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🔗 Mathematics

A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation.

For example, there is a near-equality close to the round number 1000 between powers of 2 and powers of 10:

${\displaystyle 2^{10}=1024\approx 1000=10^{3}.}$

Some mathematical coincidences are used in engineering when one expression is taken as an approximation of another.

🔗 Mathematics

In recreational mathematics, an almost integer (or near-integer) is any number that is not an integer but is very close to one. Almost integers may be considered interesting when they arise in some context in which they are unexpected.

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Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, historian, writer, essayist, social critic, political activist, and Nobel laureate. At various points in his life, Russell considered himself a liberal, a socialist and a pacifist, although he also confessed that his sceptical nature had led him to feel that he had "never been any of these things, in any profound sense." Russell was born in Monmouthshire into one of the most prominent aristocratic families in the United Kingdom.

In the early 20th century, Russell led the British "revolt against idealism". He is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege, colleague G. E. Moore and protégé Ludwig Wittgenstein. He is widely held to be one of the 20th century's premier logicians. With A. N. Whitehead he wrote Principia Mathematica, an attempt to create a logical basis for mathematics, the quintessential work of classical logic. His philosophical essay "On Denoting" has been considered a "paradigm of philosophy". His work has had a considerable influence on mathematics, logic, set theory, linguistics, artificial intelligence, cognitive science, computer science (see type theory and type system) and philosophy, especially the philosophy of language, epistemology and metaphysics.

Russell was a prominent anti-war activist and he championed anti-imperialism. Occasionally, he advocated preventive nuclear war, before the opportunity provided by the atomic monopoly had passed and he decided he would "welcome with enthusiasm" world government. He went to prison for his pacifism during World War I. Later, Russell concluded that war against Adolf Hitler's Nazi Germany was a necessary "lesser of two evils" and criticised Stalinist totalitarianism, attacked the involvement of the United States in the Vietnam War and was an outspoken proponent of nuclear disarmament. In 1950, Russell was awarded the Nobel Prize in Literature "in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought".

🔗 Mathematics 🔗 Cryptography 🔗 Cryptography/Computer science

TWINKLE (The Weizmann Institute Key Locating Engine) is a hypothetical integer factorization device described in 1999 by Adi Shamir and purported to be capable of factoring 512-bit integers. It is also a pun on the twinkling LEDs used in the device. Shamir estimated that the cost of TWINKLE could be as low as $5000 per unit with bulk production. TWINKLE has a successor named TWIRL which is more efficient.

🔗 Computing 🔗 Mathematics 🔗 Statistics 🔗 Molecular Biology 🔗 Molecular Biology/Computational Biology

In electrical engineering, statistical computing and bioinformatics, the Baum–Welch algorithm is a special case of the expectation–maximization algorithm used to find the unknown parameters of a hidden Markov model (HMM). It makes use of the forward-backward algorithm to compute the statistics for the expectation step.

🔗 Mathematics

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers (2 × 3) that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

The property of being prime is called primality. A simple but slow method of checking the primality of a given number ${\displaystyle n}$, called trial division, tests whether ${\displaystyle n}$ is a multiple of any integer between 2 and ${\displaystyle {\sqrt {n}}}$. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number has 24,862,048 decimal digits.

There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm.

Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes having just one even number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals.

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Nicole Oresme (French: [nikɔl ɔʁɛm]; c. 1320–1325 – 11 July 1382), also known as Nicolas Oresme, Nicholas Oresme, or Nicolas d'Oresme, was a French philosopher of the later Middle Ages. He wrote influential works on economics, mathematics, physics, astrology, astronomy, philosophy, and theology; was Bishop of Lisieux, a translator, a counselor of King Charles V of France, and one of the most original thinkers of 14th-century Europe.

🔗 Mathematics 🔗 Archaeology

The Ishango bone is a bone tool and possible mathematical object, dated to the Upper Paleolithic era. It is a dark brown length of bone, the fibula of a baboon, with a sharp piece of quartz affixed to one end, perhaps for engraving. It is thought by some to be a tally stick, as it has a series of what has been interpreted as tally marks carved in three columns running the length of the tool, though it has also been suggested that the scratches might have been to create a better grip on the handle or for some other non-mathematical reason. Others argue that the marks on the object are non-random and that it was likely a kind of counting tool and used to perform simple mathematical procedures.

🔗 Mathematics

Legendre's constant is a mathematical constant occurring in a formula conjectured by Adrien-Marie Legendre to capture the asymptotic behavior of the prime-counting function ${\displaystyle \pi (x)}$. Its value is now known to be exactly 1.

Examination of available numerical evidence for known primes led Legendre to suspect that ${\displaystyle \pi (x)}$ satisfies an approximate formula.

Legendre conjectured in 1808 that

${\displaystyle \pi (x)={\frac {x}{\ln(x)-B(x)}}}$

where ${\displaystyle \lim _{x\to \infty }B(x)=1.08366}$....OEIS: A228211

Or similarly,

${\displaystyle \lim _{n\to \infty }\left(\ln(n)-{n \over \pi (n)}\right)=B}$

where B is Legendre's constant. He guessed B to be about 1.08366, but regardless of its exact value, the existence of B implies the prime number theorem.

Pafnuty Chebyshev proved in 1849 that if the limit B exists, it must be equal to 1. An easier proof was given by Pintz in 1980.

It is an immediate consequence of the prime number theorem, under the precise form with an explicit estimate of the error term

${\displaystyle \pi (x)={\rm {Li}}(x)+O\left(xe^{-a{\sqrt {\ln x}}}\right)\quad {\text{as }}x\to \infty }$

(for some positive constant a, where O(…) is the big O notation), as proved in 1899 by Charles de La Vallée Poussin, that B indeed is equal to 1. (The prime number theorem had been proved in 1896, independently by Jacques Hadamard and La Vallée Poussin, but without any estimate of the involved error term).

Being evaluated to such a simple number has made the term Legendre's constant mostly only of historical value, with it often (technically incorrectly) being used to refer to Legendre's first guess 1.08366... instead.

Pierre Dusart proved in 2010

${\displaystyle {\frac {x}{\ln x-1}}<\pi (x)}$ for ${\displaystyle x\geq 5393}$, and

${\displaystyle \pi (x)<{\frac {x}{\ln x-1.1}}}$ for ${\displaystyle x\geq 60184}$. This is of the same form as

${\displaystyle \pi (x)={\frac {x}{\ln(x)-B(x)}}}$ with ${\displaystyle 1<B(x)<1.1}$.

🔗 Computing 🔗 Mathematics 🔗 Computing/Networking

In mathematics, the Cheeger constant (also Cheeger number or isoperimetric number) of a graph is a numerical measure of whether or not a graph has a "bottleneck". The Cheeger constant as a measure of "bottleneckedness" is of great interest in many areas: for example, constructing well-connected networks of computers, card shuffling. The graph theoretical notion originated after the Cheeger isoperimetric constant of a compact Riemannian manifold.

The Cheeger constant is named after the mathematician Jeff Cheeger.