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Self-adjoint operator

In mathematics, a self-adjoint operator on a complex vector space V with inner product $$\langle\cdot,\cdot\rangle$$ is a linear map A (from V to itself) that is its own adjoint. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A$∗$. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.

Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, momentum, angular momentum and spin are represented by self-adjoint operators on a Hilbert space. Of particular significance is the Hamiltonian operator $$\hat{H}$$ defined by
 * $$\hat{H} \psi = -\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi,$$

which as an observable corresponds to the total energy of a particle of mass m in a real potential field V. Differential operators are an important class of unbounded operators.

The structure of self-adjoint operators on infinite-dimensional Hilbert spaces essentially resembles the finite-dimensional case. That is to say, operators are self-adjoint if and only if they are unitarily equivalent to real-valued multiplication operators. With suitable modifications, this result can be extended to possibly unbounded operators on infinite-dimensional spaces. Since an everywhere-defined self-adjoint operator is necessarily bounded, one needs to be more attentive to the domain issue in the unbounded case. This is explained below in more detail.

Definitions
Let $$H$$ be a Hilbert space and $$A$$ an unbounded (i.e. not necessarily bounded) operator with a dense domain $$\operatorname{Dom}A \subseteq H.$$ This condition holds automatically when $$H$$ is finite-dimensional since $$\operatorname{Dom}A = H$$ for every linear operator on a finite-dimensional space.

The graph of an (arbitrary) operator $$A$$ is the set $$G(A) = \{(x,Ax) \mid x \in \operatorname{Dom}A\}.$$ An operator $$B$$ is said to extend $$A$$ if $$G(A) \subseteq G(B).$$ This is written as $$A \subseteq B.$$

Let the inner product $$\langle \cdot, \cdot\rangle$$ be conjugate linear on the second argument. The adjoint operator $$A^*$$ acts on the subspace $$\operatorname{Dom} A^* \subseteq H$$ consisting of the elements $$y$$ such that
 * $$ \langle Ax,y \rangle = \langle x,A^*y \rangle, \quad \forall x \in \operatorname{Dom} A.$$

The densely defined operator $$A$$ is called symmetric (or Hermitian) if $$A \subseteq A^*$$, i.e., if $$\operatorname{Dom} A \subseteq \operatorname{Dom} A^*$$ and $$Ax =A^*x$$ for all $$x \in \operatorname{Dom} A$$. Equivalently, $$A$$ is symmetric if and only if
 * $$ \langle Ax, y \rangle = \lang x , Ay \rangle, \quad \forall x,y\in \operatorname{Dom}A.$$

Since $$\operatorname{Dom} A^* \supseteq \operatorname{Dom} A$$ is dense in $$H$$, symmetric operators are always closable (i.e. the closure of $$G(A)$$ is the graph of an operator). If $$A^*$$ is a closed extension of $$A$$, the smallest closed extension $$A^{**}$$ of $$A$$ must be contained in $$A^*$$. Hence,
 * $$A \subseteq A^{**} \subseteq A^*$$

for symmetric operators and
 * $$A = A^{**} \subseteq A^*$$

for closed symmetric operators.

The densely defined operator $$A$$ is called self-adjoint if $$A = A^*$$, that is, if and only if $$A$$ is symmetric and $$\operatorname{Dom}A = \operatorname{Dom}A^*$$. Equivalently, a closed symmetric operator $$A$$ is self-adjoint if and only if $$A^*$$ is symmetric. If $$A$$ is self-adjoint, then $$\left\langle x, A x \right\rangle$$ is real for all $$x \in H$$, i.e.,
 * $$\langle x, Ax\rangle = \overline{\langle A x, x\rangle}=\overline{\langle x,Ax\rangle} \in \mathbb{R}, \quad \forall x \in H.$$

A symmetric operator $$A$$ is said to be essentially self-adjoint if the closure of $$A$$ is self-adjoint. Equivalently, $$A$$ is essentially self-adjoint if it has a unique self-adjoint extension. In practical terms, having an essentially self-adjoint operator is almost as good as having a self-adjoint operator, since we merely need to take the closure to obtain a self-adjoint operator.

In physics, the term Hermitian refers to symmetric as well as self-adjoint operators alike. The subtle difference between the two is generally overlooked.

Bounded self-adjoint operators
Let $$H$$ be a Hilbert space and $$A:\operatorname{Dom}(A) \to H $$ a symmetric operator. According to Hellinger–Toeplitz theorem, if $$\operatorname{Dom}(A)=H$$ then $$A$$ is necessarily bounded. A bounded operator $$A : H \to H$$ is self-adjoint if
 * $$\langle Ax, y\rangle = \langle x, Ay\rangle, \quad \forall x,y\in H.$$

Every bounded operator $$T:H\to H$$ can be written in the complex form $$T = A + i B$$ where $$A:H\to H$$ and $$B:H\to H$$ are bounded self-adjoint operators.

Alternatively, every positive bounded linear operator $$A:H \to H $$ is self-adjoint if the Hilbert space $$H$$ is complex.

Properties
A bounded self-adjoint operator $$A : H \to H$$ defined on $$\operatorname{Dom}\left( A \right) = H$$ has the following properties:
 * $$A : H \to \operatorname{Im} A \subseteq H$$ is invertible if the image of $$A$$ is dense in $$H.$$
 * $$\left\| A \right\| = \sup \left\{ |\langle x, A x \rangle| : x\in H,\,\| x \| \leq 1 \right\}$$
 * The eigenvalues of $$A$$ are real and the corresponding eigenvectors are orthogonal.
 * If $$\lambda$$ is an eigenvalue of $$A$$ then $$| \lambda | \leq \| A \|$$, where $$| \lambda | = \| A \|$$ if $$\|x\|=1$$ and $$A$$ is a compact self-adjoint operator.

Spectrum of self-adjoint operators
Let $$A:\operatorname{Dom}(A) \to H$$ be an unbounded operator. The resolvent set (or regular set) of $$A$$ is defined as
 * $$\rho(A) = \left\{\lambda \in \mathbb{C}\,:\, \exist (A - \lambda I)^{-1}\;\text{bounded and densely defined}\right\}.$$

If $$A$$ is bounded, the definition reduces to $$A - \lambda I$$ being bijective on $$H$$. The spectrum of $$A$$ is defined as the complement
 * $$\sigma(A) = \Complex \setminus \rho(A).$$

In finite dimensions, $$\sigma(A)\subseteq \mathbb{C}$$ consists exclusively of (complex) eigenvalues. The spectrum of a self-adjoint operator is always real (i.e. $$\sigma(A)\subseteq \mathbb{R}$$), though non-self-adjoint operators with real spectrum exist as well. For bounded (normal) operators, however, the spectrum is real if and only if the operator is self-adjoint. This implies, for example, that a non-self-adjoint operator with real spectrum is necessarily unbounded.

As a preliminary, define $$S=\{x \in \operatorname{Dom}A \mid \Vert x\Vert=1\},$$ $$\textstyle m=\inf_{x\in S} \langle Ax,x \rangle$$ and $$\textstyle M=\sup_{x\in S} \langle Ax,x \rangle$$ with $$m,M \in \mathbb{R} \cup \{\pm\infty\}$$. Then, for every $$ \lambda \in \Complex $$ and every $$ x \in \operatorname{Dom}A, $$
 * $$ \Vert (A - \lambda) x\Vert \geq d(\lambda)\cdot \Vert x\Vert,$$

where $$ \textstyle d(\lambda) = \inf_{r\in [m,M]} |r - \lambda|. $$

Indeed, let $$x \in \operatorname{Dom}A \setminus \{0\}.$$ By the Cauchy–Schwarz inequality,

\Vert (A - \lambda) x\Vert \geq \frac{|\langle (A - \lambda) x,x\rangle|}{\Vert x\Vert} =\left|\left\langle A\frac{x}{\Vert x\Vert},\frac{x}{\Vert x\Vert}\right\rangle - \lambda\right| \cdot \Vert x\Vert \geq d(\lambda)\cdot \Vert x\Vert. $$

If $$ \lambda \notin [m,M], $$ then $$ d(\lambda) > 0, $$ and $$ A - \lambda I $$ is called bounded below. $$ $$ $$ $$

Spectral theorem
In the physics literature, the spectral theorem is often stated by saying that a self-adjoint operator has an orthonormal basis of eigenvectors. Physicists are well aware, however, of the phenomenon of "continuous spectrum"; thus, when they speak of an "orthonormal basis" they mean either an orthonormal basis in the classic sense or some continuous analog thereof. In the case of the momentum operator $P = -i\frac{d}{dx}$, for example, physicists would say that the eigenvectors are the functions $$f_p(x) := e^{ipx}$$, which are clearly not in the Hilbert space $$L^2(\mathbb{R})$$. (Physicists would say that the eigenvectors are "non-normalizable.") Physicists would then go on to say that these "generalized eigenvectors" form an "orthonormal basis in the continuous sense" for $$L^2(\mathbb{R})$$, after replacing the usual Kronecker delta $$\delta_{i,j}$$ by a Dirac delta function $$\delta\left(p - p'\right)$$.

Although these statements may seem disconcerting to mathematicians, they can be made rigorous by use of the Fourier transform, which allows a general $$L^2$$ function to be expressed as a "superposition" (i.e., integral) of the functions $$e^{ipx}$$, even though these functions are not in $$L^2$$. The Fourier transform "diagonalizes" the momentum operator; that is, it converts it into the operator of multiplication by $$p$$, where $$p$$ is the variable of the Fourier transform.

The spectral theorem in general can be expressed similarly as the possibility of "diagonalizing" an operator by showing it is unitarily equivalent to a multiplication operator. Other versions of the spectral theorem are similarly intended to capture the idea that a self-adjoint operator can have "eigenvectors" that are not actually in the Hilbert space in question.

Multiplication operator form of the spectral theorem
Firstly, let $$(X, \Sigma, \mu)$$ be a σ-finite measure space and $$h : X \to \mathbb{R}$$ a measurable function on $$X$$. Then the operator $$T_h : \operatorname{Dom}T_h \to L^2(X,\mu)$$, defined by
 * $$T_h \psi(x) = h(x)\psi(x), \quad \forall \psi \in \operatorname{Dom}T_h,$$

where
 * $$\operatorname{Dom}T_h := \left\{\psi\in L^2(X,\mu) \;|\; h\psi \in L^2(X,\mu)\right\},$$

is called a multiplication operator. Any multiplication operator is a self-adjoint operator.

