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🔗 Mathematics 🔗 Physics

An amplituhedron is a geometric structure introduced in 2013 by Nima Arkani-Hamed and Jaroslav Trnka. It enables simplified calculation of particle interactions in some quantum field theories. In planar N = 4 supersymmetric Yang–Mills theory, also equivalent to the perturbative topological B model string theory in twistor space, an amplituhedron is defined as a mathematical space known as the positive Grassmannian.

Amplituhedron theory challenges the notion that spacetime locality and unitarity are necessary components of a model of particle interactions. Instead, they are treated as properties that emerge from an underlying phenomenon.

The connection between the amplituhedron and scattering amplitudes is at present a conjecture that has passed many non-trivial checks, including an understanding of how locality and unitarity arise as consequences of positivity. Research has been led by Nima Arkani-Hamed. Edward Witten described the work as "very unexpected" and said that "it is difficult to guess what will happen or what the lessons will turn out to be".

🔗 Mathematics

In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles S¹_a and S¹_b (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. It resides in R⁴, as opposed to in R³. To see why R⁴ is necessary, note that if S¹_a and S¹_b each exist in their own independent embedding spaces R²_a and R²_b, the resulting product space will be R⁴ rather than R³. The historically popular view that the cartesian product of two circles is an R³ torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y.

Stated another way, a torus embedded in R³ is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R⁴. The relationship is similar to that of projecting the edges of a cube onto a sheet of paper. Such a projection creates a lower-dimensional image that accurately captures the connectivity of the cube edges, but also requires the arbitrary selection and removal of one of the three fully symmetric and interchangeable axes of the cube.

If S¹_a and S¹_b each has a radius of ${\displaystyle \textstyle {\sqrt {1/2}}}$, their Clifford torus product will fit perfectly within the unit 3-sphere S³, which is a 3-dimensional submanifold of R⁴. When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space C², since C² is topologically equivalent to R⁴.

The Clifford torus is an example of a square torus, because it is isometric to a square with opposite sides identified. It is further known as a Euclidean 2-torus (the "2" is its topological dimension); figures drawn on it obey Euclidean geometry as if it were flat, whereas the surface of a common "doughnut"-shaped torus is positively curved on the outer rim and negatively curved on the inner. Although having a different geometry than the standard embedding of a torus in three-dimensional Euclidean space, the square torus can also be embedded into three-dimensional space, by the Nash embedding theorem; one possible embedding modifies the standard torus by a fractal set of ripples running in two perpendicular directions along the surface.

🔗 Mathematics

Zero to the power of zero, denoted by 0⁰, is a mathematical expression with no agreed-upon value. The most common possibilities are 1 or leaving the expression undefined, with justifications existing for each, depending on context. In algebra and combinatorics, the generally agreed upon value is 0⁰ = 1, whereas in mathematical analysis, the expression is sometimes left undefined. Computer programs also have differing ways of handling this expression.

🔗 Mathematics

In number theory, Mills' constant is defined as the smallest positive real number A such that the floor function of the double exponential function

${\displaystyle \lfloor A^{3^{n}}\rfloor }$

is a prime number, for all natural numbers n. This constant is named after William H. Mills who proved in 1947 the existence of A based on results of Guido Hoheisel and Albert Ingham on the prime gaps. Its value is unknown, but if the Riemann hypothesis is true, it is approximately 1.3063778838630806904686144926... (sequence A051021 in the OEIS).

🔗 Computer science 🔗 Mathematics 🔗 Systems 🔗 Cryptography 🔗 Cryptography/Computer science 🔗 Systems/Operations research

The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items. The problem often arises in resource allocation where the decision makers have to choose from a set of non-divisible projects or tasks under a fixed budget or time constraint, respectively.

The knapsack problem has been studied for more than a century, with early works dating as far back as 1897. The name "knapsack problem" dates back to the early works of mathematician Tobias Dantzig (1884–1956), and refers to the commonplace problem of packing the most valuable or useful items without overloading the luggage.

🔗 Computing 🔗 Computer science 🔗 Mathematics

The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified can also be solved quickly.

It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute, each of which carries a US$1,000,000 prize for the first correct solution.

The informal term quickly, used above, means the existence of an algorithm solving the task that runs in polynomial time, such that the time to complete the task varies as a polynomial function on the size of the input to the algorithm (as opposed to, say, exponential time). The general class of questions for which some algorithm can provide an answer in polynomial time is called "class P" or just "P". For some questions, there is no known way to find an answer quickly, but if one is provided with information showing what the answer is, it is possible to verify the answer quickly. The class of questions for which an answer can be verified in polynomial time is called NP, which stands for "nondeterministic polynomial time".

An answer to the P = NP question would determine whether problems that can be verified in polynomial time can also be solved in polynomial time. If it turned out that P ≠ NP, which is widely believed, it would mean that there are problems in NP that are harder to compute than to verify: they could not be solved in polynomial time, but the answer could be verified in polynomial time.

Aside from being an important problem in computational theory, a proof either way would have profound implications for mathematics, cryptography, algorithm research, artificial intelligence, game theory, multimedia processing, philosophy, economics and many other fields.

🔗 Mathematics

In mathematics, Young's lattice is a partially ordered set and a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers On quantitative substitutional analysis, developed representation theory of the symmetric group. In Young's theory, the objects now called Young diagrams and the partial order on them played a key, even decisive, role. Young's lattice prominently figures in algebraic combinatorics, forming the simplest example of a differential poset in the sense of Stanley (1988). It is also closely connected with the crystal bases for affine Lie algebras.

🔗 Computing 🔗 Computer science 🔗 Mathematics

A* (pronounced "A-star") is a graph traversal and path search algorithm, which is often used in computer science due to its completeness, optimality, and optimal efficiency. One major practical drawback is its ${\displaystyle O(b^{d})}$ space complexity, as it stores all generated nodes in memory. Thus, in practical travel-routing systems, it is generally outperformed by algorithms which can pre-process the graph to attain better performance, as well as memory-bounded approaches; however, A* is still the best solution in many cases.

Peter Hart, Nils Nilsson and Bertram Raphael of Stanford Research Institute (now SRI International) first published the algorithm in 1968. It can be seen as an extension of Edsger Dijkstra's 1959 algorithm. A* achieves better performance by using heuristics to guide its search.

🔗 Mathematics

This is a list of mathematical symbols used in all branches of mathematics to express a formula or to represent a constant.

A mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations, a different convention may be used. For example, depending on context, the triple bar "≡" may represent congruence or a definition. However, in mathematical logic, numerical equality is sometimes represented by "≡" instead of "=", with the latter representing equality of well-formed formulas. In short, convention dictates the meaning.

Each symbol is shown both in HTML, whose display depends on the browser's access to an appropriate font installed on the particular device, and typeset as an image using TeX.

🔗 Mathematics 🔗 Statistics

The Akaike information criterion (AIC) is an estimator of out-of-sample prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection.

AIC is founded on information theory. When a statistical model is used to represent the process that generated the data, the representation will almost never be exact; so some information will be lost by using the model to represent the process. AIC estimates the relative amount of information lost by a given model: the less information a model loses, the higher the quality of that model.

In estimating the amount of information lost by a model, AIC deals with the trade-off between the goodness of fit of the model and the simplicity of the model. In other words, AIC deals with both the risk of overfitting and the risk of underfitting.

The Akaike information criterion is named after the Japanese statistician Hirotugu Akaike, who formulated it. It now forms the basis of a paradigm for the foundations of statistics; as well, it is widely used for statistical inference.