...ar scheme|regular scheme]], i.e. if all its local rings are regular. For a perfect field $ k $
...d is a non-singular algebraic variety. In the case of the field of complex numbers a non-singular algebraic variety has the structure of a complex [[Analytic
3 KB (540 words) - 11:38, 12 October 2023
...were distinguished; the structure of perfect numbers (cf. [[Perfect number|Perfect number]]) was studied; and the solution in integers of the equation $ x
...mbers. Somewhat later, Eratosthenes discovered a method of obtaining prime numbers, which is still called the sieve of Eratosthenes (cf. [[Eratosthenes, sieve
10 KB (1,503 words) - 08:03, 6 June 2020
dimensional balls of equal radius (see [[Geometry of numbers|Geometry of numbers]]).
...ng all classes of boundary forms. In particular, the Voronoi algorithm for perfect forms (see [[#References|[1]]], [[#References|[3]]], [[#References|[4]]]).
3 KB (421 words) - 22:10, 5 June 2020
be the perfect [[Cantor set|Cantor set]], $ P $
the numbers $ a _ {n} $
2 KB (262 words) - 22:15, 5 June 2020
then there is an uncountable co-analytic set without perfect part, while if $ E $
...firmative solution of Luzin's problem about parts of the series of natural numbers, starting from the [[Continuum hypothesis|continuum hypothesis]] or the neg
5 KB (711 words) - 19:09, 11 January 2024
...ent (1989) one approach to finding good codes is to find curves with large numbers of rational points compared to their genus. This brings in advanced algebra
5 KB (779 words) - 07:49, 26 January 2018
...thod for reducing the solution of an equation of degree 4 over the complex numbers (or, more generally, over any field of [[Characteristic_of_a_field|characte
square brackets is a perfect square. For this the [[Quadratic equation|discriminant of the
2 KB (271 words) - 13:05, 17 December 2014
be the [[Baire space|Baire space]] (homeomorphic to the space of irrational numbers). A set $ P \subset I ^ {m} $
...aire property|Baire property]] and, if uncountable, contains a (non-empty) perfect subset; every $ A _ {3} $-
10 KB (1,492 words) - 08:08, 6 June 2020
one may restrict attention to perfect probability spaces, i.e. spaces
which occur in the general scheme, cannot occur in perfect probability
4 KB (669 words) - 17:20, 11 May 2013
...ditionally, proofs are deemed to be non-elementary if they involve complex numbers.
...partitions, of additive representations, of the approximation by rational numbers, and of continued fractions. Quite often, the solution of such problems lea
13 KB (2,043 words) - 20:28, 13 October 2014
These numbers are called the periodic correlation coefficients (in case $ a = b $,
are called perfect. Perfect sequences whose entries are $ t $
9 KB (1,351 words) - 11:02, 26 March 2023
...function $F$ on $[a,b]$ such that $F'=f$ almost everywhere, and if for any perfect set $P$ there exists a portion of $P$ on which $F$ is absolutely continuous
...approximate derivative]] is almost everywhere equal to $f$ and if, for any perfect set $P$, there exists a portion of $P$ on which $F$ is absolutely continuou
5 KB (856 words) - 09:42, 12 October 2023
error-correcting perfect codes with $ e > 1 $
is a prime power. A perfect $ e $-
5 KB (772 words) - 11:10, 23 March 2023
...mplete this part there is Luzin's theorem on subsets of the set of natural numbers, which reflects some properties of the remainder $ \beta \mathbf N \setmi
"Luzin's theorem on subsets of the set of natural numbers" states that there is a family $ \{ A _ \alpha \} _ {\alpha < \omega _
7 KB (1,075 words) - 19:29, 1 January 2021
of $ K _ {12 } $[[#References|[a9]]]; no larger numbers are known.
...o factors is a Hamilton cycle (cf. [[Hamiltonian tour|Hamiltonian tour]]). Perfect one-factorizations exist for many orders, and no order $ n $(
16 KB (2,342 words) - 09:33, 19 January 2021
tuples of numbers is the set $ \{ {\langle a _ {1} \dots a _ {n-} 1 \rangle } : {\exists b
tuples of numbers, it is possible to find a value of the parameter $ m $(
7 KB (998 words) - 19:35, 5 June 2020
...of real functions on a topological space $X$, indexed by countable ordinal numbers, which are defined inductively iterating the operation of taking pointwise
...ich takes the values $0$ on the irrational numbers and $1$ on the rational numbers, is a classical example of function which does not belong to the first Bair
5 KB (746 words) - 08:32, 18 August 2013
...itrary residue fields. There is a class field theory for local fields with perfect residue fields in terms of a certain fundamental group [[#References|[a1]]]
4 KB (548 words) - 17:43, 20 December 2014
...ications (cf. [[Perfect compactification|Perfect compactification]]) . All perfect compactifications $ bX $
itself is perfect as well) and have, like $ \beta X $,
25 KB (3,639 words) - 09:53, 26 March 2023
The numbers $ v,\ b,\ r _ {i} $,
different numbers $ \lambda _ {1} \dots \lambda _ {m} $
9 KB (1,344 words) - 16:08, 6 February 2020