Secondly, two operators $$A$$ and $$B$$ with dense domains $$\operatorname{Dom}A \subseteq H_1$$ and $$\operatorname{Dom}B \subseteq H_2$$ in Hilbert spaces $$H_1$$ and $$H_2$$, respectively, are unitarily equivalent if and only if there is a unitary transformation $$U: H_1 \to H_2$$ such that: If unitarily equivalent $$A$$ and $$B$$ are bounded, then $$\|A\|_{H_1}=\|B\|_{H_2}$$; if $$A$$ is self-adjoint, then so is $$B$$.
 * $$U\operatorname{Dom}A = \operatorname{Dom}B,$$
 * $$ U A U^{-1} \xi = B \xi, \quad \forall \xi \in \operatorname{Dom}B. $$

$$ The spectral theorem holds for both bounded and unbounded self-adjoint operators. Proof of the latter follows by reduction to the spectral theorem for unitary operators. We might note that if $$T$$ is multiplication by $$h$$, then the spectrum of $$T$$ is just the essential range of $$h$$.

More complete versions of the spectral theorem exist as well that involve direct integrals and carry with it the notion of "generalized eigenvectors".

Functional calculus
One application of the spectral theorem is to define a functional calculus. That is, if $$f$$ is a function on the real line and $$T$$ is a self-adjoint operator, we wish to define the operator $$f(T)$$. The spectral theorem shows that if $$T$$ is represented as the operator of multiplication by $$h$$, then $$f(T)$$ is the operator of multiplication by the composition $$f \circ h$$.

One example from quantum mechanics is the case where $$T$$ is the Hamiltonian operator $$\hat{H}$$. If $$\hat{H}$$ has a true orthonormal basis of eigenvectors $$e_j$$ with eigenvalues $$\lambda_j$$, then $$f(\hat{H}) := e^{-it\hat{H}/\hbar}$$ can be defined as the unique bounded operator with eigenvalues $$f(\lambda_j) := e^{-it\lambda_j/\hbar}$$ such that:
 * $$f(\hat{H}) e_j = f(\lambda_j)e_j.$$

The goal of functional calculus is to extend this idea to the case where $$T$$ has continuous spectrum (i.e. where $$T$$ has no normalizable eigenvectors).

It has been customary to introduce the following notation
 * $$\operatorname{E}(\lambda) = \mathbf{1}_{(-\infty, \lambda]} (T)$$

where $$\mathbf{1}_{(-\infty, \lambda]}$$ is the indicator function of the interval $$(-\infty, \lambda]$$. The family of projection operators E(λ) is called resolution of the identity for T. Moreover, the following Stieltjes integral representation for T can be proved:
 * $$ T = \int_{-\infty}^{+\infty} \lambda d \operatorname{E}(\lambda).$$

Formulation in the physics literature
In quantum mechanics, Dirac notation is used as combined expression for both the spectral theorem and the Borel functional calculus. That is, if H is self-adjoint and f is a Borel function,
 * $$f(H) = \int dE \left| \Psi_E \rangle f(E) \langle \Psi_E \right|$$

with
 * $$H \left|\Psi_E\right\rangle = E \left|\Psi_E\right\rangle$$

where the integral runs over the whole spectrum of H. The notation suggests that H is diagonalized by the eigenvectors ΨE. Such a notation is purely formal. The resolution of the identity (sometimes called projection-valued measures) formally resembles the rank-1 projections $$\left|\Psi_E\right\rangle \left\langle\Psi_E\right|$$. In the Dirac notation, (projective) measurements are described via eigenvalues and eigenstates, both purely formal objects. As one would expect, this does not survive passage to the resolution of the identity. In the latter formulation, measurements are described using the spectral measure of $$|\Psi \rangle$$, if the system is prepared in $$|\Psi \rangle$$ prior to the measurement. Alternatively, if one would like to preserve the notion of eigenstates and make it rigorous, rather than merely formal, one can replace the state space by a suitable rigged Hilbert space.

If f = 1, the theorem is referred to as resolution of unity:

 * $$I = \int dE \left|\Psi_E\right\rangle \left\langle\Psi_E\right|$$

In the case $$H_\text{eff} = H - i\Gamma$$ is the sum of an Hermitian H and a skew-Hermitian (see skew-Hermitian matrix) operator $$ -i\Gamma$$, one defines the biorthogonal basis set

 * $$H^*_\text{eff} \left|\Psi_E^*\right\rangle = E^* \left|\Psi_E^*\right\rangle$$

and write the spectral theorem as:

 * $$f\left(H_\text{eff}\right) = \int dE \left|\Psi_E\right\rangle f(E) \left\langle\Psi_E^*\right|$$

(See Feshbach–Fano partitioning method for the context where such operators appear in scattering theory).

Formulation for symmetric operators
The spectral theorem applies only to self-adjoint operators, and not in general to symmetric operators. Nevertheless, we can at this point give a simple example of a symmetric (specifically, an essentially self-adjoint) operator that has an orthonormal basis of eigenvectors. Consider the complex Hilbert space L2[0,1] and the differential operator
 * $$A = -\frac{d^2}{dx^2}$$

with $$\mathrm{Dom}(A)$$ consisting of all complex-valued infinitely differentiable functions f on [0, 1] satisfying the boundary conditions
 * $$f(0) = f(1) = 0.$$

Then integration by parts of the inner product shows that A is symmetric. The eigenfunctions of A are the sinusoids
 * $$f_n(x) = \sin(n \pi x) \qquad  n= 1, 2, \ldots$$

with the real eigenvalues n2π2; the well-known orthogonality of the sine functions follows as a consequence of A being symmetric.

The operator A can be seen to have a compact inverse, meaning that the corresponding differential equation Af = g is solved by some integral (and therefore compact) operator G. The compact symmetric operator G then has a countable family of eigenvectors which are complete in $L^{2}$. The same can then be said for A.

Pure point spectrum
A self-adjoint operator A on H has pure point spectrum if and only if H has an orthonormal basis {ei}i ∈ I consisting of eigenvectors for A.

Example. The Hamiltonian for the harmonic oscillator has a quadratic potential V, that is
 * $$-\Delta + |x|^2.$$

This Hamiltonian has pure point spectrum; this is typical for bound state Hamiltonians in quantum mechanics. As was pointed out in a previous example, a sufficient condition that an unbounded symmetric operator has eigenvectors which form a Hilbert space basis is that it has a compact inverse.

Symmetric vs self-adjoint operators
Although the distinction between a symmetric operator and a (essentially) self-adjoint operator is subtle, it is important since self-adjointness is the hypothesis in the spectral theorem. Here we discuss some concrete examples of the distinction.

Boundary conditions
In the case where the Hilbert space is a space of functions on a bounded domain, these distinctions have to do with a familiar issue in quantum physics: One cannot define an operator—such as the momentum or Hamiltonian operator—on a bounded domain without specifying boundary conditions. In mathematical terms, choosing the boundary conditions amounts to choosing an appropriate domain for the operator. Consider, for example, the Hilbert space $$L^2([0, 1])$$ (the space of square-integrable functions on the interval [0,1]). Let us define a momentum operator A on this space by the usual formula, setting Planck's constant equal to 1:
 * $$Af = -i\frac{df}{dx}.$$

We must now specify a domain for A, which amounts to choosing boundary conditions. If we choose
 * $$\operatorname{Dom}(A) = \left\{\text{smooth functions}\right\},$$

then A is not symmetric (because the boundary terms in the integration by parts do not vanish).

If we choose
 * $$\operatorname{Dom}(A) = \left\{\text{smooth functions}\,f \mid f(0) = f(1) = 0\right\},$$

then using integration by parts, one can easily verify that A is symmetric. This operator is not essentially self-adjoint, however, basically because we have specified too many boundary conditions on the domain of A, which makes the domain of the adjoint too big (see also the example below).

Specifically, with the above choice of domain for A, the domain of the closure $$A^{\mathrm{cl}}$$ of A is

 * $$\operatorname{Dom}\left(A^{\mathrm{cl}}\right) = \left\{\text{functions } f \text{ with two derivatives in }L^2 \mid f(0) = f(1) = 0\right\},$$

whereas the domain of the adjoint $$A^*$$ of A is
 * $$\operatorname{Dom}\left(A^*\right) = \left\{\text{functions } f \text{ with two derivatives in }L^2\right\}.$$

That is to say, the domain of the closure has the same boundary conditions as the domain of A itself, just a less stringent smoothness assumption. Meanwhile, since there are "too many" boundary conditions on A, there are "too few" (actually, none at all in this case) for $$A^*$$. If we compute $$\langle g, Af\rangle$$ for $$f \in \operatorname{Dom}(A)$$ using integration by parts, then since $$f$$ vanishes at both ends of the interval, no boundary conditions on $$g$$ are needed to cancel out the boundary terms in the integration by parts. Thus, any sufficiently smooth function $$g$$ is in the domain of $$A^*$$, with $$A^*g = -i\,dg/dx$$.

Since the domain of the closure and the domain of the adjoint do not agree, A is not essentially self-adjoint. After all, a general result says that the domain of the adjoint of $$A^\mathrm{cl}$$ is the same as the domain of the adjoint of A. Thus, in this case, the domain of the adjoint of $$A^\mathrm{cl}$$ is bigger than the domain of $$A^\mathrm{cl}$$ itself, showing that $$A^\mathrm{cl}$$ is not self-adjoint, which by definition means that A is not essentially self-adjoint.

The problem with the preceding example is that we imposed too many boundary conditions on the domain of A. A better choice of domain would be to use periodic boundary conditions:
 * $$\operatorname{Dom}(A) = \{\text{smooth functions}\,f \mid f(0) = f(1)\}.$$

With this domain, A is essentially self-adjoint.

In this case, we can understand the implications of the domain issues for the spectral theorem. If we use the first choice of domain (with no boundary conditions), all functions $$f_\beta(x) = e^{\beta x}$$ for $$\beta \in \mathbb C$$ are eigenvectors, with eigenvalues $$-i \beta$$, and so the spectrum is the whole complex plane. If we use the second choice of domain (with Dirichlet boundary conditions), A has no eigenvectors at all. If we use the third choice of domain (with periodic boundary conditions), we can find an orthonormal basis of eigenvectors for A, the functions $$f_n(x) := e^{2\pi inx}$$. Thus, in this case finding a domain such that A is self-adjoint is a compromise: the domain has to be small enough so that A is symmetric, but large enough so that $$D(A^*)=D(A)$$.

