of evolution PDEs known as transport equations. Such equations arise in a physical context whenever a quantity is “transported” in a certain direction. Some important physical examples include the mass density flow for an incompressible fluid, and the Boltzmann equation of kinetic theory. We discu...

The consumption set X is the set of all conceivable consumption bundles q, usually identified with R+ The budget set B ⊂ X is the set of affordable bundles In standard model individuals can purchase unlimited quantities at constant prices p subject to total budget y. The budget set is the Walrasian, competitive or linear budget set: B = {q ∈ R+|pq ≤ y} Notice this is a convex, closed and bounde...

Incomplete information games, local interaction games and randommatching games are all special cases of a general class of interaction games (Morris (1997)). In this paper, we use this equivalence to present a uniÞed treatment of arguments generating uniqueness in games with strategic complementarities by introducing heterogeneity in these di¤erent settings. We also report on the relation betwe...

EDITOR’S NOTE Geopersia is published biannually by the College of Science, University of Tehran, Iran. The College of Science published “Journal of Science”, which covered all fields of basic science, from 1968 to 2010. It was then decided that the College’s journals should be more specialized, and hence publication of Geopersia was launched. Geopersia aims to publish peer-reviewed original...

Background: the inappropriate use of handouts and notes is a common academic problem among students. Students' notes are too brief and incomplete in compare with references; hence they may not cover the required content for the courses. The aim of this paper is to analyze students' attitude toward reading handouts and notes instead of references. Methods: In this analytical and descriptive stud...

OBJECTIVES The delay between patient discharge and the completion of the final discharge note have prompted hospitals to consider new information technologies. This study compared the relative cost-effectiveness of an automated medical documentation system to the current system in place at a Canadian hospital. There are significant expenditures associated with the choice of medical documentatio...