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$#A+1 = 34 n = 2 $#C+1 = 34 : ~/encyclopedia/old_files/data/T092/T.0902690 Thom spectrum

$#A+1 = 20 n = 0 $#C+1 = 20 : ~/encyclopedia/old_files/data/S086/S.0806600 Spectrum of an element

Out of 22 formulas, 22 were replaced by TEX code.--> ...t of all isolated eigenvalues with finite multiplicity. H. Weyl proved (in a special case) the following result [[#References|[a4]]], which is now commo

$#A+1 = 28 n = 0 $#C+1 = 28 : ~/encyclopedia/old_files/data/L057/L.0507920 Lebesgue spectrum

$#A+1 = 62 n = 0 $#C+1 = 62 : ~/encyclopedia/old_files/data/S086/S.0806610 Spectrum of an operator

...ariants and spectral properties of the corresponding group (or semi-group) of unitary (isometric) shift operators: ...perator $A$ that is the infinitesimal generator of the one-parameter group of unitary operators $\{U_t\}$ (here $U_t=e^{itA}$, by Stone's theorem).

Out of 27 formulas, 27 were replaced by TEX code.--> Let $X$ be a CW-spectrum (see [[Spectrum of spaces|Spectrum of spaces]]) and consider

...Open-closed set|open-and-closed subset]]s of its [[Spectrum of an operator|spectrum]] $\sigma(T)$ by the formula ...n algebras of sets is one of the basic problems in the [[spectral theory]] of linear operators.

$#A+1 = 100 n = 0 $#C+1 = 100 : ~/encyclopedia/old_files/data/S110/S.1100230 Spectrum generating algebra,

$#A+1 = 16 n = 3 $#C+1 = 16 : ~/encyclopedia/old_files/data/B110/B.1100930 Brown\ANDPeterson spectrum

$#A+1 = 37 n = 1 $#C+1 = 37 : ~/encyclopedia/old_files/data/Q076/Q.0706460 Quasi\AAhdiscrete spectrum

Out of 62 formulas, 62 were replaced by TEX code.--> ...ces $\text{ker } T$ and $\operatorname{coker}T$ have finite dimension. For a semi-Fredholm operator $T$ one defines the index

''of an element of a Banach algebra'' ...ment]]). The spectral radius of an element $a$ is connected with the norms of its powers by the formula

...periodic $A(t)$ and $f(t)$ have also been solved in terms of the spectrum of the monodromy operator. ...ative theory of differential equations in Banach spaces|Qualitative theory of differential equations in Banach spaces]].

$#A+1 = 81 n = 2 $#C+1 = 81 : ~/encyclopedia/old_files/data/S086/S.0806620 Spectrum of spaces

$#A+1 = 28 n = 0 The graded algebra $ A _ {p} $

''spectral analysis of a time series'' ...f. [[Spectral decomposition of a random function|Spectral decomposition of a random function]]).

A mapping $U$ of a metric space $(X,\rho_X)$ into a metric space $(Y,\rho_Y)$ such that ...and $Y$ are real normed linear spaces, $U(X)=Y$ and $U(0)=0$, then $U$ is a linear operator.

...s $ (F_{\alpha} \mid \alpha \in \mathfrak{A}) $ ($ \mathfrak{A} $ is a set of indices), i.e. elements $ F_{\alpha} \in \Phi' $ such that for any $ \phi \ ...A \phi) = \lambda_{\alpha} {F_{\alpha}}(\phi), \qquad \alpha \in \mathfrak{A},

$#A+1 = 34 n = 0 with a periodic function $ p ( z) $;

''of a normal topological space $X$'' ...e n$. If $\mathrm{locdim}(X) \le n$ for some $n$, then the local dimension of $X$ is finite, so one writes $\mathrm{locdim}(X) < \infty$ and puts

$#A+1 = 119 n = 0 $#C+1 = 119 : ~/encyclopedia/old_files/data/P075/P.0705150 Projection spectrum

...integral operator on the space of functions which are square-summable on $[a,b]$ is A non-linear Volterra integral operator is an operator of the form

...\operatorname{Hilb}(P_k^n/k)$ is in one-to-one correspondence with the set of closed subvarieties in $P_k^n$. ...ojective over $S$. The scheme $\operatorname{Hilb}(X/S)$ is the direct sum of the schemes $\operatorname{Hilb}^P(X/S)$ over all $P\in\mathbf Q(z)$. For a

...operator]], then so is $B$; if $A$ and $B$ are bounded operators, then $\|A\|=\|B\|$. ...cient condition for the unitary equivalence of operators with a pure point spectrum.

