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...gruence with respect to the variety $\mathfrak{M}$ generated by the factor system $\mathbf{F}/\eta$.

...[[Affine coordinate system|affine coordinate system]] is determined by the factor $x_i$ such that $x_iv_i$ is the parallel projection of the position vector

...r system. Then the cross product of $G$ and $K$ with respect to the factor system $\rho$ and the mapping $\sigma$ is the set of all formal finite sums of the In the defining relations for a factor system above $\rho^{\sigma(g_1)}_{g_2,g_3}$, e.g., of course stands for the result

$#C+1 = 51 : ~/encyclopedia/old_files/data/S090/S.0900890 Subgroup system ...respect to union and intersection is called complete. A complete subgroup system is called subnormal if for any jump $ A $

...t in an expression for the density (or for the density matrix in a quantum system) in a canonical Gibbs ensemble (cf. [[Gibbs statistical aggregate|Gibbs sta 1) In a classical system, the density of the Gibbs distribution $ p( \omega ) $,

...nding sequence of subgroups (see [[Subgroup series|Subgroup series]]) each factor of which is isomorphic to an infinite cyclic group.

root system and a certain sublattice in the weight lattice that out that the system $\Sigma\subset X(T)$ of all roots of $E$ is a

...ng physical (observable) quantities related to this system. In a classical system with phase space $ \Omega $, In a quantum system described by vectors in a Hilbert space $ {\mathcal H} $,

...stability zones. A Mathieu function is even or odd, and is unique up to a factor; the second linearly-independent solution grows linearly in $ z $ these functions reduce to the [[Trigonometric system|trigonometric system]]

.... In many cases the principle can be derived as a corollary from a certain system of axioms which represents properties that are natural to demand of any "r a factor not controllable by the consumer. The choice of concrete values $ x \in X

...that if $( X , \mathcal{F} , \mu , T )$ is a [[Dynamical system|dynamical system]], then given $f \in L ^ { 1 } ( \mu )$ one can find a set of full [[Measur ==Uniform Wiener–Wintner theorem and Kronecker factor.==

A relation that connects the [[Wronskian|Wronskian]] of a system of solutions and the coefficients of an ordinary linear differential equati be an arbitrary system of $ n $

...se in statistical physics in defining a Gibbs quantum state. Let a quantum system occupying a finite volume $ V $ A Gibbs state for such a system is a state on $ \mathfrak A( {\mathcal H} _ {V} ) $

A system of multi-parameter identities giving a connection between different charact ...in the lower half-plane. Such a representation is unique up to a constant factor (cf. [[Wiener–Hopf method|Wiener–Hopf method]]), which allows one to id

uniquely up to a constant factor and itself uniquely determines $ X $. th coordinate system is the coordinate system of the $ i $-

...tant speed on a rectangular table with a convex obstacle. The state of the system (the position and velocity of the ball), at one instant of time, can be des ...em: Any flow <math>f_t</math> that is not completely predictable, has as a factor <math>B_{ct}</math> for some <math>c>0</math> (the numbers <math>c</math> i

...n solved (see [[#References|[1]]]) for approximately-finite factors (cf. [[Factor]]) of type $\mathrm{II}$ and type $\mathrm{III}_\lambda$, $0 < \lambda < 1$ ...[[#References|[6]]] that generalizes the [[entropy]] of a metric dynamical system. The entropy of automorphisms of an arbitrary $W^*$-algebra with respect to

under the projection onto the second factor is a hypersurface in $ ( \widetilde{P} {} ^ {n} ) ^ {r+} 1 $ in each system of variables. The form $ F _ {X} $

In accordance with the decimal system, the most commonly used are decimal logarithms $(a=10)$, denoted by $\lg N$ ...se to another is carried out by the formula $\log_bN=\log_aN/\log_ab$; the factor $1/\log_ab$ is called the modulus of transition from the base $a$ to the ba

...f functions|Complete system of functions]]; [[Orthogonal system|Orthogonal system]]). In spectral methods, [[Trigonometric functions|trigonometric functions] ...hod. With other types of conditions, an extra equation may be added to the system to satisfy it, or the basis functions may be modified to automatically sati

''loss system'' However, in contrast to a system with an infinite number of channels here $ q _ {n} \leq m $,

''natural system, homotopic resolution, $P$-decomposition of general type'' ...–MacLane space]]), where $\pi_n$ is some group (Abelian for $n > 1$). This system was introduced by M.M. Postnikov