Schrödinger operators with singular potentials
A more subtle example of the distinction between symmetric and (essentially) self-adjoint operators comes from Schrödinger operators in quantum mechanics. If the potential energy is singular—particularly if the potential is unbounded below—the associated Schrödinger operator may fail to be essentially self-adjoint. In one dimension, for example, the operator
 * $$\hat{H} := \frac{P^2}{2m} - X^4$$

is not essentially self-adjoint on the space of smooth, rapidly decaying functions. In this case, the failure of essential self-adjointness reflects a pathology in the underlying classical system: A classical particle with a $$-x^4$$ potential escapes to infinity in finite time. This operator does not have a unique self-adjoint, but it does admit self-adjoint extensions obtained by specifying "boundary conditions at infinity". (Since $$\hat{H}$$ is a real operator, it commutes with complex conjugation. Thus, the deficiency indices are automatically equal, which is the condition for having a self-adjoint extension.)

In this case, if we initially define $$\hat{H}$$ on the space of smooth, rapidly decaying functions, the adjoint will be "the same" operator (i.e., given by the same formula) but on the largest possible domain, namely
 * $$\operatorname{Dom}\left(\hat{H}^*\right) = \left\{ \text{twice differentiable functions }f \in L^2(\mathbb{R})\left|\left( -\frac{\hbar^2}{2m}\frac{d^2f}{dx^2} - x^4f(x)\right) \in L^2(\mathbb{R}) \right. \right\}. $$

It is then possible to show that $$\hat{H}^*$$ is not a symmetric operator, which certainly implies that $$\hat{H}$$ is not essentially self-adjoint. Indeed, $$\hat{H}^*$$ has eigenvectors with pure imaginary eigenvalues, which is impossible for a symmetric operator. This strange occurrence is possible because of a cancellation between the two terms in $$\hat{H}^*$$: There are functions $$f$$ in the domain of $$\hat{H}^*$$ for which neither $$d^2 f/dx^2$$ nor $$x^4f(x)$$ is separately in $$L^2(\mathbb{R})$$, but the combination of them occurring in $$\hat{H}^*$$ is in $$L^2(\mathbb{R})$$. This allows for $$\hat{H}^*$$ to be nonsymmetric, even though both $$d^2/dx^2$$ and $$X^4$$ are symmetric operators. This sort of cancellation does not occur if we replace the repelling potential $$-x^4$$ with the confining potential $$x^4$$.

Non-self-adjoint operators in quantum mechanics
In quantum mechanics, observables correspond to self-adjoint operators. By Stone's theorem on one-parameter unitary groups, self-adjoint operators are precisely the infinitesimal generators of unitary groups of time evolution operators. However, many physical problems are formulated as a time-evolution equation involving differential operators for which the Hamiltonian is only symmetric. In such cases, either the Hamiltonian is essentially self-adjoint, in which case the physical problem has unique solutions or one attempts to find self-adjoint extensions of the Hamiltonian corresponding to different types of boundary conditions or conditions at infinity.

Example. The one-dimensional Schrödinger operator with the potential $$V(x) = -(1 + |x|)^\alpha$$, defined initially on smooth compactly supported functions, is essentially self-adjoint for $0 < α ≤ 2$ but not for $α > 2$.

The failure of essential self-adjointness for $$\alpha > 2$$ has a counterpart in the classical dynamics of a particle with potential $$V(x)$$: The classical particle escapes to infinity in finite time.

Example. There is no self-adjoint momentum operator $$p$$ for a particle moving on a half-line. Nevertheless, the Hamiltonian $$p^2$$ of a "free" particle on a half-line has several self-adjoint extensions corresponding to different types of boundary conditions. Physically, these boundary conditions are related to reflections of the particle at the origin.

A symmetric operator that is not essentially self-adjoint
We first consider the Hilbert space $$L^2[0, 1]$$ and the differential operator

 * $$D: \phi \mapsto \frac{1}{i} \phi'$$

defined on the space of continuously differentiable complex-valued functions on [0,1], satisfying the boundary conditions
 * $$\phi(0) = \phi(1) = 0.$$

Then D is a symmetric operator as can be shown by integration by parts. The spaces N+, N− (defined below) are given respectively by the distributional solutions to the equation

 * $$\begin{align}

-i u' &= i u \\ -i u' &= -i u \end{align}$$

which are in L2[0, 1]. One can show that each one of these solution spaces is 1-dimensional, generated by the functions x → e−x and x → ex respectively. This shows that D is not essentially self-adjoint, but does have self-adjoint extensions. These self-adjoint extensions are parametrized by the space of unitary mappings N+ → N−, which in this case happens to be the unit circle T.

In this case, the failure of essential self-adjointenss is due to an "incorrect" choice of boundary conditions in the definition of the domain of $$D$$. Since $$D$$ is a first-order operator, only one boundary condition is needed to ensure that $$D$$ is symmetric. If we replaced the boundary conditions given above by the single boundary condition
 * $$\phi(0) = \phi(1)$$,

then D would still be symmetric and would now, in fact, be essentially self-adjoint. This change of boundary conditions gives one particular essentially self-adjoint extension of D. Other essentially self-adjoint extensions come from imposing boundary conditions of the form $$\phi(1) = e^{i\theta}\phi(0)$$.

This simple example illustrates a general fact about self-adjoint extensions of symmetric differential operators P on an open set M. They are determined by the unitary maps between the eigenvalue spaces
 * $$ N_\pm = \left\{u \in L^2(M): P_\operatorname{dist} u = \pm i u\right\} $$

where Pdist is the distributional extension of P.

Constant-coefficient operators
We next give the example of differential operators with constant coefficients. Let
 * $$P\left(\vec{x}\right) = \sum_\alpha c_\alpha x^\alpha $$

be a polynomial on Rn with real coefficients, where α ranges over a (finite) set of multi-indices. Thus
 * $$ \alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n)$$

and
 * $$x^\alpha = x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n}.$$

We also use the notation

 * $$D^\alpha = \frac{1}{i^{|\alpha|}} \partial_{x_1}^{\alpha_1}\partial_{x_2}^{\alpha_2} \cdots \partial_{x_n}^{\alpha_n}. $$

Then the operator P(D) defined on the space of infinitely differentiable functions of compact support on Rn by
 * $$ P(\operatorname{D}) \phi = \sum_\alpha c_\alpha \operatorname{D}^\alpha \phi$$

is essentially self-adjoint on L2(Rn).

$$

More generally, consider linear differential operators acting on infinitely differentiable complex-valued functions of compact support. If M is an open subset of Rn
 * $$P \phi(x) = \sum_\alpha a_\alpha (x) \left[D^\alpha \phi\right](x)$$

where aα are (not necessarily constant) infinitely differentiable functions. P is a linear operator
 * $$ C_0^\infty(M) \to C_0^\infty(M).$$

Corresponding to P there is another differential operator, the formal adjoint of P
 * $$ P^\mathrm{*form} \phi = \sum_\alpha D^\alpha \left(\overline{a_\alpha} \phi\right)$$

$$

Spectral multiplicity theory
The multiplication representation of a self-adjoint operator, though extremely useful, is not a canonical representation. This suggests that it is not easy to extract from this representation a criterion to determine when self-adjoint operators A and B are unitarily equivalent. The finest grained representation which we now discuss involves spectral multiplicity. This circle of results is called the Hahn–Hellinger theory of spectral multiplicity.

Uniform multiplicity
We first define uniform multiplicity:

Definition. A self-adjoint operator A has uniform multiplicity n where n is such that 1 ≤ n ≤ ω if and only if A is unitarily equivalent to the operator Mf of multiplication by the function f(λ) = λ on
 * $$L^2_\mu\left(\mathbf{R}, \mathbf{H}_n\right) = \left\{\psi: \mathbf{R} \to \mathbf{H}_n: \psi \mbox{ measurable and } \int_{\mathbf{R}} \|\psi(t)\|^2 d\mu(t) < \infty\right\}$$

where Hn is a Hilbert space of dimension n. The domain of Mf consists of vector-valued functions ψ on R such that
 * $$\int_\mathbf{R} |\lambda|^2\ \|\psi(\lambda)\|^2 \, d\mu(\lambda) < \infty.$$

Non-negative countably additive measures μ, ν are mutually singular if and only if they are supported on disjoint Borel sets.

$$

This representation is unique in the following sense: For any two such representations of the same A, the corresponding measures are equivalent in the sense that they have the same sets of measure 0.

Direct integrals
The spectral multiplicity theorem can be reformulated using the language of direct integrals of Hilbert spaces:

$$

Unlike the multiplication-operator version of the spectral theorem, the direct-integral version is unique in the sense that the measure equivalence class of μ (or equivalently its sets of measure 0) is uniquely determined and the measurable function $$\lambda\mapsto\mathrm{dim}(H_{\lambda})$$ is determined almost everywhere with respect to μ. The function $$\lambda \mapsto \operatorname{dim}\left(H_\lambda\right)$$ is the spectral multiplicity function of the operator.

We may now state the classification result for self-adjoint operators: Two self-adjoint operators are unitarily equivalent if and only if (1) their spectra agree as sets, (2) the measures appearing in their direct-integral representations have the same sets of measure zero, and (3) their spectral multiplicity functions agree almost everywhere with respect to the measure in the direct integral.

Example: structure of the Laplacian
The Laplacian on Rn is the operator
 * $$\Delta = \sum_{i=1}^n \partial_{x_i}^2.$$

As remarked above, the Laplacian is diagonalized by the Fourier transform. Actually it is more natural to consider the negative of the Laplacian −Δ since as an operator it is non-negative; (see elliptic operator).

$$

title: Self-adjoint operator


text:
Talk:Calmodulin

Untitled
"Calmodulin (CaM) is a ubiquitous..." "CaM is expressed in many cell types.." Phrases are redundant; however, it may be more informative to state CaM is expressed in all eukaryotic cells, not just many cell types.

Enzymic activity is inhibited by binding of immunophilin ligand complex (immunophilin alone does not bind) and therefore may play a part in the mechanism of action of cyclosporin A and FK506. Thought also to be involved in the control of sperm motility

 * I removed the above because it was not clear what enzyme(s) this sentence was meant to refer to and how it's relevant to calmodulin. -- Marj 18:32 Feb 23, 2003 (UTC)

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Prospective edits
Hello, I am looking to make some edits to this Wikipedia page and would love any feedback or suggestions that you think might be helpful.

First, I want to add information to the general section, giving a bit better introduction to what Calmodulin is and why it is important, why it is a good secondary messenger etc. (without creating redundancies between the introduction and the rest of the article).

I would also like to use Pymol or another structure generating program to create images that depict the overall globular structure of calmodulin and indicate where the N-terminus, C-terminus and 4 calcium binding sites are located as well as where it would bind to other larger molecules etc. I would also like to create two images side to side as a visual representation of the conformational changes in the alpha helices EF hand motif, and how that effects the binding of target proteins. And I would like to add some discussion on how exposing hydrophobic regions plays a role in its binding of target proteins.

