Textbooks in computational algebraic geometry

Analytic geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry.

The subsequent development of category theory was powered first by the computational needs of homological algebra, and later by the axiomatic needs of algebraic geometry, the field most resistant to being grounded in either axiomatic set theory or the Russell-Whitehead view of united foundations.

In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R. One can view it as a multivariate, non-linear generalization of:

The last-named theory in particular is much used by algebraic topologists as a computational tool ( e. g., for the homotopy groups of spheres ).

Geometric group theory closely interacts with low-dimensional topology, hyperbolic geometry, algebraic topology, computational group theory and differential geometry.

It can be divided into elementary number theory ( where the integers are studied without the aid of techniques from other mathematical fields ); analytic number theory ( where calculus and complex analysis are used as tools ); algebraic number theory ( which studies the algebraic numbers-the roots of polynomials with integer coefficients ); geometric number theory ; combinatorial number theory ; transcendental number theory ; and computational number theory.

His mathematical specialties were noncommutative ring theory and computational algebra and its applications, including automated theorem proving in geometry.

In mathematics and computational geometry, a Delaunay triangulation for a set P of points in a plane is a triangulation DT ( P ) such that no point in P is inside the circumcircle of any triangle in DT ( P ).

* CGAL, the Computational Geometry Algorithms Library ( CGAL ) is a software library that aims to provide easy access to efficient and reliable algorithms in computational geometry.

The term is also used as a collective term for the approach to classical, computational and relativistic geometry that makes heavy use of such algebras.

This principle is widely used in computer graphics, computational geometry and many other disciplines, to solve many proximity problems in the plane or in three-dimensional space, such as finding closest pairs in a set of points, similar shapes in a list of shapes, similar images in an image database, and so on.

Not only does this make them valuable in time-sensitive applications such as real-time applications, but it makes them valuable building blocks in other data structures which provide worst-case guarantees ; for example, many data structures used in computational geometry can be based on red – black trees, and the Completely Fair Scheduler used in current Linux kernels uses red – black trees.

Because of its enormous economic importance, CAD has been a major driving force for research in computational geometry, computer graphics ( both hardware and software ), and discrete differential geometry.

The algorithmic problem of finding the convex hull of a finite set of points in the plane or in low-dimensional Euclidean spaces is one of the fundamental problems of computational geometry.

In computational geometry, a number of algorithms are known for computing the convex hull for a finite set of points and for other geometric objects.

Numerous methods have also been proposed in the field of computational geometry.

* Normal polytopes in polyhedral geometry and computational commutative algebra

Solving collision detection problems requires extensive use of concepts from linear algebra and computational geometry.

In computational geometry, the perceptron is an algorithm for supervised classification of an input into one of two possible outputs.

Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry.

The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing ( CAD / CAM ), but many problems in computational geometry are classical in nature, and may come from mathematical visualization.

Its computational expressiveness is equivalent to that of Relational algebra.

GAP ( Groups, Algorithms and Programming ) is a computer algebra system for computational discrete algebra with particular emphasis on computational group theory.

The analytical and computational development are best effected throughout by means of matrix algebra, solving partial differential equations.

The BLAS standard for linear algebra subroutines explicitly avoids mandating any particular computational order of operations for performance reasons, and BLAS implementations typically do not use Kahan summation.

Today, parallel GPUs have begun making computational inroads against the CPU, and a subfield of research, dubbed GPU Computing or GPGPU for General Purpose Computing on GPU, has found its way into fields as diverse as machine learning, oil exploration, scientific image processing, linear algebra, statistics, 3D reconstruction and even stock options pricing determination.

Many linear algebra algorithms require significantly less computational effort when applied to triangular matrices, and this improvement often carries over to Hessenberg matrices as well.

Many linear algebra algorithms require significantly less computational effort when applied to diagonal matrices, and this improvement often carries over to tridiagonal matrices as well.

Ancient Indian Vedic civilizations are known for being skilled in geometry, algebra and computational mathematics complex enough to incorporate things like irrational numbers ( Dutta, 2002 ).

