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.
Normed vector spaceIn mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. A norm is a generalization of the intuitive notion of "length" in the physical world. If is a vector space over , where is a field equal to or to , then a norm on is a map , typically denoted by , satisfying the following four axioms: Non-negativity: for every ,. Positive definiteness: for every , if and only if is the zero vector.
Optimizing compilerIn computing, an optimizing compiler is a compiler that tries to minimize or maximize some attributes of an executable computer program. Common requirements are to minimize a program's execution time, memory footprint, storage size, and power consumption (the last three being popular for portable computers). Compiler optimization is generally implemented using a sequence of optimizing transformations, algorithms which take a program and transform it to produce a semantically equivalent output program that uses fewer resources or executes faster.
C++C++ ('si:_plVs_plVs, pronounced "C plus plus" and sometimes abbreviated as CPP) is a high-level, general-purpose programming language created by Danish computer scientist Bjarne Stroustrup. First released in 1985 as an extension of the C programming language, it has since expanded significantly over time; modern C++ currently has object-oriented, generic, and functional features, in addition to facilities for low-level memory manipulation.
Euclidean vectorIn mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a directed line segment, or graphically as an arrow connecting an initial point A with a terminal point B, and denoted by . A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier".
Chain ruleIn calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. More precisely, if is the function such that for every x, then the chain rule is, in Lagrange's notation, or, equivalently, The chain rule may also be expressed in Leibniz's notation. If a variable z depends on the variable y, which itself depends on the variable x (that is, y and z are dependent variables), then z depends on x as well, via the intermediate variable y.
Finsler manifoldIn mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski functional F(x, −) is provided on each tangent space TxM, that enables one to define the length of any smooth curve γ : [a, b] → M as Finsler manifolds are more general than Riemannian manifolds since the tangent norms need not be induced by inner products. Every Finsler manifold becomes an intrinsic quasimetric space when the distance between two points is defined as the infimum length of the curves that join them.
R (programming language)R is a programming language for statistical computing and graphics supported by the R Core Team and the R Foundation for Statistical Computing. Created by statisticians Ross Ihaka and Robert Gentleman, R is used among data miners, bioinformaticians and statisticians for data analysis and developing statistical software. The core R language is augmented by a large number of extension packages containing reusable code and documentation. According to user surveys and studies of scholarly literature databases, R is one of the most commonly used programming languages in data mining.
C++11C++11 is a version of the ISO/IEC 14882 standard for the C++ programming language. C++11 replaced the prior version of the C++ standard, called C++03, and was later replaced by C++14. The name follows the tradition of naming language versions by the publication year of the specification, though it was formerly named C++0x because it was expected to be published before 2010. Although one of the design goals was to prefer changes to the libraries over changes to the core language, C++11 does make several additions to the core language.
Whitney embedding theoremIn mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: The strong Whitney embedding theorem states that any smooth real m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real 2m-space, \R^{2m}, if m > 0. This is the best linear bound on the smallest-dimensional Euclidean space that all m-dimensional manifolds embed in, as the real projective spaces of dimension m cannot be embedded into real (2m − 1)-space if m is a power of two (as can be seen from a characteristic class argument, also due to Whitney).
