where the $\alpha_k$ are integers between 0 and $q-1$ (inclusive). In the representation of $x$ by a $q$-ary
3 KB (412 words) - 14:41, 7 March 2019
...on an interval $I$, is completely monotonic on $I$ if for all non-negative integers $n$,
4 KB (582 words) - 12:38, 23 February 2024
...$C=\partial D\subset\partial K$. For each pair $(k_+,k_-)$ of non-negative integers, not both $0$, there is a unique oriented diffeomorphism type of kinky hand
...the same [[#References|[a1]]], [[#References|[a7]]]. For any non-negative integers $k_\pm$, not both $0$, there is a Casson handle with $T_1$ having exactly $
7 KB (1,047 words) - 15:29, 4 October 2014
where the parameters <math>N,M,m</math> are non-negative integers which satisfy the condition <math>m\leq M\leq N</math>. A negative hypergeo
3 KB (447 words) - 14:48, 5 June 2017
are integers, $ 0 < \alpha \leq 1 $,
3 KB (472 words) - 22:11, 5 June 2020
...ts,p_k$ distinct prime numbers and $\alpha_1,\ldots,\alpha_k$ non-negative integers, is isomorphic to the direct product of distributive quasi-groups $Q_1,\ldo
3 KB (425 words) - 22:16, 7 January 2017
runs through the real numbers or the integers). It consists of the existence of a trajectory $ \{ T _ {t} w _ {0} \} $
3 KB (470 words) - 08:25, 6 June 2020
...$ using multiplication, for any system $U,U_1,\ldots,U_q$ of matrices over integers. The case in which $U,U_1,\ldots,U_q$ are square matrices of the same order
3 KB (462 words) - 20:23, 31 July 2014
...containing all infinite ones, let $\mathcal{O} _ { S }$ be the ring of $S$-integers and let $\mathcal{O} _ { S } ^ { * }$ be the group of $S$-units of $K$. Let
...valign="top"> N.P. Smart, "Thue and Thue–Mahler equations over rings of integers" ''J. London Math. Soc.'' , '''56''' : 2 (1997) pp. 455–462</td></tr>
7 KB (1,096 words) - 17:46, 1 July 2020
...For instance, the ordinal number of the set $ \mathbf{N} $ of all positive integers, ordered by the relation $ \leq $, is $ \omega $. The ordinal number of the
...rdinal numbers in a form resembling the decimal representation of positive integers. If $ \gamma > 1 $ and $ 1 \leq \alpha < \gamma^{\eta} $, then there exists
9 KB (1,404 words) - 18:33, 4 December 2017
are called algebraic integers. If a ring $ B $
3 KB (562 words) - 22:12, 5 June 2020
...essence may be explained as follows. Consider a given sequence of positive integers $n_1,\ldots,n_Z$ not larger than $N$, a prime number $p < \sqrt N$ and a re
3 KB (418 words) - 21:29, 11 November 2017
...A [[lattice]] $L$ is isomorphic to a subset of the totally ordered set of integers if and only if every sublattice of it is a [[Retract|retract]].
3 KB (509 words) - 06:25, 28 March 2023
...also [[Algebraic number|Algebraic number]]; [[Field|Field]]) with ring of integers $ R $
4 KB (600 words) - 17:32, 5 June 2020
the greatest common divisor of all the integers $ k $
3 KB (443 words) - 11:17, 12 May 2024
...s the number of executed operations. (In particular, for computations over integers the Turing model uses a logarithmic size measure together with bit-costs wh
...near programming|linear programming]] problem is in $\mathcal{P}$ over the integers in the Turing model (see [[#References|[a4]]]). It is not yet (1998) known
7 KB (1,161 words) - 16:46, 1 July 2020
be the ring of integers of $ K $,
4 KB (569 words) - 05:59, 11 October 2023
at these points be non-integers. Let $ a $
3 KB (479 words) - 17:43, 16 December 2020
are algebraic integers. Every character of a group $ G $
for some integers $ a _ {1} \dots a _ {n} $.
10 KB (1,488 words) - 19:39, 5 June 2020
...$-group]]s. E.g. the additive group of the ring $\mathbf{Z}_p$ of $p$-adic integers is a pro-$p$-group. In Galois theory pro-$p$-groups appear as [[Galois grou
3 KB (535 words) - 19:45, 21 October 2017
...ch of number theory with the basic aim of studying properties of algebraic integers in algebraic number fields $ K $ of finite degree over the field $ \ma
...x _{n} \omega _{n} $ where all the $ x _{i} $ run through the rational integers (i.e. $ \mathbf Z $ ). Moreover, such a representation is unique for each
28 KB (4,440 words) - 22:00, 11 December 2019
A method for adding sequences of positive integers; created by L.G. Shnirel'man in 1930. Let $ \nu ( x) \neq 0 $
3 KB (458 words) - 19:05, 20 November 2023
...two complex constants, $\operatorname{Im}c=0$; $m\geq2$, and $n\geq1$ are integers named, respectively, the order and the index of the singular point and $u$
3 KB (495 words) - 13:00, 3 October 2014
...als]] evaluated at rational arguments: Let $h$ and $k$ be relatively prime integers. If $n$ is even, then
...{ 1 } + 1 / 2$. It has also been shown [[#References|[a10]]] that for all integers $h$, $k$, $n$ with $k \neq 0$ and $n \geq 1$, $k ^ { n } ( B _ { n } ( h /
8 KB (1,206 words) - 16:55, 1 July 2020
the ring of integers, and also quotient rings of them, are Noetherian. Every [[Artinian ring]] i
3 KB (539 words) - 18:51, 3 April 2024
its ring of integers, $ B $
the maximal order is the ring of integers of $ B $,
13 KB (2,001 words) - 02:12, 1 March 2022
...ymmetric way. The classical semi-simple rings and all rings of residues of integers $ \mathbf Z /( n) $
4 KB (550 words) - 08:13, 6 June 2020
of non-negative integers satisfying the condition $ \lambda _ {1} \geq \dots \geq \lambda _ {n}
4 KB (519 words) - 16:18, 5 February 2022
that is defined for any set of non-negative integers $ n _ {1} \dots n _ {k} $
4 KB (522 words) - 13:11, 6 January 2024
are not integers and the relevant derivatives are continuous, then a Nikol'skii space is a [
