with integral rational coefficients $ a _ {i _ {1} } \dots a _ {i _ {n} } $
rational points of the [[Algebraic variety|algebraic variety]] defined by the system
8 KB (1,148 words) - 17:46, 4 June 2020
is an algebra over the field of rational numbers and the module of differentials $ \Omega _ {R/S} ^ {1} $
5 KB (785 words) - 19:35, 5 June 2020
...homogeneous solutions). Examples are the rings of integers and of rational numbers, algebraic number fields, and finite rings. For such a ring $ R $,
10 KB (1,471 words) - 14:52, 30 May 2020
are complex numbers, $ p $
are non-negative rational numbers, $ F _ {n} ( x) \not\equiv 0 $,
7 KB (1,019 words) - 08:02, 6 June 2020
...tion]]; [[Pole (of a function)|Pole (of a function)]]; [[Rational function|Rational function]]).
is the least among the numbers for which $ a ^ {n} = 1 $.
13 KB (2,001 words) - 02:12, 1 March 2022
is rational and diagonalizable, so $ \mathfrak g $
is the group of rational characters of $ T $ )
9 KB (1,402 words) - 11:29, 17 December 2019
are rational numbers, $ n/2 \leq r _ {0} < \dots < r _ {k} \rightarrow + \infty $.
...n="top"> V.I. Arnol'd, "Remarks on the stationary phase method and Coxeter numbers" ''Russian Math. Surveys'' , '''28''' : 5 (1973) pp. 19–48 ''Uspekhi Mat.
8 KB (1,104 words) - 08:08, 4 March 2022
...cyclopediaofmath.org/legacyimages/m/m064/m064430/m06443076.png" /> are odd
numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.o
...yclopediaofmath.org/legacyimages/m/m064/m064430/m064430159.png" /> and the
rational points on the real axis are often referred to as cusps.
39 KB (5,287 words) - 17:07, 7 February 2011
if $G$ is the field of rational numbers or the field of residues modulo a prime number, and
5 KB (741 words) - 06:05, 22 November 2018
...ecause of poles of the coefficients $c_j ( \lambda )$. These solutions are rational functions of $\lambda$ with possible poles at the poles of $c _ { 1 } ( \la
...special cases are given first. Let $\mathbf{N}$ denote the set of natural numbers starting at $1$ (i.e., excluding $0$). Note that neither of the special cas
13 KB (1,803 words) - 07:40, 4 February 2024
...nt coefficients, every $ \ell $-adic cohomology theory for various prime numbers $ \ell $
modulo a suitable (rational, algebraic, numerical, etc.) equivalence relation, or $ C ( X) = K ( X) $
8 KB (1,244 words) - 10:52, 16 March 2023
...or $H _ { \phi }$ has finite rank if and only if $\mathcal{P} - \phi$ is a rational function. Moreover, $\operatorname{rank} H _ { \phi } = \operatorname { deg
...or a function $\phi$ on $\bf T$ in $\operatorname{BMO}$ one can define the numbers $\rho _ { n } ( \phi )$ by
18 KB (2,757 words) - 00:48, 15 February 2024
over the field of rational numbers. An isogeny $ f: A \rightarrow A _ {1} $
5 KB (774 words) - 09:04, 8 April 2023
These were concerned with index numbers, a critique of Elderton and [[Pearson, Karl|Pearson]] (see below), reviews
he wrote a long, prize-winning essay on Index Numbers.
17 KB (2,523 words) - 19:38, 21 March 2023
are numbers.