$#A+1 = 36 n = 0 A topological space $ M $

$#A+1 = 106 n = 0 $#C+1 = 106 : ~/encyclopedia/old_files/data/M062/M.0602540 Markov spectrum problem

...l operators is one of the few well-developed spectral theories for a class of non-self-adjoint operators. For the latter, see [[#References|[a3]]]. ...r–Shaw inequality: If $T$ is a hyponormal operator (i.e. $[T^*,T]\geq 0$), of finite rational cyclicity $r(T)$, then:

Out of 30 formulas, 30 were replaced by TEX code.--> ...all infinitely smooth functions with compact support in $\Omega$. This is a symmetric operator, and the associated quadratic form (with the same domain

Out of 60 formulas, 60 were replaced by TEX code.--> ...]] and [[#References|[a4]]]), in studying homotopy classes of mappings out of various classifying spaces ([[#References|[a3]]], [[#References|[a10]]] and

$#A+1 = 29 n = 0 A quantum-mechanical system consisting of a nucleus of mass $ M $

...rom an automorphism $S$ of a [[Measure space|measure space]] $(X,\nu)$ and a measurable function $f$ (defined on $X$ and taking positive values)'' ...s measure. The motion takes place in such a way that the second coordinate of $(x,s)$ increases with unit speed until it reaches the value $f(x)$, and th

Out of 57 formulas, 56 were replaced by TEX code.--> ...gether with a strong excision axiom (i.e. $p : ( X , A ) \rightarrow ( X / A , * )$ induces an isomorphism in homology).

$#A+1 = 59 n = 1 ...e operator on a Hilbert space is a [[Linear operator|linear operator]] $ A $

$#A+1 = 52 n = 0 ...this family contained in such subspaces. More precisely, let $ {\mathcal A} $

$#A+1 = 47 n = 0 ''(in group theory)''

$#A+1 = 125 n = 0 ...the spectrum and its main parts, spectral multiplicity and the asymptotics of eigenvalues.

$#A+1 = 52 n = 0 A mapping $ A $

$#A+1 = 36 n = 0 ...6 : ~/encyclopedia/old_files/data/S086/S.0806350 Spectral decomposition of a linear operator

...tal in solid state quantum physics, this conjecture becomes in mathematics a problem in [[Spectral theory|spectral theory]] for a Schrödinger operator (cf. also

Out of 88 formulas, 84 were replaced by TEX code.--> ''BDF theory''

$#A+1 = 24 n = 0 of a discrete-time stationary stochastic process such that 1) the first $ q $

A complete sup-lattice $Q$ together with an associative product $\otimes$ sat a \otimes \left({ \bigvee_i b_i }\right) = \bigvee_i a \otimes b_i

$#A+1 = 98 n = 2 A [[Hilbert space with an indefinite metric|Hilbert space with an indefinite

...tive group scheme is a group scheme $G$ over a basis scheme $S$, the value of which on any ...tori]]. A generalization of algebraic tori in the framework of the theory of group schemes is the following

$#A+1 = 33 n = 0 + ( a + b \cos 2z) u = 0,\ \

$#A+1 = 13 n = 0 A [[Linear operator|linear operator]] $ U $

$#A+1 = 32 n = 6 ...ce and one time dimension [[#References|[a11]]]. It is defined in terms of a two-component quantized fermion field $ \Psi _ {k} ( x,t ) $(

Out of 73 formulas, 72 were replaced by TEX code.--> ...and with spectral lines present in the light originating in a distant part of the Universe.

$#A+1 = 91 n = 0 ...of a certain function algebra $ A $ (cf. [[Algebra of functions|Algebra of functions]]) into the algebra $ L ( X) $

$#A+1 = 132 n = 0 ...e introduction of a canonical structure. M. Sato has constructed the sheaf of micro-functions on the cotangent sphere bundle $ S ^ {*} M $

$#A+1 = 29 n = 0 ...ually, it is this probabilistic branch which has made morphology into such a powerful methodology. Other main references are [[#References|[a4]]], [[#Re

...isconnected spaces). Examples of $Y$ include a discrete topological space, a rational straight line, analytic manifolds, and Lie groups over fields with ...themselves as dynamic characteristics of a physical system, in the models of which the spatial elements are provided with non-commutative binary operati

Out of 227 formulas, 223 were replaced by TEX code.--> ...s in the spectral theory of operators (cf. also [[Spectral theory|Spectral theory]]).