Another important change that I would like to make is to add sections about the importance of calmodulin in specific physiological systems such as muscle contraction, long-term synaptic potentiation (short-term/long-term memory), metabolism, inflammation, apoptosis, intracellular movements, and immune response. These sections will have details about the way the calmodulin plays a role in specific bodily functions and what other molecules it interacts with during these processes.

Thank you! Anordquist (talk) 22:47, 9 February 2016 (UTC)

In terms of using Pymol, have you considered using Chimera for the conformational changes? I specifically think about how calmodulin is affected by lead in it's conformations.Honestbean (talk) 02:49, 28 January 2018 (UTC)

Caps
Why the capitalised letters in the names of proteins (calmodulin, troponin C etc.)? — Preceding unsigned comment added by Marmio (talk • contribs) 20:30, 4 February 2017 (UTC)
 * Conventional naming of subtypes. Klbrain (talk) 08:38, 6 January 2019 (UTC)

Changes to "structure" and "mechanism" sections
I just rewrote the "structure" and "mechanism" sections. Irrelevant text has been removed, the section has been shortened, and many references have been added. I kept the images the same for now but might go back and upload different ones later. WeirdNAnnoyed (talk) 16:47, 28 July 2020 (UTC)

And I just cleaned up the reference list; due to edits to the sections above there were many redundant entries. WeirdNAnnoyed (talk) 23:24, 14 August 2020 (UTC)

Calmodulin-like protein
Would it be apt to include a section about calmodulin-like proteins and their role in recent research, as some references include how this alludes to their role? Alexniy (talk) 02:55, 26 April 2024 (UTC)

 * I think that would be appropriate, as long as length doesn't get to be too much. If these other proteins are the subject of significant research, they should have their own articles and be linked to this one.  But a short section might be a good idea. WeirdNAnnoyed (talk) 14:12, 26 April 2024 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Calmodulin


text:
Lamus

Lamus may refer to:

 * Lamus River (Limonlu Çayı)
 * Lamus (city), ancient city on the river
 * Lamus (Isauria), town of ancient Isauria
 * Lamus (see), Catholic titular see
 * Eduardo Cote Lamus (1928–1964), Colombian politician and poet
 * Acrossocheilus lamus, a species of ray-finned fish

title: Lamus


text:
Telepylos

Telepylos or Telepylus (Τηλέπυλος Tēlépylos, meaning "far-off port" or "big-gated" ) was the mythological city of the Laestrygonians.

Mythology
In the Odyssey it is described as the rocky stronghold of Lamos. When Odysseus reaches the city in the Odyssey, he sends three scouts to explore the island. They come across the king, a giant cannibal, who then eats one of the men, causing the other scouts to run away. Most of Odysseus' men are killed in the incident, but his boat is moored outside the Laestrygonians' harbour. He is able to sail away, without the bombardment of rocks received by the rest of the fleet who did moor within the harbour. Only forty-five men escape.

It has been identified with Mezapo, located on the Mani Peninsula. Iman Jacob Wilkens makes a less likely identification: the harbour of Havana, Cuba, believing that Ulysses had in fact crossed the Atlantic Ocean. "The harbour, about which on both sides a sheer cliff runs continuously, and projecting headlands opposite to one another stretch out at the mouth, and the entrance is narrow, ..., and the ships were moored within the hollow harbour, for therein no wave ever swelled, great or small, but all about was a bright calm..."

Locations
In Greek mythology, the name Telepylos is mentioned in the Odyssey (k 82, ps 318) the city or country of the Laistrygons ("laistrygonii"). The name, from tele- = far and the door, perhaps according to some authors has the meaning of "eurypylos, megalopylos", or "macropylos" (Eustathius: "at a distance from each other, but next to the doors or at the length " ). In the view of the later ancient Greeks (e.g. Thucydides, 6.2), Telepilos was located in Sicily, while the Romans believed it to be located in mainland Italy, in Formia ( today Mola di Gaeta). The significant Laestrygonian associations with the Talayotic culture or pre-Talayotic culture in Balearic Islands have recently been admitted, and the interest in the description -let's think of one of the well-known ports of the Mediterranean-, allows us to believe that is a first and almost perfect port of Mahón. Jasen Boko establishes Omiš as the most probable location, for its unique landscape that corresponds to the verses.

title: Telepylos


text:
Talk:Saint David

Do we need a citation for a calendar calculation?
The note about his death being "generally accepted" as 589 may need a citation, but the "citation needed" flag appears to be on the statement that 1 March 589 was a Tuesday, which is a simple mathematical fact about the Julian calendar that can be readily checked any number of ways. It seems odd to require documentation for such a thing. If the Welsh somehow presaged Pope Gregory's new calendar 1000 years early and made 1 March 589 a Sunday, that would indeed require a citation.. Markjreed (talk) 12:16, 1 March 2015 (UTC)
 * Looking into it, it seems that 589 is but one of several dates, another favourite is 601, but nothing is certain; looking at the wording, it appears that someone felt that saying that he was born on a Tuesday would therefore fix the year as 589, and someone else appropriately queried this. I have reworded slightly, and put in a cite, but more work is needed in this area.  SilkTork  ✔Tea time  19:47, 1 March 2016 (UTC)

Semi-protection
This article has been semi-protected. Semi-protection prevents edits from unregistered users (IP addresses), as well as edits from any account that is not autoconfirmed (is at least four days old and has at least ten edits to Wikipedia) or confirmed. Such users can request edits to this article by proposing them on this talk page, using the template if necessary to gain attention. New users may also request the confirmed user right by visiting Requests for permissions.  SilkTork  ✔Tea time  19:50, 1 March 2016 (UTC)

Semi-protected edit request on 28 February 2019
TYPO: In the third sentence of the second paragraph in the "Legend" section, change the word "practised" to "practiced" Quietlyartistic (talk) 18:56, 28 February 2019 (UTC)
 * Red information icon with gradient background.svg Not done: This is correct spelling. See WP:ENGVAR RudolfRed (talk) 20:03, 28 February 2019 (UTC)

typo for "Calixtus"
"Quick facts" section shows "Calixtus", links to page with spelling "Callixtus" Aa599 (talk) 17:37, 16 March 2020 (UTC)

 * (Requester means infobox). The target article is inconsistent with itself, even preferring the one-L version in most of the article body, so I'm unsure what should be done with this. –Deacon Vorbis (carbon • videos) 20:34, 16 March 2020 (UTC)
 * Yes check.svg Done It was a mis-spelling of Pope Callixtus II in the infobox who was listed in the "canonized_by    =" field. In most of these cases the mis-spelling is setup as a redirect to the correct article. I just simply replaced the incorrect spelling (Pope Calixtus II) which is a redirect with the correct article title & spelling Pope Callixtus II.    Alucard 16  ❯❯❯ chat?    21:40, 18 March 2020 (UTC)
 * Are you sure though? My point was that if you actually look at that page, almost every use except the actual title and the first mention uses one L rather than 2. Not a huge deal for me, I just wasn't sure if you understood why I was unsure in the first place.  –Deacon Vorbis (carbon • videos) 21:48, 18 March 2020 (UTC)
 * I was unsure as well until I clicked on the wiki-link for Pope Calixtus II and realized it kept redirecting to Pope Callixtus II so I just chalked it up to a mis-spelling.   Alucard 16  ❯❯❯ chat?    07:09, 19 March 2020 (UTC)

How did David become the patron saint?
How, and when, did St David become adopted as the patron saint of Wales? This seems to me to be a fundamental piece of information that is missing from the article. FreeFlow99 (talk) 17:20, 7 October 2020 (UTC)
 * Wales and Patron saint disappoint too. Catholic Encyclopedia says a patron is "one who has been assigned by a venerable tradition, or chosen by election". The later statement "An authentic catalogue of patron saints of countries of the world has yet to be made" makes election seem the less likely. Sparafucil (talk) 10:26, 8 October 2020 (UTC)

Canonisation date
What does "David was officially recognised at the Holy See by Pope Callixtus II in 1120" mean? Elsewhere it is stated that he was canonised in 1123. Does this refer to something different or is the year incorrect? 195.213.169.161 (talk) 15:58, 1 March 2023 (UTC)

"Now" used incorrectly
The word "now" is used incorrectly in the text. I'd fix it, but there's no "Edit" button. 90.202.97.45 (talk) 22:48, 1 March 2024 (UTC)

title: Talk:Saint David


text:
La Madeleine, Paris

The Church of Sainte-Marie-Madeleine (French: L'église Sainte-Marie-Madeleine), or less formally, La Madeleine, is a Catholic parish church on Place de la Madeleine in the 8th arrondissement of Paris. It was planned by Louis XV as the focal point of the new Rue Royal, leading to the new Place Louis XV, the present Place de la Concorde. It was dedicated in 1764 by Louis XV, but work halted due to the French Revolution. Napoleon Bonaparte had it redesigned in the Neoclassical style to become a monument to the glory of his armies. After his downfall in 1814 construction as a church resumed, but it was not completed until 1842. The building is surrounded on all four sides by columns in the Corinthian style. The interior is noted for its frescoes on the domed ceiling, and monumental sculptures by François Rude, Charles Marochetti and other prominent 19th-century French artists.

The exterior and interior of the church are undergoing a major project of cleaning and restoration, which began in 2020 and is scheduled for completion in 2024.

First church
The neighbourhood, then at the edge of Paris, was annexed to the city in 1722. An earlier church of Saint-Marie-Madeleine was built in the 13th century on avenue Malesherbes, but was considered too small for the growing neighbourhood. Louis XV authorised the construction of a new, larger church, with a view along Rue Royale toward the new Place Louis XV, now Place de la Concorde. In 1763 the King laid the first stone for a new church, designed by Pierre Contant d'Ivry and Guillaume-Martin Couture.

The first design for the new church by Pierre Contant d'Ivry proposed a large dome atop a building in the form of Latin cross, similar to the Les Invalides church designed by Jules Hardouin Mansart. D'Ivry died in 1777 and was replaced by his pupil Guillaume Martin Couture. Couture abandoned the first plan, demolished much of the early work. and went to work on a simpler, more classical design, modelled after an ancient Greek or Roman temple.

Proposed monument to Napoleon's Army and railroad station, then church again
The construction of the new church was abruptly halted in 1789 by the French Revolution, with only the foundations and grand classical portico completed. After the execution of Louis XVI in 1793, his body was transported to the old Church of the Madeleine, which was still standing until 1801. The King's body was thrown onto bed of quicklime at the bottom of a pit and covered by one of earth, the whole being firmly and thoroughly tamped down. Louis XVI's head was placed at his feet. On 21 January 1815 Louis XVI and Marie Antoinette's remains were moved to a new tomb in the Basilica of Saint-Denis.

