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🔗 Mathematics

In knot theory, Conway notation, invented by John Horton Conway, is a way of describing knots that makes many of their properties clear. It composes a knot using certain operations on tangles to construct it.

🔗 Mathematics 🔗 Books

Proofs from THE BOOK is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book."

🔗 Computer science 🔗 Mathematics

In combinatory logic for computer science, a fixed-point combinator (or fixpoint combinator),^: p.26 is a higher-order function (i.e. a function which takes a function as argument) that returns some fixed point (a value that is mapped to itself) of its argument function, if one exists.

Formally, if ${\displaystyle {\textsf {fix}}}$ is a fixed-point combinator and the function ${\displaystyle f}$ has one or more fixed points, then ${\displaystyle {\textsf {fix}}\ f}$ is one of these fixed points, i.e.

${\displaystyle f\ ({\textsf {fix}}\ f)={\textsf {fix}}\ f\ .}$

Fixed-point combinators can be defined in the lambda calculus and in functional programming languages and provide a means to allow for recursive definitions.

🔗 Mathematics 🔗 Physics

The Fresnel integrals S(x) and C(x) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (erf). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:

The simultaneous parametric plot of S(x) and C(x) is the Euler spiral (also known as the Cornu spiral or clothoid).

🔗 Mathematics

Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D

${\displaystyle Df(x)={\frac {d}{dx}}f(x)\,,}$

and of the integration operator J

${\displaystyle Jf(x)=\int _{0}^{x}f(s)\,ds\,,}$

and developing a calculus for such operators generalizing the classical one.

In this context, the term powers refers to iterative application of a linear operator D to a function f, that is, repeatedly composing D with itself, as in ${\displaystyle D^{2}(f)=(D\circ D)(f)=D(D(f))}$.

For example, one may ask for a meaningful interpretation of:

${\displaystyle {\sqrt {D}}=D^{\frac {1}{2}}}$

as an analogue of the functional square root for the differentiation operator, that is, an expression for some linear operator that when applied twice to any function will have the same effect as differentiation. More generally, one can look at the question of defining a linear functional

${\displaystyle D^{a}}$

for every real-number a in such a way that, when a takes an integer value n ∈ ℤ, it coincides with the usual n-fold differentiation D if n > 0, and with the −nth power of J when n < 0.

One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator D is that the sets of operator powers { D^a |a ∈ ℝ } defined in this way are continuous semigroups with parameter a, of which the original discrete semigroup of { Dⁿ | n ∈ ℤ } for integer n is a denumerable subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics.

Fractional differential equations, also known as extraordinary differential equations, are a generalization of differential equations through the application of fractional calculus.

🔗 Biography 🔗 Internet culture 🔗 Mathematics 🔗 Biography/science and academia

Cleo was the pseudonym of an anonymous mathematician active on the mathematics Stack Exchange from 2013 to 2015, who became known for providing precise answers to complex mathematical integration problems without showing any intermediate steps. Due to the extraordinary accuracy and speed of the provided solutions, mathematicians debated whether Cleo was an individual genius, a collective pseudonym, or even an early artificial intelligence system.

During the poster's active period, Cleo posted 39 answers to advanced mathematical questions, primarily focusing on complex integration problems that had stumped other users. Cleo's answers were characterized by being consistently correct while providing no explanation of methodology, often appearing within hours of the original posts. The account claimed to be limited in interaction due to an unspecified medical condition.

The mystery surrounding Cleo's identity and mathematical abilities generated significant interest in the mathematical community, with users attempting to analyze solution patterns and writing style for clues. Some compared Cleo to historical mathematical figures like Srinivasa Ramanujan, known for providing solutions without conventional proofs. In 2025, Cleo was revealed to be Vladimir Reshetnikov, a software developer originally from Uzbekistan.

🔗 Computing 🔗 Computer science 🔗 Mathematics 🔗 Computing/Software 🔗 Computing/Computer science

In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs.

It is a generalization of a syntactic analogy between systems of formal logic and computational calculi that was first discovered by the American mathematician Haskell Curry and logician William Alvin Howard. It is the link between logic and computation that is usually attributed to Curry and Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by L. E. J. Brouwer, Arend Heyting and Andrey Kolmogorov (see Brouwer–Heyting–Kolmogorov interpretation) and Stephen Kleene (see Realizability). The relationship has been extended to include category theory as the three-way Curry–Howard–Lambek correspondence.

🔗 Mathematics

Graham's number is an immense number that arises as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. It is named after mathematician Ronald Graham, who used the number in conversations with popular science writer Martin Gardner as a simplified explanation of the upper bounds of the problem he was working on. In 1977, Gardner described the number in Scientific American, introducing it to the general public. At the time of its introduction, it was the largest specific positive integer ever to have been used in a published mathematical proof. The number was published in the 1980 Guinness Book of World Records, adding to its popular interest. Other specific integers (such as TREE(3)) known to be far larger than Graham's number have since appeared in many serious mathematical proofs, for example in connection with Harvey Friedman's various finite forms of Kruskal's theorem. Additionally, smaller upper bounds on the Ramsey theory problem from which Graham's number derived have since been proven to be valid.

Graham's number is much larger than many other large numbers such as Skewes' number and Moser's number, both of which are in turn much larger than a googolplex. As with these, it is so large that the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume, possibly the smallest measurable space. But even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of that number—and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe. Thus Graham's number cannot be expressed even by power towers of the form ${\displaystyle a^{b^{c^{\cdot ^{\cdot ^{\cdot }}}}}}$.

However, Graham's number can be explicitly given by computable recursive formulas using Knuth's up-arrow notation or equivalent, as was done by Graham. As there is a recursive formula to define it, it is much smaller than typical busy beaver numbers. Though too large to be computed in full, the sequence of digits of Graham's number can be computed explicitly through simple algorithms. The last 12 digits are ...262464195387. With Knuth's up-arrow notation, Graham's number is ${\displaystyle g_{64}}$, where

🔗 Mathematics

In the mathematical fields of set theory and extremal combinatorics, a sunflower or ${\displaystyle \Delta }$-system is a collection of sets whose pairwise intersection is constant. This constant intersection is called the kernel of the sunflower.

The main research question arising in relation to sunflowers is: under what conditions does there exist a large sunflower (a sunflower with many sets) in a given collection of sets? The ${\displaystyle \Delta }$-lemma, sunflower lemma, and the Erdős-Rado sunflower conjecture give successively weaker conditions which would imply the existence of a large sunflower in a given collection, with the latter being one of the most famous open problems of extremal combinatorics.

🔗 Biography 🔗 Mathematics 🔗 France 🔗 Biography/science and academia

Évariste Galois (; French: [evaʁist ɡalwa]; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem that had been open for 350 years. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra.

Galois was a staunch republican and was heavily involved in the political turmoil that surrounded the French Revolution of 1830. As a result of his political activism, he was arrested repeatedly, serving one jail sentence of several months. For reasons that remain obscure, shortly after his release from prison, Galois fought in a duel and died of the wounds he suffered.