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...egree 1. In the case of several variables there are absolutely irreducible polynomials of arbitrarily high degree, for example, any polynomial of the form $f(x_1, ...lynomials of arbitrarily high degree; for example, $x^n+px+p$, where $n>1$ and $p$ is a prime number, is irreducible in $\mathbf Q[x]$ by Eisenstein's cri

...]$, then $c(g_1g_2) = c(g_1)c(g_2)$. In particular, a product of primitive polynomials is a primitive polynomial. ...p $E^*$ where $E$ is the extension of $F$ by the roots of $f$, cf [[Galois field structure]].

...many elements. The field of rational numbers is contained in every number field. ...re $\alpha$ is a fixed complex number and $H(x)$ and $F(x)$ range over the polynomials with rational coefficients.

...,|A|\rangle$, with $\Omega$ the signature of $L$ (cf. [[Model theory|Model theory]]). ...n every $\alpha$-saturated extension of $A$ (cf. also [[Model theory|Model theory]]).

Please remove this comment and the {{TEX|auto}} line below, A relation between integral polynomials $ a( x) $

in which the coefficients located at equal distances from the beginning and from the end are equal: $a_i=a_{n-i}$. A reciprocal equation of degree $2n$ [[Category:Field theory and polynomials]]

''finite field'' A field with a finite number of elements. First considered by E. Galois [[#Referenc

''of a field $K$'' An algebraic field extension (cf. [[Extension of a field]]) $L$ of $K$ satisfying one of the following equivalent conditions:

$#C+1 = 56 : ~/encyclopedia/old_files/data/C027/C.0207580 Cyclotomic polynomials, Please remove this comment and the {{TEX|auto}} line below,

A polynomial $f$ with coefficients in a field or a commutative associative ring $K$ with a unit, which is a [[Symmetric f The symmetric polynomials form the algebra $S(x_1,\ldots,x_n)$ over $K$.

The ''resultant of two polynomials $f(x)$ and $g(x)$'' is the element of the field $Q$ defined by the formula:

...$ are variables and $A,B,\dots,D$ (the ''coefficients'' of the polynomial) and $k,l,\dots,t$ (the ''exponents of the powers'', which are non-negative inte ...same way it is possible to introduce or omit terms with zero coefficients and, in each individual term, zero powers. When the polynomial has one, two or

...ment $w \in E$ that is simultaneously free in $E/K$ for every intermediate field $K$. ...ments of maximal multiplicative order, cf. [[Primitive element of a Galois field]]), see [[#References|[a4]]]

...distinct primes, $\alpha_i \ge 0\,\,\ldots\,\,\nu_i \ge 0$, $i=1,\ldots,s$ and $\delta_i = \min\{\alpha_i,\ldots,\nu_i\}$, then ...actor, with a number of steps bounded by the smaller of the degrees of the polynomials: cf [[Euclidean ring]]

Gröbner bases are certain sets of multivariate polynomials with field coefficients. The importance of Gröbner bases stems from the fact that ...uced by structurally easy algorithms to the construction of Gröbner bases; and

...alois theory]]), and of stating the conditions which ensure the existence (and non-existence) of such an extension over $k$. ...ith the given Galois group. Such equations exist for the symmetric groups, and also for the alternating groups. I. Schur constructed equations for the alt

$#C+1 = 79 : ~/encyclopedia/old_files/data/A012/A.0102800 Appell polynomials Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, A statement obtained by K. Hensel [[#References|[1]]] in the creation of the theory of $ p $-

Please remove this comment and the {{TEX|auto}} line below, of discrete valuations on the field of fractions (cf. [[Fractions, ring of|Fractions, ring of]]) $ K $

...mi-group]] with unit (i.e. [[Monoid]]) satisfying the [[cancellation law]] and in which any non-invertible element $a$ is decomposable into a product of i a = b_1 \cdots b_k\ \ \text{and}\ \ a = c_1 \cdots c_l

Please remove this comment and the {{TEX|auto}} line below, ...a unit element (cf. [[Associative rings and algebras]]) in which all right and left ideals are principal, i.e. have the form $ aR $