Under the Revolutionary government, a debate began on the future purpose of the building. Proposals included a library, a public ballroom, and a marketplace. The new building of the National Assembly, in the Palais Bourbon, at the other end of the former Rue Royale, was given a classical colonnade to match the already completed portico of church. The new Emperor, Napoleon Bonaparte, was crowned in 1804 and in 1806 settled the debate. In 1806 he declared that the church would become "A Temple to the Glory of the Grand Army". While on a military campaign in Poland, he personally chose the design of a new architect, Pierre-Alexandre Vignon (fr: Pierre-Alexandre Vignon), over the design that was recommended to him by the Academy of Architecture. The plan of Vignon took the form of a classical temple with Corinthian columns on all four sides. The work began anew, with new foundations but preserving the classical columns that had already been raised.

Completion of the church (1855)
After the fall of Napoleon in 1814, the new King, Louis XVIII, resumed construction on the unfinished church, which he intended to make an Expiatory chapel for the sins of the Revolution and the execution of Louis XVI. However, this idea was dropped, and the new church was instead dedication to Mary Magdalene, or the Madeleine, a follower of Jesus who witnessed both the Crucifixion and the Resurrection of Christ.

The architect Vignon died in 1828 before completing the project and was replaced by Jacques-Marie Huvé. A new competition was set up in 1828–29 to determine the design for sculptures for the pediment. The design chosen was "The Last Judgment", depicting Saint Mary Magdalene kneeling to pray for sinners, by Philippe Joseph Henri Lemaire. The new government of the July Monarchy decided to go ahead with the church, despite financial difficulties. in 1830 they declared that it would be dedicated to national reconciliation. The vaults were finally completed in 1831.

Work on the church was largely completed during the reign of King Louis-Philippe, between 1830 and 1848. in 1837 a proposal was brought forward to convert church into the first railroad station in Paris, but this was abandoned as expensive and impractical. The church was finally inaugurated on July 24, 1842, the day of Saint Mary-Magdalene.

The new church became popular with musicians. The funeral of Chopin at the Church of the Madeleine in Paris was delayed almost two weeks, until 30 October 1849. Chopin had requested that Mozart's Requiem be sung. The Requiem had major parts for female voices, but the Church of the Madeleine had never permitted female singers in its choir. The church finally relented, on condition that the female singers remain behind a black velvet curtain.

During the Paris Commune of 1871, the curé of the church, Abbé Deguerry, was one of those arrested and held hostage by the Commune. He was executed alongside Georges Darboy, the Archbishop of Paris and four other hostages on 24 May, during the Semaine sanglante, as French government troops were bloodily retaking the city and executing Communard defenders.

Besides Chopin, musicians and artists whose funerals were held at the church include Jacques Offenbach, Charles Gounod, Camille Saint-Saëns, Coco Chanel, Joséphine Baker, Charles Trenet, Dalida and Johnny Hallyday.

Inspiration
The design of the church by Vignon was an example of the Neo-Classical style, and was a type of building called a Périptére or Peripeterus; a structure having rows of classical columns on all four sides, not just on the facade. Notable examples included the Olympeion, the Temple of Olympian Zeus, the largest temple in ancient Athens, located below the Parthenon,  and the much smaller Roman Maison Carrée in Nîmes in France one of the best-preserved of all Roman temples. The Madeleine is one of the rare large neo-classical buildings to imitate the whole external form of an ancient temple, rather than just the portico front. Its fifty-two Corinthian columns, each 20 m high, surround the building.

South facade and Fronton
The inscription on the frieze over the entrance reads in Latin: D⸱O⸱M⸱SVB⸱INVOC⸱S⸱M⸱MAGDALENAE, that is Deo Optimo Maximo sub invocatione Sanctae Mariae Magdalenae ("To God all-powerful and Very Great, under the invocation of Saint Mary Magdalen.")

The pediment sculpture of the Last Judgement is by Philippe Joseph Henri Lemaire. Lemaire's sculpture also has a prominent place on the Arc de Triomphe. In the sculpture, Christ is in the centre, presiding over the Last Judgement, flanked by two angels. On the right is the Archangel Michael, with a group of figures representing the Vices, who will be refused entry to heaven. To the left are the Virtues, escorting those admitted to heaven. Mary Magadelen is shown kneeling with those refused entrance to heaven, expressing her repentance.

Bronze Doors
The bronze doors of the south portal have reliefs illustrating the Ten Commandments. They measure 108 by. The artist was Henri de Triqueti (1804–1874), who was only thirty years old when he won the commission. His main influence was the doors made by Ghiberti for the Baptistry of Florence, as well as those found on Pantheon in Rome and Christian basilicas in Pisa, Rome, Verona and Venice. Their size is exceptional; they are larger than the doors of the Saint Peter's Basilica in Rome; but they were also designed to be thinner and lighter. Unlike many bronze church doors, they have no gilding, just the color of the bronze. The doors earned Triqueti a place as a royal sculptor for the projects of King Louis-Philippe, including sculpture in Napoleon's tomb.

Statues in niches of the Colonnade
Another feature of the exterior is a series of statues of Saints, made by different sculptors, alternating women and men, arranged on the outside walls along the portico, within the colonnade. The original plan by Vignon had only bare walls on the exterior, but the new architect, Huvé, proposed a series of thirty-three statues in niches. The selection of Saints was largely made by the Orleans family of King Louis Philippe and his family. The two most prominent places, by the south entrance, were given to traditional French Saints, Louis IX and Saint Philip, Bernard of Clairvaux, as well as Saint Joan of Arc and Saint Genevieve, the Patron Saint of Paris. Women saints alternate with men. At the north or rear end of the church, the heads of four of the statues were knocked off by the explosion of a German shell during the First World War, in 1918.

Parts
The plan of the church was inspired more by the classical Roman architecture, particularly the baths, than by traditional church architecture. Inside, the church designed by Huvé is composed of a single long space, without a transept. It is divided into three wide arched bays, each with a dome, with circular skylights that provide limited illumination. All the walls and arches and the ceiling are covered with decoration, largely composed of colored marble in intricate geometric forms, and frequently gilded.

The choir dome - "The History of Christianity"
The cul-de-four or half-dome over the choir of the church is decorated with a painting by Jules-Claude Ziegler (1804-1856) which depicts major events in the history of Christianity, with an emphasis on France. At the top is the figure of Christ with Apostles and Mary Magdalene. In the foreground are Napoleon Bonaparte and Pope Pius VII signing the Concordat of 1801, which, following the French Revolution, marked the reconciliation of the French church and state and allowed Catholic churches to re-open in France. Ziegler was a pupil of Ingres, and painted the figures with the same realism and animation. The work took four years to complete.

Mosaic of Lameire - "Christianity of France" and "The Ecstasy of Mary Magdalene"
Below the "History of Christianity" and above the altar is later, unusual work; a wide ceramic mosaic depicting Christ with a group of Saints who had connections with France. This was conceived between 1888 and 1893 by Father LeRebours, the curate of the church, and is in the Neo-Byxantine style, very different from the rest of the art in the choir. It painted by Charles-Joseph Lameire, and transformed into ceramic by the Sevres Porcelain workshops in Paris. In the dim light inside of the church, the gilded ceramic tiles catch the light, sometimes making it the most visible art work in the church.

The Christ of the Resurrection is the central figure in the mosaic, accompanied by the first disciples and missionaries who lived and preached in Gaul, including the patron saint of the church, Mary Magdalene; Saint Martha, sister of Mary Magdalene, buried in Tarascon; Saint Lazare, who founded the first church in Marseille; Saint Genevieve, patron saint of Paris; Saint Trophyme, a disciple of Saint Paul and founder of the church in Arles. The figure of Saint Front of Perigeaux, founder of the church in Rocamadour, who is given the features of the artist, Lameire); Saint Ursin, founder of the church in Bourges, who is given the features of the architect Charles Garnier, and others.

Below the mosaic is a row of Corinthian columns which form a theatrical background behind the altar. and a marble stairway leading up to the altar. Behind the altar is a monumental sculpture, "The Ecstasy of Mary Magdalene", by Carlo Marochetti (1805-1868), depicting Mary Magdalene, kneeling in prayer, as she is transported into heaven by three Angels.

Sculpture in the Vestibule - "The Baptism of Christ"
In the vestibule at the south end of the church, is another monumental sculpture, "The Baptism of Christ" by François Rude (1784-1855). Rude was already famous for a work he made in 1836, "The Departure of the Volunteers of 1795", prominently featured on the Arc de Triomphe.

Decoration
The decoration of the interior was completed in a relatively short period, under King Louis-Philippe, and is noted for its unusual harmony.

The organs
The church has had a long association with music and musicians. The funeral of Chopin took place in the church, and the composers Camille Saint-Saëns and Gabriel Fauré each held the title of the church organist. The church has a celebrated pipe organ, located in the tribune over the south entrance to the church. It is contained in a very ornate case with sculpted angels, spires, and other ornament harmonised with the decor. The organ was built by Aristide Cavaillé-Coll in 1845. It was restored by Cavaillé-Coll's successor Charles Mutin in 1927, who also extended the manuals to 56 notes. Tonal modifications were carried out by Roethinger, Danion-Gonzalez, and Dargassies in 1957, 1971 and 1988 respectively. :.  A smaller organ from the same period is located in the choir.

Organists

 * 1842–1846 Charles-Alexandre Fessy
 * 1847–1858 Louis James Alfred Lefébure-Wély
 * 1858–1877 Camille Saint-Saëns
 * 1877–1896 Théodore Dubois
 * 1896–1905 Gabriel Fauré
 * 1905–1934 Henri Dallier
 * 1935–1962 Édouard Mignan
 * 1962–1968 Jeanne Demessieux
 * 1969–1979 Odile Pierre
 * 1979– François-Henri Houbart (2011)

The foyer
In the basement of the Church (entrance on the Flower Market side) is the Foyer de la Madeleine. Typical of various foyers run by religious and civic groups throughout France, the Madeleine is the home of a restaurant in which, for a yearly subscription fee, one can dine under the vaulted ceilings on a three-course French meal served by volunteers for a nominal price. The walls of the Foyer are often decorated by local artists.

Bibliography (In French)

 * Dumoulin, Aline; Ardisson, Alexandra; Maingard, Jérôme; Antonello, Murielle; Églises de Paris (2010), Éditions Massin, Issy-Les-Moulineaux, ISBN 978-2-7072-0683-1

title: La Madeleine, Paris


text:
Iamb (poetry)

An iamb or iambus is a metrical foot used in various types of poetry. Originally the term referred to one of the feet of the quantitative meter of classical Greek prosody: a short syllable followed by a long syllable (as in καλή "beautiful (f.)"). This terminology was adopted in the description of accentual-syllabic verse in English, where it refers to a foot comprising an unstressed syllable followed by a stressed syllable (as in abóve). Thus a Latin word like íbī, because of its short-long rhythm, is considered by Latin scholars to be an iamb, but because it has a stress on the first syllable, in modern linguistics it is considered to be a trochee.

