The group $\langle 4,3,2 \rangle$ abstractly presented as:
...\langle 3,3,2 \rangle$ as a subgroup of index 2. It occurs as a subgroup of the [[unit quaternion]]s.
612 bytes (88 words) - 20:50, 23 November 2023
The exceptional group $G_4$ or $\langle 3,3,2 \rangle$, abstractly presented as:
...] $A_4$ as quotient by the centre and the [[quaternion group]] of order 8 as a quotient.
764 bytes (115 words) - 20:46, 23 November 2023
The group $\langle 5,3,2 \rangle$ abstractly presented as:
It is finite of order 120. It occurs as a subgroup of the [[unit quaternion]]s.
520 bytes (74 words) - 14:29, 12 November 2023
...s classes of numbers; [[conjunction]], [[disjunction]] and [[implication]] of propositions.
...A \times A$ (as for example division by zero is not defined). Properties of binary operations which occur in many contexts include
1 KB (193 words) - 19:47, 13 November 2016
...uling problems, the problem of minimizing the makespan is as hard to solve as the [[travelling salesman problem]].
309 bytes (46 words) - 12:27, 17 February 2021
...d at the barycentre. Geometrically, the barycentre is at the [[centroid]] of the body.
628 bytes (99 words) - 20:44, 5 December 2023
...ontain elliptic points as well. Thus, an elliptic paraboloid consists only of elliptic points, and is an affine minimal surface.
295 bytes (46 words) - 09:29, 27 June 2014
...a''' is a branch of [[mathematics]]. The term is used in combinations such as [[homological algebra]], [[commutative algebra]], [[Linear-algebra(2)|linea
An [[algebraic system]], consisting of a set with a family of algebraic operations.
788 bytes (106 words) - 11:22, 9 April 2017
A term with a variety of meanings.
...l function]], such as $\exp(x+y) = \exp(x)\exp(y)$ or of a [[power]], such as $x^{yz} = (x^y)^z$.
531 bytes (94 words) - 18:21, 19 December 2014
...hen a computable real number is considered in the framework of some system of constructive mathematics (see [[Constructive analysis|Constructive analysis
388 bytes (57 words) - 17:04, 7 February 2011
...ously with its negation. In the language of propositional calculus the law of contradiction is expressed by
This formula is derivable in classical as well as in intuitionistic [[Constructive propositional calculus|constructive propos
421 bytes (57 words) - 13:51, 30 December 2018
...nly if $P$ is balanced as an $R$-module, projective and finitely generated as an $\mathrm{End}_R P$-module.
646 bytes (112 words) - 20:23, 10 October 2017
...al{A}$-set (Suslin's criterion); and 2) it can be represented as the union of non-intersecting components (Luzin's criterion).
356 bytes (55 words) - 21:57, 19 December 2014
...s are taken as $ h_i $ or as $ m_i $. Thus, let the measurements of trunks of 1000 firs give the following results:
<tr> <td colname="1" style="background-color:white;" colspan="1">number of trunks</td> <td colname="2" style="background-color:white;" colspan="1">100
3 KB (376 words) - 00:30, 11 June 2013
...multiplication $(m,r) \cdot (n,s) = (mn, ms + nr + rs)$ which has $(1,0)$ as a multiplicative identity.
...ra $A$ over a field $K$ is an algebra over $K$ which is unital as a ring. As with rings, any $K$-algebra $A$ can be embedded in a unital $K$-algebra $A^
1 KB (199 words) - 18:38, 13 November 2023
...ignature]]. A Gödelization could be understood as encoding of the elements of $A$ by natural numbers {{Cite|W90}}.
...lon C \longrightarrow A$. More precisely, $\beta=\nu^{-1}$ is the inverse of $\nu$ in this case. For a sensible signature $\Sigma$ and a term-generated
2 KB (241 words) - 07:50, 21 March 2023
...omains in the plane have been constructed with an indecomposable continuum as common boundary.
815 bytes (126 words) - 08:39, 12 April 2014
...ts of two digits" one says the term "12" is used autonymously (as the name of itself).