From the point of view of abstract algebra, the material is divided between symmetric function theory, field theory, Galois theory, and computational considerations including numerical analysis.

Most recently, it led Hestenes to formulate conformal geometric algebra, a new approach to computational geometry.

This is not only an example of terse array programming from the coding point of view but also from the computational efficiency perspective, which in several array programming languages benefits from quite efficient linear algebra libraries such as ATLAS or LAPACK.

Since the SGS is critical for many algorithms in computational group theory, computer algebra systems typically rely on the Schreier – Sims algorithm for efficient calculations in groups.

Gurevich: "... Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine ... according to Savage, an algorithm is a computational process defined by a Turing machine ".

For some such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise.

Algorithm analysis is an important part of a broader computational complexity theory, which provides theoretical estimates for the resources needed by any algorithm which solves a given computational problem.

Image showing shock waves from NASA's X-43A hypersonic research vehicle in flight at Mach 7, generated using a computational fluid dynamics algorithm. On September 30, 1935 an exclusive conference was held in Rome with the topic of high velocity flight and the possibility of breaking the sound barrier.

In one application, it is actually a benefit: the password-hashing method used in OpenBSD uses an algorithm derived from Blowfish that makes use of the slow key schedule ; the idea is that the extra computational effort required gives protection against dictionary attacks.

Since the desired effect is computational difficulty, in theory one would choose an algorithm and desired difficulty level, thus decide the key length accordingly.

An algorithm's key length is distinct from its cryptographic security, which is a logarithmic measure of the fastest known computational attack on the algorithm, also measured in bits.

A key distinction between analysis of algorithms and computational complexity theory is that the former is devoted to analyzing the amount of resources needed by a particular algorithm to solve a problem, whereas the latter asks a more general question about all possible algorithms that could be used to solve the same problem.

" Difficult ", in this sense, is described in terms of the computational resources needed by the most efficient algorithm for a certain problem.

The algorithm, named after its inventor, Jay Earley, is a chart parser that uses dynamic programming ; it is mainly used for parsing in computational linguistics.

Investigators are typically supported for projects in data acquisition, analysis, algorithm development and dissemination in computational phylogenetics and phyloinformatics.

A similar fact takes place for particular computational tasks, like the search problem, for which Grover's algorithm is optimal.

* Too high computational complexity of algorithm

Modern advances in software and computational power have even eliminated the need to use grids or tracking marks – the software analyzes the relative motion of colored pixels against other colored pixels and calculates the ' motion ' to create a camera-motion algorithm which can be used in compositing software to match the motion of composited elements to a moving background plate.

The original DES cipher's key size of 56 bits was generally sufficient when that algorithm was designed, but the availability of increasing computational power made brute-force attacks feasible.

This formulation is especially useful for implementing a numerical convolution on a computer: The standard convolution algorithm has quadratic computational complexity.

In complexity theory, computational problems that are co-NP-complete are those that are the hardest problems in co-NP, in the sense that they are the ones most likely not to be in P. If there exists a way to solve a co-NP-complete problem quickly, then that algorithm can be used to solve all co-NP problems quickly.

Bresenham also published a Run-Slice ( as opposed to the Run-Length ) computational algorithm.

A minimum T-join can be obtained using a weighted matching algorithm that uses O ( n < sup > 3 </ sup >) computational steps.

A classic result in computational geometry was the formulation of an algorithm that takes O ( n log n ).

Rey takes a few tentative steps towards the daunting task of trying to describe an algorithm by which sensory experiences ( inputs ) can be translated into abstract mental representations ( elements of a Language of Thought ) which can then be subjected to computational processes and so produce new representations and human behaviors ( outputs ).

DNSCurve uses techniques from elliptic curve cryptography to give a vast decrease in computational time over the RSA public-key algorithm used by DNSSEC, and uses the existing DNS hierarchy to propagate trust by embedding public keys into specially formatted ( but backward-compatible ) DNS records.