In computational complexity theory, EXPSPACE is the set of all decision problems solvable by a deterministic Turing machine in exponential space, i.e., in space, where is a polynomial function of . Some authors restrict to be a linear function, but most authors instead call the resulting class ESPACE. If we use a nondeterministic machine instead, we get the class NEXPSPACE, which is equal to EXPSPACE by Savitch's theorem.
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB.
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing and hexagonal close packing arrangements. The density of these arrangements is around 74.05%.
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions or to non-Euclidean spaces such as hyperbolic space.
Russell Graham Impagliazzo is a professor of computer science at the University of California, San Diego, specializing in computational complexity theory.
In computational complexity theory, the unique games conjecture is a conjecture made by Subhash Khot in 2002. The conjecture postulates that the problem of determining the approximate value of a certain type of game, known as a unique game, has NP-hard computational complexity. It has broad applications in the theory of hardness of approximation. If the unique games conjecture is true and P ≠ NP, then for many important problems it is not only impossible to get an exact solution in polynomial time, but also impossible to get a good polynomial-time approximation. The problems for which such an inapproximability result would hold include constraint satisfaction problems, which crop up in a wide variety of disciplines.
ACM SIGACT or SIGACT is the Association for Computing Machinery Special Interest Group on Algorithms and Computation Theory, whose purpose is support of research in theoretical computer science. It was founded in 1968 by Patrick C. Fischer.
The following tables list the computational complexity of various algorithms for common mathematical operations.
In abstract algebra, the triple product property is an identity satisfied in some groups.
Ran Raz is a computer scientist who works in the area of computational complexity theory. He was a professor in the faculty of mathematics and computer science at the Weizmann Institute. He is now a professor of computer science at Princeton University.
In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms are a central subroutine in theoretical and numerical algorithms for numerical linear algebra and optimization, so finding the fastest algorithm for matrix multiplication is of major practical relevance.
Christopher Umans is a professor of Computer Science in the Computing and Mathematical Sciences Department at the California Institute of Technology. He is known for work on algorithms, computational complexity, algebraic complexity, and hardness of approximation.
Amit Sahai is an Indian-American computer scientist. He is a professor of computer science at UCLA and the director of the Center for Encrypted Functionalities.
Maryna Sergiivna Viazovska is a Ukrainian mathematician known for her work in sphere packing. She is a full professor and Chair of Number Theory at the Institute of Mathematics of the École Polytechnique Fédérale de Lausanne in Switzerland. She was awarded the Fields Medal in 2022.
In affine geometry, a cap set is a subset of where no three elements sum to the zero vector. The cap set problem is the problem of finding the size of the largest possible cap set, as a function of . The first few cap set sizes are 1, 2, 4, 9, 20, 45, 112, ....
Virginia Vassilevska Williams is a theoretical computer scientist and mathematician known for her research in computational complexity theory and algorithms. She is currently the Steven and Renee Finn Career Development Associate Professor of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology. She is notable for her breakthrough results in fast matrix multiplication, for her work on dynamic algorithms, and for helping to develop the field of fine-grained complexity.
Urmila Mahadev is an American mathematician and theoretical computer scientist known for her work in quantum computing and quantum cryptography.
In mathematics, a dijoin is a subset of the edges of a directed graph, with the property that contracting every edge in the dijoin produces a strongly connected graph. Equivalently, a dijoin is a subset of the edges that, for every dicut, includes at least one edge crossing the dicut. Here, a dicut is a partition of the vertices into two subsets, so that each edge that has an endpoint in both subsets is directed from the first subset to the second.
Tali Kaufman is an Israeli theoretical computer scientist whose research topics have included property testing, expander graphs, coding theory, and randomized algorithms with sublinear time complexity. She is a professor of computer science at Bar-Ilan University, and a fellow of the Israel Institute for Advanced Studies.