...d without the quotation marks. Autonymous use of a term merges the meaning of the term with the term itself: it is both the object itself and the name by
2 KB (318 words) - 14:25, 30 December 2018
The same as a [[Degenerate distribution|degenerate distribution]].
...]]; the latter is defined as a [[Measure|measure]] $\mu$ on the Borel sets of $\mathbf R$ such that $\mu(\mathbf R)<1$.
393 bytes (58 words) - 21:48, 11 April 2014
A [[Metric|metric]] $\rho$ which is defined for any two points $x$, $y$ of a [[Metric space|metric space]] that can be connected by a [[rectifiable cu
where $s_\rho$ is the length of the curve in the metric $\rho$. A [[Riemannian metric]] induces an internal
1 KB (231 words) - 12:31, 9 November 2014
...rticular the limits of $T$ and its derivatives as $x' \rightarrow x$, such as
...ed in the [[calculus of variations]] and in the [[Relativity theory|theory of relativity]].
815 bytes (132 words) - 22:12, 29 November 2014
...ted to a certain form of the concept of infinity in mathematics — the idea of a potential infinity.
...., but it does not imply the idea of the existence of the natural sequence as an actual "infinite object" .
1 KB (196 words) - 17:15, 7 February 2011
...x)$ are called asymptotically equal as $x\to x_0$ if in some neighbourhood of the point $x_0$ (except possibly at $x_0$ itself)
...onsideration are defined). If $g(x)$ does not vanish in some neighbourhood of $x_0$, this condition is equivalent to the requirement
2 KB (363 words) - 19:10, 7 July 2014
...nd the real number $\mu>0$ are parameters. The [[characteristic function]] of the Erlang distribution has the form
...variables having the same exponential distribution with parameter $n\mu$. As $n\to\infty$, the Erlang distribution tends to the [[degenerate distributio
2 KB (279 words) - 19:37, 4 December 2016
...extensively employed in Ancient Egypt and, as a result, received the name of Egyptian fractions.
468 bytes (68 words) - 21:30, 15 November 2014
...ons are comparatively rare (in practice) and, as a rule, arise as mixtures of probability distributions.
498 bytes (65 words) - 16:55, 7 February 2011
...ll $x$ satisfying the condition $|x-x_0|<\delta$. This fact can be written as follows:
...inition of the limit of a function. In fact, the limit of the function $f$ as $x\to x_0$ is finite and equal to $A$ if and only if
1 KB (208 words) - 09:21, 14 December 2012
...elated to a certain form of the idea of infinity in mathematics — the idea of the so-called actual infinity.
...of all non-negative integers — the [[Natural sequence|natural sequence]] — as a mathematical object.
2 KB (297 words) - 17:03, 7 February 2011
''of a set $X$''
A subset $A$ of $X$ for which the [[relative complement]] $X \setminus A$ is finite.
476 bytes (81 words) - 10:14, 22 October 2016
''of rings, algebras''
The same as [[tensor product]].
104 bytes (13 words) - 12:34, 30 December 2014
''of a [[graph]] $G $''
...x]] $I$, then the incidence matrix $I[S]$ consists precisely of those rows of $I$ corresponding to vertices in $S$.
330 bytes (65 words) - 14:49, 10 January 2016
...ng operator (cf. [[Generating operator of a semi-group|Generating operator of a semi-group]]).
467 bytes (65 words) - 17:22, 7 February 2011
...part of the homology group with as domain of coefficients the group $\ZZ$ of integers, if this homology group is finitely generated. Named after E. Bett
...top">[1]</TD> <TD valign="top"> H. Seifert, W. Threlfall, "A textbook of topology" , Acad. Press (1980) (Translated from German)</TD></TR><TR><TD
792 bytes (117 words) - 18:34, 11 April 2023
...f [[tiling]]s, '''protoset''' is sometimes used as abbreviation for "''set of [[prototile]]s''".
111 bytes (17 words) - 18:46, 11 February 2012
...ential-geometric structures on fibre spaces that later simply became known as connections. (See [[Connection|Connection]].)
