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In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. [1] Limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.
If J = 1, the category with a single object and morphism, then a diagram of shape J is essentially just an object X of C. A cone to an object X is just a morphism with codomain X . A morphism f : Y → X is a limit of the diagram X if and only if f is an isomorphism .
In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers.Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the ...
In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series.
The cap set problem is the problem of finding the size of the largest possible cap set, as a function of . [1] The first few cap set sizes are 1, 2, 4, 9, 20, 45, 112, ... (sequence A090245 in the OEIS). Caps are defined more generally as subsets of a finite affine or projective space with no three in a line. [2]
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares.It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [2]
Put into words, property 1 means, that the intervals are nested according to their index. The second property formalizes the notion, that interval sizes get arbitrarily small; meaning, that for an arbitrary constant ε > 0 {\displaystyle \varepsilon >0} one can always find an interval (with index N {\displaystyle N} ) with a length strictly ...
The distribution of X 1 + ⋯ + X n / √ n need not be approximately normal (in fact, it can be uniform). [38] However, the distribution of c 1 X 1 + ⋯ + c n X n is close to (,) (in the total variation distance) for most vectors (c 1, ..., c n) according to the uniform distribution on the sphere c 2 1 + ⋯ + c 2 n = 1.