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. where the two initial polynomials are $D _ { 0 } ( x , a ) = 2$ and $D _ { 1 } ( x , a ) = x$. A second closed form is given by

In the field of complex numbers a quadratic equation has two solutions, expressed by rad When $b^2>4ac$ both solutions are real and distinct, when $b^2<4ac$, they are complex (complex-conjugate) numbers, whe

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. for the polynomials $\sigma ( z )$, $\omega ( z )$, $\operatorname { deg } \omega ( z ) < \oper

Please remove this comment and the {{TEX|auto}} line below, ...case of the coefficients and roots being elements of an arbitrary [[Field|field]] may also be considered.

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...L ( X )$ be the set of all linear operators with domains and ranges in $X$ and let

...ative ring]] and let $(A,B)$ be a pair of matrices of sizes $(n \times n)$ and $(n \times m)$, respectively, with coefficients in $R$. The pole assignment In both cases, state feedback (see [[Automatic control theory]]), $u \mapsto u + Fx$, changes the pair $(A,B)$ to $(A+BF,B)$.

...le$, where $\Omega$ denotes the signature of $T$ (cf. [[Model theory|Model theory]]; [[Structure(2)|Structure]]). If $T$ is substructure complete, then it is ...admit a root in a given interval, cf. also [[Real closed field|Real closed field]]).

...//www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l ...math.org/legacyimages/h/h110/h110290/h11029054.png" />-Galois extension if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofma

...in [[#References|[a2]]]. Let $p$ denote the characteristic of the residue field $Kv$ if it is a positive prime; otherwise, set $p=1$. Hypothesis A requires ...$ has a solution in $Kv$. Requirement ii) implies that $Kv$ is a [[perfect field]]. Using [[Galois cohomology|Galois cohomology]], G. Whaples [[#References|

...ariables with coefficients in some fixed field or ring. Let $K$ be a field and $F_3(x_0,\ldots,x_n)$ a cubic form with coefficients in $K$ (one calls it a ...f cubic forms over an [[algebraically closed field]] $K$ is reduced to the theory of cubic hypersurfaces (see [[#References|[1]]]).

The theory of eliminating unknowns from systems of algebraic equations. More precisely ...$x_1,\dots,x_k$, one also consider the homogeneous problem in elimination theory (the inhomogeneous problem is trivial in this case): Find the projection on

Please remove this comment and the {{TEX|auto}} line below, consisting of the principal ideals. The divisor class group is Abelian and is usually denoted by $ C ( A) $.

Please remove this comment and the {{TEX|auto}} line below, are called the roots of the recurrence, and the $ n _ {i} $

...es the existence of algebraic numbers of degree $n$. All rational numbers, and only such numbers, are algebraic numbers of the first degree. The number $i ...aic numbers; this means that the set of all algebraic numbers is a [[Field|field]]. A root of a polynomial with algebraic coefficients is an algebraic numbe

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ''factoring polynomials''

Please remove this comment and the {{TEX|auto}} line below, of Laurent polynomials in the variables $ t _ {i} $.

Please remove this comment and the {{TEX|auto}} line below, and $ w( x) $

...Bernoulli numbers are the values of the [[Bernoulli polynomials|Bernoulli polynomials]] at $x=0$: $B_n=B_n(0)$; they also often serve as the coefficients of the L. Euler in 1740 pointed out the connection between Bernoulli numbers and the values of the Riemann zeta-function $\zeta(s)$ for even $s=2m$:

...th.org/legacyimages/m/m110/m110160/m1101601.png" /> of the ordered [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org ...ctions (cf. [[Analytic function|Analytic function]]) for which there exist polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l

...rentiation: differential rings, differential modules, differential fields, and differential algebraic varieties. ...of the main objects of differential algebra is the algebra of differential polynomials $ {\mathcal F} \{ Y _{1} \dots Y _{n} \} $ ,

Please remove this comment and the {{TEX|auto}} line below, and satisfies the relation usually referred to as the [[Leibniz rule]]

...the theory of field extensions (cf. [[Extension of a field|Extension of a field]]). Let $ K $ be some extension of a field $ k $ .