150 bytes (19 words) - 17:18, 7 February 2011
$#C+1 = 8 : ~/encyclopedia/old_files/data/D032/D.0302650 Diophantine problems of additive type
...[[Additive problems|additive problems]], i.e. as problems of decomposition of an integer $ n $(
2 KB (277 words) - 19:35, 5 June 2020
...ar polyhedra (Platonic solids); if the edge length is $a$, then the volume of the octahedron is
...n nature as a crystal of chrome alum. In terms of its circumradius as unit of measurement, its six vertices have Cartesian coordinates
2 KB (244 words) - 14:08, 30 July 2014
''of a curve $ y = f(x) $
A straight line the distance of which from the point $ (x, f(x)) $
3 KB (410 words) - 18:48, 5 April 2020
Any topological space for which $X$ serves as a retract (cf. [[Retract of a topological space]]).
...fact no name for this concept. The standard Western usage of dominating is as follows: A space $X$ is said to dominate a space $Y$ if and only if there a
476 bytes (82 words) - 17:20, 29 September 2016
...o the given test" . The term "testing" is basically used in combinations as "repeated testing" , "independent testing" , "testing in a Markov chain"
649 bytes (94 words) - 07:34, 2 November 2014
...f the image $\mathrm{im}\, d$ are called ''boundaries''. The ''homology'' of $C$ is the quotient $\ker d / \mathrm{im}\,d$.
513 bytes (78 words) - 14:52, 8 April 2023
...tained as the extension of the cyclic group of order $2$ by a cyclic group of order $2n$. It has the presentation $\langle n,2,2 \rangle$ and [[group pr
It may be realised as a subgroup of the [[unit quaternion]]s.
601 bytes (92 words) - 20:50, 23 November 2023
...ithmetic functions that can be stated as answers to problems on the number of integral points (cf. [[Integral point|Integral point]]) in some manifolds,
...isor problems]] (P.G.L. Dirichlet) are classical (starting points) as well as their numerous generalizations.
1 KB (188 words) - 17:31, 30 April 2014
...approximation of functions. Nowadays the phrase "the constructive theory of functions" is seldom used.
...G. Alexits (ed.) S.B. Stechkin (ed.) , ''Proc. Conf. Constructive Theory of Functions'' , Akad. Kindó (1969) pp. 19–26</TD></TR></table>
2 KB (252 words) - 17:17, 7 February 2011
See [[Algebraic systems, quasi-variety of|Algebraic systems, quasi-variety of]].
...ally called Horn sentences. Correspondingly, a quasi-variety is also known as a universal Horn class; it is also called a quasi-primitive class.
607 bytes (77 words) - 17:15, 7 February 2011
''quadratic variance, standard deviation, of quantities $x_1,\dots,x_n$ from $a$''
The square root of the expression
2 KB (285 words) - 14:22, 30 December 2018
...ations includes, first of all, ordinary differential equations and systems of the form
...l as their corresponding difference equations, usually belong to the class of evolution equations.
3 KB (443 words) - 15:35, 14 February 2020
...ncountable metrizable space contains an uncountable subspace that, as well as its complement, is totally imperfect.
A subset of $\mathbf R$ such that it and its complement are totally imperfect is usuall
732 bytes (104 words) - 12:02, 5 July 2014
''in the theory of partial differential equations''
The same as a [[Characteristic|characteristic]].
101 bytes (13 words) - 16:57, 7 February 2011
''of a topological space $X$''
...as the same [[homotopy type]] as does $X$. See also [[Retract]]; [[Retract of a topological space]].
536 bytes (94 words) - 19:45, 5 February 2016
...egardless of its origin. A syntactic language is used to study this aspect of formal systems.
...a formal system, two languages are connected: one is the research language of the formal system itself (the object language) and the other is that in whi
2 KB (394 words) - 17:28, 7 February 2011
''of a stochastic process''
The same as the [[Correlogram|correlogram]].
73 bytes (10 words) - 17:04, 7 February 2011
...function (or that of some derivative of it) or a majorant of it may figure as $\phi(\tau)$.