A ...nected by the relation $\norm{x^* x} = \norm{x}^2$ for any element $x \in A$. $\Cstar\!$-algebras were introduced in 1943

Out of 22 formulas, 22 were replaced by TEX code.--> ''Riesz theory of compact operators''

Out of 3 formulas, 3 were replaced by TEX code.--> A [[Measurable flow|measurable flow]] ( $ K $-

Out of 168 formulas, 167 were replaced by TEX code.--> ...k } ) _ { j , k \geq 0 }$. Then $T$ is bounded if and only if there exists a function $\phi \in L ^ { \infty }$ on the unit circle $\bf T$ such that

$#A+1 = 179 n = 0 $#C+1 = 179 : ~/encyclopedia/old_files/data/A011/A.0101370 Algebra of functions,

$#A+1 = 38 n = 0 A periodic variation with time of a physical magnitude, which may be written in analytical form as

$#A+1 = 263 n = 0 ...of special functors from the category of pairs of spaces into the category of graded Abelian groups.

Out of 82 formulas, 82 were replaced by TEX code.--> ...y. $T = T _ { 0 } \otimes T _ { 1 }$ is called the canonical decomposition of $T$.

Out of 192 formulas, 188 were replaced by TEX code.--> ...ows that $E _ { i } ^ { * } \xi = \xi ^ { \prime }$. Indeed, each $E_i$ is a partial isometry, satisfying $E _ { i } ^ { * } E _ { j } + E _ { j } E _ {

...em. The differential equation of motion of a linear system with one degree of freedom in an environment with viscous friction and with harmonic external $$a\ddot q+b\dot q+cq=H\sin(pt+\delta),$$

$#A+1 = 106 n = 0 An iterative algorithm for finding a solution to a linear equation

$#A+1 = 67 n = 1 A manner for uniquely finding solutions to equations analogous to the [[Helmh

Out of 43 formulas, 42 were replaced by TEX code.--> ...0$ and the multiplicity of the zero-eigenvector of $M$ is one (uniqueness of the vacuum).

$#A+1 = 38 n = 0 ...he case of isotropic scattering of quanta, diffusing without absorption in a stellar atmosphere.

Out of 50 formulas, 48 were replaced by TEX code.--> ...ry homotopy theory for CW-spaces (cf. also [[Cohomology]]; [[Cohomology of a complex]]; [[CW-complex|CW-complex]]).

$#A+1 = 92 n = 0 ...with a real symmetric kernel (cf. [[Kernel of an integral operator|Kernel of an integral operator]]):

$#A+1 = 54 n = 0 ''of a dynamical system''

$#A+1 = 65 n = 0 A topological space associated with a vector (or sphere) bundle or spherical fibration.

Out of 75 formulas, 75 were replaced by TEX code.--> ...anach spaces, and Banach and Fréchet algebras. Throughout, $E$ will denote a complex [[Banach space|Banach space]] with open unit ball $B_E$.

.... It satisfies the trace condition: $\mathrm{res}(AB) = \mathrm{res}(BA)$. A very important property is that it can be expressed as an integral (local f \mathrm{res} A = \int_{S^*M} \tr a_{-n}(x,\xi) d\xi

$#A+1 = 23 n = 0 $#C+1 = 23 : ~/encyclopedia/old_files/data/R077/R.0707070 Radiative transfer theory

Out of 53 formulas, 53 were replaced by TEX code.--> ...applications in commutative algebra, algebraic geometry and algebraic $K$-theory; see [[#References|[a5]]] and [[#References|[a6]]].

Out of 174 formulas, 170 were replaced by TEX code.--> ...n applies here, mutatis-mutandis, [[#References|[a7]]]. Thus, each element of $\mathcal{X}$ may be reconstructed from its spectral parts just as ordinary

A [[Self-adjoint operator|self-adjoint operator]] generated by a differential expression x \in (a, b),

$#A+1 = 43 n = 0 A bounded [[Linear operator|linear operator]] $ A $

$#A+1 = 74 n = 0 A commutative local Noetherian ring $ A $,

...erator and in the study of its spectrum, which, in this case, has a number of ‘good’ properties. Examples of compact operators are the Fredholm [[Integral operator|integral operator]]

$#A+1 = 17 n = 0 A [[Linear operator|linear operator]] $ A $

$#A+1 = 220 n = 0 $#C+1 = 220 : ~/encyclopedia/old_files/data/S086/S.0806520 Spectral theory

$#A+1 = 71 n = 4 ...m from the [[Stable homotopy group|stable homotopy group]] of the spectrum of $ \mathop{\rm SO} $

A system with one degree of freedom whose oscillations are described by the equation ...$T=2\pi/\omega$, does not depend on the amplitude. The potential energy of a harmonic oscillator depends quadratically on $x$:

$#A+1 = 270 n = 2 $#C+1 = 270 : ~/encyclopedia/old_files/data/S086/S.0806530 Spectral theory of differential operators

$#A+1 = 332 n = 0 $#C+1 = 332 : ~/encyclopedia/old_files/data/Q076/Q.0706250 Qualitative theory of differential equations in Banach spaces

Out of 100 formulas, 100 were replaced by TEX code.--> ...the subspace corresponding to the continuous spectrum into itself, and, as a consequence, the model becomes solvable in the sense described below.