the analogous assertion is true for these classes with any rational number $ r > 0 $(
5 KB (791 words) - 10:58, 29 May 2020
elliptic curve with distinguished point $Q_0$, then any rational mapping
curve over an arbitrary field $k$. If the set of $k$-rational points
19 KB (3,251 words) - 20:37, 19 September 2017
...tion does one associate the set of all real numbers, or the set of natural numbers, as a single object of study? Modern physical conceptions do not give a goo
...ind an extremum for functions of a very restricted class (polynomials with rational coefficients in several variables), and, what is essential, the theorem ind
24 KB (3,697 words) - 22:13, 5 June 2020
...the foundations of mathematical analysis, which unifies the theory of real numbers (cf. [[Real number|Real number]]), the theory of limits, the theory of [[Se
of numbers, is associated by some rule a number $ y $,
19 KB (2,844 words) - 08:57, 13 January 2024
which admits no rational parametrization, but which may be uniformized by means of elliptic function
are rational functions in the Weierstrass $ \wp $-function and its derivative, with co
14 KB (2,068 words) - 02:08, 18 July 2022
and all sufficiently large numbers $ n $,
...ace. A non-singular projective algebraic surface over the field of complex numbers is a compact four-dimensional oriented real manifold; in particular, the in
26 KB (3,736 words) - 13:08, 8 February 2020
The standard example of $(K,\infty,A)$ is given by a rational function field $K=\F_q(T),$, $\infty$ being the usual place at infinity, $A
...d $\Q$ of rationals. This also works for general base rings $A$ with class numbers $>1$; here the situation resembles the theory of complex multiplication of
19 KB (3,204 words) - 20:11, 14 April 2012
...cal]], if $K$ contains a transcendental element over the field of rational numbers.
5 KB (872 words) - 22:00, 29 April 2012
Over the field of complex numbers $ \mathbf C $
The field of rational functions $ k(X) $
5 KB (791 words) - 01:37, 15 January 2022
...$\mathbf{Q}(\zeta_n,a^{1/n})$, where $\mathbf{Q}$ is the field of rational numbers and $a \in \mathbf{Q}$.
5 KB (938 words) - 20:00, 18 September 2017
...eger|Remainder of an integer]]). In order to express the congruence of the numbers $ a $
with just one of the numbers $ 0 \dots m-1 $;
19 KB (3,006 words) - 06:05, 26 May 2024
...\operatorname { cos } ( n \operatorname { arccos } x )$. Over the complex numbers, if $u = e ^ { i \alpha }$ so that $x = u + 1 / u = 2 \operatorname { cos }
...a )$, and linear polynomials $a x + b$ (here, $a$ and $b$ may be rational numbers). The first proof of this conjecture was given by M. Fried [[#References|[a
15 KB (2,207 words) - 16:45, 1 July 2020
study of linear algebraic groups over the field of complex numbers was
complex numbers $\mathbb C$ can be regarded as an analytic subgroup (cf.
14 KB (2,165 words) - 22:05, 5 March 2012
...tant $c_3$ (cf. also [[Distribution of prime numbers|Distribution of prime numbers]]).
where $m,m_1,\ldots,m_s \in \mathbb{Z}$ and $p_1,\ldots,p_s$ are fixed prime numbers. Till recently, the proofs of this theorem suffered from a common drawback
5 KB (784 words) - 20:40, 18 October 2014
...l axioms must thereupon be recognizable in the arithmetic of this field of numbers. In this way the desired proof for the compatibility of the geometrical axi
::''Is Hilbert's second problem about the real numbers or the natural numbers?''
15 KB (2,329 words) - 14:32, 19 March 2023
...th for rotation numbers which can be very rapidly approximated by rational numbers, even if the transformation $ S $
is rational or irrational. Thus, in the absence of equilibrium positions there exist on
12 KB (1,841 words) - 18:33, 5 June 2020
is some set of complex numbers, and $ \alpha $
may be one of various spaces of sequences of complex numbers. In connection with the class $ H ^ \infty $
8 KB (1,230 words) - 08:02, 6 June 2020
...ative algebra with a unit over $ k $ . In general, the group of $ k $ -rational points of an [[Algebraic group|algebraic group]], defined over $ k $ , is
6 KB (902 words) - 17:45, 12 December 2019
i) In the graded case: Is the Poincaré series rational?
The Poincaré series of a graph is a rational function of the form $ P _ {G} ( z) = q( z) p( z) ^ {-1} $
19 KB (2,908 words) - 20:20, 12 January 2024
...t of all open intervals (it is sufficient to take only open intervals with rational end points). The remaining open sets are unions of such intervals.