...f. [[Approximation of functions, direct and inverse theorems|Approximation of functions, direct and inverse theorems]]).
2 KB (326 words) - 09:43, 26 April 2014
Out of 20 formulas, 20 were replaced by TEX code.-->
...ports. It can be extended as a sheaf morphism to an imbedding of the space of distributions $\mathcal{D} ^ { \prime } ( \Omega )$.
2 KB (342 words) - 16:45, 1 July 2020
...nd [[Non-Euclidean geometries|non-Euclidean geometries]]. The trigonometry of a sphere in Euclidean space is called [[Spherical trigonometry|spherical tr
639 bytes (81 words) - 17:07, 7 February 2011
...which any even natural number larger than 2 can be represented as the sum of two prime numbers.
...oth the Goldbach–Euler problem and the more general problem of solvability of the linear Diophantine equation
1 KB (201 words) - 20:18, 14 October 2014
...ins unchanged in all its applications. The number of possible applications of a given rule, however, is in principle unlimited.
...ithms, whether $P_0$ is deducible from $P_1,\ldots,P_l$ by one application of the rule.
3 KB (431 words) - 19:19, 17 October 2014
...his inequality appeared as late as 1884 (without any reference to the work of Bunyakovskii).
2. W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)
2 KB (248 words) - 09:09, 31 May 2016
...tertiary ideals are the same as [[primary ideal]]s (cf. [[Additive theory of ideals]]; [[Primary decomposition]]).
...there exists a tertiary radical, $\mathrm{ter}(Q)$, the largest ideal $T$ of $R$ such that, for any ideal $B$,
2 KB (297 words) - 19:23, 5 October 2017
...h any node and connect it to the node nearest to it. Given a connected set of nodes, take the unconnected node that is nearest to one in the connected se
...ideas. See [[#References|[a1]]] for more details and for the justification of such methods.
2 KB (387 words) - 15:16, 20 November 2014
The same as the [[Antinomy|antinomy]] of Epimenides.
65 bytes (10 words) - 12:47, 27 August 2014
A polynomial in several variables all terms of which are of the same degree.
...om each form one can obtain a multilinear form by a certain process, known as the [[Polarization identity|polarization process]].
1 KB (237 words) - 07:31, 19 January 2016
A [[topological space]] which cannot be expressed as a non-trivial union of disjoint [[closed set|closed subset]]s
173 bytes (25 words) - 14:45, 3 January 2016
...a magic magnetic gate through which an armed person could not pass as well as about compasses.
...are the [[Maxwell equations|Maxwell equations]] describing the interaction of electric and magnetic fields in physical media. See also [[Electromagnetism
1 KB (211 words) - 14:43, 1 May 2014
...metric completion of a metric totally-bounded space is compact. The image of a totally-bounded space under a uniformly continuous mapping is a totally-b
...TD valign="top"> A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock (1957–1961)
2 KB (349 words) - 13:16, 12 December 2013
...y|boundary-layer theory]] developed, as well as applications in the theory of heat transfer, diffusion, processes in semi-conductors, etc.
...development of this concept, as well as its applications, are in the field of singular perturbations. For a partial review one may consult [[#References|
2 KB (264 words) - 11:57, 27 September 2012
...is considered as the way information is transmitted in the nervous system of animals. Cardiac, muscle and some endocrine cells also display action poten
...sient departure from equilibrium that is not propagated. After the passing of an action potential the axon apparently returns to its equilibrium state bu
2 KB (251 words) - 08:13, 21 June 2014
...age]] used for denoting a [[Logical operation|logical operation]] by means of which a new statement can be obtained from given statements. The most impor
...is usually not introduced as an independent propositional connective, but as an abbreviation:
2 KB (311 words) - 15:56, 14 February 2020
...logic the following [[Truth table|truth table]] applies for the operation of negation:
772 bytes (119 words) - 07:34, 12 August 2014
$#C+1 = 16 : ~/encyclopedia/old_files/data/C024/C.0204220 Complexification of a vector space
by extending the field of scalars. The space $ V ^ {\mathbf C } $
1 KB (197 words) - 17:46, 4 June 2020
...the same as [[Algebra|algebra]]. The phrase frequently occurs in the title of general texts on modules, rings, fields, vector spaces, etc.
