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In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.

The Weierstrass function has historically served the role of a pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions of smoothness. These types of functions were denounced by contemporaries: Henri Poincaré famously described them as "monsters" and called Weierstrass' work "an outrage against common sense", while Charles Hermite wrote that they were a "lamentable scourge". The functions were difficult to visualize until the arrival of computers in the next century, and the results did not gain wide acceptance until practical applications such as models of Brownian motion necessitated infinitely jagged functions (nowadays known as fractal curves).

In mathematics, the Bernoulli numbers B_n are a sequence of rational numbers which occur frequently in number theory. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by ${\displaystyle B_{n}^{-{}}}$ and ${\displaystyle B_{n}^{+{}}}$; they differ only for n = 1, where ${\displaystyle B_{1}^{-{}}=-1/2}$ and ${\displaystyle B_{1}^{+{}}=+1/2}$. For every odd n > 1, B_n = 0. For every even n > 0, B_n is negative if n is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials ${\displaystyle B_{n}(x)}$, with ${\displaystyle B_{n}^{-{}}=B_{n}(0)}$ and ${\displaystyle B_{n}^{+}=B_{n}(1)}$ (Weisstein 2016).

The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Kōwa. Seki's discovery was posthumously published in 1712 (Selin 1997, p. 891; Smith & Mikami 1914, p. 108) in his work Katsuyo Sampo; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine (Menabrea 1842, Note G). As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.

Shor's algorithm is a polynomial-time quantum computer algorithm for integer factorization. Informally, it solves the following problem: Given an integer ${\displaystyle N}$, find its prime factors. It was invented in 1994 by the American mathematician Peter Shor.

On a quantum computer, to factor an integer ${\displaystyle N}$, Shor's algorithm runs in polynomial time (the time taken is polynomial in ${\displaystyle \log N}$, the size of the integer given as input). Specifically, it takes quantum gates of order ${\displaystyle O\!\left((\log N)^{2}(\log \log N)(\log \log \log N)\right)}$ using fast multiplication, thus demonstrating that the integer-factorization problem can be efficiently solved on a quantum computer and is consequently in the complexity class BQP. This is almost exponentially faster than the most efficient known classical factoring algorithm, the general number field sieve, which works in sub-exponential time — ${\displaystyle O\!\left(e^{1.9(\log N)^{1/3}(\log \log N)^{2/3}}\right)}$. The efficiency of Shor's algorithm is due to the efficiency of the quantum Fourier transform, and modular exponentiation by repeated squarings.

If a quantum computer with a sufficient number of qubits could operate without succumbing to quantum noise and other quantum-decoherence phenomena, then Shor's algorithm could be used to break public-key cryptography schemes, such as the widely used RSA scheme. RSA is based on the assumption that factoring large integers is computationally intractable. As far as is known, this assumption is valid for classical (non-quantum) computers; no classical algorithm is known that can factor integers in polynomial time. However, Shor's algorithm shows that factoring integers is efficient on an ideal quantum computer, so it may be feasible to defeat RSA by constructing a large quantum computer. It was also a powerful motivator for the design and construction of quantum computers, and for the study of new quantum-computer algorithms. It has also facilitated research on new cryptosystems that are secure from quantum computers, collectively called post-quantum cryptography.

In 2001, Shor's algorithm was demonstrated by a group at IBM, who factored ${\displaystyle 15}$ into ${\displaystyle 3\times 5}$, using an NMR implementation of a quantum computer with ${\displaystyle 7}$ qubits. After IBM's implementation, two independent groups implemented Shor's algorithm using photonic qubits, emphasizing that multi-qubit entanglement was observed when running the Shor's algorithm circuits. In 2012, the factorization of ${\displaystyle 15}$ was performed with solid-state qubits. Also, in 2012, the factorization of ${\displaystyle 21}$ was achieved, setting the record for the largest integer factored with Shor's algorithm.

Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, yet many mathematical problems, both major and minor, still remain unsolved. These unsolved problems occur in multiple domains, including physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph, group, model, number, set and Ramsey theories, dynamical systems, partial differential equations, and more. Some problems may belong to more than one discipline of mathematics and be studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems (such as the list of Millennium Prize Problems) receive considerable attention.