Please remove this comment and the {{TEX|auto}} line below, and equality $ \mathop{\rm rk} L = \mathop{\rm dim} L $

...tive algebraic groups [[#References|[4]]], [[#References|[5]]], and in the theory of formal groups [[#References|[6]]]. Let $ A $ which are added and multiplied in accordance with the following rules: $$

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...stance, if $K \subseteq L$ is a field extension (cf. also [[Extension of a field]]), then $L$ is a Fatou extension of $K$.

$#C+1 = 48 : ~/encyclopedia/old_files/data/R077/R.0707310 Random field, generalized, Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, and endowing $ U ^ {*} $,

Please remove this comment and the {{TEX|auto}} line below, of modules and any homomorphism $ \beta : P \rightarrow C $

...nd numerical; for example, in the (numerical) solution of linear equations and eigenvalue problems. A few well-known factorizations are listed below. ...th $m\ge n$ over $\C$. Then there exist a unitary $(m\times m)$-matrix $Q$ and a right-triangular $m\times n$-matrix $R$ such that $A=QR$. Here, a right-t

''Associative rings and algebras'' are rings and algebras with an associative multiplication, i.e., sets with

and the solution to this equation may be obtained by using Ferro's result; he also solved the equation $x^3+px+q$ ($(p>0$, $q>0$), and

Please remove this comment and the {{TEX|auto}} line below, ...ce in which the modulus is a prime number. A distinguishing feature of the theory of congruences modulo a prime number is the fact that the residue classes m

...articular, in a ring of polynomials in a finite number of variables over a field or over a ring of integers any ideal is generated by a finite number of ele be an irreducible polynomial over the field $ \mathbf Q $

...among the widely available systems is REDUCE, which runs on many platforms and is still being further developed. ...1983. Maple was released in 1986, while Scratchpad II (later called AXIOM) and Mathematica appeared by 1988.

a linear algebraic group over some field, related to a semi-simple Chevalley basis of the algebra $\fg$, and let $\fg_\Z$ be its linear

A [[Differential field|differential field]] $ F \supset F _{0} $ and $ F _{0} $;

...ring, there exists a one-to-one correspondence between varieties of rings and the $T$-ideals of a countably-generated free ring. If for two varieties of ...bra with a finite number of generators over a field of characteristic zero and if $M_2\not\subseteq\mathfrak M$, then $\mathfrak M$ is a Specht variety.

$#C+1 = 70 : ~/encyclopedia/old_files/data/D033/D.0303860 Double and dual numbers Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, of polynomials over a field $ F $,

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...{ q }$ which contains $V$ (cf. also [[Extension of a field|Extension of a field]]). Then the monic separable polynomial whose roots are precisely the eleme

...lowing more general assertion, which was stated without proof by Lindemann and was proved by K. Weierstrass in 1885, is known as the Lindemann–Weierstra Let $a_1,\dots,a_m$ be non-zero algebraic numbers and let $\alpha_1,\dots,\alpha_m$ be pairwise distinct algebraic numbers; then

...$f_1, \ldots, f_k$, then the curve $L$ defined by (1) is called reducible and is the union of the curves $L_1, \ldots, L_k$ (the components of $L$) defin ...cible curve. Two irreducible plane real algebraic curves, one of order $n$ and the other of order $m$, intersect in at most $mn$ points (Bezout's theorem)

Please remove this comment and the {{TEX|auto}} line below, and of a rank lower than $ t $,

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...{ 0 }$ is a [[Polynomial|polynomial]] in $X$ with coefficients in $\bf Z$ and there exists an $x \in \widetilde{\mathbf{Z}}$ such that $f ( x )$ is a uni

Please remove this comment and the {{TEX|auto}} line below, be a field which is complete with respect to an [[ultrametric]] valuation $ | \cdot