...balls for which the radius and the coordinates of the centre are rational numbers. A topology is often defined by some (natural) standard procedure on a set
21 KB (3,431 words) - 05:43, 18 December 2016
the field of rational functions in one variable over an algebraically
classification of rational varieties over not algebraically closed
17 KB (2,785 words) - 22:37, 23 December 2014
...s,i_n } x_1^{i_1}\cdots x_n^{i_n}$ over $A$ such that there exist positive numbers $r_1,\ldots,r_n$ and $C$ such that $\Vert c_{ i_1,\ldots,i_n } \Vert \le C
6 KB (1,093 words) - 08:26, 16 March 2023
...of a topological space|weight]] all coincide. The non-coincidence of these numbers is an indication of the non-metrizability of the corresponding space.
...ce is metrizable by a complete metric: an example is the space of rational numbers. A space is metrizable by a complete metric if and only if it is metrizable
9 KB (1,484 words) - 08:00, 6 June 2020
...0 } ^ { \infty }$ with a given function $S ( z )$ in the Schur class. The numbers are defined in terms of a sequence of Schur functions which is constructed
...}$ are called the Schur parameters of $S ( z )$. Every sequence of complex numbers of modulus at most one and having the property that if some term has unit m
14 KB (2,163 words) - 09:53, 11 November 2023
into the field of complex numbers. Conversely, every multiplicative linear functional on a commutative Banach
consists precisely of the numbers of the form $ \phi ( a) $.
18 KB (2,806 words) - 03:47, 25 February 2022
For instance, the set of all natural numbers $ \mathbf Z _ {0} $
...peration of immediate succession and its iteration. The set of all natural numbers $ >1 $
28 KB (4,564 words) - 07:37, 26 March 2023
Here the numbers $ s $
are rational functions of $ x _ {1}, \dots, x _ {n} $,
32 KB (4,863 words) - 08:08, 12 August 2022
of complex numbers, on which the moduli of the Gel'fand transforms $ \widehat{a} $
of uniform limits of rational functions on a compactum $ X $
10 KB (1,551 words) - 08:37, 26 March 2023
...dea is due to H. Whitney [[#References|[5]]], may be used to calculate the rational cohomology of a simplicial complex $ K $.
with polynomials with rational coefficients as coefficients when written in barycentric coordinates. The
27 KB (4,062 words) - 01:31, 7 May 2022
adic numbers $ \mathbf Q _ {p} $
...prescribed poles and zeros lying in a given open subset of the set of all rational points of the curve over $ {\widetilde{\mathbf Q} } _ {p} $.
12 KB (1,775 words) - 11:58, 4 April 2020
...t. In particular, the differential ring of everywhere-defined differential rational functions on an affine differential algebraic set is not a differential coo
...for the approximation of differential algebraic functions by differential rational functions. However, the analogue of the Thue–Siegel–Roth theorem has no
30 KB (4,468 words) - 18:44, 17 December 2019
...ii [[#References|[5]]] has studied the problem of Riemann: Let $A(t)$ be a rational function of $t$ and let the singularities of the fundamental matrix $X(t)$
where $R(t)$ is a rational function, and the equation
16 KB (2,410 words) - 11:15, 28 January 2020
...in the decimal system, while the possible inputs are ordered pairs of such numbers. It is, in general, not assumed that the result is necessarily obtained: th
for which the set of its possible inputs is the set of natural numbers.
18 KB (2,796 words) - 19:10, 11 December 2020
...(cf. [[Continued fraction|Continued fraction]]): Given a sequence of real numbers $ \{ \mu _ {n} \} $,
be the set of all sequences of real numbers $ \{ \mu _ {n} \} $
33 KB (4,743 words) - 01:58, 17 June 2022
...permits $s$ real and $2t$ complex imbeddings in the field $\C$ of complex numbers, then
...=\infty$. For a quadratic field $\Q(\sqrt{d})$, where $d$ is a square-free rational integer, $d\ne 1$, one has the formulas
16 KB (2,947 words) - 08:53, 9 December 2016
...presentations of $G ( K )$. This makes possible a "modular theory" for the rational representations of these groups analogous to Brauer's modular representatio
.... also [[Young tableau|Young tableau]]). The $\lambda$-tableau where these numbers are inserted in order along the rows downward, is called the initial $\lamb
33 KB (5,081 words) - 10:26, 11 November 2023
are the spatial wave numbers, which travel in all directions. Computationally, the domain is of finite s
For (a1), a wave with wave numbers $ \xi $,
10 KB (1,394 words) - 16:08, 1 April 2020
...ages thus defined are referred to as (rational) transductions and inverse (rational) transductions. If $ \mathop{\rm ST}\nolimits $
...ngs) is again a rational transduction (respectively, a gsm mapping). Every rational transduction $ \tau $
28 KB (4,202 words) - 08:53, 1 February 2020
Let $p$ be a prime number and let $k$ be a finite extension of the rational number field $\mathbf{Q}$. A $\mathbf{Z} _ { p }$-extension of $k$ is an ex
...erences|[a17]]] gave a new proof of this using the $\Gamma$-transform of a rational function.