161 bytes (25 words) - 15:20, 1 May 2014
...es as an estimator of the [[auto-correlation]] (auto-correlation function) of a [[Time series|time series]]. Let $ x _ {1} \dots x _ {N} $
be a time series. The serial correlation coefficient of order $ k $
2 KB (327 words) - 08:24, 20 January 2024
...$a>0$ white and $b>0$ black balls. The numbers $c$ and $d$, the parameters of the urn model, may also be negative.
...ons. The negative binomial distribution and the Poisson distribution arise as limit distributions from certain urn models.
2 KB (335 words) - 07:02, 27 May 2023
An argument $x$ of a function $f$ that satisfies the condition
...e-sided derivative exists at a smooth point, an ordinary derivative exists as well.
569 bytes (102 words) - 06:04, 22 August 2014
The same as the [[Antinomy|antinomy]] of the "village barber".
76 bytes (12 words) - 12:46, 27 August 2014
Convergence
of a sequence
of random variables <img align="absmiddle" border="0" src="https://www.encyclo
...rgence is so called because condition (*) is equivalent to the convergence
of the distribution functions <img align="absmiddle" border="0" src="https://w
2 KB (315 words) - 16:40, 6 February 2012
...learly any field of characteristic zero is a binomial ring, as is the ring of integers $\mathbf{Z}$.
663 bytes (103 words) - 20:19, 20 November 2023
...r hand the relation of symmetry may be taken as a basis for the axiomatics of plane Euclidean geometry (see [[#References|[5]]]).
<TR><TD valign="top">[2]</TD> <TD valign="top"> V.F. Kagan, "Foundations of geometry" , '''1''' , Moscow-Leningrad (1949) (In Russian)</TD></TR>
3 KB (380 words) - 20:59, 25 October 2014
.... An example of a Sturm curve is the trajectory of the focus of a parabola as it rolls along the $x$-axis — a [[Catenary|catenary]].
...><TD valign="top">[a1]</TD> <TD valign="top"> J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)</TD></TR><TR><TD valign="top
955 bytes (145 words) - 11:44, 5 July 2014
...ucible module are also completely reducible. The [[lattice]] of submodules of a module $M$ is a [[lattice with complements]] if and only if $M$ is comple
...reducible it is sufficient for it to be completely reducible when regarded as a left (right) module over itself.
1 KB (194 words) - 12:47, 13 December 2015
...of the Schreier system be represented as a reduced word in the generators of the group:
...$g'$ should also belong to this system (the element $g'$ can be considered as the reduced word obtained from $g$ by deleting its last letter). The elemen
1 KB (211 words) - 12:17, 19 August 2014
One of the best known optimal stopping problems (see also [[Stopping time|Stopping
...s with the girls yet unseen. A rule is asked for that maximizes the chance of actually selecting the best girl.
2 KB (291 words) - 18:52, 11 December 2020
The same as [[Continued fraction|continued fraction]], i.e. an expression of the form
are finite or infinite sequences of complex numbers or functions. For continued fractions one uses the notation
6 KB (886 words) - 17:03, 7 February 2011
''double ratio, anharmonic ratio, of four points $M_1$, $M_2$, $M_3$, $M_4$ on a straight line''
...s well as the cross ratio of four points, one may consider the cross ratio of four straight lines passing through a point. This ratio, which is denoted b
2 KB (296 words) - 09:34, 13 April 2014
...d $P_3$, while $P_6$ is obtained from $P_3$ and $P_5$, may be written down as follows:
...tree is also given an analysis, i.e. each line is written with the number of the respective rule; axioms are written with their numbers.