...f Lie algebras, i.e., $ \sigma $ is a $ \mathbb{k} $-linear transformation and $ \sigma([x,y]) = \sigma(x) \sigma(y) - \sigma(y) \sigma(x) $ for all $ x,y # For every associative $ \mathbb{k} $-algebra $ A $ with a unit element and every $ \mathbb{k} $-algebra homomorphism $ \alpha: \mathfrak{g} \to A $ su

Please remove this comment and the {{TEX|auto}} line below, ...structure. While the theory of analytic surfaces forms part of the general theory of complex manifolds, the two-dimensional case is treated separately, since

Please remove this comment and the {{TEX|auto}} line below, ...o account both the geometry of the domain on which the function is defined and the behaviour of the function while approaching the boundary, so that the s

...dges, which may be replaced by a triangle (A. Adler, 1890); a straightedge and a circle with a marked centre given in the plane of the figure (Poncelet– ..., $k \in \mathbf{N}$. For example, the regular $n$-gon is constructible if and only if each odd prime power dividing $n$ is a [[Fermat prime]] $2^{2^m} +

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...fference set|Difference set]] (Vol. 3). For an extensive discussion of the theory of Abelian difference sets, see also [[#References|[a1]]], Chap. VI.

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...\mathbf{C} \backslash \{ 0 , \infty \}$ which have algebraic poles at $0$ and $\infty$. A basis of the algebra is given by the set

[[Category:Linear and multilinear algebra; matrix theory]] consisting of $m$ rows and $n$ columns, the

In the most general sense, ''Galois theory'' is a theory dealing with mathematical

see [[Algebraic K-theory|Algebraic $ K $- theory]]).

Please remove this comment and the {{TEX|auto}} line below, In the simplest and most often used case, a [[Hilbert space|Hilbert space]] consisting of infin

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...concept of an arithmetical semi-group $G$ (cf. [[Abstract analytic number theory]]; [[Semi-group]]).

Please remove this comment and the {{TEX|auto}} line below, ...domains where the [[Ray method|ray method]] cannot be employed because the field of rays suffers from a singularity in one sense or another. Consider, for e

Please remove this comment and the {{TEX|auto}} line below, and $ X $ (cf. [[Chern class|Chern class]]). It can be used to calculate the

Please remove this comment and the {{TEX|auto}} line below, with coefficients in some field $ K $,

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...p$, the appropriate generalization is not just to take $x$, $y$ and $z$ as polynomials over $\mathbf{F}$ in $T$. In any event, in characteristic zero, or for $n$

...nnected with solving inequalities in integers — Diophantine inequalities — and also with solving equations in integers (cf. [[Diophantine equations|Diopha ...d03260013.png" />-adic number fields and Diophantine approximations in the field of power series).

Please remove this comment and the {{TEX|auto}} line below, ...l tool in many applications, particularly communication science and coding theory.

Please remove this comment and the {{TEX|auto}} line below, with identity over the field $ \mathbf C $

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...lead to abstract generalizations and analogues of ordinary analytic number theory, which may then be applied in a unified way to further enumeration question

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ''Galois field (update)''

...obabilistically checkable" is somewhat misleading, since both holographic and traditional proofs can be checked either deterministically or probabilistic ...oice. This chance never exceeds $50$%, regardless of the nature of errors, and vanishes as $1/2^k$ with $k$ independent rounds.

''(in the classical sense), invariant theory'' ...dies on the theory of binary quadratic forms posed the problem of studying polynomials in the coefficients of the form $ ax ^{2} + 2 b xy + c y ^{2} $

Please remove this comment and the {{TEX|auto}} line below, The field concerned with best rational approximation to power series. Let

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. In [[#References|[a2]]], M. Eichler conceived the "Eichler cohomology theory" (but not the designation) while studying "generalized Abelian integrals"

where $i_1,\dots,i_n$ is a permutation of the numbers $1,\dots,n$ and $k$ is the number of inversions of the permutation $1\mapsto i_1,\dots,n\ma ...eld (especially a number field), a ring of functions (especially a ring of polynomials) or a ring of integers.

