$#A+1 = 34 n = 2
$#C+1 = 34 : ~/encyclopedia/old_files/data/T092/T.0902690 Thom spectrum
3 KB (438 words) - 08:25, 6 June 2020
$#A+1 = 20 n = 0
$#C+1 = 20 : ~/encyclopedia/old_files/data/S086/S.0806600 Spectrum of an element
3 KB (528 words) - 08:22, 6 June 2020
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
4 KB (564 words) - 17:00, 1 July 2020
$#A+1 = 28 n = 0
$#C+1 = 28 : ~/encyclopedia/old_files/data/L057/L.0507920 Lebesgue spectrum
3 KB (415 words) - 15:28, 28 February 2022
$#A+1 = 62 n = 0
$#C+1 = 62 : ~/encyclopedia/old_files/data/S086/S.0806610 Spectrum of an operator
5 KB (744 words) - 08:22, 6 June 2020
...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).
5 KB (710 words) - 12:58, 16 July 2014
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
3 KB (432 words) - 16:46, 1 July 2020
...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.
1 KB (174 words) - 18:24, 22 April 2016
$#A+1 = 100 n = 0
$#C+1 = 100 : ~/encyclopedia/old_files/data/S110/S.1100230 Spectrum generating algebra,
12 KB (1,660 words) - 08:22, 6 June 2020
$#A+1 = 16 n = 3
$#C+1 = 16 : ~/encyclopedia/old_files/data/B110/B.1100930 Brown\ANDPeterson spectrum
4 KB (633 words) - 08:47, 26 March 2023
$#A+1 = 37 n = 1
$#C+1 = 37 : ~/encyclopedia/old_files/data/Q076/Q.0706460 Quasi\AAhdiscrete spectrum
7 KB (978 words) - 05:56, 19 March 2022
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...ces $\text{ker } T$ and $\operatorname{coker}T$ have finite dimension. For a semi-Fredholm operator $T$ one defines the index
5 KB (812 words) - 17:00, 1 July 2020
''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
2 KB (345 words) - 14:46, 21 August 2014
...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]].
1 KB (219 words) - 17:03, 12 August 2014
$#A+1 = 81 n = 2
$#C+1 = 81 : ~/encyclopedia/old_files/data/S086/S.0806620 Spectrum of spaces
9 KB (1,281 words) - 14:40, 21 March 2022
$#A+1 = 28 n = 0
The graded algebra $ A _ {p} $
4 KB (522 words) - 08:23, 6 June 2020
''spectral analysis of a time series''
...f. [[Spectral decomposition of a random function|Spectral decomposition of a random function]]).
2 KB (261 words) - 17:19, 7 February 2011
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.
2 KB (433 words) - 12:44, 18 August 2014
...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},
3 KB (517 words) - 00:44, 16 September 2015
$#A+1 = 34 n = 0
with a periodic function $ p ( z) $;
5 KB (759 words) - 22:10, 5 June 2020
''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
2 KB (304 words) - 21:59, 9 December 2014
$#A+1 = 119 n = 0
$#C+1 = 119 : ~/encyclopedia/old_files/data/P075/P.0705150 Projection spectrum
14 KB (2,029 words) - 19:53, 3 February 2021
...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
2 KB (370 words) - 19:05, 27 April 2014
...\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
3 KB (484 words) - 05:48, 17 April 2024
...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.
3 KB (374 words) - 11:04, 22 August 2014
$#A+1 = 36 n = 0
A topological space $ M $
4 KB (616 words) - 19:33, 20 January 2021
$#A+1 = 106 n = 0
$#C+1 = 106 : ~/encyclopedia/old_files/data/M062/M.0602540 Markov spectrum problem
10 KB (1,413 words) - 07:59, 6 June 2020
...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:
6 KB (970 words) - 07:33, 18 March 2023
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
4 KB (579 words) - 19:34, 7 February 2024
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
7 KB (996 words) - 11:15, 20 January 2021
$#A+1 = 29 n = 0
A quantum-mechanical system consisting of a nucleus of mass $ M $
5 KB (757 words) - 19:32, 22 March 2024
...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
2 KB (367 words) - 14:04, 30 August 2014
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).
8 KB (1,134 words) - 17:46, 1 July 2020
$#A+1 = 59 n = 1
...e operator on a Hilbert space is a [[Linear operator|linear operator]] $ A $
7 KB (1,026 words) - 08:07, 6 June 2020
$#A+1 = 52 n = 0
...this family contained in such subspaces. More precisely, let $ {\mathcal A} $
9 KB (1,382 words) - 08:22, 6 June 2020
$#A+1 = 47 n = 0
''(in group theory)''
4 KB (568 words) - 14:10, 31 December 2020
$#A+1 = 125 n = 0
...the spectrum and its main parts, spectral multiplicity and the asymptotics of eigenvalues.
