A combinatorial proof is a method of proving a mathematical identity or theorem by demonstrating it through a counting argument, often involving the enumeration of sets or counting the same quantity in two different ways. Instead of relying on algebraic manipulations and formal symbolic manipulation, combinatorial proofs use combinatorial arguments to show that two expressions count the same object or quantity.

A De Bruijn sequence is a cyclic sequence containing a particular set of symbols in such a way that every possible subsequence of a given length appears exactly once. Specifically, for a sequence of length \( n \) over an alphabet of size \( k \), a De Bruijn sequence is a cyclic sequence of length \( k^n \) in which every possible string of length \( n \) made up of the symbols from the alphabet occurs as a contiguous subsequence.

Double counting is a combinatorial proof technique used to show that two different expressions count the same quantity. The idea is to count the same set or scenario in two distinct ways. If both methods give the same total, it can help establish identities or combinatorial equalities. ### Steps in Double Counting: 1. **Identify the Set**: Choose a specific set or mathematical object that can be counted in two different ways.

Lots of features, but slow because written in Python. A faster version may be csvtools. Also some annoyances like obtuse header handing and missing features like grep + cut in one go: csvgrep and select column in csvkit.

Enumerations of specific permutation classes refer to the systematic counting and characterization of permutations that belong to a defined class or family based on certain properties. A permutation is an arrangement of a set of elements, typically represented as a sequence. In combinatorial mathematics, particularly in the study of permutations, a permutation class is defined as a set of permutations that can be characterized by a restriction, such as avoiding certain "forbidden" permutations or following particular combinatorial patterns.

The term "exponential formula" can refer to several different concepts, depending on the context. Here are a few interpretations: 1. **Exponential Growth/Decay Formula**: This formula is often used in mathematics and the sciences to model processes that grow or decay at a rate proportional to their current value.

A closed standard: stackoverflow.com/questions/81656/where-do-i-find-the-current-c-or-c-standard-documents. Nice.

The Enrico Fermi Award is a prestigious award presented by the U.S. Department of Energy (DOE) in honor of the Italian-American physicist Enrico Fermi. Established in 1956, the award recognizes individuals for their outstanding contributions to the field of nuclear science and related disciplines, particularly those that have advanced the understanding and application of nuclear energy.

Fritz Strassmann was a German physicist who is best known for his work in nuclear chemistry and for his role in the discovery of nuclear fission. He was born on February 22, 1902, and passed away on April 22, 1980.

The one true game engine!

A circumscribed sphere, also known as a circumsphere, is a sphere that completely encloses a geometric figure, such as a polyhedron or a set of points, in three-dimensional space. The defining property of a circumscribed sphere is that all the vertices (corners) of the figure are located on the surface of the sphere.

Concurrent lines are geometrical lines that intersect at a single point. In a plane, if three or more lines are concurrent, they all meet at one common point, which is referred to as the point of concurrency. A classic example of concurrent lines can be found in triangles, where the three medians (lines drawn from each vertex to the midpoint of the opposite side) are concurrent at a point called the centroid.

"Confocal" generally refers to a type of microscopy or imaging technique that is used to increase the optical resolution and contrast of a micrograph by using a spatial pinhole to block out-of-focus light. The most common application of confocal technology is in confocal laser scanning microscopy (CLSM), which allows for the collection of three-dimensional images of specimens by scanning them with a focused laser beam.

A diagonal is a line segment that connects two non-adjacent vertices of a polygon or polyhedron. In simpler terms, it is a line drawn from one corner (vertex) of a shape to another corner that is not next to it. For example: - In a **rectangle**, there are two diagonals that connect opposite corners. - In a **square**, the diagonals also connect opposite corners and are equal in length.

Diameter is a protocol designed for authentication, authorization, and accounting (AAA) in computer networks. It is an evolution of the older RADIUS (Remote Authentication Dial-In User Service) protocol. Diameter offers several enhancements and improvements over RADIUS, making it more suitable for managing AAA needs in modern networks, especially in environments like telecommunications and mobile networks.