if and only if for any non-negative integers $ k $
11 KB (1,572 words) - 19:46, 12 January 2024
...If $p(x)=x^2$ or $p(x)=x^3$, one obtains sequences of squares or cubes of integers, i.e. special cases of arithmetic series of the second and third orders. If
4 KB (566 words) - 07:28, 26 March 2023
3) if $k$ and $n$ are integers which satisfy $2^n\leq k<2^{n+1}$, then $w_k$ changes sign once on interval
3 KB (467 words) - 09:42, 27 November 2018
functors of their rings of integers, and also between the values of the $ L $-
...<TD valign="top"> B. Harris, G. Segal, "$K_i$ groups of rings of algebraic integers" ''Ann. of Math.'' , '''101''' : 1 (1975) pp. 20–33 {{MR|0387379}} {{ZBL|
8 KB (1,120 words) - 20:06, 31 October 2023
is a collection of non-negative integers, then $ c _ \omega $
be a set of non-negative integers. Let $ S _ \omega ( c _ {1} \dots c _ {n} ) $
13 KB (1,910 words) - 19:40, 7 January 2024
...D valign="top"> J.H. Conway, N.J.A. Sloane, "Self-dual codes over the integers modulo $4$" ''J. Combin. Th. A'' , '''62''' (1993) pp. 31–45</TD></TR>
3 KB (459 words) - 17:32, 17 April 2016
runs over all non-negative integers less than $ k $
3 KB (502 words) - 14:27, 17 March 2023
...and $k,l,\dots,t$ (the ''exponents of the powers'', which are non-negative integers) are constants. The individual expressions
...of polynomials similar to that played by the prime numbers in the ring of integers. For example, the following theorem holds: If a product $PQ$ is divisible b
9 KB (1,497 words) - 10:44, 27 June 2015
are integers, and $ n $
3 KB (485 words) - 19:43, 5 June 2020
with coefficients in the ring of integers $ \mathbf Z $
5 KB (636 words) - 22:11, 5 June 2020
Positive integers are written by repeating these numerals. When a letter is followed by a let
11 KB (1,241 words) - 08:12, 6 June 2020
...$A$ and $B$ of finite $k$-dimension are equivalent if there exist positive integers $m$ and $n$ such that the tensor products $A \otimes_k M_m(k)$ and $B \otim
...ditive group of rational numbers and $\mathbb{Z}$ is the additive group of integers. This fact is of importance in local class-field theory.
7 KB (1,232 words) - 12:12, 30 December 2015
the ring of integers, then mod- $ R $
4 KB (572 words) - 08:01, 6 June 2020
are non-negative integers and $ M \leq N $,
4 KB (602 words) - 22:11, 5 June 2020
...[2]</TD> <TD valign="top"> A. Mostowski, "On definable sets of positive integers" ''Fund. Math.'' , '''34''' (1947) pp. 81–112</TD></TR></table>
4 KB (548 words) - 15:41, 22 June 2020
...rithmetic theory of cubic forms over [[number field]]s (and their rings of integers) is still (1987) rather poorly developed in comparison with the rich and me
3 KB (526 words) - 11:44, 12 October 2023
runs through the set of integers and each $ b _ {k} \in A $.
4 KB (566 words) - 10:58, 29 May 2020
...very ring can naturally be considered as an operator ring over the ring of integers.
4 KB (634 words) - 07:02, 30 March 2024
...indexed by $i$ running over the natural numbers (one-sided, superword) or integers (two-sided, bi-infinite word).
3 KB (544 words) - 07:12, 10 November 2023
are zero or negative integers, the series (2) terminates after a finite number of terms, and the hypergeo
are integers, can be obtained by iterated application of Gauss' relations. The following
12 KB (1,576 words) - 01:14, 21 January 2022
...dditive group of a graded algebra (considered as a module over the ring of integers) is a positively [[Graded module|graded module]]. As an example of a graded
4 KB (576 words) - 19:42, 5 June 2020
...uares: Any natural number can be represented as the sum of four squares of integers. This was established by J.L. Lagrange [[#References|[1]]]. For a generaliz
4 KB (562 words) - 21:05, 11 October 2014
where $g(c_i)$ and $h(c_i)$ are integers. Thus, $g(c_i)$ divides $f(c_i)$. Choosing arbitrary divisors $d_i$ of the
3 KB (574 words) - 18:14, 14 June 2023
...m p ^ { r - 1 } } ( \operatorname { mod } p ^ { 3 r } )$ for all positive integers $m$, $r$ and all prime numbers $p \geq 5$. Another congruence is $a _ {( p
3 KB (525 words) - 20:51, 23 January 2024
...sums actually lie in the ring $\mathbf{Z} [ \zeta _ { e } ]$ of algebraic integers in the cyclotomic field ${\bf Q} [ \zeta _ { { e } } ]$ (cf. also [[Cyclot
<tr><td valign="top">[a8]</td> <td valign="top"> B. Schmidt, "Cyclotomic integers and finite geometry" ''J. Amer. Math. Soc.'' (to appear)</td></tr>
9 KB (1,331 words) - 19:36, 13 February 2024
are integers (a special case of Abel's theorem, see [[Abelian function|Abelian function]
are integers such that $ \alpha \delta - \beta \gamma = n $,
9 KB (1,292 words) - 19:08, 20 January 2022
4 KB (577 words) - 10:26, 10 December 2023
...hat of a diagonalizable group. If $M={\mathbb Z}$ is the additive group of integers, then $D_S(M)$ coincides with the multiplicative group scheme $G_{m,S}$.
4 KB (629 words) - 20:08, 15 December 2020
...one of the following sets: the set of natural numbers, the set of rational integers, the set of integral ideals of a given algebraic number field, a lattice in
4 KB (608 words) - 08:18, 4 November 2023
In number theory one considers equations over the integers, the study of the solutions of which forms the subject of the theory of [[D
4 KB (692 words) - 13:38, 31 July 2014
...whose states form a single zero class is the symmetric random walk on the integers:
4 KB (466 words) - 07:59, 6 June 2020
...ng of algebraic integers|Local-global principles for the ring of algebraic integers]]).
