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CAP theorem (theoretical computer science) CPCTC (triangle geometry) Cameron–Erdős theorem (discrete mathematics) Cameron–Martin theorem (measure theory) Cantor–Bernstein–Schroeder theorem (set theory, cardinal numbers) Cantor's intersection theorem (real analysis) Cantor's isomorphism theorem (order theory)
Note that consistency as defined in the CAP theorem is quite different from the consistency guaranteed in ACID database transactions. [4] Availability Every request received by a non-failing node in the system must result in a response. This is the definition of availability in CAP theorem as defined by Gilbert and Lynch. [1]
Computational real algebraic geometry is concerned with the algorithmic aspects of real algebraic (and semialgebraic) geometry. The main algorithm is cylindrical algebraic decomposition. It is used to cut semialgebraic sets into nice pieces and to compute their projections. Real algebra is the part of algebra which is relevant to real algebraic ...
The technical statement appearing in Nash's original paper is as follows: if M is a given m-dimensional Riemannian manifold (analytic or of class C k, 3 ≤ k ≤ ∞), then there exists a number n (with n ≤ m(3m+11)/2 if M is a compact manifold, and with n ≤ m(m+1)(3m+11)/2 if M is a non-compact manifold) and an isometric embedding ƒ: M → R n (also analytic or of class C k). [15]
They are used particularly in algebraic geometry and related fields. A key result known as Chevalley's theorem in algebraic geometry shows that the image of a constructible set is constructible for an important class of mappings (more specifically morphisms) of algebraic varieties (or more generally schemes). In addition, a large number of ...
In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single plane. If the plane passes through the center of the sphere (forming a great circle), so that the height of the cap is equal to the radius of the sphere, the spherical cap ...
An example of a theorem of Euclidean geometry which cannot be so formulated is the Archimedean property: to any two positive-length line segments S 1 and S 2 there exists a natural number n such that nS 1 is longer than S 2. (This is a consequence of the fact that there are real-closed fields that contain infinitesimals. [5])
Littlewood's three principles are quoted in several real analysis texts, for example Royden, [2] Bressoud, [3] and Stein & Shakarchi. [4] Royden [5] gives the bounded convergence theorem as an application of the third principle. The theorem states that if a uniformly bounded sequence of functions converges pointwise, then their integrals on a ...
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