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Resolution in the context of structural biology is the ability to distinguish the presence or absence of atoms or groups of atoms in a biomolecular structure. Usually, the structure originates from methods such as X-ray crystallography , electron crystallography , or cryo-electron microscopy .
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution [1]) is an exact sequence of modules (or, more generally, of objects of an abelian category) that is used to define invariants characterizing the structure of a specific module or object of this category ...
Memorial in Jena, Germany to Ernst Karl Abbe, who approximated the diffraction limit of a microscope as = , where d is the resolvable feature size, λ is the wavelength of light, n is the index of refraction of the medium being imaged in, and θ (depicted as α in the inscription) is the half-angle subtended by the optical objective lens (representing the numerical aperture).
The contributions of these mathematicians and others, such as Weierstrass, developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis. Around the same time, Riemann introduced his theory of integration , and made significant advances in complex analysis.
Limit of a function (ε,_δ)-definition of limit, formal definition of the mathematical notion of limit; Limit of a sequence; One-sided limit, either of the two limits of a function as a specified point is approached from below or from above; Limit inferior and limit superior; Limit of a net; Limit point, in topological spaces; Limit (category ...
Augustin-Louis Cauchy in 1821, [6] followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit. The modern notation of placing the arrow below the limit symbol is due to G. H. Hardy, who introduced it in his book A Course of Pure Mathematics in 1908. [7]
The definition of limit given here does not depend on how (or whether) f is defined at p. Bartle [9] refers to this as a deleted limit, because it excludes the value of f at p. The corresponding non-deleted limit does depend on the value of f at p, if p is in the domain of f. Let : be a real-valued function.
Mathematical puzzles require mathematics to solve them. Logic puzzles are a common type of mathematical puzzle. Conway's Game of Life and fractals , as two examples, may also be considered mathematical puzzles even though the solver interacts with them only at the beginning by providing a set of initial conditions.