15 KB (2,309 words) - 06:58, 13 February 2024
...ic to the field $\mathbb{Z}/p\mathbb{Z}$ of residue classes of the ring of integers modulo $p$. In any fixed [[Algebraic closure|algebraic closure]] $\Omega$ o
4 KB (749 words) - 18:32, 2 November 2014
there exist integers $ q _ {i} $,
4 KB (615 words) - 22:15, 5 June 2020
are non-negative integers,
4 KB (524 words) - 04:11, 6 June 2020
adic integers (for any prime $ p $);
4 KB (674 words) - 17:31, 5 June 2020
of integers.
4 KB (643 words) - 19:38, 5 June 2020
...$( X _ { n } ) _ { n \in {\bf Z} ^ { d }}$, indexed by the $d$-dimensional integers ($d \geq 1$). The definition of $\mathcal{T}$ is the same as before, but no
4 KB (552 words) - 16:56, 1 July 2020
...lpha _ { 1 } , \ldots , \alpha _ { n } )$, all $\alpha_j$ are non-negative integers, $z = ( z_ 1 , \dots , z _ { n } )$, $z ^ { \alpha } = z _ { 1 } ^ { \alpha
7 KB (1,065 words) - 09:52, 11 November 2023
...dius $\epsilon > 0$ centred at $a + i b$, where $a$, $b$ range over all integers, satisfies this criterion and hence is in $\operatorname{BMOA}$.
5 KB (707 words) - 19:01, 22 January 2024
...pute functions defined on an infinite domain of constructive objects (e.g. integers, words over a finite alphabet, finite graphs, infinite trees, etc., cf. [[C
3) Abstract computers which usually process words or integers and have an elaborate structure determined by practical requirements, havin
9 KB (1,295 words) - 17:46, 4 June 2020
a discretely valued complete field; the ring of integers is $ k[[ X ]] $,
8 KB (1,203 words) - 10:36, 20 January 2024
is the [[Eilenberg–MacLane space|Eilenberg–MacLane space]] of the group of integers $ \mathbf Z $
4 KB (616 words) - 19:33, 20 January 2021
are integers, is solvable, and even gave an asymptotic formula for the number of solutio
5 KB (759 words) - 08:29, 6 June 2020
is the group of integers under addition (cf. also [[Homology|Homology]]). A group which satisfies th
4 KB (665 words) - 08:29, 6 June 2020
are integers with $ n $
4 KB (623 words) - 08:03, 6 June 2020
...ther than numbers; thus, it does calculations in exact mode. For instance, integers are regarded as strings of digits of arbitrary length and $\sqrt 3$ is rega
...s handled by these systems are polynomials over effective rings (e.g., the integers, the rational numbers, algebraic number fields, finite rings), strings of c
9 KB (1,292 words) - 20:51, 18 September 2016
...inite alphabet $A$ and assume a [[total order]] on the letters. For given integers $p$ and $n$, every sufficiently long word in $A^*$ has either a factor whic
3 KB (562 words) - 14:15, 12 November 2023
...prime number theorem]]), and the construction of a theory of quotients of integers that is essentially equivalent to the theory of rational numbers. With the
9 KB (1,351 words) - 20:43, 26 November 2016
...bers from the [[Uniform distribution|uniform distribution]]: A sequence of integers is initialized with a value $z_0$ and continued as
...ncrement, and $z_0$, the starting number, are suitably chosen non-negative integers. Three choices of $m$, $a$ and $r$ are common on most computers:
11 KB (1,696 words) - 19:30, 8 February 2024
...is the quotient group of the group of all real numbers by the subgroup of integers, and is the largest of all homological dimensions.
coincides with the cohomological dimension for the group of integers.
12 KB (1,894 words) - 19:35, 5 June 2020
Examples of principal ideal rings include the ring of integers, the ring of polynomials $ F ( x) $
5 KB (880 words) - 19:00, 9 January 2024
exist for all non-negative integers $ \nu _ {1} \dots \nu _ {k} $
5 KB (661 words) - 12:22, 17 January 2021
are integers. The dimension polynomial is a birational invariant of the field, that is,
5 KB (651 words) - 08:36, 1 July 2022
Suppose that a set of integers $ ( d _ {1} \dots d _ {n} ) $
6 KB (931 words) - 20:20, 15 March 2023
for all but a finite number of integers $ n $.
6 KB (913 words) - 19:44, 16 January 2024
$n>1$, is solvable (or unsolvable) in integers. Questions on the distribution of power residues and non-residues have been
4 KB (628 words) - 19:38, 19 December 2014
...unity if and only if $|z|=1$ and $\arg z=2\pi m/n$, where $m$ and $n$ are integers, i.e. if and only if
4 KB (680 words) - 13:40, 30 December 2018
...ginated in [[#References|[a1]]], Chapt. II, Sect. 3. Denoting the positive
integers by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
10 KB (1,395 words) - 06:44, 9 October 2016
be positive integers. If $ a _{1} \dots a _{s} $
...with zero-sum solutions to systems of equations and inequalities over the integers. Surveys of this can be found in [[#References|[a2]]] and [[#References|[a4
10 KB (1,573 words) - 17:25, 28 January 2020
...problems is as follows [[#References|[4]]]: Uniformly with respect to the integers $ k \geq 2 $,
for any integers $ k \geq 2 $,
11 KB (1,578 words) - 11:58, 26 March 2023
...f matrices $(a_{ij})\in \GL_n(\Z_p)$ (where $\Z_p$ is the ring of $p$-adic integers), such that $a_{ij}\in p\Z_p$ for $i>j$, and let $N$ be the subgroup of mon
4 KB (647 words) - 09:57, 30 January 2014
...last theorem is the claim that $x^n+y^n=z^n$ has no solutions in non-zero integers for $n>2$. For $n=2$ there is an infinity of solutions, such as: $3^2+4^2=5
...$\mathcal E_{a,b,c}$ given by $y^2=x(x-a)(x+b)$, where $a+b=c$ in non-zero integers. This Frey curve $\mathcal E_{a,b,c}$ is an [[Elliptic curve|elliptic curve
8 KB (1,419 words) - 15:20, 17 March 2023
...rs for the module of all homogeneous solutions). Examples are the rings of integers and of rational numbers, algebraic number fields, and finite rings. For suc
10 KB (1,471 words) - 14:52, 30 May 2020
...ps is of course the multiplicative semi-group $\mathbb{N}$ of all positive integers $\{1,2,3,\ldots\}$, with its subset $P_{\mathbb{N}}$ of all rational prime
...K$ of all non-zero ideals in the [[Ring|ring]] $R = R(K)$ of all algebraic integers in a given [[Algebraic number|algebraic number]] field $K$, with $|I| = \ma
24 KB (3,738 words) - 07:41, 7 February 2024
...n a union of subsets $B _ { \kappa }$, labelled by unordered $n$-tuples of integers $\kappa$. Each of these consists of all loops with $\kappa$ as the set of p
4 KB (630 words) - 17:01, 1 July 2020
...one resulting from the binomial theorem) without having to use division by integers.