Out of 196 formulas, 195 were replaced by TEX code.--> ...lf-adjoint representations of Lie algebras (or non-unitary representations of discrete groups).

...clopediaofmath.org/legacyimages/l/l059/l059340/l0593404.png" />, is called a linear operator from <img align="absmiddle" border="0" src="https://www.enc ...ce is fundamental in linear algebra, plays a role in very diverse branches of mathematics and physics, above all in analysis and its applications.

''of a topological space'' ...ra are mapped into arbitrary spaces, while the latter is based on mappings of arbitrary spaces into polyhedra, and is especially useful whenever the appl

$#A+1 = 86 n = 0 .... de Vries [[#References|[1]]] to describe wave propagation on the surface of shallow water. It can be interpreted using the inverse-scattering method, w

$#A+1 = 43 n = 0 ''of a linear (dissipative) operator $ A $

Out of 95 formulas, 95 were replaced by TEX code.--> ...ntation]] on Bergman space, cf. also [[Bergman spaces|Bergman spaces]], is of special interest in [[Harmonic analysis|harmonic analysis]]).

$#A+1 = 51 n = 0 $#C+1 = 51 : ~/encyclopedia/old_files/data/R081/R.0801590 Resolution of the identity

$#A+1 = 27 n = 0 ''of eigen values''

$#A+1 = 30 n = 0 defined on the (closed) [[Linear closure|linear span]] of a basis $ \{ e _ {k} \} _ {k \geq 1 } $

$#A+1 = 101 n = 2 ...nce of restrictions imposed on the dimensions of spaces) for the analogues of ordinary homotopy and cohomotopy groups in the suspension category — for

A second-order linear ordinary differential equation ...ume complex values. The substitution $y=ux^{-1/2}$ yields the reduced form of equation \eqref{(1)}:

$#A+1 = 29 n = 0 A compact invariant set $ L $

A branch of algebra, dealing mainly with the study of the so-called $K$-functors ($K_0, K_1$, etc., cf. ...ra appear in topology. A vector space can be regarded as a special case of a

Out of 33 formulas, 33 were replaced by TEX code.--> ...la is called the Riesz integral representation and Riesz representation of a superharmonic function

$#A+1 = 85 n = 1 A [[Semi-group|semi-group]] $ e ^ {- tH } $

$#A+1 = 71 n = 0 ...[#References|[1]]], Appendix A). The modern formulation is as follows. Let a function $ f(z) $,

$#A+1 = 38 n = 0 A recursive algorithm realizing a sequence of point-to-set mappings $ A _ {k} : V \rightarrow V $,

$#A+1 = 201 n = 0 .../old_files/data/F042/F.0402070 Functional equation, methods of solution of a

Out of 28 formulas, 28 were replaced by TEX code.--> ...es|[a8]]] (see also [[Spectral theory of compact operators|Spectral theory of compact operators]]).

Out of 98 formulas, 97 were replaced by TEX code.--> ''colligation of operators, node of operators, operator node''

$#A+1 = 49 n = 0 ...ode in which words are encoded letter by letter, and words in the alphabet of three symbols $ \{ \cdot , - , \wedge \} $(

$#A+1 = 394 n = 0 ...encyclopedia/old_files/data/L059/L.0509160 Linear differential equation in a Banach space

Out of 75 formulas, 73 were replaced by TEX code.--> Let $a$ be a complex-valued function defined on the complex unit circle $\bf T$, with [[

A ''scheme'' is a ...nsists of a topological space $X$ (the underlying space of the scheme) and a

Out of 76 formulas, 76 were replaced by TEX code.--> ...trami equation]]) and, more generally, second-order self-adjoint operators of Laplace type on $C ^ { \infty } ( E )$, that is, with leading symbol given

...rems. For defining the stability concepts one must rely on the second type of boundary value problem. ...ctional-differential equation this compactness condition is not necessary. A Lipschitz-type condition will assure the existence and uniqueness even for