16 KB (2,424 words) - 08:22, 6 June 2020
$#A+1 = 52 n = 0
A mapping $ A $
4 KB (571 words) - 06:29, 30 May 2020
$#A+1 = 36 n = 0
...6 : ~/encyclopedia/old_files/data/S086/S.0806350 Spectral decomposition of a linear operator
5 KB (793 words) - 08:22, 6 June 2020
...tal in solid state quantum physics, this conjecture becomes in mathematics a problem in
[[Spectral theory|spectral theory]] for a Schrödinger operator (cf. also
8 KB (1,239 words) - 16:33, 4 December 2017
Out of 88 formulas, 84 were replaced by TEX code.-->
''BDF theory''
10 KB (1,450 words) - 17:44, 1 July 2020
$#A+1 = 24 n = 0
of a discrete-time stationary stochastic process such that 1) the first $ q $
6 KB (747 words) - 18:22, 14 January 2021
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
13 KB (2,065 words) - 18:59, 12 December 2023
$#A+1 = 98 n = 2
A [[Hilbert space with an indefinite metric|Hilbert space with an indefinite
9 KB (1,299 words) - 08:07, 6 June 2020
...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
4 KB (629 words) - 20:08, 15 December 2020
$#A+1 = 33 n = 0
+ ( a + b \cos 2z) u = 0,\ \
5 KB (803 words) - 07:59, 6 June 2020
$#A+1 = 13 n = 0
A [[Linear operator|linear operator]] $ U $
2 KB (255 words) - 08:27, 6 June 2020
$#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 ) $(
8 KB (1,151 words) - 17:23, 12 May 2024
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.
9 KB (1,326 words) - 07:37, 24 November 2023
$#A+1 = 91 n = 0
...of a certain function algebra $ A $ (cf. [[Algebra of functions|Algebra of functions]]) into the algebra $ L ( X) $
11 KB (1,521 words) - 02:23, 16 June 2022
$#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 $
13 KB (1,805 words) - 19:06, 9 January 2024
$#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
6 KB (839 words) - 07:59, 6 June 2020
...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
5 KB (682 words) - 16:58, 2 July 2014
Out of 227 formulas, 223 were replaced by TEX code.-->
...s in the spectral theory of operators (cf. also [[Spectral theory|Spectral theory]]).
23 KB (3,372 words) - 17:44, 1 July 2020
A
...nected by the relation $\norm{x^* x} = \norm{x}^2$ for any element $x \in A$. $\Cstar\!$-algebras were introduced in 1943
15 KB (2,316 words) - 21:05, 20 April 2012
Out of 22 formulas, 22 were replaced by TEX code.-->
''Riesz theory of compact operators''
2 KB (301 words) - 17:00, 1 July 2020
Out of 3 formulas, 3 were replaced by TEX code.-->
A [[Measurable flow|measurable flow]] ( $ K $-
7 KB (1,037 words) - 16:58, 1 July 2020
Out of 168 formulas, 167 were replaced by TEX code.-->
...k } ) _ { j , k \geq 0 }$. Then $T$ is bounded if and only if there exists a function $\phi \in L ^ { \infty }$ on the unit circle $\bf T$ such that
14 KB (2,017 words) - 06:44, 15 February 2024
$#A+1 = 179 n = 0
$#C+1 = 179 : ~/encyclopedia/old_files/data/A011/A.0101370 Algebra of functions,
11 KB (1,877 words) - 16:10, 1 April 2020
$#A+1 = 38 n = 0
A periodic variation with time of a physical magnitude, which may be written in analytical form as
5 KB (715 words) - 19:43, 5 June 2020
$#A+1 = 263 n = 0
...of special functors from the category of pairs of spaces into the category of graded Abelian groups.
23 KB (3,297 words) - 19:41, 5 June 2020
Out of 82 formulas, 82 were replaced by TEX code.-->
...y. $T = T _ { 0 } \otimes T _ { 1 }$ is called the canonical decomposition of $T$.
6 KB (897 words) - 20:02, 27 February 2021
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 _ {
19 KB (2,677 words) - 17:44, 1 July 2020
...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
$$a\ddot q+b\dot q+cq=H\sin(pt+\delta),$$
2 KB (305 words) - 07:04, 27 May 2023
$#A+1 = 106 n = 0
An iterative algorithm for finding a solution to a linear equation
16 KB (2,300 words) - 19:29, 17 January 2024
$#A+1 = 67 n = 1
A manner for uniquely finding solutions to equations analogous to the [[Helmh
7 KB (1,032 words) - 19:27, 26 March 2023
Out of 43 formulas, 42 were replaced by TEX code.-->
...0$ and the multiplicity of the zero-eigenvector of $M$ is one (uniqueness of the vacuum).
