Welcome to Math Snacks, a series of interesting and profound mathematical results in bite-sized form!

This snack explores several interesting properties of an infinite graph.

*Definitions:*

A **graph** *G* is a pair (*V*, *E*) of vertices and edges. A **Rado Graph **(or **Erdős–Rényi Graph**)** **is a random infinite graph constructed as follows:

LetVbe a countably infinite set of vertices {v1, v2, …, vn,…}. For each pair of vertices, flip a (fair or unfair) coin. Connect the vertices with an edge if and only if the coin lands on heads.

Graphs *G1* and *G2* are said to be…

Welcome to Math Snacks, a series of interesting and profound mathematical results in bite-sized form!

This snack reveals that not all infinities are the same.

*Definitions:*

A **power set** of a set S (denoted ℘(S)) is the set of all possible distinct subsets of S (sets that can be formed from the elements of S). For instance, consider the set S = {0, 1, 2}. Then ℘(S) = {{}, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}}.

The **cardinality** of a set is the number of elements in that set. The cardinality of S is denoted…

Welcome to Math Snacks, a series of interesting and profound mathematical results in bite-sized form!

This snack concerns the Fibonacci Sequence and a surprising way to define it.

*Definitions:*

A function is **real-valued and continuous** if, roughly speaking, its graph is a single unbroken curve whose domain is the entire real number line.

A **recurrence relation** is an equation that recursively defines a sequence of values. A simple example is the following: the first term of a recurrence relation *F* is 0, and the *v-*th term of *F*** **is the (

This snack concerns random walks on *n*-dimensional lattices.

*Definitions:*

For our purposes, a** lattice** is a regular arrangement of points in Euclidean space. In this snack, a 1D integer lattice can be thought of as the integer line … -2, -1, 0, 1, 2, …, a 2D integer lattice can be thought of as the integer grid of all integer tuples (i, j), …, an *n-*dimensional integer lattice can be thought of as all integer tuples of dimension *n*.

Also for our purposes, a **random…**

Ste·ga·no·graph·y / stegəˈnägrəfi / (noun): the practice of concealing messages or information within other nonsecret text or data.

**Synopsis:** it’s possible to effectively conceal sensitive information inside language without raising eavesdropper suspicion using deep learning. This has important implications for secure communication.

*Edit 09/06/20:** Check out the end of the article for a machine-generated summary! More to come in that direction soon. :)*

Steganography has been used for ages to communicate information in a hidden manner. A natural question: how is this different from cryptography? Isn’t cryptography a very well-studied field in which two individuals aim to share information with…

Co-Founder of AccessBell, MSCS/MBA at Stanford; past: Ciphense, Marlin Protocol, IBM Research, Apfyt. Musician, composer, researcher, pennyboarder, builder