In mathematics, the upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given...

values under f. Then b is an upper bound for S, and the least upper bound must be a root of f. The Bolzano–Weierstrass theorem for R states that every sequence...

an important role in polyhedral combinatorics: according to the upper bound theorem, proved by Peter McMullen and Richard Stanley, the boundary Δ(n,d)...

and in time matching the worst-case output complexity given by the upper bound theorem in higher dimensions. As well as for finite point sets, convex hulls...

order n. Ramsey's theorem states that such a number exists for all m and n. By symmetry, it is true that R(m, n) = R(n, m). An upper bound for R(r, s) can...

theorem or collection of theorems. The least-upper-bound property states that every nonempty subset of real numbers having an upper bound (or bounded...

this theorem provides only an upper bound of the number of points, which is almost always reached. This bound is often referred to as the Bézout bound. Bézout's...

supremum of S {\displaystyle S} exists, it is unique, and if b is an upper bound of S {\displaystyle S} , then the supremum of S {\displaystyle S} is...

example is of dimension d = 4 and has f0 = 8 vertices. The upper bound theorem gives upper bounds for the numbers fi of i-faces of any simplicial d-sphere...

In probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function...

Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . The theorem states that each infinite bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} has a...

mathematics, the extreme value theorem states that if a real-valued function f {\displaystyle f} is continuous on the closed and bounded interval [ a , b ] {\displaystyle...

over an interval with a variable upper bound. Conversely, the second part of the theorem, the second fundamental theorem of calculus, states that the integral...

mcgill.ca. Retrieved 29 October 2018. Seidel, Raimund (1995). "The upper bound theorem for polytopes: an easy proof of its asymptotic version". Computational...

and computer science. It is sometimes called Holevo's bound, since it establishes an upper bound to the amount of information that can be known about a...

decomposition into finitely many chains, or when there exists a finite upper bound on the size of an antichain, the sizes of the largest antichain and of...

converges to its smallest upper bound, its supremum. Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its infimum. In particular...

of W(r, k) for most values of r and k. The proof of the theorem provides only an upper bound. For the case of r = 2 and k = 3, for example, the argument...

In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under...

partially ordered set in which every chain has a least upper bound Szpilrajn extension theorem – Mathematical result on order relations Tarski finiteness –...

(and the theorem) does not say anything about the limit of the difference of the two functions as x increases without bound. Instead, the theorem states...

Hartley. The Shannon–Hartley theorem states the channel capacity C {\displaystyle C} , meaning the theoretical tightest upper bound on the information rate...

Shiganov, I.S. (1986). "Refinement of the upper bound of a constant in the remainder term of the central limit theorem". Journal of Soviet Mathematics. 35 (3):...

In mathematical analysis, the intermediate value theorem states that if f {\displaystyle f} is a continuous function whose domain contains the interval...

In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented...

subject to some constraints, e.g., all particles must be bound to a finite volume. The theorem is commonly discussed in the context of ergodic theory,...

} This bound cannot be improved, except in terms of the implicit constants in its big O notation. An equivalent formulation of the theorem is the following...

a comparison, one may want upper bounds or lower bounds of the accuracy. A lower bound is typically described by a theorem like "for every element α of...

2,3,\dotsc ,n\}|}{n}}>0.} Szemerédi's theorem asserts that a subset of the natural numbers with positive upper density contains an arithmetic progression...

vice versa. The weak duality theorem states that the objective value of the dual LP at any feasible solution is always a bound on the objective of the primal...