4 KB (663 words) - 08:28, 20 January 2024
...r symmetric bilinear, form over an ordered field is a pair of non-negative integers $ ( p, q) $,
4 KB (651 words) - 18:47, 13 January 2024
...ulated as follows: Let $J_{k,n}(N)$ be the number of solutions in positive integers $x_1,\ldots,x_n$ of the equation
...r $n=2$ by showing that any positive integer is the sum of four squares of integers. D. Hilbert (1909) was the first to give a general solution to Waring's pro
27 KB (4,516 words) - 18:38, 18 October 2014
be integers. Then a field $ F $
...|non-standard analysis]]; in particular, non-standard models of the reals, integers, etc. can be obtained as ultrapowers of $ \mathbf R $,
10 KB (1,605 words) - 16:57, 13 January 2024
are non-negative integers, and $ a _ {1} \dots a _ {p} $,
5 KB (645 words) - 08:01, 6 June 2020
...t and output of arithmetic, logical and text data. Arithmetic data include integers, real (with ordinary and enhanced precision) and usually also complex numbe
4 KB (631 words) - 19:39, 5 June 2020
respect to the norm of $\fp$, and let $A_\fp$ be the ring of integers of
4 KB (757 words) - 21:21, 22 November 2014
algebraic integers of an imaginary quadratic field $k$ and $f\in {\mathbb N}$; or
is a subring of finite index in the ring of algebraic integers of the
19 KB (3,251 words) - 20:37, 19 September 2017
...nd a family of them. When a NEF is concentrated on the set of non-negative integers, its elements are sometimes called "power series" distributions, since th
...) ^ { p }$ (see [[#References|[a4]]]. Trivially, it contains all positive integers. The model
12 KB (1,855 words) - 17:45, 1 July 2020
are non-negative integers and $ \alpha _ {0} + \dots + \alpha _ {k} = m $,
5 KB (717 words) - 08:28, 6 June 2020
is simply the ordered set of integers $ (k _{1} \dots k _{r} ) $ ,
5 KB (737 words) - 10:19, 17 December 2019
be a sequence of positive integers. From $ \mathbf n $
4 KB (631 words) - 08:14, 6 June 2020
...alpha_i$ are the principal values of the logarithms. If there are rational integers $b_1,\ldots,b_n$, $|b_i| < B$, such that
5 KB (776 words) - 08:31, 23 November 2023
...laim that $x ^ { n } - y ^ { n } = z ^ { n }$ has no solutions in non-zero integers for $n > 2$ (see also [[Fermat last theorem|Fermat last theorem]]). Howe
4 KB (702 words) - 16:45, 1 July 2020
3) If $a$ and $b$ are integers, $b\ge 1$, if $r$ is a real number, and if
...expansion of a number $r$ into a regular continued fraction, then for any integers $a$ and $b$ it follows from $b>0$, $\d_n \ne a/b$ and
11 KB (1,649 words) - 11:41, 4 September 2013
...\dim X=\dim_\mathbf ZX$ if $\dim X<\infty$ and $\mathbf Z$ is the group of integers. It was subsequently noted by L.S. Pontryagin that homological dimensions w
5 KB (741 words) - 06:05, 22 November 2018
...all [[#References|[a24]]] proved that there exist infinitely many abundant integers $n \equiv a ( \operatorname { mod } b )$ (with $a$ and $b$ given). Let $k$
...expressed as the sum of two abundant numbers. Actually, this is valid for integers $> 20162$, see [[#References|[a27]]].
15 KB (2,194 words) - 05:54, 15 February 2024
be the ring of integers of an algebraic number [[Field|field]] $ F $(
5 KB (731 words) - 08:21, 26 March 2023
are integers and $ \Delta _ {j} $
is the constant sheaf of integers. Thus, if $ H ^ {1} ( M, {\mathcal O} ) = H ^ {2} ( M, \mathbf Z ) = 0
16 KB (2,209 words) - 11:03, 26 March 2023
...s/s085/s085000/s0850009.png" /> and if there exists a sequence of rational
integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
16 KB (2,130 words) - 07:52, 11 December 2016
into the ring of integers $ \mathbf Z $.
5 KB (659 words) - 06:06, 23 April 2024
are integers, namely, the multiplicities with which the $ \pi _ \alpha $
6 KB (828 words) - 09:30, 26 March 2023
...{ n } + a _ { 1 } x ^ { n - 1 } + \dots + a _ { n }$, where the $a_k$ are integers, $a _ { n } \neq 0$ and the roots of $P ( x )$ other than $\theta$ all lie
5 KB (723 words) - 19:03, 23 January 2024
and the integers $ n > 0 $,
6 KB (838 words) - 08:08, 6 June 2020
in the sense of Weyl). For integers $ r = 1, 2 \dots $
5 KB (791 words) - 10:58, 29 May 2020
The simplest algebraic operations — arithmetic operations on positive integers and positive rational numbers — can be encountered in the oldest mathemat
...e best textbooks of that time, L. Euler's Introduction to algebra includes integers, ordinary and decimal fractions, roots, logarithms, algebraic equations of
17 KB (2,478 words) - 16:09, 1 April 2020
composed of the integers from 1 up to $ n ^ {2} $
5 KB (762 words) - 19:27, 12 January 2024
...ng machines computing functions from non-negative integers to non-negative integers.) Let $ T $
16 KB (2,413 words) - 16:10, 1 April 2020
...n of a positive integer $n$ is a decomposition of $n$ as a sum of positive integers. For example, the partitions of $4$ read: $4$, $3+1$, $2+2$, $2+1+1$, $1+1+
4 KB (718 words) - 15:32, 11 November 2023
which runs over the non-negative integers, is called the dimension of $ H ^ { n } ( G, A) $.