{{Cite|Ch}} on conics and was systematized and used to great effect by H. Schubert in {{Cite|Sc}}. The justification of Schubert's enumerative calculus and the verification of the numbers he obtained was the contents of Hilbert's 1

Please remove this comment and the {{TEX|auto}} line below, The branch of algebra studying the properties of commutative rings and objects relating to them (ideals, modules, valuations, etc., cf. [[Ideal|Id

Please remove this comment and the {{TEX|auto}} line below, and $ m $

and $n$ are natural numbers; the $a_{ij}$ ($i=1,\dots,m$; $j=1,\dots,n$) are ca coefficients at the unknowns and are given; the $b_i$ ($i=1,\dots,m$) are called

Please remove this comment and the {{TEX|auto}} line below, An algebra over a field for which certain polynomial identities are true.

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. The geometry of the Yang–Mills equations (cf. [[Yang–Mills field|Yang–Mills field]]) led to deep purely mathematical insight, some of which is given below. F

and the structure of an associative graded [[Co-algebra|co-algebra]] $ \delta and $ B $

Please remove this comment and the {{TEX|auto}} line below, of this sequence and only at these points. This function may be constructed as a [[Canonical pro

Please remove this comment and the {{TEX|auto}} line below, ...4]]] on the other. Another approach has been undertaken by Yu. Berezanskii and collaborators [[#References|[a1]]].

$#C+1 = 92 : ~/encyclopedia/old_files/data/R082/R.0802480 Rings and algebras Please remove this comment and the {{TEX|auto}} line below,

...The discriminant is symmetric with respect to the roots of the polynomial and may therefore be expressed in terms of the coefficients of this polynomial. ...ula|Cardano formula]]) is $-27 q^2-4p^3$. If $f(x)$ is a polynomial over a field of characteristic zero, then

Baxter algebras originated in the following problem in fluctuation theory: Find the distribution functions of the maxima $\max\{0, S_1, \ldots, S_n \ ...n(t)$ is the [[characteristic function]] of $\max\{0, S_1, \ldots, S_n \}$ and $\psi_k(t)$ is the characteristic function of $\max\{0, S_k \}$. Spitzer's

....encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346010.png" /> is a field with respect to the usual [[pointwise operation]]s followed by redefinition ...fmath.org/legacyimages/m/m063/m063460/m06346022.png" /> coincides with the field of fractions of the ring <img align="absmiddle" border="0" src="https://www

[[Field|field]] $L$ of characteristic $p>0$ is an exotic $A$-module structure on the addi ...omorphism rings, the existence of analytic "Weierstrass uniformizations" , and the moduli theories (modular varieties, modular forms; cf. also

Please remove this comment and the {{TEX|auto}} line below, and is called the stabilizer or stationary subgroup of the point $ x $

Please remove this comment and the {{TEX|auto}} line below, over a field $ K $

and for $k_i \ge 1$, and a corresponding multinomial theorem. The Wick exponential is defined as

...r series in the associative but non-commutative variables $ u $ and $ v $, and $ \widehat{L} $ is the closure of $ L $ in $ \widehat{A} $. Then the mappin ...ormula provides an expression for $ u \circ v $ as a power series in $ u $ and $ v $:

as the sum of a semi-simple and a nilpotent endomorphism that commute with each other: $ g = g _{s} + g _ and $ g _{n} $

over a field $ k $ the adjoint transformation) and $ \Lambda _{p} ( x ,\ y ) $

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...\rangle$ denote an [[Inner product|inner product]] on $\mathbf{C} ^ { n }$ and let $\| .\|$ be the induced [[Norm|norm]].

...erous applications in algebraic geometry, class field theory and cobordism theory. A formal group over a field $ k $

Please remove this comment and the {{TEX|auto}} line below, and $ \Delta _ {2} $);

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...be determined by means of computations involving finitely many steps, i.e. field operations in $k$. Any $R$-module for $R : = k [ X _ { 1 } , \dots , X _ {

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...ty of $C$, see [[#References|[a1]]]. For an interplay between the geometry and the algebra, see [[#References|[a2]]].