1 KB (203 words) - 19:15, 16 April 2014
...is called a [[formally real field]]. The field of real numbers is a model of a formally real field. More generally, every ordered field is formally real
...osed; or 3) every positive element in $k$ is a square and every polynomial of odd degree over $k$ has a root in $k$. Every formally-real field has a real
2 KB (375 words) - 19:58, 7 December 2014
''comparison of functions, $ O $-
...ehaviour of a function with respect to another function in a neighbourhood of some point (this point may be infinite).
5 KB (805 words) - 17:44, 31 March 2020
...integers $(a>b)$. The Pythagorean numbers can be interpreted as the sides of a right-angled triangle (cf. [[Pythagoras theorem]]).
...valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XIII</TD></TR>
761 bytes (116 words) - 07:30, 10 December 2016
The ellipse of least surface area obtained as the intersection of a one-sheet [[Hyperboloid|hyperboloid]] with a plane perpendicular to its a
145 bytes (22 words) - 17:06, 7 February 2011
...identifying isomorphic groups with each other, one arrives at the concept of an abstract group, etc.
...><TD valign="top">[2]</TD> <TD valign="top"> A.A. Markov, "On the logic of constructive mathematics" , Moscow (1972) (In Russian)</TD></TR></table>
2 KB (216 words) - 17:08, 7 February 2011
...esian square $\mathbf{N} \times \mathbf{N}$, where $\mathbf{N}$ is the set of non-negative integers, with respect to the operation
...icyclic semi-group form a chain, which is ordered with respect to the type of the positive numbers. The bicyclic semi-group is bisimple (cf. [[Simple sem
2 KB (332 words) - 18:22, 16 January 2018
...(a,b)$ of elements of a given set $A$. A binary relation is a special case of a [[Relation|relation]]. Let $R\subseteq A\times A$. If $(a,b)\in R$, then
..., the nil relation and the universal relation in the set $A$. The diagonal of the set $A\times A$, i.e. the set $\Delta=\{(a,a)\colon a\in A\}$, is the e
4 KB (625 words) - 08:47, 26 November 2023
A subscheme of a scheme $ X $
defined by a quasi-coherent sheaf of ideals $ J $
1 KB (214 words) - 17:44, 4 June 2020
...$|x-x_0|<\delta$ the inequality $|f(x)|>M$ holds. This fact may be written as follows:
...x<-\delta$. The study of infinitely-large functions may be reduced to that of infinitely-small functions (cf. [[Infinitely-small function|Infinitely-smal
1 KB (191 words) - 12:21, 28 August 2014
...ally ordered rings $R_\lambda$ (cf. [[Totally ordered set]]). Each element of a vector ring is a vector $\mathbf{a} = (\ldots,a_\lambda,\ldots)$with coor
...provided with various linear extensions of its partial order, may be taken as $R_\lambda$.
755 bytes (120 words) - 21:01, 22 December 2014
...mages/b/b015/b015220/b01522016.png" />; this property is known
as the step
of the bar induction.
...conditions 1) through 4) above are met, one can deduce from the principle
of bar induction that <img align="absmiddle" border="0" src="https://www.encyc
5 KB (703 words) - 17:01, 7 February 2011
...d by curves; the closure condition implies that if $A,B,C,D$ lie on curves of the third family, then so do $E$ and $F$.
...t form as a system of defining equations, regarding $z$ as the "product" of $x$ and $y$: $z=xy$,
2 KB (385 words) - 09:42, 26 March 2023
...nalysis|discrete analysis]] sometimes serves as a synonym for the concepts of "finite mathematics" and "discrete mathematics" .
1,014 bytes (137 words) - 11:26, 28 June 2014
A collection $\mathfrak G$, respectively $\mathfrak F$, of subsets of a set $X$, satisfying the following properties:
$1$. The set $X$, as well as the empty set $\emptyset$, are elements of the collection $\mathfrak G$, respectively $\mathfrak F$.
1 KB (214 words) - 06:39, 13 October 2014
...f a differential-functional equation is often understood to be the synonym of an ordinary differential equation with deviating or distributed arguments (
Instead of differential-functional equation one often uses the phrase functional-diffe
1 KB (145 words) - 19:04, 31 March 2017