Etymology
R. S. P. Beekes has suggested that the ἴαμβος iambos has a Pre-Greek origin. An old hypothesis is that the word is borrowed from Phrygian or Pelasgian, and literally means "Einschritt", i. e., "one-step", compare dithyramb and thriambus, but H. S. Versnel rejects this etymology and suggests instead a derivation from a cultic exclamation. The word may be related to Iambe, a Greek minor goddess of verse, especially scurrilous, ribald humour. In ancient Greece iambus was mainly satirical poetry, lampoons, which did not automatically imply a particular metrical type. Iambic metre took its name from being characteristic of iambi, not vice versa.

Accentual-syllabic use

In accentual-syllabic verse and in modern linguistics an iamb is a foot that has the rhythmic pattern:

Using the 'ictus and x' notation (see systems of scansion for a full discussion of various notations) we can write this as:

The word 'attempt' is a natural iamb:

In phonology, an iambic foot is notated in a flat representation as (σ'σ) or as foot tree with two branches W and S where W = weak and S = strong.

Iambic pentameter is one of the most commonly used measures in English and German poetry, for instance it can be found Shakespeare's Sonnets. A line of iambic pentameter comprises five consecutive iambs.

Iambic trimeter is the metre of the spoken verses in Greek tragedy and comedy, comprising six iambs—as one iambic metrum consisted of two iambs. In English accentual-syllabic verse, iambic trimeter is a line comprising three iambs.

Less common iambic measures include iambic tetrameter (four iambs per line) and iambic heptameter, sometimes called the "fourteener" (seven iambs per line). Lord Byron's also "She Walks in Beauty" exemplifies iambic tetrameter; iambic heptameter is found in Australian poet A. B. "Banjo" Paterson's "The Man from Ironbark". Related to iambic heptameter is the more common ballad verse (also called common metre), in which a line of iambic tetrameter is succeeded by a line of iambic trimeter, usually in quatrain form. Samuel Taylor Coleridge's The Rime of the Ancient Mariner is a classic example of this form.

The reverse of an iamb is called a trochee.

Types of meter
Key:
 * Non-bold = unstressed syllable
 * Bold = stressed syllable

Dimeter
Iambic dimeter is a meter referring to a line consisting of two iambic feet.

The way a crow

Shook down on me.... (Robert Frost, "Dust of Snow")

Trimeter
Iambic trimeter is a meter referring to a line consisting of three iambic feet.

We romped until the pans

Slid from the kitchen shelf; (Theodore Roethke, "My Papa's Waltz")

The only news I know

Is bulletins all day (Emily Dickinson, "The Only News I Know")

Tetrameter
Iambic tetrameter is a meter referring to a line consisting of four iambic feet:

She walks in beauty, like the night

Of cloudless climes and starry skies; (Lord Byron, "She Walks in Beauty")

Pentameter
Iambic Pentameter is a meter referring to a line consisting of five iambic feet:

 * To strive, to seek, to find, and not to yield. (Alfred Tennyson, "Ulysses")

 * Shall I compare thee to a summer's day? (William Shakespeare, Sonnet 18)

(Although, it could be argued that this line in fact reads: Shall I compare thee to a summer's day? Meter is often broken in this way, sometimes for intended effect and sometimes simply due to the sound of the words in the line. Where the stresses lie can be debated, as it depends greatly on where the reader decides to place the stresses. Although in this meter the foot is no longer iambs but trochees.)

 * A horse! A horse! My kingdom for a horse! (William Shakespeare, Richard III)

 * They also serve who only stand and wait. (John Milton, When I Consider How My Light is Spent)

Hexameter
Iambic hexameter is a meter referring to a line consisting of six iambic feet. In English verse, "alexandrine" is typically used to mean "iambic hexameter"

 * Ye sacred Bards, that to ¦ your harps ' melodious strings

 * Sung th'ancient Heroes' deeds (the monuments of Kings) (Michael Drayton, Poly-Olbion)

Heptameter
Iambic Heptameter is a meter referring to a line consisting of seven iambic feet:

 * I s'pose the flats is pretty green up there in Ironbark. (A. B. Paterson, The Man from Ironbark)

Sound change
Through iambic shortening, a word with the shape light–heavy or short–long changes to become light–light; for example, ibī changes to ibi with two short syllables. In modern linguistics this change is sometimes referred to as "trochaic shortening", since íbī has a stress on the first syllable and is thus in modern linguistic terms a trochee.

title: Iamb (poetry)


text:
Secret Affair

Secret Affair are a mod revival band, formed in 1978 and disbanded in 1982 during which period their work is predominantly best-known. They reformed in 2002 and have since then produced an album in 2012.

Career
Formed after the demise of the CBS Records signed power pop band New Hearts, singer Ian Page and guitarist David Cairns spent the second half of 1978 writing songs that would form the basis of the first two Secret Affair albums.

After spending January 1979 demoing songs, Page and Cairns enlisted the services of bassist Dennis Smith from the power pop band Advertising and Young Bucks drummer Seb Shelton. Saxophone player Dave Winthrop, originally playing with the group in his capacity as a session musician, would join as a full-time member of the band later in the year. After the demise of the group in 1982, Cairns went on to form a band called Flag, with Archie Brown from The Bureau and signed a recording contract in the US. He subsequently teamed up with Scottish singer Alan King in Walk on Fire, writing the majority of the band's material and playing keyboards on tours. Signing to MCA in the US, the band released the album Blind Faith in 1990 and toured supporting Foreigner, Nils Lofgren and Dan Reed Network, before Cairns joined Gibson Guitars USA as Special Projects and Events manager. Page released two solo singles before leaving the music industry. Having departed the band in late 1980, drummer Seb Shelton most notably joined up with Kevin Rowland's Dexys Midnight Runners and enjoyed success with them between 1980 and 1982.

Reunion
In June 2002, Page, Cairns, Smith and Winthrop, with drummer Paul Bultitude, reunited for three gigs, including a performance at London's Shepherd's Bush Empire. They returned in June the following year, playing at The Scala in Islington to promote the release of Time For Action: The Anthology (Sanctuary), a major CD retrospective of hits, rarities and previously unreleased tracks. The concert was filmed and released on DVD.

In 2006, Page and Cairns attempted to mend their sometimes fractious relationship and started work on a fourth Secret Affair album, recording five new songs for Peer Music. The album remains unfinished, but in 2010 Captain Mod Records released Secret Affair, The Singles Collection which featured two of the Peer Music tracks, "Soho Dreams" and "Land of Hope".

Page and Cairns reformed Secret Affair as a touring band with a new line-up. In 2009 they have played regularly around the UK and further afield, having played their first ever concerts in Dublin, Madrid, Moscow and Tokyo.

In July 2011, Secret Affair re-recorded "Time For Action" for Save The Children. The latter spearheaded their campaign for health workers in poor countries, and over 42,000 people signed the petition in the UK which was taken to David Cameron in September 2011, resulting in funds for the cause endorsed and approved by the United Nations.

On 5 December 2011, Dave Cairns appeared as a guest on BBC Two programme, Never Mind The Buzzcocks.

Secret Affair released their fourth studio album, Soho Dreams after an absence of 30 years on their label, I-SPY Records, on 10 September 2012, distributed by Code7/Plastic Head and supported by a 13-date UK tour ending at The 229 Club, London, on 24 November. The band continue to tour throughout 2013 and perform their end of year show at the Islington Assembly Hall, 7 December.

On 5 September 2014, Secret Affair begin their '35th Anniversary Tour' of the UK in Gloucester at the Guildhall, followed by Stockton on Tees and ending in Manchester at the Band on the Wall venue, on 13 December 2014, celebrating the 1979 release of their debut album, Glory Boys. In November that year, Sony/BMG/Captain Mod Records released Est 1979; The Secret Affair 35th Anniversary Box Set featuring all four albums from 1979 to the present day. In August 2016, I-SPY Records released Secret Affair's cover of the Northern soul classic "Do I Love You (Indeed I Do)" on 7-inch vinyl only. In September that year the band appeared on Vintage TV, episode 28, performing live in concert three tracks, "Do I Love You' (Indeed I Do)", "I Don't Need No Doctor" and "My World".

Secret Affair celebrated their 40th anniversary in 2019, with a 14 date April and May tour of the UK.

Band members

 * Current members
 * Ian Page - vocals (1979-1982, 2002–present)
 * David Cairns - guitars (1979-1982, 2002–present)

 * Former members
 * Dave Winthrop - saxophone (1979-1982, 2002-2008)
 * Dennis Smith - bass (1979-1982, 2002-2003)
 * Chris Bennett - drums (1979)
 * Seb Shelton - drums (1979–1980)
 * Paul Bultitude - drums (1980-1982, 2002-2003)
 * Simon Hanson - drums (2012, 5 tracks on the 'Soho Dreams' album.)

 * Current touring musicians
 * Steve Rinaldi - trombone (2006–present)
 * Russ Baxter - drums (2008–present)
 * Ed Pearson - bass (2011–present)
 * Stevie Watts - hammond organ (2016–present)
 * John O'Neill - saxophone (2015–present)
 * Tim Pannell - trumpet (2012–present)

 * Former touring musicians
 * Andy Brush - saxophone (2008–2012)
 * Martyn Blagden - trumpet (2008–2012)
 * Sean Kelly - bass (2008–2011)
 * Bryn Barkham - hammond organ (2009–2011)
 * Stephen Wilcock - saxophone (2012–2015)
 * Andy Fairclough - hammond organ (2011–2016)

Studio albums

 * Glory Boys (1979) – UK #41
 * Behind Closed Doors (1980) – UK #48
 * Business as Usual (1982) – UK #84
 * Soho Dreams (2012)

Live albums

 * Live at the Bridge (1999)
 * Time for Action (2012)
 * Time for Action - Live red vinyl  (2022)

Compilation albums

 * Time for Action – The Very Best Of (1997)
 * Time for Action – The Anthology (2003)
 * Mod Singles Collection (2010)
 * 35th Anniversary Box Set (2014)

Other sources

 * Sleevenotes to the CD reissues of Glory Boys, Behind Closed Door and Business As Usual, by Chris Hunt

title: Secret Affair


text:
Spondee

A spondee (Latin: spondeus) is a metrical foot consisting of two long syllables, as determined by syllable weight in classical meters, or two stressed syllables in modern meters. The word comes from the Greek σπονδή, spondḗ, 'libation'.

Libations
Sometimes libations were accompanied by hymns in spondaic rhythm, as in the following hymn by the Greek poet Terpander (7th century BC), which consists of 20 long syllables:

Ζεῦ πάντων ἀρχά,

πάντων ἀγήτωρ,

Ζεῦ, σοὶ σπένδω

ταύτᾱν ὕμνων ἀρχάν.

"Zeus, Beginning of all things,

Leader of all things,

Zeus, I make a libation to Thee

this beginning of (my) hymns."

In hexameter poetry
However, in most Greek and Latin poetry, the spondee typically does not provide the basis for a metrical line in poetry. Instead, spondees are found as irregular feet in meter based on another type of foot.

For example, the epics of Homer and Virgil are written in dactylic hexameter. This term suggests a line of six dactyls, but a spondee can be substituted in most positions. The first line of Virgil's Aeneid has the pattern dactyl-dactyl-spondee-spondee-dactyl-spondee:

 * – ᴗ ᴗ | – ᴗ ᴗ | – – | – – | – ᴗ ᴗ | – –
 * arma virumque canō, Trōiae quī prīmus ab ōrIs
 * 'I sing of arms and of the man, who first from the shores of Troy...'

Most of Virgil's lines, like the above, are a mixture of dactyls and spondees. However, sometimes he will begin a line with three or four spondees for special effect, such as the following, which describes how Aeneas and his companion made their way slowly down a dark passage into the Underworld. In this line all the feet are spondaic except the fifth:

 * – – | – – | – – | – – | – ᴗ ᴗ | – –
 * ībant obscūrī sōlā sub nocte per umbram
 * 'They began moving in the darkness beneath the lonely night through the shadow'

Spondees can also add solemnity, as in the following lines where Dido, Queen of Carthage, curses Aeneas after he has abandoned her. The first line begins with three spondees, the second with four:

 * – – | – – | – – | – ᴗ ᴗ | – ᴗ ᴗ | – –
 * – – | – – | – – | – – | – ᴗ ᴗ | – –
 * Sōl, quī terrārum flammīs oper(a) omnia lūstrās,
 * tūqu(e) hār(um) interpres cūrār(um) et cōnscia Iūnō,...
 * 'O Sun, who surveyest all the works of the world with thy flames,
 * and Thou, interpreter and witness of these sorrows, Juno...'

Only two hexameter lines in Latin poetry use spondees throughout the verse. One is by Ennius:
 * ollī respondit rēx Albāī Longāī
 * 'to him replied the King of Alba Longa'
 * 'to him replied the King of Alba Longa'

The other is in an elegiac couplet in the last poem of Catullus (116), perhaps mocking the poetic style of his addressee:

 * – – | – ᴗ ᴗ | – || – ᴗ ᴗ | – ᴗ ᴗ | –
 * quī tē lēnīrem nōbīs, neu cōnārēre
 * tēl(a) īnfesta meum mitter(e) in usque caput
 * 'with which I could soften your attitude towards me, and so that you would not try
 * to keep hurling hostile missiles at my head'
 * to keep hurling hostile missiles at my head'

Spondees in English verse
In Latin and Greek meter spondees are easily identified because the distinction between long and short syllables is unambiguous. In English meter indisputable examples are harder to find because metrical feet are identified by stress, and stress is a matter of interpretation.

For example, the first part of this line from Shakespeare's Troilus and Cressida (in iambic pentameter) would normally be interpreted as two spondees: ''Crý, crý! Tróy búrns, or élse let Hélen gó.''

The effect of spondees in verse is often to slow the line down and to represent slow movement. Thus Alexander Pope writes, in a poem illustrating how the sound of the words should imitate their meaning: When Ajax strives, some Rock's vast Weight to throw,

The Line too labours, and the Words move slow; In the first line above, most of the syllables, even those in weak positions, are long and heavy: "A-jax strives some Rock's vast weight"; only the last foot, "to throw", is a true iamb. The final foot of the second line "move slow" is another spondee replacing an iamb.

John Masefield also uses spondees effectively in the line: Dirty British / coaster with a / salt-caked / smoke-stack Here the last four syllables make two spondees, contrasting with the eight short syllables in the first two feet. The length and weight of the last four syllables derives partly from the fact that all of them are closed by one or more consonants, and partly from the fact that all of them are stressed.

Another Masefield poem, Sea Fever (1902), which includes spondees contains these lines: And the wheel's kick and the wind's song and the white sail's shaking,

And a grey mist on the sea's face, and a grey dawn breaking.

title: Spondee


text:
User talk:Montrealais/Talk archive 3

 * Talk archive 1
 * Talk archive 2
 * Talk archive 4
 * Talk archive 5
 * Talk archive 6

John Ralston Saul
Dear Montre: Nice cat! I wanted to thank you for answering my question on the John Saul matter. I will proceed to write about the mystery writer, probably using a parenthesis.

Thanks and God bless you.

Sincerely yours, Antonio Life of the Party Martin''

 * You're welcome. Actually I don't think a parenthesis is necessary as long as you put a disamb notice at the top. – Montréalais 15:48, 4 Aug 2003 (UTC)

Moldova, Angles
Hi, I saw you deleted Moldova from List of etymologies of country subdivision names with the comment moldova is an independent state. Actually, Moldova/Moldovia region is now divided between Romania and the state of Moldova (also called Bessarabia), so there is currently a Romanian region called Moldova.

Also about the ethymology of England, I heard they were called after the Angles – a West Germanic people that came from Angul (hook), a district in Schleswig (now Angeln).

 * The explanation I always heard is that their name came from that root, but because they were fishers. I think the notion that they came from a "hook shaped land" requires us to believe that they named themselves after looking at an overhead map, rather as if the ancient pre-roman Italics had decided to call themselves Bootfolk.
 * you're right :-)
 * As for Moldova, the article linked to was about the state and made no mention of this Romanian Moldova
 * Yes, it should have linked to Moldovia.
 * (do they have the same sort of issues as Greece has with FYROM?)
 * No, it was basically the same teritory, until the Eastern side of Moldova was occupied by the Soviets in the 1948 and was declared independent in the 90s after the fall of the Soviet Union (as Moldova).
 * The article linked to in the subnational entities article should probably be either Moldavia or a new article about the subnational region of Romania. – Montréalais

Gentlemen! The British Isle was indeed invaded by a wave of Angles, Saxons, and Jutes. All from the Jutland Penninsula. Angeln is indeed in "Schleswig" – but then there was no Germany... So these people were closely related to the other two groups and spoke a language related to Frisian and Plattdeutsch – hence the Anglo basis of the English language. Plus the name of the poor people and of their language which since became the numero uno in world communication. Through most of history, Angeln (and all of Slesvig for that matter) was Danish. Most of the toponymics in Angeln are very Danish indeed. But it seems to me that the Danes came from the East and actually conquered Jutland – imposing their language on the poor Jutes – including Angeln – after the great invasion of Britain. Enough of that. Moldova is not only the name of a country and a Romanian region (formerly forming one principality) but also of a river. And the name of the river seems to be of... Gothic origin – related to English "mold": fertile soil. Romanian (incl. Moldavian) toponymics is an odd science. Virtually all old names (rivers in particular) are non-Latin. One is, as mentioned, Gothic. VW —Preceding unsigned comment added by 80.161.120.82 (talk) 22:40, 30 January 2008 (UTC)

Press release
I understand that your absence from the mailing list(s) may be in the interest of maintaining sanity. The following by Anthere, therefore, may not have come to your attention. I am very excited by the first press release for wikimedia. Incidently, I thank the one who suggested a multilingual release (I think it is Mav, but messages get a lost among 95 digests...well..). Whether the english wikipedia has a release for 200000 is good and

should be done, but a common release for the fundation is ... the good way imho.

Anticipating :

When is decided that the time is the good one for the press release, could we organise a sort of schedule ? Each time the english wiki has a press release, everything is slowed down, last time the german release also had a clear impact on availability. Should not we a bit try to schedule an order among wikis, or some rough dates, so we can smooth the peak ?

Eclecticology (and other canadian friends)...could you look for sending the release to a major canadian press web site for the french wikipedia ? We need more canadians on the fr [;-)] As I am on the west coast and you are in Montreal you may be in a much better position to make appropriate contact with the French media.  ☮ Eclecticology 19:05, 2003 Aug 8 (UTC)

 * Given the existence of email, I don't see why it wouldn't be simpler for one person just to email the Quebec newspapers at the same time as the French, Belgian, etc., ones, but if I'm needed for some reason I'll do it. – Montréalais 19:10, 8 Aug 2003 (UTC)

 * Hello Matt. The thing is, the pb is not sending the mail or courriel, but rather where to send it. I do not know your newspapers or your websites. I know the french major ones, the american and british major ones. Not the belgium, or the swiss, or the algerian or marrocans and not the canadians :-) We have a partial translation of the press release right now. We'll make it nice, with if possible a list of good french speaking articles. I would like that coverage is not limited to France though. I really would like so. Can you help us provide a couple of reference ? We can sent then afterward or you, as you wish, but please, help relocate the french speaking wikipedia somewhere else than France. Tu comprend ? User:Anthere

Detailed Canada maps
Colouring those seemingly tiny Arctic islands in Image:Provmap-nu.png must be quite a feat! It actually looks better coloured than just outlined. :-p --Menchi 19:53, Aug 10, 2003 (UTC)

Not really ^_^ all you have to do is select the rough borders, deselect the sea, and colour in the rest. – Montréalais

I really like the maps you done for the Canadian provinces. Awesome. dave 21:37, 10 Aug 2003 (UTC)

Photo comments
I think your photos are nicely composed (and of very useful subjects) but your colours on occasion seem washed out and the sky is glary. Are your photos a little overexposed? --Robert Merkel 00:41, 11 Aug 2003 (UTC)

 * No. They're exposed for the darker parts of the photos, e.g. at Central Park they're exposed for the masonry and at Quebec City for the buildings in the foreground.  The options would be to take the photos at a different time of day (e.g. closer to dusk) or to take them with a filter to reduce contrast.  Dark subjects with a bright background make for tricky metering, and even professionals get it wrong sometimes: much of John Sayles' Sunshine State, starring black actors in mid-day Florida, was washed out at the top, and blinding in the background.  Koyaanis Qatsi 03:24, 11 Aug 2003 (UTC)

 * I'm doing better these days; what I do is instead of lightening the whole photo in Photoshop, I select out the sky and just lighten the darker subject. Some examples will be available in a month or two on my website, http://www.metrodemontreal.com/ . Which photos did you notice this problem on? If it seem appropriate I might try to find the original and try to fix it. – Montréalais 04:05, 11 Aug 2003 (UTC)
 * P.S. Just took a look at the Central Park photo. I took it in January, and the Quebec City ones in November, so it's quite possible the sky really was that colour. – Montréalais

Mammals
Hi Montréalais,

On December 9, 2002, on Talk:Carolus Linnaeus you wrote:

 * " The group "mammalia" are named for their breasts because he wanted to encourage women to breast-feed their infants. "
 * Didn't I see this on Bad jokes and other deleted nonsense? What's the scoop?