...vertex corresponds a system of equations, chosen in a special way from the system of inequalities \eqref{eq:2}, \eqref{eq:3}, so the computational procedure ...trices. At the heart of many of them lies the idea of decomposition of the system of restrictions into subsystems for each of which it is necessary to solve

...tions, then the product $\prod_{i\in I} X_i$ can be made into an algebraic system of the same signature: For functions $f_1,\ldots,f_k : I \rightarrow X$ and For an arbitrary factor of a direct product $X = \prod_{i\in I} X_i$ there exists a natural project

...gcd } ( a ^ { ( q ^ { i } - 1 ) / 2 } - 1 , f _ { i } )$ is a non-trivial factor of $f_i$. To describe the cost of these methods, one uses fast arithmetic, ...mbinations of the irreducible factors modulo $p ^ { k }$ to recover a true factor in $\mathbf{Q} [ x ]$. This works quite well in practice, at least for poly

...for investigating the singular points of an [[Autonomous system|autonomous system]] of second-order ordinary differential equations is a singular point of the system (1), that is, $ f ( O) = 0 $,

...e types|Voronoi lattice types]]) is non-trivial if and only if the scaling factor of the tiling is a Pisot number [[#References|[a4]]].

The construction of cross products with the aid of [[factor system]]s {{Cite|Ch}} results in a cohomological interpretation of Brauer groups. ...t]]; [[Extension of a group]]. The latter article contains the notion of a system of factors.

such that any factor of $ \mathfrak A _{i} $ one of which contains some variable as factor, while the other contains the negation of this variable (under the conditio

onto the factor $ X $, Having chosen a coordinate system (Cartesian, polar or any other coordinates), the numerical points $ ( x,

and the volumes of two similar bodies are proportional to the cube of the factor of the corresponding similarity. and let an affine coordinate system $ ( x ^ {1} \dots x ^ {n} ) $

define the same (quantum mechanical) state if they differ by a phase factor: $ \psi _ {2} = \mathop{\rm exp} ( ia) \psi _ {1} $. ...quantum measurements [[#References|[a1]]] overcomplete subsystems of this system indexed by a lattice in $ \mathbf C $

...able manifold is called a Lie pseudo-group of transformations defined by a system $ S $ that satisfy the system $ S $.

...s not complete in $H ^ { 2 } ( \mu )$ (cf. also [[Complete system|Complete system]]). ...2 } = \mu ^ { \prime } ( \theta )$. Therefore it is also called a spectral factor of the weight function $\mu ^ { \prime }$. Other asymptotic formulas were o

5) basic states, elementary excitations (in the case of a quantum system). ...udied. Statistical mechanics, depending on the method used to describe the system, is divided into classical and quantum mechanics.

the [[Root system|root system]] of $ G $ induced by left-multiplication on the first factor of $ G ( K ) \times A $.

3) a scalar factor can be taken out of the scalar product: is called the [[Minkowski space|Minkowski space]]. In any system of $ n $

Statistics used in a system of non-interacting particles obeying the laws of classical mechanics (a cla must be divided by a factor $ N ! $:

...take into account the effects of the influence of state degeneracy of the system. ...orresponds to a single energy level more than one independent state of the system; the average $ \langle A \rangle $

...results of the productive activity of economic systems at various levels; factor models of productivity; systems of demand functions for groups of consumers ...solution of problems of one class, as well as the tendency to formulate a system of recommendations by means of which the hypotheses concerning the models o

A linear dynamical input-output system used here is uniquely determined by the discount factor $ \lambda $,

$#C+1 = 106 : ~/encyclopedia/old_files/data/D110/D.1100230 Discrete event system ...t driven as opposed to time driven, i.e. the behaviour of a discrete event system is governed only by occurrences of different types of events over time rath

are called the roots of the root system $ R $. A root system $ R $

The solution to the system the error in the initial approximation can be reduced in norm by a factor $ \epsilon ^ {-1} $

...g so (s)he must take into consideration such factors as the purpose of the system, the conditions under which it is to operate, and also economic factors. A ...tical models in reliability theory. Assume that the state of an industrial system is defined by a point $ x $

...tions. This probability density can be expressed, apart from a normalizing factor, by means of a functional integral over paths of the process. The correspon

...ro divisors and let $f$ be a non-degenerate sesquilinear form. Then, for a system of elements $v_1,\dots,v_n$ from $E$ to be free it is necessary and suffici ...ined invariantly, while its discriminant is defined up to a multiplicative factor which is the square of a non-zero element of $F$. The algebra $A$ is separa

...number of places with bounded gaps. In other words, the symbolic dynamical system generated by the Thue–Morse sequence is minimal (see, for instance, [[Dynamical system|Dynamical system]]).

...its of measurement and has the form of a monomial. For example, in the CGS system the dimension formulas contain three arguments: the length unit $L$, the ti In the CGS system the formula for any quantity $N$ of a mechanical, thermal or electromagneti

...of which to the argument multiplies the value of the function by a certain factor. In other words, one has the identities (in $ z $): is the moduli system or system of periods and quasi-periods of $ \theta ( z) $.

...invariant]] of the Lie group. The latter can be found as a solution to the system of partial differential equations ...s, that is, groups that admit one and only one invariant (up to a constant factor). In particular, simple transitive groups are of this type.

...ider theory of integrable systems (cf. also [[Integrable system|Integrable system]]); some of the connections are explained in [[#References|[a3]]]. One can ...a loop group (in fact a central extension when the action on the conformal factor $\Omega$ is included).

in the data. Thus, the factor $ k ( A ) $, is the main factor affecting computational accuracy, since by setting $ \| {\delta A } \| \l

and leaving the third factor alone. Thus, $ R ^ {12} = R \otimes \mathop{\rm id} $, ...etely-integrable Hamiltonian systems (cf. [[Hamiltonian system|Hamiltonian system]]; [[Soliton|Soliton]]), cf. [[#References|[a12]]], and the results of [[#R

...#References|[a10]]] that it is necessary to assume that one can completely factor an arbitrary polynomial over $k$. ...f _ { 1 } , \dots , f _ { l } ) \in R ^ { l }$ of the related homogeneous system of equations

where the product is time ordered, i.e. a factor with a larger $t_j$ stands to the left of all those with smaller $t_j$; ...(see [[#References|[a7]]] for the details). The result can be applied to a system of quasi-linear partial differential equations

...of uniform and rectilinear motion relative to the first inertial reference system. ...l experiment alone. A moving reference system is obtained from a reference system, taken to be at rest, by means of a coordinate transformation. It follows f

give a system of generators for $ G ( k) $ in the group system. In the case $ \mu = 1 $

...pe of interaction with the medium, but to allow for absorption by a weight factor according to the rule $ \rho _ {n} = q \rho _ {n-} 1 $. ...n with increased mean run and by compensating this by an extra exponential factor in the weight.

...system with voice and data traffic demanding transmission; a manufacturing system with several work stations; patients arriving at a doctor's office; vehicle ...ith a probabilistic model (also known as a stochastic model) of a queueing system. If we analyze such models using mathematical techniques we can derive its

...he error of a splitting iterative method is reduced by each iteration by a factor bounded by $ \rho ( M ^ {- 1 } N ) $. in (a1) by some preconditioning matrix so that the transformed system $ {\widetilde{A} } {\widetilde{x} } = {\widetilde{b} } $

...likely that this will have an interpretation as the entropy of a dynamical system but this remains as of yet (1998) a future development. ...td> <td valign="top"> D.W. Boyd, "Two sharp inequalities for the norm of a factor of a polynomial" ''Mathematika'' , '''39''' (1992) pp. 341–349 {{MR|12032

...at the inner-outer factorization $f = \theta . g$ of $f$. Here, the right factor $g$ is an outer function; such functions are characterized operator-theoret ...fer function $\theta ( z ) = d + c z ( I - z A ) ^ { - 1 } b$ of a unitary system

are given and the factor $ k $ remains undetermined before the observations are taken, then this factor and the variance of $ \overline{Y}\; $

..._ { 2 } - \lambda _ { 2 } \sigma _ { 1 } + \tilde { \gamma }$, called (for system-theoretic reasons, see below) the input and the output determinantal repres ...near difference and differential equations alluded to above. Actually, the system can even be solved explicitly using theta-functions, yielding explicit form

..., $A$ has the Artin approximation property if and only if for every finite system of polynomial equations $f$ over $A$ the set of its solutions in $A$ is den ...over $A$ having a solution in $A ^ { \prime }$ can be enlarged to a finite system of polynomial equations over $A$ having a solution in $A ^ { \prime }$, for

of the displacement vector, if a Cartesian coordinate system is chosen so that the $ xy $- In the case of a system of $ N $

...usually equivalent to considering a new measure for the first algorithmic system. ...nsignificant increase in space complexity (usually, an additional constant factor).

...e maximal submodule of $\Delta ( \lambda )$. The corresponding irreducible factor module is denoted by $L ( \lambda )$. It turns out that the set of all $L ( ...oint action of $\mathfrak h $ on $\frak g$ determines a [[Root system|root system]] $\Phi$ with basis $\Sigma$ and, corresponding to the choice of $\Sigma$,

...corresponding line elements on the surfaces. The choice of the coordinate system is governed (apart from the simplicity of the description of the line eleme is the factor by which the ellipsoid has been reduced. The relative scales correspond to

The Mizohata operator can be obtained (up to a factor) by pulling back the standard Cauchy–Riemann operator on $\mathbf{C}$ via ...rty of the functions and distributions annihilated by a locally integrable system of complex vector fields" ''Ann. of Math.'' , '''113''' (1981) pp. 387–42

...function (the Lagrangian), and the conservation laws for the corresponding system of Euler–Lagrange equations indicates the omission of the corresponding factor), that is, an $ ( n - 1 ) $-

...on of both the spatial and the temporal aspects. As a pattern changes, the system should be able to measure and store this change. How can it do that? is a constant known factor. The learning session having a duration $ T $,

...effects: $\mu$ is the general mean, the $\alpha$s are the main effects for factor $\mathbf{A}$, the $\beta$s for $\operatorname{B}$, and the $\gamma$s are th ...out there are three types of two-factor interactions and one type of three-factor interactions. Layouts of this type are called factorial, and completely cro

...s of a computation procedure on the accuracy of the computed solution to a system of algebraic equations. The most commonly employed technique for a priori e ...satisfy (1), but it can be expressed as the exact solution of a perturbed system

used for modeling how a system moves from one state to another in time. Transitions between state are rand the move into a new state, given the current state of the system.

is a normalization factor: $ \int _ {0} ^ \infty | y( x , i \lambda _ {k} ) | ^ {2} dx = 1 $. ..."> N.I. Akhiezer, "A continuous analogue of orthogonal polynomials on a system of intervals" ''Soviet Math. Dokl.'' , '''2''' : 6 (1961) pp. 1409–14

runs over a system of neighbourhoods of the identity in $ G $; ...nder right translation. Moreover, in this case one may choose the constant factor such that $ \mu (G) = 1 $.

...m]]s of weight 4 and 6, respectively, that are identical, up to a constant factor, with the forms defined by [[Eisenstein series]] of lowest weights. In that is determined by the linear system of cubic curves passing through the 6 points. There are 27 straight lines o

...didi, J. Eliashberg and W.S. DeSarbo [[#References|[a10]]]. They model the system dynamics as: $e ^ { a }$ is the scale factor;

...gnized when P. Shor (1994) discovered that they can be used to efficiently factor large numbers — a problem believed to be hard for classical deterministic ...t of classical information is the bit, which can be understood as an ideal system in one of two states or configurations, usually denoted by $0$ and $1$. The

in such a case (1) is a linear hyperbolic system), or may depend on $ w = w ( x, t ) $( a quasi-linear system). In the latter case, let the system of equations (1) be reduced to divergence form:

is a contour (or system of contours) in the complex plane in which both variables $ z $ or given class of functions) is the determining factor from the point of view of the successful application of the method of integ

...ines are particular cases of control systems (cf. [[Control system|Control system]]). They arose in connection with the analysis of the concept of an algorit ...dels all simulate each other with polynomial overhead in time and constant factor overhead in space (provided space measures are defined correctly). As a con

...escription, which form a fairly extensive discipline with a well-developed system of ideas and techniques of calculation, were presented. Modes of statistica ...sses|statistical analysis of stochastic processes]]; and [[Factor analysis|factor analysis]]. New considerations in regression analysis have appeared (see al

...]]) in the form of conservation laws in an inertial orthonormal coordinate system: 7) Representation of a complicated system of linear or non-linear algebraic equations as recurrence relations (the me

An axiom system for higher cohomology operations is constructed in [[#References|[12]]]. called its $ n $-th Postnikov factors (see [[Postnikov system|Postnikov system]]). For example, for the sphere $ S ^ {n} $,

...d on computational number theory and is at most as secure as it is hard to factor large integers (but it could be weaker). Both hardware and software impleme ...en. In this context, it may be preferable to use a public key distribution system [[#References|[a6]]] instead of a public-key cryptosystem.