Out of 64 formulas, 64 were replaced by TEX code.--> ...= ( V ^ { + } , V ^ { - } )$ (i.e. a pair $V = ( V ^ { + } , V ^ { - } )$ of real or complex vector spaces together with trilinear mappings $V ^ { \pm }

Out of 93 formulas, 91 were replaced by TEX code.--> ...ctral analysis]] of the [[Schrödinger equation|Schrödinger equation]] with a periodic potential. In time, this method has been very well-developed and a

$#A+1 = 79 n = 0 $#C+1 = 79 : ~/encyclopedia/old_files/data/E035/E.0305170 Eigen values of integral operators, numerical methods

$#A+1 = 57 n = 1 ...7 : ~/encyclopedia/old_files/data/D032/D.0302070 Differential equations on a torus,

Out of 50 formulas, 50 were replaced by TEX code.--> ...number field $K$ can be generated by Pisot numbers, even by Pisot units ($a _ { n } = 1$).

...c equation determines the distribution function of the dynamical states of a single particle, which usually depends on time, coordinates and velocity. ...mann equation if one uses a power series expansion in terms of the density of the gas.

...ewicz [[#References|[1]]] in the context of problems on the classification of continuous mappings. Homotopy groups are defined for any $ n \geq 1 $ . ...y type|Homotopy type]]), to a greater or lesser extent, to the computation of certain homotopy groups.

$#A+1 = 57 n = 0 ...tion, to which the implicit function theorem applies. For a local ring $ A $

$#A+1 = 43 n = 0 $#C+1 = 43 : ~/encyclopedia/old_files/data/F040/F.0400640 Floquet theory

$#A+1 = 540 n = 4 ''cobordism theory''

$#A+1 = 129 n = 0 $#C+1 = 129 : ~/encyclopedia/old_files/data/C023/C.0203840 Complete problem of eigen values,

$#A+1 = 11 n = 1 ...s, an additive set function defined on the algebra of all cylinder subsets of the function space such that its restriction to any $ \sigma $-

$#A+1 = 51 n = 3 A group scheme that is finite and flat over the ground scheme. If $ G $

Out of 3 formulas, 3 were replaced by TEX code.--> be a complex linear space on which a Hermitian sesquilinear form $ [ \cdot , \cdot ] $

$#A+1 = 27 n = 3 ...e theorem]], proved in 1961, gives necessary and sufficient conditions for a [[Multiplicative arithmetic function|multiplicative arithmetic function]]

$#A+1 = 75 n = 0 ...files/data/D032/D.0302060 Differential equations, infinite\AAhorder system of,

$#A+1 = 43 n = 3 is a bounded regular [[Analytic function|analytic function]] in the unit disc $

$#A+1 = 55 n = 1 ...of group schemes (cf. [[Group scheme|Group scheme]]) with a finite kernel. A morphism $ f: G \rightarrow G _ {1} $

Out of 158 formulas, 158 were replaced by TEX code.--> ...peratorname {Hol}( \mathcal{D} )$ the set of all holomorphic self-mappings of $\mathcal{D}$ (cf. also [[Analytic function|Analytic function]]).

$#A+1 = 298 n = 0 $#C+1 = 298 : ~/encyclopedia/old_files/data/G044/G.0404350 Geometry of numbers,

...he original proof appeared in [[#References|[a7]]], a proof in the context of finite group cohomology appears in [[#References|[a1]]]. ...\textbf{Z}/p)^I$ for some indexing set $I$, where $\textbf{Z}/p$ is cyclic of order $p$). Then Serre's theorem asserts that there exist non-trivial $\mod

$\newcommand{\calA}{\mathcal{A}}$ A sequence appearing in numerous fields and discovered and rediscovered in di

$#A+1 = 440 n = 0 A simplicial object in the category of sets $ \mathop{\rm Ens} $(

$#A+1 = 25 n = 0 ...s|ill-posed problems]] that are stable with respect to small perturbations of the initial data (see also [[Regularization method|Regularization method]])

$#A+1 = 70 n = 3 ...neralization of the concept of a commutative [[Formal group|formal group]] of finite height. The homomorphism induced by multiplication by the prime numb

$#A+1 = 194 n = 0 A [[stochastic process]] $ X( t) $

$#A+1 = 53 n = 0 ...es is very important. An analogous role is played by the representation of a function $ f $

...analysis — the theory of Fourier series and Fourier integrals — underwent a rapid development, stimulated by physical problems, in the 18th century and .../h0464201.png" /> be a complex-valued square-summable function on a circle of unit length (or on the segment <img align="absmiddle" border="0" src="https

''metric theory of dynamical systems'' [[Category:Ergodic theory]]