For arbitrary integers $ m $
16 KB (2,427 words) - 09:48, 26 March 2023
adic integers, the ring $ k [[ T ]] $
5 KB (800 words) - 19:36, 5 June 2020
is the completion of the ring of integers $ \mathbf Z $
with Galois group isomorphic to the group of $ l $-adic integers $ \mathbf Z _ {l} $.
12 KB (1,769 words) - 11:19, 26 March 2023
...sot $E$-sequence $E ( a _ { 0 } , a _ { 1 } )$ is the sequence of positive integers $\{ a _ { n } \}$ defined for $0 < a _ { 0 } < a _ { 1 }$ by the recursion
5 KB (774 words) - 20:10, 24 January 2024
be integers such that
Ramsey's theorem: Let there be given integers $ q _ {1} \dots q _ {l} $
13 KB (1,942 words) - 08:13, 6 June 2020
...power of 2. In this connection, a (single) arithmetic error in the ring of integers $ \mathbf Z $
8 KB (1,189 words) - 09:46, 26 March 2023
...of the binomial coefficients and the Newton binomial formula for positive integers $ n $
...L. Euler. In his papers on the partitioning and decomposition of positive integers into summands he laid down the beginnings of one of the basic methods of ca
28 KB (4,167 words) - 17:45, 4 June 2020
...e equation $x^n+y^n=z^n$ (the Fermat equation) has no solution in non-zero integers $x,y,z$. It was stated by P. Fermat in about 1630 in the margins of his cop
has no solutions in non-zero relatively-prime integers $x,y,z$. One can also assume that $x$ and $y$ are relatively prime to $p$.
16 KB (2,596 words) - 09:27, 13 February 2024
...cal N P$-hard (cf. [[NP|$\cal N P$]]), even if distances are rounded up to integers and it is required only to decide whether a tour exists whose total length
4 KB (704 words) - 12:30, 17 February 2021
are integers $ \geq 1 $,
6 KB (821 words) - 01:34, 26 April 2022
...]] for the relation between the divisor class group of a ring of algebraic integers and Abelian field extensions.
5 KB (820 words) - 19:36, 5 June 2020
be positive integers such that $ 0 \leq n \leq s \leq 2n - 1 $.
8 KB (1,138 words) - 09:31, 19 January 2021
...oup of class two is uniquely determined by its group ring over the ring of integers, and that a countable Abelian $p$-group is uniquely determined by its group
5 KB (872 words) - 22:00, 29 April 2012
are integers, and $ u _ {m} ^ {n} $
6 KB (908 words) - 14:48, 7 April 2020
are mutually prime integers and $ q \geq 2 $,
6 KB (861 words) - 13:02, 6 January 2024
are integers.
5 KB (794 words) - 19:38, 5 June 2020
...-groups) need not have minimal ideals (the simplest example is the ring of integers) and a minimal ideal in a ring, if it exists, need not be unique. The sum o
6 KB (961 words) - 18:14, 7 May 2016
...an arbitrary [[Polynomial|polynomial]] over a [[Field|field]] or over the integers can be factored into irreducible factors, essentially uniquely (cf. also [[
...m extremely well, both in theory and practice. Of course, factorization of integers and thus also in $\mathbf Z [ x ]$ remains difficult; much of cryptography
12 KB (1,739 words) - 13:11, 26 March 2023
...the so-called Kloosterman sums, which appear in the study of partitions of integers and in the more general Hardy–Littlewood–Ramanujan methods. From the po
but also appear in complex-conjugate pairs. Furthermore, as algebraic integers, they are stable under the action of the Galois group of the rational numbe
12 KB (1,769 words) - 08:31, 26 March 2023
on the positive integers. For example, for irrational $ \alpha $
6 KB (911 words) - 12:00, 22 December 2019
...nce to $\mu_k(f)$. There exists an absolute constant $C$ such that for all integers $k$ and all integer-valued additive arithmetic functions $f(m)$ such that $
...ore precise information on $\omega(n)$. Denote by $N(x;a,b)$ the number of integers in the interval $[3,x]$ for which the inequalities
10 KB (1,718 words) - 04:55, 8 August 2018
is the group of integers. Furthermore, if the polyhedron is finite, the integer group, which is an A
6 KB (897 words) - 08:54, 25 April 2022
...as the discriminant of the lattice $M$. If $M$ coincides with the ring of integers of the field $K$, the discriminant of $M$ is simply called the discriminant
...be a finite separable field extension. Let $A_F$ and $A_E$ be the rings of integers (principal orders) of $F$ and $E$, respectively. Let $\def\fm{ {\mathfrak m
16 KB (2,947 words) - 08:53, 9 December 2016
is the ring of integers in the imaginary quadratic number field $ \mathbf Q ( \sqrt {- d } ) $.
5 KB (775 words) - 07:59, 26 March 2023
and not merely positive integers). If one chooses for $ X $
7 KB (1,086 words) - 17:45, 4 June 2020
are integers.