19 KB (2,876 words) - 05:38, 15 February 2024
...s been verified under various additional assumptions. Namely, if $F$ has a rational inverse (O.H. Keller) and, more generally, if the field extension $\mathbf{
...}$ or $\operatorname { deg } F _ { 2 }$ is a product of at most two prime numbers (H. Applegate, H. Onishi). Finally, if there exists one line $l \subset \ma
7 KB (1,112 words) - 05:20, 15 February 2024
are rational integers. The modular group can be identified with the quotient group $
real numbers (respectively, integers) and $ ad - bc = 1 $.
7 KB (1,031 words) - 18:33, 13 January 2024
...of Kummer was that if one appends to the ordinary numbers certain "ideal" numbers (in the same way as one appends points at infinity in projective geometry),
...Kronecker to extend the theory of divisors to arbitrary rings of algebraic numbers. In Dedekind's theory (1882) the role of the integral elements of a field e
16 KB (2,400 words) - 17:45, 4 June 2020
where $p$ runs through all prime numbers.
...f the problem of the [[Distribution of prime numbers|distribution of prime numbers]]. However, the most deeply intrinsic properties of $\zeta(s)$ were discove
45 KB (7,251 words) - 02:20, 29 June 2022
...ormed ring]]. For a description of all valuations of the field of rational numbers, see [[#References|[4]]].
of real numbers. In this case the mapping $ x \mapsto \mathop{\rm exp} ( - v ( x) ) $
14 KB (2,135 words) - 08:27, 6 June 2020
...ith a "set of points in a coordinate space" . If the coordinates are real numbers, and if the space is two- or three-dimensional, the situation may be clearl
In the case of algebraic geometry over the field of complex numbers, every algebraic variety is simultaneously a complex-analytic, differentiab
29 KB (4,414 words) - 17:20, 17 December 2019
...a given sequence $\{ c _ { n } \} _ { n = - \infty } ^ { \infty }$ of real numbers is concerned with finding real-valued, bounded, monotone non-decreasing fun
The rational functions $( - z ) P _ { n } ( - z ) / Q _ { n } ( - z )$ are the convergen
9 KB (1,334 words) - 19:52, 6 February 2024
over the field of rational numbers equals the number of conjugacy classes of cyclic subgroups of the group. If
10 KB (1,488 words) - 19:39, 5 June 2020
are any numbers not simultaneously equal to zero and such that
...ari, "Inequalities for the coefficients of univalent functions" ''Arch. Rational Mech. and Anal.'' , '''34''' : 4 (1969) pp. 301–330</TD></TR><TR><TD v
9 KB (1,232 words) - 07:25, 26 March 2023
rational point $ D $,
..., k) $[[#References|[6]]], [[#References|[10]]]. In the general case these numbers are different, but $ { \mathop{\rm ord} } ( D) $
7 KB (1,109 words) - 16:59, 1 July 2020
1) The real numbers $t_\text{min}$ and $t_\text{max}$ are the minimum and the maximum of the fu
...^ { 3 }$ intersects the knot $K$ in some number, say $n_j$, of points. The numbers $n_j$ are constants if $( t _ { 1 } , \dots , t _ { m } )$ belongs to a fix
10 KB (1,481 words) - 19:05, 23 January 2024
defined over the complex numbers. For smooth projective $ X $
equal (up to a non-zero rational number) to the first non-vanishing coefficient of the $ L $-
25 KB (3,559 words) - 19:33, 7 February 2024
...1 }$ and $Z_2$ are said to be numerically equivalent if their intersection numbers are equal, $( D . Z _ { 1 } ) = ( D . Z _ { 2 } ) \in \bf R$ for any Cartie
