codethrasher

The Cosmological Principle states that at large enough scales (>100Mpc) the universe’s matter distribution is isotropic and homogeneous and should not produce irregularities in the large-scale structure of the universe over the course of its evolution.

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Table of Contents Chapter 1 Predicates/Expressions Functions v.

The Structure and Interpretation of Computer Programs (SICP)

A manifold is a topological space that locally resembles Euclidean space near each point.

This definition is more clearly described sybollically: \begin{equation} \forall a,b \in X(aRb\,\iff\,bRa) \end{equation} Tao’s definition: Given any two objects \(x\) and \(y\) of the same type, if \(x = y\), then \(y = x\).

A relation \(R\) over a set \(X\) is transitive if for all elements x, y, z in \(X\), whenever \(R\) relates \(x\) to \(y\) and \(y\) to \(z\), then \(R\) also relates \(x\) to \(z\).

A binary relation \(R\) over a set \(X\) is reflexive if it relates every \(x \in X\) to itself.

The Feynman Technique

Articles Black Holes as Primordial Dark Matter Papers The Inflationary Universe (Guth, 1980) Concepts Configuration Space Cosmic Inflation Equations of Motion Galilean Transformation Generalized Coordinates Homogeneous Inertial Frames Isotropy Newtonian S…

Branches Physics Articles Phosphine, Life, and Venus (Biology, Chemistry) Black Holes as Primordial Dark Matter (Physics) Papers The Inflationary Universe: Guth, 1980 (Physics) Concepts Chemical Oxidation Abiotic Processes Cosmic Inflation Newtons Laws of…

Concepts Relations Cardinality Binary Relation Symmetric Relation Transitive Relation Cartesian Product Reflexive Relation Surjective Function

The Feynman Technique

Real Analysis Differential Geometry Linear Algebra Topology Set Theory General Concepts Bijective Function Injective Function Surjective Function Transitive Relation Symmetric Relation Locality Peano Axioms

Articles Phosphine, Life, and Venus

Articles Phosphine, Life, and Venus

For any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

Concepts Metric Space Open Balls Neighborhood Manifold Topological Space

An ordered pair \((X, \tau)\), where \(X\) is a set and \(\tau\) is a collection of subsets of \(X\) satisfying: The empty set (\(\emptyset\)) and \(X\) belong to \(\tau\) Any arbitrary (in)finite union of members of \(\tau\) still belongs to \(\tau\) The…

A function is surjective or “onto” if For every \(y \in Y\), there exists \(x \in X\) such that \(f(x) = y\).

Concepts Null Hypothesis

Open/Closed/Flat TODO

Definition A relation \(R\) from the elements of set \(A\) to the elements of set \(B\) is a subset of \(A \times B\).

The theoretical foundation which underlies calculus Peano Axioms Surjective Function Bijective Function Injective Function Cartesian Product

The Principle of Relativity states that: The laws of physics take the same form in every intertial frame.

Source: Phosphine, Life, and Venus Regarding the released report that Venus was found to “possibly harbor life” (see this Nature article) What’s Phosphine?

A natural number is any element of the set \(\mathbb{N} = \{0, 1, 2, 3…\}\) I 0 is a natural number.

A type of chemical reaction in which electrons are lost

Let \((X, d)\) be a metric space.

The Problem People used to believe that the Universe was infinite (in age and size) and static.

Denoted H_0 (subscript 0) The position that there is no relationship among two measured phenomenon or among groups.

I In an intertial frame of reference (i.e. a frame of reference undergoing zero acceleration), an object at rest stays at rest (or, similarly, keeps its initial velocity) unless acted upon by an outside force.

Newtonian Spacetime differs from Relativistic spacetime in that time is considered an absolute (i.e.

The set of points surrounding a point \(p\) in a set \(V\), such that \(p \in \mathbb{R}^{n}\), within a radius \(\epsilon > 0\) See also: Open Balls

A set and a function which defines the distance between two points on that set An ordered pair \((X, d)\), where \(X\) is an arbitrary set and \(d\) is a metric (distance-defining function).

Locality refers to: A property \(P\), of a point \(x\), which holds true near every point around \(x\).

Isotropy means that direction is not preferred, and that there is uniformity in any orientation.

A function is injective (“one-to-one”) if \(x \neq x^{\prime} \Longrightarrow f(x) \neq f(x^{\prime})\) i.e.

Above are two frames in Cartesian coordinates, \(S\) and \(S^{\prime}\).

Homogeneous means that location is not preferred, that the physics are the same at any point.

Generalized coordinates are useful when calculating Lagrangians or Hamiltonians.

Concepts Inertial Frames Newtons Laws of Motion

In Newton’s formulation of spacetime, time is an absolute; meaning that every observer experiences the same flow of time.

Overview In the Big Bang model there are multiple parameters which appear to be “fine tuned” in that, small changes to a given parameter have massive effects on the evolution of the universe as a whole.

Linear Motion (under constant \(\hat{\mathbf{a}}\)) \begin{equation} \mathbf{v} = \mathbf{a}t + \mathbf{v_{0}}\tag{1} \end{equation} \begin{equation} \mathbf{r} = \mathbf{r_{0}} + \mathbf{v_{0}}t + \frac{1}{2}\mathbf{a}t^{2}\tag{2} \end{equation} \begin{e…

Emacs notes (doom emacs) Org handy cheatsheet: https://orgmode.org/orgcard.pdf To get a TODO item created on the same level as the one in which you’re working (org-insert-todo-heading) Add tag: To add a tag to a heading (* line) SPC m q (or C-c C-q) then…

Any coordinate system which cannot describe an entirety of a space As an example, polar coordinates \(r,\psi)\) are degenerate at \((0, \psi)\)

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