8 KB (976 words) - 17:46, 4 June 2020
Let $S$ be a fixed set of integers. Let $b _ { j }$ be fixed complex numbers and let $z_j$ be complex numbers
8) Given a finite set $S$ of integers, fixed complex numbers $b _ { j }$, $d \in [ 0,3 ]$, and two generalized po
40 KB (5,729 words) - 17:51, 5 May 2024
...th the $p$-adic valuation $\nu$ with $\nu(p)=1$ and let $R$ be its ring of integers (cf. [[Extension of a field|Extension of a field]]; [[Norm on a field|Norm
5 KB (919 words) - 04:20, 15 February 2024
...lpha _ { 1 } , \ldots , \alpha _ { n } )$, all $\alpha_j$ are non-negative integers, $| \alpha | = \alpha _ { 1 } + \ldots + \alpha _ { n }$, $z ^ { \alpha } =
7 KB (996 words) - 09:31, 3 February 2024
...} )$, i.e., $\lambda$ is a weakly decreasing sequence of $l$ non-negative integers $\lambda _ { i }$ called parts. Associated with any partition is an [[alter
.... For the example tableau, $\pi_T = 3111324$. Also, a sequence of positive integers $\pi = w _ { 1 } \dots w _ { n }$ is a lattice permutation or ballot sequen
14 KB (2,001 words) - 10:09, 11 November 2023
are integers, then interchanging in (2) the positions of $ \alpha $
8 KB (1,027 words) - 08:38, 14 July 2022
...> <TD valign="top"> E.L. Post, "Recursively enumerable sets of positive integers and their decision problems" ''Bull. Amer. Math. Soc.'' , '''50''' (1944)
6 KB (862 words) - 19:26, 22 January 2016
...{Z} _ { p }$, where $\mathbf{Z} _ { p }$ is the additive group of $p$-adic integers. Then there is a sequence of fields
...of $A _ { n } ( k )$. Iwasawa [[#References|[a8]]] proved that there exist integers $\lambda _ { p } ( K / k ) \geq 0$, $\mu _ { p } ( K / k ) \geq 0$ and $\nu
19 KB (2,876 words) - 05:38, 15 February 2024
For any integers $ a, b, c $(
7 KB (1,143 words) - 07:00, 6 July 2023
into positive integers $ m _{i} $ ,
6 KB (816 words) - 16:46, 17 December 2019
of integers of the field $ K $,
5 KB (793 words) - 16:10, 1 April 2020
...tribution|Multi-dimensional distribution]]) are defined similarly: For any integers $ k _ {i} \geq 0 $,
7 KB (1,019 words) - 08:53, 21 January 2024
be a sequence of real numbers (indexed by the non-negative integers). A regularization of $ \{ a _ {n} \} $
7 KB (1,067 words) - 12:49, 12 March 2021
...io $\lambda_i/\lambda_j$ is irrational, then for all $\eta\geq0$ there are integers $x_1,\ldots,x_s$, not all zero, such that $\lvert x_1\lambda_1+\cdots+x_s\l
5 KB (868 words) - 21:24, 18 November 2016
...s an instruction labelled by this label. Fundamental data types are lines, integers and real numbers, names, and models. The basic type is the line, and all ot
6 KB (915 words) - 08:14, 6 June 2020
...arbitrary commutative associative ring $R$, for example, over the ring of integers, are also discussed. The accepted notation for the ring of polynomials in a
6 KB (1,011 words) - 06:01, 30 September 2023
...ps is of course the multiplicative semi-group $\mathbf{N}$ of all positive integers $\{ 1,2,3, \ldots \}$, with its subset $P_{\mathbf{N}}$ of all rational pri
...i-group $G_K$ of all non-zero ideals in the ring $R=R(K)$ of all algebraic integers in a given
14 KB (2,037 words) - 09:45, 11 November 2023
are arbitrary non-negative integers. If all terms of the series (1) are non-negative, the sequence of its parti
7 KB (1,082 words) - 19:36, 5 June 2020
each run through all the integers in an independent manner, is known as the lattice (of points) $ \Lambda $
then there exist integers $ x _ {1} \dots x _ {n} $,
28 KB (4,011 words) - 11:37, 19 January 2022
is the maximal ideal of the ring of integers of an algebraic closure of $ K $ .
is the ring of integers of a local field of residue characteristic $ p $
17 KB (2,537 words) - 22:38, 15 December 2019
...be the multiplication operation on real numbers, the addition operation on integers, etc. Suppose that the first of these interpretations is used for $\cdot$.
7 KB (1,091 words) - 18:50, 14 August 2014
Let $n$, $d$ be positive integers with $d \geq 2$ and $\alpha$ an algebraic number with $0 < | \alpha | < 1$.
6 KB (816 words) - 20:26, 5 December 2023
...lume element in $\Omega$. Here, all $q_j$ and $s _ { j }$ are non-negative integers. Note that the integral moments $c _ { q }$ depend on $f$ and $\Gamma$, but
6 KB (978 words) - 17:44, 1 July 2020
are integers. There are positive constants $ c $,
6 KB (922 words) - 06:29, 30 May 2020
are non-negative integers, $ | \alpha | = \alpha _ {1} + \dots + \alpha _ {n} $,
7 KB (987 words) - 18:05, 22 May 2024
...arrow \mathbf{F} _ { p }$, where $\mathbf{F} _ { p }$ denotes the field of integers modulo $p$. Using the [[Dirichlet theorem|Dirichlet theorem]] on prime numb
...ickson polynomials have been obtained for the ring $\mathbf{Z} _ { n }$ of integers modulo $n$ as well as for the Galois ring $\operatorname{GR} ( p ^ { r } ,
15 KB (2,207 words) - 16:45, 1 July 2020
...les (the (disjoint) cycle decomposition of a permutation); the sequence of integers
7 KB (1,003 words) - 08:24, 6 June 2020
Any ring can be regarded as an algebra over the ring of the integers by taking the product $ n a $ (where $ n $
...uirements of the theory of algebraic equations, and residue-class rings of integers in number theory. The general concept of a ring did not exist.
21 KB (3,225 words) - 09:25, 13 July 2022
where $\Z$ is the set of all integers, then
6 KB (1,047 words) - 11:37, 13 February 2024
is the quotient group of the real numbers by the integers, associates to each group its group of characters.
8 KB (1,241 words) - 19:40, 5 June 2020
...king" well-defined because the $\Gamma$ function has poles in all negative integers. However, given the [[Laurent series]] of $\Gamma$ at such poles and the fa
6 KB (1,108 words) - 07:54, 26 March 2023
...a types can be viewed as Herbrand universa. For instance, the non-negative integers $ 0,1,2, \dots $
.... For instance, consider the simple case of addition over the non-negative integers. This operation is determined by the following two recursion equations:
15 KB (2,311 words) - 04:11, 6 June 2020
...ion (cf. [[Diophantine equations]]). In the simplest case of a solution in integers $(x_0,\ldots,x_n)$ of a Diophantine equation, the height is a function of t
6 KB (893 words) - 08:43, 1 May 2023
...uence $x ( n )$, $n \in \mathbf{Z}$, that is identically zero for negative integers, is defined as
9 KB (1,351 words) - 19:32, 2 February 2024
has a solution in integers $ a _ {0} \dots a _ {n - 1 } $,
of integers with condition (2) satisfies condition (1).
8 KB (1,143 words) - 08:03, 6 June 2020
...the first time that $S _ { n } = - J$ and decides $H _ { 0 }$ for positive integers $J$ and $K$. In this case $S _ { 1 } , S _ { 2 } , \ldots$ is a [[Random wa
8 KB (1,155 words) - 19:33, 2 February 2024
where the integers $ h _ {i} , h _ {i} ^ \prime $
7 KB (963 words) - 14:55, 7 June 2020
are coprime integers, and is determined by the rational number $ r = {p / q } $.