If you're still curious, the answer is on Carolus Linnaeus now.

Cheers! Cyan 23:13, 13 Aug 2003 (UTC)

-

heartfelt apology
I am sincerly sorry about what happened, but I may have inadvertently destroyed one of your edits on the 2003 US-Canada blackout. I prostrate myself. I will offer a public apology with full explication at the relevant talk page. Respectfully. -- Cimon Avaro on a pogostick

 * blink* *blink* No worries, happens all the while. – Montréalais 23:18, 14 Aug 2003 (UTC)

---

image comments
Is it ok if I put your image link here: Brilliant_pictures ? Fantasy 11:05, 15 Aug 2003 (UTC)

 * By all means. Thank you! – Montréalais 14:21, 15 Aug 2003 (UTC)

-

Spanish autonomías
Hi, there!

Thanks for all your work on the autonomous communities of Spain. I don't understand why you care so much about that topic, but thanks anyway :-) — Miguel

 * Porque me ha gustado tanto visitar España; entonces es un tema que me apasiona :) ¡espera que haya más! – Montréalais 03:18, 19 Aug 2003 (UTC)

 * Comme je suis sûr tu sais, en Espagne on s'intéresse aussi au Canada, en ce qui concerne le Québec et l'organisation de l'état fédéral. Trouves-tu des points de comparaison entres les deux pays? -- Miguel

 * Je n'en sais pas assez de détails sur la balance de pouvoirs en Espagne mais l'idée du fédéralisme assymétrique est une que prône mon parti politique quant au Québec. L'Espagne offre un exemple interessant dans son traitement des autonomies 'historiques'. – Montréalais 19:34, 19 Aug 2003 (UTC)

The Welsh nation (in Principality)
Hi!

I get a bit curious by your notion of Wales as a subnational entity. I doesn't fit with my understanding. Neither the one I've acquired in school, nor what I've learned through mates at work.

About Asturia I've absolutely no clue, so I don't comment on that. :-) --Ruhrjung 06:26, 23 Aug 2003 (UTC)

 * Perhaps I've fudged some relevant political subtlety, but the article on the UK lists Wales as one of the countries that make up the UK. – Montréalais 15:58, 23 Aug 2003 (UTC)

 * Not political but national. ;-) --Ruhrjung 03:46, 13 Sep 2003 (UTC)

St. or Saint?
Hi, I just responded to your (very very old :)) comment on Louis St. Laurent...I've never seen it as Saint Laurent either, and the National Archives biography has St. Laurent. Some other articles have St. expanded though, like Saint Lawrence River, so I'm not sure. Adam Bishop 23:50, 26 Aug 2003 (UTC)

 * Responded to you at Talk:Louis Stephen Saint Laurent. – Montréalais 23:55, 26 Aug 2003 (UTC)

--

Montreal metro map
About the map of montreal metro you just uploaded – I was going to say that even if the drawing is by you, the design itself will be (C) montreal metro, so there's a problem. then I read your user page & you say you worked there -- did you by any chance design the original? if so, from another metrophile ... wow, cool! :-) -- Tarquin 20:08, 28 Aug 2003 (UTC)

 * Thanks for the concern, but I don't think it will be. There are several important differences between my design and theirs, chief among them is that mine is done right. (The new maps with Longueuil-U de S are typeset hideously!)
 * I am (mercifully) not responsible for theirs; mine is designed for my personal website, http://www.metrodemontreal.com/ . I did work at the STM (doing other stuff) for a few months in summer 2002; it was a lot of fun.
 * - Montréalais

I've uploaded a version saved from Gimp explicitly without the gamma info; it looks a little cleaner than the version with correction for the correction. :) (Photoshop, apparently, saves the gamma wrong and you can't disable it as far as I can see.) It's also reduced to 8-bit color for better compression. --Brion 07:19, 29 Aug 2003 (UTC)

 * Looks faboo. Arigatou gozaiimasu! – Montréalais 15:57, 29 Aug 2003 (UTC)

Spanish province names
Wikipedia convention for city names is generally not to have the name of the country, province, state, etc. in the article name (Canada & the US are exeptions). If for some reason, there is reason to restore the province names in this perticular case, it's easy to revert to your last version. – Efghij 02:48, Aug 31, 2003 (UTC)

 * Very well, let's put it to the Wikipedians: Wikipedia talk:Naming conventions (city names)/Spain. In the meantime, may I ask you to please not alter any more articles until there is a consensus?- Montréalais 02:56, 31 Aug 2003 (UTC)

Fair enough. No point in going through the trouble of correcting all those articles if there's a possiblity of changing the standard. – Efghij 03:10, Aug 31, 2003 (UTC)

Dorval Airport
Sorry for moving the Montreal Dorval article incorrectly. I've never moved a page before. Thanks to you, now I know how. dave 02:31, 12 Sep 2003 (UTC)

 * No worries, I learned how the same way originally ^_^ – Montréalais 05:15, 12 Sep 2003 (UTC)

Ontario general election, 2003
Hi, are you planning on creating an article for the Ontario election like you did for the PEI one? I was thinking about making one, but I won't have time to do it tomorrow, at least. But if you were going to do it anyway, I won't worry about it :) Adam Bishop 01:59, 2 Oct 2003 (UTC)

 * On its way. – Montréalais 14:38, 2 Oct 2003 (UTC)

Please make sure that additions to the main page fit the 800x600 screen. (Add one, take one. I had to take one for you.) Cheers, Jiang 21:52, 2 Oct 2003 (UTC)

McGuinty isn't really premier yet, is he? Eves would still be premier until McGuinty is sworn in, whenever that happens. Adam Bishop 02:35, 3 Oct 2003 (UTC)

 * If McGuinty is struck by lightning before he's sworn in, I'll change it. – Montréalais 02:37, 3 Oct 2003 (UTC)

--

TTC
Hi, Montréalais. I was mucking about with your list of Toronto subway stations because like most Torontonians I don't consider the LRT part of the subway. I then found out that the TTC apparently considers it part, though -- god only knows why, since it's entirely above ground and inaccessible to subway trains, so you have to transfer from the LRT to the subway. Anyway, I tried to restore the article as best I could and add some clarification about the LRT, but perhaps you should check that I haven't unwittingly vandalized it. Trontonian 20:26, 20 Oct 2003 (UTC)

 * And now I have added brief notes about where the stations are. I think it makes the article more useful, but it clutters it up. If you don't like it or had other plans for the article, revert it and let me know -- I'll put the sites in a separate article. Trontonian

 * If it's OK with you, I'll reformat the list as a table, using your format from the list of Montreal metro stations. Trontonian

 * I converted it to a table. It's not as pretty as yours, but I may work on it tomorrow. Trontonian 03:42, 21 Oct 2003 (UTC)

title: User talk:Montrealais/Talk archive 3


text:
Amphibrach

An amphibrach is a metrical foot used in Latin and Greek prosody. It consists of a long syllable between two short syllables. The word comes from the Greek ἀμφίβραχυς, amphíbrakhys, "short on both sides".

English
In English accentual-syllabic poetry, an amphibrach is a stressed syllable surrounded by two unstressed syllables. It is rarely used as the overall meter of a poem, usually appearing only in a small amount of humorous poetry, children's poetry, and experimental poems. The individual amphibrachic foot often appears as a variant within, for instance, anapaestic meter.

It is the main foot used in the construction of the limerick, as in "There once was / a girl from / Nantucket." It was also used by the Victorians for narrative poetry, e.g. Samuel Woodworth's poem "The Old Oaken Bucket" (1817) beginning "How dear to / my heart are / the scenes of / my childhood." W. H. Auden's poem "O where are you going?" (1931) is a more recent and slightly less metrically-regular example. The amphibrach is also often used in ballads and light verse, such as the hypermetrical lines of Sir John Betjeman's poem "Meditation on the A30" (1966).

Russian
Amphibrachs are a staple meter of Russian poetry. A common variation in an amphibrachic line, in both Russian and English, is to end the line with an iamb, as Thomas Hardy does in "The Ruined Maid" (1901): "Oh did n't / you know I'd / been ru in'd / said she".

Polish
Amphibrachic metre is very popular in Polish literature. It can be found in romantic poetry in some works by Adam Mickiewicz and Juliusz Słowacki. The best known example is "Deszcz jesienny" (English: "Autumn rain"; 1908) by Leopold Staff. The amphibrach is believed to be suitable for lullabies. Usually Polish amphibrachic lines have feminine endings (for example the tetrametre ◡–◡◡–◡◡–◡◡–◡) but some poets experiment with masculine lines as well. In the poem "Walc" (English: "Waltz") from Czesław Miłosz's 1945 collection Ocalenie the first line is feminine, ◡–◡◡–◡◡–◡◡–◡, the second masculine, ◡–◡◡–◡◡–◡◡–, and so on. Jan Bolesław Ożóg experimented with irregular amphibrachic verse with different numbers of feet in successive lines. An example of this way of making verse is the poem Jemioła (Mistletoe), included in the book with the same title (1966).

Dr. Suess
Some books by Dr. Seuss contain many lines written in amphibrachs, such as these from If I Ran the Circus (1956):

 * All ready / to put up / the tents for / my circus.
 * I think I / will call it / the Circus / McGurkus.

 * And NOW comes / an act of / Enormous / Enormance!
 * No former / performer's / performed this / performance!

Famous Blue Raincoat
Much of Leonard Cohen's song "Famous Blue Raincoat" (1971) is written in amphibrachs - e.g. the first verse (apart from the first foot of the third line, which is a spondee):

 * It's four in / the morning, / the end of / December
 * I'm writing / you now just / to see if / you're better
 * New York / is cold, but / I like where / I'm living
 * There's music / on Clinton / Street all through / the evening.

Versace
Migos' 2013 hit " Versace" popularized rapping in (amphibrachic) triplets: Versace / Versace / Medusa / head on me / like I'm 'Lum / inati

I know that / you like it / Versace / my neck and / my wrist is / so sloppy

Versace / Versace / I love it / Versace / the top of / my Audi

My plug is / John Gotti / he give me / the ducks and / I know that / they mighty

title: Amphibrach