$#A+1 = 50 n = 0 A Room square of side $ n $,

$#A+1 = 38 n = 0 ...nly if it is completely regular (see [[Regular element|Regular element]]). A semi-group $ S $

$#A+1 = 239 n = 2 ...n with the theory of nuclear spaces (cf. [[Nuclear space|Nuclear space]]), A. Grothendieck carried over the concept to locally convex spaces (see [[#Ref

$#A+1 = 91 n = 0 ...= 91 : ~/encyclopedia/old_files/data/A110/A.1100400 Adjoint semi\AAhgroup of operators

$#A+1 = 140 n = 0 ...0 : ~/encyclopedia/old_files/data/S086/S.0806360 Spectral decomposition of a random function,

$#A+1 = 125 n = 4 ...ttering, and others. Certain self-adjoint problems, in which by separation of variables an operator-valued function $ {\mathcal L} ( \lambda ) $

$#A+1 = 33 n = 10 A mapping $ {q _ {N} } : X \rightarrow {X _ {N} ^ {+} } $

$#A+1 = 76 n = 0 ''of a system of ordinary differential equations

...ematical interpretations, enabling one to use methods from different areas of mathematics. ...sidered from the point of view of $ C^{\ast} $-algebras and representation theory.

...er external effects are also considered in shell theory, for example, that of heat. ...the non-linear terms may be neglected. The problem reduces to the solution of the linear system

$#A+1 = 203 n = 1 A [[Hilbert space|Hilbert space]] $ E $

$#A+1 = 88 n = 0 ...ing a generalized manifold) with coefficients in a locally constant system of groups (modules) $ {\mathcal G} $,

Out of 48 formulas, 48 were replaced by TEX code.--> ...hogonal array|orthogonal array]] of strength $t = 2$. For detailed studies of nets, transversal designs and orthogonal arrays, see [[#References|[a1]]] a

Out of 145 formulas, 145 were replaced by TEX code.--> ''Fermi–Thomas theory''

$#A+1 = 118 n = 1 ...118 : ~/encyclopedia/old_files/data/B017/B.0107290 Boundary (in the theory of uniform algebras)

Out of 32 formulas, 32 were replaced by TEX code.--> ...Darboux transformation is functionally parametrized by a pair of solutions of the equation and the transform vanishes if the solutions coincide.

$#A+1 = 244 n = 1 ...nsion of a topological space]] which is a [[Compact space|compact space]]. A compactification exists for any topological space, and any $ T _ {1} $-

$#A+1 = 236 n = 0 One of the [[Interpolation|interpolation]] problems in the real or complex domain.

$#A+1 = 145 n = 5 A problem in which it is required to reconstruct a function (a potential) $ q $

A natural generalization of the concept of the [[Picard variety]] $ \mathfrak P (X) $ for a smooth [[algebraic variety]] $ X $

$#A+1 = 70 n = 0 .... For this reason the Virasoro algebra plays a key role in conformal field theory.

Out of 187 formulas, 184 were replaced by TEX code.--> One of the [[Interpolation|interpolation]] problems in the complex domain.

Out of 213 formulas, 204 were replaced by TEX code.--> ..._ { K } = \operatorname { max } _ { z \in K } | P ( z ) |$ be the sup-norm of $P \in \mathcal{P}$ on $K$. The set

Out of 39 formulas, 39 were replaced by TEX code.--> ...statistics and time-series analysis, and also offers a very wide spectrum of applications.

Out of 75 formulas, 73 were replaced by TEX code.--> ...proper quotients). There are two (non-equivalent) ways to define the class of branch groups.

$#A+1 = 107 n = 1 $#C+1 = 107 : ~/encyclopedia/old_files/data/A110/A.1100660 Arens multiplication,

$#A+1 = 187 n = 0 ...gral equations of the second kind, amounting to performing a finite number of operations on numbers.

$#A+1 = 366 n = 3 ...tors into others (most often — into themselves). By a cohomology operation of type $ ( n , m ; \pi , G ) $,

$#A+1 = 221 n = 1 ''of a [[Topological group|topological group]]''

Out of 73 formulas, 73 were replaced by TEX code.--> ...e closely connected to important fields of geometric analysis, e.g., index theory, geometric quantization and several complex variables.

...number of degrees of freedom. Quantum field theory unifies the description of fields and particles, which, in classical physics, appear as two distinct e ...the example of an electromagnetic field, as this is the only field having a clear content, both in the classical and the quantum case.

$#A+1 = 85 n = 0 ...in general, finite-dimensional extremal problems, based on optimization of a functional on finite-dimensional subspaces or manifolds.