7 KB (1,038 words) - 11:33, 26 March 2023
are integers and $ 2 \pi i $
11 KB (1,593 words) - 19:37, 5 June 2020
...the solutions are looked for either in that field or else in its rings of integers. The theory of algebraic functions developed in parallel with the theory of
be a form with rational coefficients, and let there exist a solution in integers $ ( x _ {0} , y _ {0} ) $
24 KB (3,602 words) - 11:48, 26 March 2023
...n $ with coefficients in the ring $ \mathbf Z _{p} $ of $ p $ -adic integers (with the topology induced by that determined by the $ p $ -adic norm of
8 KB (1,117 words) - 20:03, 27 February 2021
is a collection of complex integers (a lattice domain) and that the Green function $ G _ {z} ( z, z ^ \prim
8 KB (1,168 words) - 20:13, 12 January 2024
are called integers (it is assumed that $ +a = a $;
...tural number, are known as rational numbers or fractions. They include all integers. The number $ (a)1 $
26 KB (4,086 words) - 09:51, 4 April 2020
of non-negative integers satisfying $ \lambda _ {1} \geq \dots \geq \lambda _ {n} $
14 KB (1,916 words) - 08:00, 5 May 2022
...ional ruled surfaces $F_n=P(\mathcal O_{P^1}+\mathcal O_{P^1}(n))$ for all integers $n\geq2$ and $n=0$.
7 KB (1,005 words) - 21:13, 21 November 2018
integers $n_1,\dots,n_k$ such that $n_1+2n_2+\cdots +kn_k = n$, and they are related
7 KB (1,115 words) - 16:05, 22 January 2021
...h includes the study of operations performed on numbers. The properties of integers form the subject of number theory (cf. [[Elementary number theory|Elementar
...a of that period was of a high standard. Indian mathematicians operated on integers and fractions using methods very similar to our own. They solved many probl
20 KB (3,091 words) - 18:48, 5 April 2020
theory]], is also an exterior algebra (over the ring of integers).
7 KB (1,013 words) - 19:38, 5 June 2020
are integers, to find a similar expression for $ e $.
8 KB (1,278 words) - 19:39, 5 June 2020
...let $R^*$ be its multiplicative group. $K_0(R) = \Z$ is then the group of integers, $K_1(R) = R^*/[R^*,R^*]$; and $K_2(\Z)$ is the cyclic group of order two.
...ems for the functors $K_i(R)$ and $G_i(R)$. In fact, if $A$ is the ring of integers or the ring of polynomials over a finite field, and $R$ is an $R$-order and
14 KB (2,405 words) - 22:14, 10 January 2015
...><TR><TD valign="top">[a2]</TD> <TD valign="top"> E.K. van Douwen, "The integers and topology" K. Kunen (ed.) J.E. Vaughan (ed.) , ''Handbook of Set-Theor
7 KB (1,075 words) - 19:29, 1 January 2021
is the set of integers.
8 KB (1,242 words) - 07:04, 6 May 2022
...also [[Orthogonal basis|Orthogonal basis]]) for $H$. For any non-negative integers $n _ { 1 } , n _ { 2 } , \dots$ such that $n _ { 1 } + n _ { 2 } + \ldots =
9 KB (1,318 words) - 17:43, 1 July 2020
positive integers if there are no special requirements, and $ n $
7 KB (1,029 words) - 10:58, 29 May 2020
...Namely, let $d \geq 1$ and $\{ h _ { i } \} _ { 0 \leq i \leq d - 1 }$ be integers. Then there exists a Buchsbaum local ring $A$ with $\operatorname { dim } A
...\alpha _ { d } ^ { n _ { d } }$ forms a $d$-sequence in any order for all integers $n _ { i } \geq 1$. Such a sequence is called an unconditioned strong $d$-s
27 KB (4,003 words) - 17:43, 1 July 2020
with the integers $ \{ 1 \dots \rho ( t ) \} $
8 KB (1,232 words) - 08:28, 12 November 2023
...say, assuming the validity of the continuum hypothesis, for any triple of integers $ n $,
9 KB (1,337 words) - 06:29, 30 May 2020
...have become their constituent parts. Problems related to the properties of integers, which subsequently brought about number theory, were typical of this perio
9 KB (1,264 words) - 16:56, 15 April 2012
is the subgroup of the integers (or real numbers modulo 1). In Euclidean space $ X \subset \mathbf R ^
9 KB (1,321 words) - 17:45, 4 June 2020
...solvable elementary theories are those of addition of natural numbers and integers, of Abelian groups, of $ p $-
9 KB (1,436 words) - 19:37, 5 June 2020
...d J. Tannery (1894) constructed a theory of rational numbers from pairs of integers.
The system of integers $ \mathbf Z = \langle Z , + , \cdot , 0 , 1 \rangle $
23 KB (3,482 words) - 08:03, 6 June 2020
are integers. The subgroup $ Q (R) $
and it can be described as the set of algebraic integers of the cyclotomic field generated by a cubic root of unity with norm $ 1
22 KB (3,351 words) - 19:14, 21 December 2019
...verline { m } = \{ m _ { n } \} _ { n = 0 } ^ { \infty }$ be a sequence of integers $\geq 2$, called a branch index. Let $T _ { \overline{m} }$ be a sphericall
8 KB (1,322 words) - 13:08, 10 February 2024
...al segment of the ordinal numbers or as the set of (positive and negative) integers, and also systems in which all formulas are objects of a well-defined type,
8 KB (1,261 words) - 20:30, 13 December 2019
copies of the group of integers $ \mathbf Z $;
are integers. In applications, branched coverings are also of importance. Consider a map
20 KB (3,075 words) - 08:26, 6 June 2020
the ring of integers, a ring of functions, or an arbitrary associative
integers. Then a matrix $A\in \M_{m,n}(K)$ can be written in the form
18 KB (3,377 words) - 17:54, 2 November 2013
has a solution in integers $ x,\ y,\ z $,
9 KB (1,344 words) - 16:08, 6 February 2020
...cyclopediaofmath.org/legacyimages/m/m064/m064430/m0644305.png" /> are real integers (this group is called the [[Modular group|modular group]]). The transformat
39 KB (5,287 words) - 17:07, 7 February 2011
...tation with a highest weight vector]]). Given an $n$-tuple of non-negative integers $\def\L{\Lambda}\def\l{\lambda}\L=(\l_1,\dots,\l_n)$, the integrable highes
..._0$ for $i=1,\dots,r$, where the $a_{0i}, a_{i0}$ are certain non-positive integers, and one puts
15 KB (2,514 words) - 20:27, 15 November 2017
are coprime integers, $ 0 \leq \nu < \mu $).
