Intersection numberIn mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem. The intersection number is obvious in certain cases, such as the intersection of the x- and y-axes in a plane, which should be one.
Plane curveIn mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions. A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form for some specific function f.
Computational scienceComputational science, also known as scientific computing, technical computing or scientific computation (SC), is a division of science that uses advanced computing capabilities to understand and solve complex physical problems. This includes Algorithms (numerical and non-numerical): mathematical models, computational models, and computer simulations developed to solve sciences (e.
Computational physicsComputational physics is the study and implementation of numerical analysis to solve problems in physics. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science. It is sometimes regarded as a subdiscipline (or offshoot) of theoretical physics, but others consider it an intermediate branch between theoretical and experimental physics - an area of study which supplements both theory and experiment.
Algebraic curveIn mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation h(x, y, t) = 0 can be restricted to the affine algebraic plane curve of equation h(x, y, 1) = 0.
CurveIn mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line is [...] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which [...
Cubic plane curveIn mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation F(x, y, z) = 0 applied to homogeneous coordinates (x:y:z) for the projective plane; or the inhomogeneous version for the affine space determined by setting z = 1 in such an equation. Here F is a non-zero linear combination of the third-degree monomials x^3, y^3, z^3, x^2 y, x^2 z, y^2 x, y^2 z, z^2 x, z^2 y, xyz These are ten in number; therefore the cubic curves form a projective space of dimension 9, over any given field K.
Intersection theoryIn mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form. There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov-Witten theory and the extension of intersection theory from schemes to stacks.
Computational chemistryComputational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of molecules, groups of molecules, and solids. It is essential because, apart from relatively recent results concerning the hydrogen molecular ion (dihydrogen cation, see references therein for more details), the quantum many-body problem cannot be solved analytically, much less in closed form.
Computational mathematicsComputational mathematics is an area of mathematics devoted to the interaction between mathematics and computer computation. A large part of computational mathematics consists roughly of using mathematics for allowing and improving computer computation in areas of science and engineering where mathematics are useful. This involves in particular algorithm design, computational complexity, numerical methods and computer algebra. Computational mathematics refers also to the use of computers for mathematics itself.
Elliptic curveIn mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K^2, the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions (x, y) for: for some coefficients a and b in K. The curve is required to be non-singular, which means that the curve has no cusps or self-intersections.
Computational economicsComputational Economics is an interdisciplinary research discipline that involves computer science, economics, and management science. This subject encompasses computational modeling of economic systems. Some of these areas are unique, while others established areas of economics by allowing robust data analytics and solutions of problems that would be arduous to research without computers and associated numerical methods.
TangentIn geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f(c), where f is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
Differentiable curveDifferential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus.
IP PBXAn IP PBX ("Internet Protocol private branch exchange") is a system that connects telephone extensions to the public switched telephone network (PSTN) and provides internal communication for a business. An IP PBX is a PBX system with IP connectivity and may provide additional audio, video, or instant messaging communication utilizing the TCP/IP protocol stack. Voice over IP (VoIP) gateways can be combined with traditional PBX functionality to allow businesses to use their managed intranet to help reduce long distance expenses and take advantage of the benefits of a single network for voice and data (converged network).
Voice over IPVoice over Internet Protocol (VoIP), also called IP telephony, is a method and group of technologies for voice calls, the delivery of voice communication sessions over Internet Protocol (IP) networks, such as the Internet. The broader terms Internet telephony, broadband telephony, and broadband phone service specifically refer to the provisioning of voice and other communications services (fax, SMS, voice messaging) over the Internet, rather than via the public switched telephone network (PSTN), also known as plain old telephone service (POTS).
Audio over IPAudio over IP (AoIP) is the distribution of digital audio across an IP network such as the Internet. It is used increasingly to provide high-quality audio feeds over long distances. The application is also known as audio contribution over IP (ACIP) in reference to the programming contributions made by field reporters and remote events. Audio quality and latency are key issues for contribution links. In the past, these links have made use of ISDN services but these have become increasingly difficult or expensive to obtain.