$#A+1 = 219 n = 0 The part of modern mathematical analysis in which the basic purpose is to study functio

...ncepts of general topology are the concepts of a [[topological space]] and a [[continuous mapping]], introduced by F. Hausdorff in 1914. ...the space, in some way related to its topology. The topological invariance of the homology groups may serve as an example.

...rst-order sentences true in a given structure forms a complete first-order theory $ T $. For example, the models of the theory of the complex field $ ( \mathbf C ,\ +,\ \cdot ,\ 0,\ 1) $

Out of 62 formulas, 62 were replaced by TEX code.--> ...(cf. also [[Block design|Block design]]) of size $k$ (that is, $k$-subsets of the point set), such that any two distinct points are on precisely $\lambda

Out of 163 formulas, 163 were replaced by TEX code.--> ..., deals with wave scattering by obstacles when $S$ is smooth, for example, a $C^{ 1 , \lambda }$-surface, $0 &lt; \lambda \leq 1$. (See [[#References|[

$#A+1 = 80 n = 1 A [[Semi-group|semi-group]] in which every element is regular (see [[Regular

Out of 47 formulas, 46 were replaced by TEX code.--> ...the vacuum state there exist subspaces associated with one-particle states of mass zero.

$#A+1 = 481 n = 1 $#C+1 = 481 : ~/encyclopedia/old_files/data/P110/P.1100120 Pesin theory

$#A+1 = 434 n = 3 ...a kind of problems involved in the study of inequalities between the norms of the same function in different classes (normed spaces). One is usually conc

$#A+1 = 134 n = 1 A [[characteristic class]] defined for complex vector bundles. A Chern class of the complex vector bundle $ \xi $

$#A+1 = 115 n = 2 ''for a matrix eigen value problem''

$#A+1 = 113 n = 0 ''of a solution of a linear system''

$#A+1 = 84 n = 0 $#C+1 = 84 : ~/encyclopedia/old_files/data/A110/A.1100500 Algebraic Diophantine equations

$#A+1 = 60 n = 2 A relativistically-invariant equation in two space-time variables, of the form

Out of 71 formulas, 65 were replaced by TEX code.--> ...lly extended systems, and as a paradigm for finite-dimensional dynamics in a partial differential equation.

Out of 170 formulas, 166 were replaced by TEX code.--> ...ntinuous complex-valued functions on $G$ vanishing at infinity. For $u \in A _ { p } ( G )$ one sets

Out of 202 formulas, 199 were replaced by TEX code.--> ...analysis, abstract]]). They play an important role in the duality theories of these groups.

Out of 227 formulas, 227 were replaced by TEX code.--> ...rier hyperfunctions was proposed by M. Sato at the same time as his theory of hyperfunctions, in which the transformed objects can be interpreted natural

Out of 113 formulas, 113 were replaced by TEX code.--> ...a|Jordan algebra]] over the field of real or complex numbers, endowed with a complete norm $|.|$ satisfying

$#A+1 = 44 n = 0 ...4 : ~/encyclopedia/old_files/data/Q076/Q.0706380 Quasi\AAhaverages, method of

Out of 49 formulas, 49 were replaced by TEX code.--> The following meta-theorem: It is not possible for a non-trivial function and its [[Fourier transform|Fourier transform]] to be

$#A+1 = 199 n = 2 $#C+1 = 199 : ~/encyclopedia/old_files/data/E037/E.0307040 Extension of an operator

$#A+1 = 235 n = 0 ''logarithmic norm, norm on a field''

$#A+1 = 75 n = 0 ...else on a space dual to a space of this type. A differential expression is a mapping $ \lambda $

...s|[1]]] in 1936 in the context of the description of unramified extensions of $ p $- ...ces|[5]]], and in the theory of formal groups [[#References|[6]]]. Let $ A $

Out of 78 formulas, 77 were replaced by TEX code.--> ...tion $u \in C ^ { 2 } ( \Omega ) \cap C ^ { 0 } ( \overline { \Omega } )$ (a Dirichlet eigenfunction) satisfying the following Dirichlet boundary value

$#A+1 = 203 n = 1 A system

$#A+1 = 499 n = 0 ''of a topological space $ X $''

...ume of information processing that is involved. Therefore for the solution of these problems, alongside analytic methods, numerical simulation on compute ...ime scales of the processes being studied in the atmosphere and the method of describing them.

...hematics. In conjunction with algebra, topology forms a general foundation of mathematics, and promotes its unity. ...th of the class of geometric objects that fall within the sphere of action of its laws.