...here exist an element $v ^ { \prime } \in \overline { N E } ( X / S )$ and numbers $r _ { j } \in {\bf R} _ { \geq 0 }$, which are zero except for finitely ma
11 KB (1,780 words) - 14:48, 3 February 2024
In the case of Lie groups over the field $ \mathbf C $ of complex numbers the main result of the local classification is that every simply-connected
...on it, and any analytic homomorphism of $ G $ to an algebraic group is rational. The corresponding algebra of regular functions on $ G $ coincides with
11 KB (1,458 words) - 18:15, 12 December 2019
The numbers $c ( n )$ are the Fourier coefficients of $f$. The modular form is a cusp f
...framework for such results. Ramanujan also conjectured that for all prime numbers $p$ one has the inequality $| \tau ( p ) | \leq 2 p ^ { 11 / 2 }$. P. Delig
13 KB (1,907 words) - 07:36, 22 March 2023
...not have isolated points (an example is the space $\mathbf{Q}$ of rational numbers). All zero-dimensional spaces are completely regular. Zero-dimensionality i
8 KB (1,230 words) - 13:00, 3 January 2016
partly because it contains statements on the theory of numbers which
first among them. This was that the set of real numbers on the unit
14 KB (2,139 words) - 18:37, 8 March 2024
...w that the kernel is isomorphic to the additive group of $n$-adic rational numbers. Thus, such groups are meta-Abelian (cf. [[Meta-Abelian group|Meta-Abelian
...eferences|[a5]]], while the residually-finite Baumslag–Solitar groups have rational growth [[#References|[a2]]], [[#References|[a6]]] but, as noted earlier, ar
18 KB (2,803 words) - 16:46, 1 July 2020
of rational characters of the torus $ T $ ,
...e of finite-dimensional semi-simple Lie algebras over the field of complex numbers. A Weyl group may also be defined for an arbitrary splittable semi-simple f
9 KB (1,394 words) - 16:55, 17 December 2019
...e a countable subset of $ X $, and let $ p_1, p_2, \dots $ be non-negative numbers. Then the function
...initely-additive measure is a measure. For example, if $ X $ is the set of rational points of the segment $ [0,1] $, $ \mathcal{P} $ is the semi-ring of all po
46 KB (7,065 words) - 19:30, 1 January 2021
is a rational function of the form $ f ( t) ( 1 - t ) ^ {-} d( A) $,
Other homological invariants are the so-called Betti numbers $ b _ {i} ( A) $
9 KB (1,456 words) - 22:17, 5 June 2020
of complex numbers the cohomology spaces $ H ^{*} (X _{s} ) $
is the multiplicative group of the field of complex numbers, considered as a real algebraic group, while $$
12 KB (1,699 words) - 09:49, 20 December 2019
where $c_j$, $a_{ij}$ and $b_i$ are given numbers.
...e devoted to work in one or other technical mode. The problem is to make a rational distribution of the time spent in working in the various technical modes, t
12 KB (1,882 words) - 22:25, 1 September 2016
...rations — arithmetic operations on positive integers and positive rational numbers — can be encountered in the oldest mathematical texts, which indicates th
...plicit numbers. It was assumed, however, that the symbols stood for actual numbers: integers or fractions. A brief table of the contents of one of the best te
17 KB (2,478 words) - 16:09, 1 April 2020
real numbers, $ n \geq 2 $,
is rational, then there are arbitrarily-small perturbations which will destroy the toru
8 KB (1,212 words) - 17:33, 7 June 2020
...an algebraic variety. A morphism of $k$-schemes $\Spec k\to X$ is called a rational point of the $k$-scheme $X$; the set of such points is denoted by $X(k)$.