10 KB (1,350 words) - 12:59, 28 April 2024
are integers.
11 KB (1,603 words) - 16:08, 1 April 2020
of integers, where the multi-index $ i _ {1} \dots i _ {k} $
A set of such integers $ \{ b _ {i _ {1} } \dots b _ {i _ {k} } \} $
42 KB (6,290 words) - 19:33, 17 January 2024
...ion coinciding with $f$ at each point $2 \pi k / N$, where $k , N > 0$ are integers, $k < N$, and $f _ { N }$ is linear on intervals. Suppose $f \in \mathcal{A
10 KB (1,493 words) - 08:15, 15 February 2024
...from the vertex set $V$ of a digraph $G = ( V , E )$ into the non-negative integers ${\bf Z} ^ { 0 }$ defined inductively by: $g = \operatorname { mex } g ( F
13 KB (1,999 words) - 16:52, 18 February 2024
if and only if there are integers $ a , b , c , d \in \mathbf Z $,
10 KB (1,413 words) - 07:59, 6 June 2020
are symmetrically additive for all pairs of integers $ m, n \geq 2 $
9 KB (1,370 words) - 22:15, 5 June 2020
...ww.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642057.png" /> of
integers. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.o
....encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420242.png" /> and
integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
66 KB (9,085 words) - 17:28, 31 March 2020
and any collection of non-negative integers $ n _ {1} \dots n _ {k} $
11 KB (1,537 words) - 19:39, 5 June 2020
is a chain with the order type of the negative integers [[#References|[2]]].
10 KB (1,515 words) - 08:10, 6 June 2020
..._{1} +m _{3} \omega _{3} $ , and $ m _{1} ,\ m _{3} $ run through all integers except $ m _{1} = m _{3} = 0 $ . The function $ \wp (z) $ is an even el
11 KB (1,535 words) - 13:55, 18 May 2023
...ell as a discrete analogue applicable to distributions concentrated on the integers.
9 KB (1,390 words) - 07:41, 4 February 2024
are non-negative integers (cf. [[Cartan theorem|Cartan theorem]] on the highest weight vector). For a
10 KB (1,344 words) - 08:11, 6 June 2020
is a category the objects of which are all integers plus the null object $ N $,
10 KB (1,515 words) - 18:19, 31 March 2020
...: n=0,1,...\}$, starting at an integer $x$, is a stochastic process on the integers, given by $S_0=x$, $S_n = x +X_1+...+X_n (n\geq1)$, where $X_n, n\geq1$, is
For $c\leq x\leq d$ integers, the probability $\pi(x)$ that a simple random walk, starting at $x$, reach
11 KB (1,725 words) - 19:55, 5 March 2024
The space $W^l_p(\Omega)$ can also be defined for negative integers $l$. Its elements are usually generalized functions, that is, linear functi
8 KB (1,334 words) - 17:47, 30 November 2012
...)$-matrix ($n \geq 2$), which uses Cassini ovals instead of discs. For any integers $i$ and $j$ ($1 \leq i , j \leq n$) with $i \neq j$, the $( i , j )$th Cass
10 KB (1,639 words) - 07:05, 15 February 2024
1) Any Abelian group $M$ is a module over the ring of integers $\ZZ$. For $a \in \ZZ$ and $m \in M$ the product $am$ is defined as the res
...umber field, then other $G$-modules are: the additive group of the ring of integers of $K$, the group of units of $K$, the group of divisors and the
23 KB (3,918 words) - 04:31, 23 July 2018
...= \sum _ { k = 1 } ^ { r } a _ { k } D _ { k }$, where $a_k$ are positive integers and $D _ { k }$ are smooth irreducible divisors such that any subset of the
8 KB (1,253 words) - 15:30, 1 July 2020
...tion numbers, which are real numbers when their geometric counterparts are integers. Examples of such real numbers are the heights of points and subvarieties,
8 KB (1,219 words) - 21:00, 13 July 2020
or through the integers in this interval. In statistical problems, the distribution $ P ^ {T} $
runs through the integers of $ [ 0, T] $
27 KB (3,850 words) - 14:55, 7 June 2020
are positive integers, $ e( \alpha ) $
are negative integers, $ e( \alpha ) $
32 KB (4,602 words) - 04:46, 7 January 2022
for certain integers $ N _ {i} $.
12 KB (1,694 words) - 06:42, 26 March 2023
is spanned for all sufficiently large positive integers $ s $.
9 KB (1,417 words) - 16:09, 1 April 2020
For all positive integers $ n $
13 KB (1,981 words) - 20:02, 17 December 2019
...n matrices $W_+$ and $W_-$, also without poles in $\R$ or at $\infty$, and integers $k_1\le \cdots \le k_m$ such that
9 KB (1,294 words) - 22:47, 28 February 2015
are any given integers such that $ r \geq p $,
10 KB (1,586 words) - 09:40, 26 March 2023
For any function $t ( n )$ from the positive integers to itself, the complexity class $\operatorname {DTIME}[t(n)]$ is the set of
...nstructible function is used. Recall that a function $s$ from the positive integers to itself is space constructible (respectively, time constructible) if and
22 KB (3,250 words) - 17:43, 1 July 2020
are integers; the real axis is invariant with respect to modular fractional-linear mappi
13 KB (1,875 words) - 13:58, 17 March 2023
Here, ${\bf Z}_+$ is the set of non-negative integers; $x _ { i j } ^ { h } \in \mathbf{R} ^ { n _ { 1 } }$, respectively $x _ {
15 KB (2,237 words) - 17:01, 1 July 2020
...varphi } \otimes 1 _ { n } \|$ is finite, where the supremum runs over all integers $n \geq 1$ and $1_n$ is the identity operator from the $C ^ { * }$-algebra
14 KB (2,163 words) - 19:56, 8 February 2024
be either the ring of integers $ \mathbf Z $
which are positive integers $ > 1 $
51 KB (7,267 words) - 07:39, 14 January 2024