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29 lip 2023 · Let’s first explore the range of possibilities with models for different amounts of mass in the universe and for different contributions by dark energy. In some models—as we shall see—the universe expands forever.
Let’s first explore the range of possibilities with models for different amounts of mass in the universe and for different contributions by dark energy. In some models—as we shall see—the universe expands forever.
Let’s first explore the range of possibilities with models for different amounts of mass in the universe and for different contributions by dark energy. In some models—as we shall see—the universe expands forever.
We will choose to model the matter and energy in the universe by a perfect fluid. We discussed perfect fluids in Section One, where they were defined as fluids which are isotropic in their rest frame.
I describe two approaches to modelling the universe, the one having its origin in topos theory and differential geometry, the other in set theory. The first is synthetic differential geometry. Traditionally, there have been two methods of deriving the theorems of geometry: the analytic and the synthetic.
The range of arcsecant: y∈ [0; π/2)∪ ( π/2; π]. Arcsecant is a non-periodic function. The arcsecant increases and is continuous on the interval x∈ (-∞; -1] and x∈ [1, + ∞), since the secant function (x= secy) is strictly increasing and continuous in the intervals [0; π/2) and (π/2;π]
Using 0<=x<pi/2 and pi<=x<3pi/2 as a range for arcsecant simplifies its derivative to 1/ (x*sqrt (x^2-1)). Using the range 0<=x<pi/2 and pi/2<x<=pi, the derivative becomes 1/ (abs (x)*sqrt (x^2-1)), with abs (x) being the absolute value function. Slightly more tedious to handle.
