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Formal definition. Formally, a modulus of continuity is any increasing real-extended valued function ω : [0, ∞] → [0, ∞], vanishing at 0 and continuous at 0, that is. lim 0. {\displaystyle \lim _ {t\to 0}\omega (t)=\omega (0)=0.} Moduli of continuity are mainly used to give a quantitative account both of the continuity at a point, and of ...
In this case, Y is the set of real numbers R with the standard metric d Y (y 1, y 2) = |y 1 − y 2 |, and X is a subset of R. In general, the inequality is (trivially) satisfied if x 1 = x 2. Otherwise, one can equivalently define a function to be Lipschitz continuous if and only if there exists a constant K ≥ 0 such that, for all x 1 ≠ x 2,
Modular graph. In graph theory, a branch of mathematics, the modular graphs are undirected graphs in which every three vertices x, y, and z have at least one median vertex m(x, y, z) that belongs to shortest paths between each pair of x, y, and z. [1] Their name comes from the fact that a finite lattice is a modular lattice if and only if its ...
Modulus and characteristic of convexity. In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex " the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ε - δ definition of uniform convexity as the modulus of continuity does to the ε - δ ...
Definition of uniform continuity. is called uniformly continuous if for every real number there exists a real number such that for every with , we have . The set for each is a neighbourhood of and the set for each is a neighbourhood of by the definition of a neighbourhood in a metric space.
Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones ...
The absolute value of a number may be thought of as its distance from zero. In mathematics, the absolute value or modulus of a real number , denoted , is the non-negative value of without regard to its sign. Namely, if is a positive number, and if is negative (in which case negating makes positive), and . For example, the absolute value of 3 is ...
Moduli of algebraic curves. In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered ...