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Riemann Hypothesis – uncountable
Riemann Hypothesis
The conjecture that the zeta function has no non-trivial zeros off of the line Re(z) = 1/2.
Riemann integral
See integral.
Riemann sum
Let f be a real-valued function defined on the closed interval [a, b], and let D be a partition of [a, b], i.e., a = x0 < x1 < ... < xn = b, and where Dxi is the width of the i th subinterval. If c i is any point in the i th subinterval, then the sum
is called the Riemann sum of f for the partition D.
root
An nth root of a real or complex number x is a number which when multiplied by itself n times yields x.
Of a polynomial p: A number x such that p(x) = 0.
Roth’s Theorem
(Klaus Friedrich Roth, 1955) Given a real algebraic number a, consider the least upper bound m(a) of all numbers m for which there are infinitely many rational numbers p/q such that
Then for all a, m(a) = 2. This result improves on earlier theorems of Joseph Liouville, Axel Thue, and Carl Ludwig Siegel regarding the approximation of irrational numbers by rational numbers.
semi-finite measure
See measure.
series
ARTICLE
A series is an infinite sum, where the nth summand is the nth term of a sequence. A series is usually denoted using “sigma notation,” i.e.,
The index n may begin with 0, 1, or k for any natural number k, as a matter of convenience. The nth partial sum Sn of a series is the (finite) sum of the first n terms of the series. A series is said to converge if and only if its sequence of partial sums {S 1, S 2, . . . , Sn, . . . } is a convergent sequence. There are several important types of series and several tests for the convergence of a series. Additionally, most useful functions have Taylor series representations, which makes them very important in the study of differential equations. See the article for a complete description.
signed measure
Given a set X together with a s-algebra of sets M defined on it, a signed measure on (X, M) is an extended real-valued function m with domain M satisfying: - The signed measure of the empty set is zero.
- The signed measure m assumes at most one of the values +/- infinity.
- (Countable additivity) Given a countable sequence of disjoint sets in M, the signed measure of the union of the sequence is equal to the sum of the signed measures of the sets in the sequence, where this sum converges absolutely if the signed measure of the union is finite.
Technically speaking, every measure is a signed measure; ordinary (i.e., nowhere negative) measures are sometimes called positive measures.
sphere
A closed surface, all points of which are equidistant from a given point, called the center.
In 3-dimensional Euclidean space, the equation of a sphere of radius r and center (h, j, k) is
The term sphere may also refer to the solid bounded by this surface, and the interior is then called the open sphere of radius r.
More generally, a sphere may be defined as the set of points in n-dimensional space (or any metric space) equidistant from a given point. The unit sphere in n-dimensional space is typically denoted S n - 1. Thus, the unit sphere in ordinary 3-space is denoted S2, and the unit circle in the plane is denoted S1.
Stone-Weierstrass Theorem
If X is a compact space and C(X) denotes the space of all continuous functions on X, and A is an algebra of functions in C(X) which separates the points of C(X) and which contains a constant function f not identically zero, then A is dense in C(X).
sup
Abbreviation of supremum.
supremum
The supremum of any subset of a linearly ordered set is the least upper bound of the subset. In particular, the supremum of any set of numbers is the smallest number in the set which is greater than or equal to every number in the set. In a complete linear order the supremum of any bounded set always exists.
Cf. infimum, least upper bound axiom.
surjection
A surjective function, i.e., a function that maps at least one element of its domain to each element of its range.
Cf. injection, bijection.
surjective
A function f from a set X to a set Y is surjective, also called “onto,” if to each element y of Y there is an element x of X such that f maps x to y, i.e., f (x)=y. Compare: injective, bijective.
Taylor series
Given a function having derivatives of all orders, the Taylor series of the function is given by
where f (k)(a) is the kth derivative of f at a. A function is equal to its Taylor series if and only if its error term Rn can be made arbitrarily small, where
It can be shown that
for some c between a and x.
totally bounded
Given a metric space X, a subset E of X is called totally bounded if for every e greater than zero there is a finite covering of E by open spheres in X whose radius is less than e.
Cf. bounded.
totally ordered set
A set with a total order defined on it.
total variation
(For total variation of a signed measure, see the Jordan Decomposition Theorem.) If f is a function on the real numbers with range in the complex numbers, then the function Tf given by
is called the total variation function of f, where here the supremum is being taken over all finite partitions of the real line up to x. If the limit of this function as x goes to infinity is finite, then f is said to be of bounded variation. The space of all such functions is usually denoted by BV. Functions which are increasing and bounded are in BV, and differentiable functions whose derivative is bounded are in BV.
transcendental number
A number which is not an algebraic number, i.e., that is not the root of any polynomial with rational coefficients. It is known that e and p (pi) are transcendental. Since the algebraic numbers are countable and the set of all real numbers is uncountable, this means that the set of transcendental numbers is uncountably large as well.
Tychonoff’s Theorem
If {Xa}, a in some index set A, is any family of compact topological spaces, then the cartesian product of the Xa, with the product topology, is compact.
uncountable
A set is uncountable (uncountably infinite) if it is infinite but not countable, i.e., no complete one-to-one match-up of the set with the set of natural numbers (finite ordinals) can be performed. Georg Cantor proved that the set of real numbers is uncountably infinite. (This is sometimes called the “non-denumerability of the continuum”).
Related MiniText: Infinity -- You Can't Get There From Here...
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