...complex, or strong, topology on $X(\C)$, the fundamental group, the Betti numbers, etc. The desire to find something similar for arbitrary schemes and the fa
12 KB (2,191 words) - 13:20, 24 November 2013
Then the numbers
can be described explicitly by rational functions of the values $ \mu $
18 KB (2,750 words) - 19:29, 16 January 2024
is a non-increasing sequence of positive numbers and $ \{ f _ {n} \} $,
are orthonormal systems. The numbers $ s _ {n} = s _ {n} ( T) $
24 KB (3,593 words) - 18:51, 13 January 2024
...ubgroups of classical Lie groups — groups of units of quadratic forms with rational coefficients, groups of units of simple algebras over $\mathbf{Q}$, groups
...ice in $H_\mathbf{R}$ it is necessary and sufficient for $H$ not to permit rational homomorphisms into the group $\mathbf{C}^*$, defined over $\mathbf{Q}$ (thi
20 KB (3,146 words) - 02:52, 23 July 2018
...ences|[a9]]]), for the aforementioned solutions are given over the complex numbers;
...belong to special classes of functions (cf. [[#References|[a1]]]), such as rational, solitonic, elliptic, bispectral;
8 KB (1,099 words) - 08:01, 20 February 2023
is invertible [[#References|[7]]], in particular over the field of rational numbers, and, if $ \Delta (0) = +1 $,
12 KB (1,694 words) - 06:42, 26 March 2023
...1,2,3, \ldots \}$, with its subset $P_{\mathbf{N}}$ of all rational prime numbers $\{ 2,3,5,7,\ldots \}$. Here one may define the norm of an integer $|n|$ to
...E.A. Bender and Wormald [[#References|[a1]]] considered the corresponding numbers $\mathcal{P} _ { V } ^ { \# } ( n )$, $\mathcal{P} _ { \text{F} } ^ { \# }
14 KB (2,037 words) - 09:45, 11 November 2023
...e natural numbers as well as adding any two natural numbers, since natural numbers can be regarded, for example, as words of the form $ 0 , 0 \mid , 0 \mid
...nce of words. The introduction of more complex structures such as the real numbers and functions on them, etc., are realized in constructive mathematics by th
20 KB (2,913 words) - 07:40, 14 August 2023
...the usual order relation is an elementary subsystem of the system of real numbers with the usual order relation.
...the signature of $A$ is $(+,\,.\,,0,1,<,U)$, $|A|$ is the set of all real numbers, $U(A)$ the set of all integers, and $+,\,.\,,0,1,<$, are defined in the u
17 KB (2,788 words) - 15:19, 16 December 2020
...[[Lattice of points|Lattice of points]]; [[Geometry of numbers|Geometry of numbers]]). The original insight of E. Bombieri and Iwaniec into the second spacing
...roximated, via [[Taylor series|Taylor series]], by a cubic polynomial with rational quadratic coefficient $a /q$.
14 KB (2,083 words) - 17:01, 1 July 2020
adic numbers and let $ \mathbf F _ {p} (( t)) $
fields are also called quasi-algebraically closed. The rational functions in one variable over an algebraically closed field form a $ C _
10 KB (1,605 words) - 16:57, 13 January 2024
where $B_{2n}$ are the [[Bernoulli numbers]]. It implies the equality
...ntal functions which do not satisfy any linear differential equation with rational coefficients (Hölder's theorem).
10 KB (1,513 words) - 09:50, 5 December 2012
There is a set of rational numbers $a_i$, known as the Seiberg–Witten invariants, which can be obtained by c
...nite number of classes (known as basic classes) $\kappa_i \in H^2(M)$ with rational coefficients $a_i$ (called the Seiberg–Witten invariants), resulting in t
16 KB (2,663 words) - 10:57, 13 February 2024
...ity laws gain a statistical expression on the strength of the law of large numbers (probabilities are realized approximately in the form of frequencies, and e
The statistical distribution corresponding to this partition is given by the numbers (frequencies) $ n _ {1} \dots n _ {r} $(
21 KB (3,113 words) - 12:48, 13 January 2024
Furthermore, the corresponding characteristic numbers, which are elements of the rings $ h ^ {*} ( \mathop{\rm pt} ) $
cf. [[Chern class]]), it follows that the Chern numbers (cf. [[Chern number]]) completely determine the unitary cobordism class of
42 KB (6,290 words) - 19:33, 17 January 2024
Irrationality and transcendence of certain numbers.
The numbers in question are of the form $\alpha^\beta$ with $\alpha$ an [[algebraic num
29 KB (4,109 words) - 19:54, 18 March 2018
...hat is, consuming only polynomial resources) algorithm for factoring large numbers and for computing discrete logarithms. It implied that widely-used public-k
...tary transition operators are rational (or, in general, computable complex numbers), then there is no difference between classical and quantum computability.