The medieval Cistercian numerals, or "ciphers" in nineteenth-century parlance, were developed by the Cistercian monastic order in the early thirteenth century at about the time that Arabic numerals were introduced to northwestern Europe. They are more compact than Arabic or Roman numerals, with a single glyph able to indicate any integer from 1 to 9,999.

Digits are based on a horizontal or vertical stave, with the position of the digit on the stave indicating its place value (units, tens, hundreds or thousands). These digits are compounded on a single stave to indicate more complex numbers. The Cistercians eventually abandoned the system in favor of the Arabic numerals, but marginal use outside the order continued until the early twentieth century.

In computer science and graph theory, the Canadian traveller problem (CTP) is a generalization of the shortest path problem to graphs that are partially observable. In other words, the graph is revealed while it is being explored, and explorative edges are charged even if they do not contribute to the final path.

This optimization problem was introduced by Christos Papadimitriou and Mihalis Yannakakis in 1989 and a number of variants of the problem have been studied since. The name supposedly originates from conversations of the authors who learned of a difficulty Canadian drivers had: traveling a network of cities with snowfall randomly blocking roads. The stochastic version, where each edge is associated with a probability of independently being in the graph, has been given considerable attention in operations research under the name "the Stochastic Shortest Path Problem with Recourse" (SSPPR).

A Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and aperiodic means that shifting any tiling with these shapes by any finite distance, without rotation, cannot produce the same tiling. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s.

There are several different variations of Penrose tilings with different tile shapes. The original form of Penrose tiling used tiles of four different shapes, but this was later reduced to only two shapes: either two different rhombi, or two different quadrilaterals called kites and darts. The Penrose tilings are obtained by constraining the ways in which these shapes are allowed to fit together. This may be done in several different ways, including matching rules, substitution tiling or finite subdivision rules, cut and project schemes, and coverings. Even constrained in this manner, each variation yields infinitely many different Penrose tilings.

Penrose tilings are self-similar: they may be converted to equivalent Penrose tilings with different sizes of tiles, using processes called inflation and deflation. The pattern represented by every finite patch of tiles in a Penrose tiling occurs infinitely many times throughout the tiling. They are quasicrystals: implemented as a physical structure a Penrose tiling will produce diffraction patterns with Bragg peaks and five-fold symmetry, revealing the repeated patterns and fixed orientations of its tiles. The study of these tilings has been important in the understanding of physical materials that also form quasicrystals. Penrose tilings have also been applied in architecture and decoration, as in the floor tiling shown.

Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones.

The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory.

Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its definitions and methods are inspired by previous work in constructive analysis and proof theory. The use of second-order arithmetic also allows many techniques from recursion theory to be employed; many results in reverse mathematics have corresponding results in computable analysis. Recently, higher-order reverse mathematics has been introduced, in which the focus is on subsystems of higher-order arithmetic, and the associated richer language.

The program was founded by Harvey Friedman (1975, 1976) and brought forward by Steve Simpson. A standard reference for the subject is (Simpson 2009), while an introduction for non-specialists is (Stillwell 2018). An introduction to higher-order reverse mathematics, and also the founding paper, is (Kohlenbach (2005)).

Ronald Lewis Graham (born October 31, 1935) is an American mathematician credited by the American Mathematical Society as being "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He has done important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness.

He is the Chief Scientist at the California Institute for Telecommunications and Information Technology (also known as Cal-(IT)²) and the Irwin and Joan Jacobs Professor in Computer Science and Engineering at the University of California, San Diego (UCSD).

Terence Chi-Shen Tao (born 17 July 1975) is an Australian-American mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory. As of 2015, he holds the James and Carol Collins chair in mathematics at the University of California, Los Angeles.

Tao was a recipient of the 2006 Fields Medal and the 2014 Breakthrough Prize in Mathematics. He is also a 2006 MacArthur Fellow. Tao has been the author or co-author of 275 research papers.

Tao is the second mathematician of Han Chinese descent to win the Fields medal after Shing-Tung Yau, and the first Australian citizen to win the medal.