Out of 165 formulas, 161 were replaced by TEX code.--> ...operators in function theory; it has many applications in different fields of mathematics and applied mathematics.

A [[Topological algebra|topological algebra]] $A$ over the field of complex numbers whose topology is ...hich converts $A$ into a [[Banach space|Banach space]], the multiplication of the

Out of 153 formulas, 148 were replaced by TEX code.--> Inequalities concerning the negative eigenvalues of the Schrödinger operator (cf. also [[Schrödinger equation|Schrödinger eq

$#A+1 = 77 n = 3 ...of gravity, such that the equations of motion are integrable in the sense of Kowalewski. The latter means that the system admits solutions, expressible

$#A+1 = 9 n = 0 ...when choosing statistical models for vertical-heterogeneous random fields of turbulence and [[Internal waves|internal waves]] (cf. also [[Turbulence, ma

Out of 54 formulas, 54 were replaced by TEX code.--> ...ions (cf. also [[Evolution equation|Evolution equation]]). For this circle of ideas, see e.g. [[#References|[a3]]], [[#References|[a4]]], [[#References|[

$#A+1 = 17 n = 0 The branch of algebra studying the properties of commutative rings and objects relating to them (ideals, modules, valuations

$#A+1 = 202 n = 0 is a unit vector, $ \phi = \phi ( x, \Omega , v) $

$#A+1 = 123 n = 0 ...t case, a cybernetic system is reduced to a single element. An element $ A $

$#A+1 = 261 n = 0 A Banach algebra $ A $

$#A+1 = 138 n = 0 ...ble "perturbation theory" . The results obtained by using the two classes of methods complement each other.

A set together with a given transitive [[group action]]. More precisely, $ M $ is a homogeneous space with group $ G $

$#A+1 = 89 n = 0 .../encyclopedia/old_files/data/D034/D.0304260 Dynamic problems of elasticity theory

$#A+1 = 37 n = 0 ...onstructive trend in mathematics) is often applied in a more general sense of the word, referring to this Soviet approach to constructive mathematics. In

$#A+1 = 307 n = 0 A function related to integral representations of solutions of boundary value problems for differential equations.

$#A+1 = 302 n = 0 $#C+1 = 302 : ~/encyclopedia/old_files/data/A110/A.1100010 A priori and a posteriori bounds in matrix computations

$#A+1 = 302 n = 0 ...yclopedia/old_files/data/S087/S.0807450 Statistical problems in the theory of stochastic processes

$#A+1 = 34 n = 0 ...clopedia/old_files/data/A013/A.0103570 Astrophysics, mathematical problems of

A Drinfel'd $A$-module, (where $A$ is an appropriate [[Ring|ring]]) over a

$#A+1 = 263 n = 0 ...each point. These concepts of curvature are connected with the examination of deviations which are small to the second order; hence the object in questio

Out of 225 formulas, 223 were replaced by TEX code.--> ...be a finitely-generated $A$-module with $\dim_AM = s$. Then $M$ is called a Buchsbaum module if the difference

$#A+1 = 203 n = 1 ...nces|[2]]]) and in studying the singularities of the fundamental solutions of hyperbolic equations (see [[#References|[1]]], [[#References|[2]]], [[#Refe

...pyrighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribut Squares, W. J. Youden created many designs tailored to the needs of the

...Polish mathematician W. Sierpiński, about forty years after the discovery of the Cantor set. ...length of the original is removed. This process is then repeated with each of the remaining triangles ad infinitum (see Fig.a1).

$#A+1 = 469 n = 2 $#C+1 = 469 : ~/encyclopedia/old_files/data/S084/S.0804140 Semi\AAhgroup of operators

$#A+1 = 85 n = 0 ...= 85 : ~/encyclopedia/old_files/data/D032/D.0302180 Differential geometry of manifolds

$#A+1 = 144 n = 0 $#C+1 = 144 : ~/encyclopedia/old_files/data/P072/P.0702430 Perturbation theory

...is stable on the spaces $(Z,U)$, i.e.: For every $\epsilon > 0$ there is a $\delta(\epsilon) > 0$ such that for any $u_1, u_2 \in U$ it follows from ...to apply approximation methods to such problems. This put the expediency of studying ill-posed problems in doubt.

...on of 'quantum computation' exist and are still unsolved. Other properties of the quantum Turing machine like the question, whether universality does hol ===Quantum Aspects of Computation===

Out of 229 formulas, 217 were replaced by TEX code.--> ...y theory, isomonodromy deformations, quantum cohomology and [[K-theory|$K$-theory]], Gromov–Witten invariants, Witten–Dijkgraaf–Verlinde–Verlinde equ