15 KB (2,154 words) - 17:45, 1 July 2020
be the numbers of linearly independent solutions of the homogeneous integral equation $
associated with it. The numbers $ k $
17 KB (2,509 words) - 03:12, 13 January 2022
...$-fold composite (or superposition) of $f_1,\dots,f_n$. For example, every rational function in any number of variables is a composite of the four arithmetic o
In 1954 A.G. Vitushkin proved [[#References|[10]]] that if natural numbers $m,n,m_1$, and $n_1$ satisfy the inequality $(m/n)>(m_1/n_1)$, then it is p
11 KB (1,799 words) - 21:31, 10 December 2018
...nu$. By allowing $l_{i j}$ to be indeterminates over the field of rational numbers, the generating function version of the matrix tree theorem is obtained [[#
10 KB (1,636 words) - 08:14, 15 February 2024
is a finite extension of the field of rational $ p $-adic numbers $ \mathbf Q _ {p} $;
17 KB (2,620 words) - 07:48, 13 February 2024
...the left is the first non-trivial coefficient of the $j$-function, and the numbers on the right are the dimensions of the smallest irreducible representations
...me form of moonshine for finite subgroups of the automorphism group of any rational vertex operator algebra obeying a technical (and probably redundant) "C2 c
12 KB (1,765 words) - 09:46, 10 November 2023
...l numbers ... using set-theoretic constructions, starting from the natural numbers." <ref>[[Arithmetization]], ''Encyclopedia of Mathematics''. See also Lakof
# the creation of the theory of real numbers.
91 KB (14,186 words) - 16:58, 23 November 2023
...nalytic cases one can replace the real field $\RR$ by the field of complex numbers $\CC$.</ref>.
...$\l_1/\l_2=-\beta_2/\beta_1$, $\gcd(\beta_1,\beta_2)=1$, is a nonpositive rational number, then the corresponding identity $\left<\beta,\l\right>=0$ implies '
37 KB (5,881 words) - 19:10, 24 November 2023
...th reaction, $b_l$ is the "reaction constant" for that reaction, and the numbers $a_{jl}$ are "stoichiometric parameters" , specifying the amount of specie
...s of nonlinear diffusion equations to travelling front solutions" ''Arch. Rational Mech. Anal.'' , '''65''' (1977) pp. 335–361</TD></TR><TR><TD valign="to
12 KB (1,852 words) - 13:56, 26 July 2014
He introduced the concept of "rational subgroup" to define the method and
the "law of large numbers" imparts the appearance of infinite precision.
11 KB (1,624 words) - 12:48, 25 March 2023
...Aleksandrov–Čech homology functor (over the field $\mathbf{Q}$ of rational numbers; cf. also [[Aleksandrov–Čech homology and cohomology|Aleksandrov–Čech
16 KB (2,462 words) - 06:56, 15 February 2024
algebra; G. Bergman proved that numbers in $ ( 1 , 2 ) $
...geometry; cf. also [[Analytic space|Analytic space]]; [[Rational function|Rational function]]; [[Analytic set|Analytic set]]; [[Cohomological dimension|Cohomo
38 KB (5,928 words) - 19:35, 5 June 2020
...in), where the corresponding dichotomy algebraic versus transcendental (or rational versus irrational) is given a concrete meaning (see also [[#References|[a20
26 KB (3,837 words) - 19:21, 17 March 2024
An algebraic vector bundle on a variety defined over the field of complex numbers $ \mathbb{C} $ may be regarded both as an [[Vector bundle, analytic|analyti
...the fibers of the determinant mapping $ \det: U(r,d) \to U(1,d) $ are uni-rational varieties; if $ r $ and $ d $ are co-prime, then $ U(r,d) $ uniquely determ
14 KB (2,169 words) - 08:12, 14 December 2016
...aic theory of invariants) that studies algebraic expressions (polynomials, rational functions or families of them) that change in a specified way under non-deg
are real or complex numbers, it is converted to the form$$
22 KB (3,406 words) - 07:08, 6 May 2022
is a set of real numbers, and suppose that $ \phi : \mathfrak x \times X \times T \rightarrow X $
...hile the solution of bimatrix games can always brought about by the use of rational operations.