9 KB (1,385 words) - 20:57, 8 February 2024
$#A+1 = 38 n = 0
...he case of isotropic scattering of quanta, diffusing without absorption in a stellar atmosphere.
6 KB (887 words) - 20:31, 25 February 2021
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]]).
8 KB (1,246 words) - 19:51, 24 December 2023
$#A+1 = 92 n = 0
...with a real symmetric kernel (cf. [[Kernel of an integral operator|Kernel of an integral operator]]):
13 KB (1,912 words) - 16:30, 6 January 2024
$#A+1 = 54 n = 0
''of a dynamical system''
11 KB (1,535 words) - 17:57, 29 October 2020
$#A+1 = 65 n = 0
A topological space associated with a vector (or sphere) bundle or spherical fibration.
6 KB (876 words) - 08:25, 6 June 2020
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 $B_E$.
8 KB (1,191 words) - 15:19, 1 July 2020
.... 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
2 KB (395 words) - 18:13, 25 September 2017
$#A+1 = 23 n = 0
$#C+1 = 23 : ~/encyclopedia/old_files/data/R077/R.0707070 Radiative transfer theory
8 KB (1,248 words) - 18:31, 13 January 2024
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]]].
8 KB (1,239 words) - 17:01, 1 July 2020
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...n applies here, mutatis-mutandis, [[#References|[a7]]]. Thus, each element of $\mathcal{X}$ may be reconstructed from its spectral parts just as ordinary
14 KB (2,151 words) - 17:43, 1 July 2020
A [[Self-adjoint operator|self-adjoint operator]] generated by a differential expression
x \in (a, b),
9 KB (1,365 words) - 12:24, 10 February 2020
$#A+1 = 43 n = 0
A bounded [[Linear operator|linear operator]] $ A $
3 KB (507 words) - 05:59, 20 January 2022
$#A+1 = 74 n = 0
A commutative local Noetherian ring $ A $,
7 KB (1,158 words) - 17:45, 4 June 2020
...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]]
2 KB (399 words) - 05:44, 7 December 2016
$#A+1 = 17 n = 0
A [[Linear operator|linear operator]] $ A $
3 KB (375 words) - 08:13, 6 June 2020
$#A+1 = 220 n = 0
$#C+1 = 220 : ~/encyclopedia/old_files/data/S086/S.0806520 Spectral theory
24 KB (3,593 words) - 18:51, 13 January 2024
$#A+1 = 71 n = 4
...m from the [[Stable homotopy group|stable homotopy group]] of the spectrum of $ \mathop{\rm SO} $
9 KB (1,201 words) - 20:31, 16 January 2024
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 $x$:
2 KB (359 words) - 11:41, 3 August 2014
$#A+1 = 270 n = 2
$#C+1 = 270 : ~/encyclopedia/old_files/data/S086/S.0806530 Spectral theory of differential operators
32 KB (4,393 words) - 08:22, 6 June 2020
$#A+1 = 332 n = 0
$#C+1 = 332 : ~/encyclopedia/old_files/data/Q076/Q.0706250 Qualitative theory of differential equations in Banach spaces
26 KB (3,943 words) - 20:22, 16 January 2024
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.
16 KB (2,493 words) - 16:55, 1 July 2020
Out of 196 formulas, 195 were replaced by TEX code.-->
...lf-adjoint representations of Lie algebras (or non-unitary representations of discrete groups).
24 KB (3,136 words) - 20:00, 24 November 2023
...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.
67 KB (9,247 words) - 17:12, 29 October 2017
''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
9 KB (1,350 words) - 11:39, 25 October 2014
$#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
14 KB (2,037 words) - 20:24, 16 January 2024
$#A+1 = 43 n = 0
''of a linear (dissipative) operator $ A $
3 KB (453 words) - 20:40, 16 January 2024
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]]).
10 KB (1,477 words) - 06:10, 26 March 2024
$#A+1 = 51 n = 0
$#C+1 = 51 : ~/encyclopedia/old_files/data/R081/R.0801590 Resolution of the identity
4 KB (554 words) - 08:11, 6 June 2020
$#A+1 = 27 n = 0
''of eigen values''
3 KB (460 words) - 08:00, 6 June 2020
$#A+1 = 30 n = 0
defined on the (closed) [[Linear closure|linear span]] of a basis $ \{ e _ {k} \} _ {k \geq 1 } $
3 KB (457 words) - 19:55, 27 February 2021
$#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
11 KB (1,442 words) - 13:45, 8 June 2020
A second-order linear ordinary differential equation
...ume complex values. The substitution $y=ux^{-1/2}$ yields the reduced form of equation \eqref{(1)}:
7 KB (1,130 words) - 20:00, 22 March 2024
$#A+1 = 29 n = 0
A compact invariant set $ L $
10 KB (1,485 words) - 04:11, 6 June 2020