Please remove this comment and the {{TEX|auto}} line below, and for each point $ x _ {t} $

...algebra|Lie algebra]]; [[Alternative rings and algebras|Alternative rings and algebras]]; [[Jordan algebra|Jordan algebra]]; [[Mal'tsev algebra|Mal'tsev ...mple algebras that are far from associative — in the class of all algebras and in the class of all commutative (anti-commutative) algebras. For these clas

Please remove this comment and the {{TEX|auto}} line below, and each element $ h \in X $

...erences|[a1]]] have their origins in physics in earlier work of N. Seiberg and E. Witten [[#References|[a2]]], [[#References|[a3]]]. ...ection|connection]] $A$ on a line bundle $L$ over $M$ and a "local spinor field" $\psi$. The Seiberg–Witten equations are then

A complex number is a number of the form $z=x+iy$, where $x$ and $y$ are real numbers (cf. [[Real number|Real number]]) and $i=\def\i{\sqrt{-1}}\i$ is the so-called imaginary unit,

...up is historically one of the first examples of abstract algebraic systems and served, in many respects, as a model for the restructuring of other mathema 1) the operation is associative, i.e. $(ab)c = a(bc)$ for any $a$, $b$ and $c$ in $G$;

Please remove this comment and the {{TEX|auto}} line below, over a field $ K $.

Please remove this comment and the {{TEX|auto}} line below, ...re sought for in the algebraic integers of some algebraic extension of the field $ \mathbf Q $

...these theorems usually become invalid, and the Grothendieck group $K_0(R)$ and the Whitehead group $K_1(R)$ are, in a certain sense, a measure of their de ...jects may be studied with the aid of the homotopy theory of vector bundles and of topological

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...ns and the computational steps consist of applying local unitary operators and measurements as permitted by quantum mechanics.

The tensor product of two unitary modules $V_1$ and $V_2$ over an associative commutative ring $A$ with a unit is the $A$-modul ...product is uniquely defined up to a natural isomorphism. It always exists and can be constructed as the quotient module of the free $A$-module $F$ genera

Please remove this comment and the {{TEX|auto}} line below, The origin is a stationary point and the Hamiltonian evaluated at the origin is

Please remove this comment and the {{TEX|auto}} line below, There exist two approaches to the moduli problem: an algebraic and an analytic one. The algebraic approach is connected with studies of the fi

Please remove this comment and the {{TEX|auto}} line below, and $ f $

Please remove this comment and the {{TEX|auto}} line below, times covariant [[tensor analysis|tensor field]] on $ M $.

Please remove this comment and the {{TEX|auto}} line below, ..., was the first to use the letters of the alphabet to denote the constants and the variables in a problem. Most of the present-day symbols of algebra were

Please remove this comment and the {{TEX|auto}} line below, be a given [[Bilinear form|bilinear form]] (not necessarily symmetric) over a field $ P $.

Please remove this comment and the {{TEX|auto}} line below, or in a space of constant curvature), and also to mappings; moreover, symmetrization is constructed so that $ F ^ {

$#C+1 = 104 : ~/encyclopedia/old_files/data/R080/R.0800280 Recursive model theory, Please remove this comment and the {{TEX|auto}} line below,

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...se of an algebraic group this reduction is carried out with respect to the field of definition of the group.

Please remove this comment and the {{TEX|auto}} line below, ...nt problem in the real domain is due to T.J. Stieltjes (1894). He proposed and solved the following problem in connection with the study of continued frac

...rget <math>\mathbb{P}^n</math> are defined by <math>(n+1)</math>-tuples of polynomials in the homogeneous coordinate variables of <math>\mathbb{P}^1</math>. For e ...{P}^1</math> and rescaling yields a <math>(5d+1)</math>-dimensional smooth and irreducible space. Fix a choice of ''smooth'' rational curve <math>C</math>

Please remove this comment and the {{TEX|auto}} line below, and, finally, on the specific nature of the problem to be solved (for example,

Please remove this comment and the {{TEX|auto}} line below, dimensional [[Affine space|affine space]] over the field $ \mathbf K $.