25 KB (4,000 words) - 19:41, 5 June 2020
then the subring of rational functions $ K[ x, P ^ {-1} ] $
consist of strictly negative rational numbers. See also [[#References|[a6]]] for this. The roots of $ b( s) $
28 KB (4,182 words) - 19:30, 19 January 2024
defines a rational mapping from $ A $
of complex numbers is a [[Complex torus|complex torus]]
18 KB (2,511 words) - 06:25, 26 March 2023
is an entire rational function, or a polynomial; if $ a = \infty $
is a rational function. For a transcendental meromorphic function $ f(z) $
66 KB (9,825 words) - 01:45, 23 June 2022
...etric with non-positive two-dimensional curvature, then the characteristic numbers of the form
is an arbitrary rational Pontryagin–Hirzebruch class of the manifold $ M $
17 KB (2,459 words) - 07:32, 26 February 2022
...onverse is false, since a link may be represented by braids with different numbers of strings. In addition, the braids $ \omega \sigma _ {n - 1 } $
...raic geometry as complements to the discriminant of versal deformations of rational singularities (see [[#References|[12]]], [[#References|[13]]]).
24 KB (3,637 words) - 08:41, 26 March 2023
...hroughout we will denote by $q$ the value $p^{r/n}$, which is in general a rational number but is an integer when $F$ is a Frobenius endomorphism. Note that mo
...l$ denote an algebraic closure of the field of [[P-adic number|$\ell$-adic numbers]] where $\ell$ is a prime number distinct from $p$. For $i \in \mathbb{Z}$
48 KB (8,458 words) - 18:22, 13 August 2023
are given numbers. Then the solution of the problem at any interior point is expressed by the
...><TR><TD valign="top">[7]</TD> <TD valign="top"> G.I. Petrashen', "On a rational method of solving problems in the dynamical theory of elasticity in the cas
20 KB (3,044 words) - 19:36, 5 June 2020
...\phi : X _ { 0 } ( N ) \rightarrow E$. Let $S _ { 0 }$ be the set of prime numbers containing $p$ and the primes where $E$ has bad reduction. For each prime n
...agin, D.Y. Logacev, "Finiteness of Shafarevich–Tate group and the group of rational points for some modular Abelian varieties" ''Algebra i Anal.'' , '''1''' (1
19 KB (2,901 words) - 17:41, 25 November 2023
...if the identity \eqref{e:power_series} holds for some sequence of complex numbers $\{a_n\}$), then $f$ is complex-differentiable everywhere in $U$ and indeed
...etitions in the sequence and $\{\mu_n\}$ is a sequence of positive natural numbers, then $f$ can be chosen so that it vanishes at each $z_n$ with order $\mu_n
61 KB (9,850 words) - 19:04, 20 January 2022
...case. Kashiwara proved that the roots of the $ b $-function are rational numbers. If $ f : ( \mathbf C ^ {n+1} , 0 ) \rightarrow ( \mathbf C , 0 ) $
24 KB (3,511 words) - 07:03, 10 May 2022
...$\{1,2,3,\ldots\}$, with its subset $P_{\mathbb{N}}$ of all rational prime numbers $\{2,3,5,7,\ldots\}$. Here one may define the norm of an integer $|n|$ to b
24 KB (3,738 words) - 07:41, 7 February 2024
...entation of the "state space" of a given problem as a domain in a space of numbers $ \mathbf R ^ {n} $.
This gives one the possibility to describe by a set of numbers the coordinates of the corresponding point (the coordinate method). In the
30 KB (4,462 words) - 07:59, 6 June 2020
and, in particular, the equality of the Betti numbers
dimensional Betti numbers are equal, as are the $ p $-
64 KB (9,418 words) - 12:44, 8 February 2020
see [[#References|[1]]], [[#References|[2]]]), or its Frobenius, rational or quasi-natural normal form (see [[#References|[4]]]). In contrast to the
is the field of complex numbers or, more generally, any algebraically closed field.) Then every one of the
51 KB (7,267 words) - 07:39, 14 January 2024
of rational numbers into the Hausdorff space $ \mathbf R $
57 KB (8,236 words) - 19:41, 20 January 2021