In the study of motion of liquids in which viscosity is a significant factor, a number of mathematical models — representing simplifications of the co in a liquid-filled space from the system of linear equations

...cription of the system under study using some specially selected "ideal" system which can be correctly and completely studied. One of the criteria of appli ...celestial mechanics, in the context of the motions of planets in the solar system. Since the planets are very remote from each other, and since their mass is

are non-negative integers. Every factor $ e _ {j} ^ {n _ {ij} } $ ...st natural normal form are of substantial importance in linear control and system theory [[#References|[a1]]], [[#References|[a2]]]. Here one studies systems

...number of parameters, and to find then the actual space implicitly using a system of equations ( "state equations" ), or to parametrize the space locally ( " are homeomorphisms in a system $ \Gamma $

...y began in the 1950s. The development of algebraic geometry was a decisive factor in this approach. The simultaneous study of number and function fields, whi ...he set of equations is large compared with the degree of the equation, the system usually has a solution. However, this is very difficult to prove for any sp

It is defined up to a factor by the formula it is called the Beltrami equation (or Beltrami system). For example, the problem of conformally mapping one domain $ D $

...ongruence relations of algebraic systems (cf. [[Algebraic system|Algebraic system]]) in a variety, the Arguesian law is equivalent to the modular law. and each subdirectly indecomposable factor of $ L $

...inear algebra (cf. [[Iteration methods|Iteration methods]]) applied to the system of linear equations resulting from a full discretization of Radon's integra ...m the standard Radon transform by the presence of an exponential weighting factor under the integral sign. For the use of the Radon transform in the area of

of groups containing a fundamental system of neighbourhoods of the unit element that are invariant under inner automo defined up to a positive factor, and the set of quasi-equivalence classes of trace-admissible quotient repr

...concentrated at one point or to a measure which is equal, up to a positive factor, to the direct product $ \prod _{ {i \in I}} (U _{i} ,\ {\mathcal U} _{i ...patible probability measures on each finite product is called a projective system of measure spaces, and the corresponding probability measure on $ \prod {

...burden, i.e. the number of flops required to solve an $n \times n$ linear system of equations, can become prohibitively large. This is one reason why one se ...diag } \{ d _ { 0 } , \dots , d _ { n - 1 } \}$, and the lower triangular factor $L$ is normalized in such a way that the $\{ d _ { i } \}$ appear on its ma

...suitable for obtaining invariants of geometric objects. Thus, the complete system of invariants of a Riemannian metric, found by E. Christoffel and R. Lipsch ...efined as a manifold with a Finsler metric. The Lagrangian of a mechanical system is an example of such a metric. Finsler manifolds also appear in optics in

...ath.org/legacyimages/c/c026/c026010/c026010188.png" /> is the state of the system and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.o ...aofmath.org/legacyimages/c/c026/c026010/c026010402.png" /> is the discount factor. The one-stage cost with terminal cost <img align="absmiddle" border="0" sr

...the Reynolds stresses. The simplest method for closing the Freedman–Keller system of equations is to represent the $ \tau _ {jl} $ ...larity becomes incomplete and the structure function acquires a correction factor $ ( r/L) ^ {m} $

...e [[Fan|fan]] theorem. A consequence of this is, e.g., that in the natural system of concepts every intuitionistic real-valued function defined on a closed s Another factor which has influenced the revival of interest in intuitionism, mentioned abo

Consider the system of equations ...valign="top"> X.L. Meng, S. Schilling, "Fitting full-information item factor models and an empirical investigation of bridge sampling" ''J. Amer. Stati

...of variable magnitudes into mathematics by R. Descartes was the principal factor in the creation of differential calculus. Differential and integral calculu be the plane curve defined in an orthogonal coordinate system by the equation $ y = f ( x) $

====Cauchy-Riemann system==== The system \eqref{e:del_bar} is equivalent to the system of $2n$ first-order partial differential equations

...ghest-order differentials and is independent of the choice of a coordinate system; cf. also [[Principal part of a differential operator|Principal part of a d ...$ being the [[Fundamental class|fundamental class]] of $T ^ { * } M$. (The factor $( - 1 ) ^ { n }$ reflects the choice of the [[Orientation|orientation]] of