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...f knots, links or three-dimensional manifolds (cf. also [[Knot theory|Knot theory]]; [[Link|Link]]; [[Three-dimensional manifold|Three-dimensional manifold]]

If $m$ is a positive integer and $a_1,\ldots,a_{2m-1}$ is a sequence of elements from the [[Cyclic group|cyc zeros and $ m - 1 $

...s the behaviour of number-theoretic functions, and the theory of algebraic and transcendental numbers. ...one of the most interesting and most difficult problems in analytic number theory. The first result on the problem of the distribution of primes was obtained

Please remove this comment and the {{TEX|auto}} line below, ...pment of mathematics. The practical activities of mankind on the one hand, and the internal demands of mathematics on the other, have determined the devel

$#C+1 = 254 : ~/encyclopedia/old_files/data/K055/K.0505600 Knot theory Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, ...nsists in the representation of the information sequence in a special form and in the introduction of additional information ([[Redundancy|redundancy]]) i

...ction|Dirichlet $L$-function]]) form the basis of modern [[analytic number theory]]. In addition to Riemann's zeta-function one also distinguishes the genera which converges absolutely and uniformly in any bounded domain of the complex $s$-plane for which $\sigma\

...ay one defines the concepts of a subalgebra, an ideal, a quotient algebra, and a homomorphism of Lie algebras. A Lie algebra $L$ is said to be commutative ...tant case is that in which $k$ is a field (especially when $k=\R$ or $\C$) and $L$ is a vector space (of finite or infinite dimension) over $k$.

Please remove this comment and the {{TEX|auto}} line below, and are defined to be equal to $ p _ {i} ( \xi ) = ( - 1 ) ^ {i} c _ {2i} (

Please remove this comment and the {{TEX|auto}} line below, ...white noise analysis has developed into a viable framework for stochastic and infinite-dimensional analysis [[#References|[a4]]]–[[#References|[a6]]],

Please remove this comment and the {{TEX|auto}} line below, and $ M $

Please remove this comment and the {{TEX|auto}} line below, ...iples and results concerning the concepts listed above. In particular, the theory of generalized displacement operators has substantial applications in abstr

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...L} _ { 2 } ( \mathbf{Z} )$ with as representation the space of homogeneous polynomials of degree $n$,

Please remove this comment and the {{TEX|auto}} line below, A method for approximately solving non-linear (and linear) functional and operator equations $ y = P( x) $,

Please remove this comment and the {{TEX|auto}} line below, ''cobordism theory''

Please remove this comment and the {{TEX|auto}} line below, ...this theory with abstract harmonic analysis, i.e. with part of the general theory of topological algebras (cf. [[Harmonic analysis, abstract|Harmonic analysi

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...the notion of a quantum group (cf. also [[Quantum groups|Quantum groups]]) and a [[Super-group|super-group]], in which the role of Bose–Fermi statistics

does not contain the null vector, and it generates $ V $; and such that the endomorphism $ s _ \alpha : \ x \mapsto x-\alpha ^{*} (x)

...important classes: the [[Real analytic function|real analytic functions]] and the complex analytic functions, which are commonly called ''holomorphic fun ...'s) the theory of analytic functions forms the main subject of the general theory of functions of a complex variable.

..._p}$ of the finite [[field]] $\mathbb{F}_p$ of prime [[Characteristic of a field|characteristic]] $p > 0$. Furthermore let $F$ be a generalised Frobenius en ...cial linear group $\SL_n(q) = \SL_n(\mathbb{F}_q)$ defined over the finite field of order $q$. The quotient group $\PSL_n(q) = \SL_n(q)/Z(\SL_n(q))$, where

Please remove this comment and the {{TEX|auto}} line below, ...) under the condition that these collections contain the identity mappings and are closed with respect to successive composition (or product) of mappings.

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...discrete data, then called the log-linear model; see [[#References|[a6]]], and [[#References|[a10]]].