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Commodore (rank)

Commodore is a senior naval rank used in many navies which is equivalent to brigadier or brigadier general and air commodore. It is superior to a navy captain, but below a rear admiral. It is either regarded as the most junior of the flag officers rank or may not hold the jurisdiction of a flag officer at all depending on the officer's appointment. Non-English-speaking nations commonly use the rank of flotilla admiral, counter admiral, or senior captain as an equivalent, although counter admiral may also correspond to rear admiral lower half abbreviated as RDML.

Traditionally, "commodore" is the title for any officer assigned to command more than one ship, even temporarily, much as "captain" is the traditional title for the commanding officer of a single ship even if the officer's official title in the service is a lower rank. As an official rank, a commodore typically commands a flotilla or squadron of ships as part of a larger task force or naval fleet commanded by an admiral. A commodore's ship is typically designated by the flying of a broad pennant, as compared to an admiral's flag.

"Commodore" is typically regarded as a one-star rank with a NATO code of OF-6, known in the U.S. as "rear admiral (lower half)", but whether it is regarded as a flag rank varies among countries.

It is sometimes abbreviated as "Cdre" in British Royal Navy, "CDRE" in the US Navy, "Cmdre" in the Royal Canadian Navy, "COMO" in the Spanish Navy and in some navies speaking the Spanish language, or "CMDE" as used in the Indian Navy and in navies of several other countries.

Etymology
The rank of commodore derives from the French commandeur, which was one of the highest ranks in the orders of knighthood, and in military orders the title of the knight in charge of a commande (a local part of the order's territorial possessions).

History
The Dutch Navy also used the rank of commandeur from the end of the 16th century for a variety of temporary positions, until it became a conventional permanent rank in 1955. The Royal Netherlands Air Force has adopted the English spelling of "commodore" for an equivalent rank.

In the Royal Navy, the position was introduced in the 17th century to combat the cost of appointing more admirals—a costly business with a fleet as large as the Royal Navy's at that time.

The rank of commodore was at first a position created as a temporary title to be bestowed upon captains who commanded squadrons of more than one vessel. In many navies, the rank of commodore was merely viewed as a senior captain position, whereas other naval services bestowed upon the rank of commodore the prestige of flag officer status.

United States
In 1899, the substantive rank of commodore was discontinued in the United States Navy, but revived during World War II in both the U.S. Navy and U.S. Coast Guard. It was discontinued as a rank in these services during the postwar period, but as an appointment, the title "commodore" was then used to identify senior U.S. Navy captains who commanded squadrons of more than one vessel or functional air wings or air groups that were not part of a carrier air wing or carrier air group. Concurrently, until the early 1980s, U.S. Navy and U.S. Coast Guard captains selected for promotion to the rank of rear admiral (lower half), would wear the same insignia as rear admiral (upper half), i.e., two silver stars for collar insignia or sleeve braid of one wide and one narrow gold stripe, even though they were actually only equivalent to one-star officers and paid at the one-star rate.

To correct this inequity, the rank of commodore as a single star flag officer was reinstated by both services in the early 1980s. This immediately caused confusion with those senior U.S. Navy captains commanding destroyer squadrons, submarine squadrons, functional air wings and air groups, and so on, who held the temporary "title" of commodore while in their major command billet. As a result of this confusion, the services soon renamed the new one-star rank as commodore admiral (CADM) within the first six months following the rank's reintroduction. However, this was considered an awkward title and the one-star flag rank was renamed a few months later to its current title of rear admiral (lower half), later abbreviated by the U.S. Navy and U.S. Coast Guard as RDML.

The "title" of commodore continues to be used in the U.S. Navy and Coast Guard for those senior captains in command of organizations consisting of groups of ships or submarines organized into squadrons; air wings or air groups of multiple aviation squadrons other than carrier air wings (the latter whose commanders still use the title "CAG"); explosive ordnance disposal (EOD), mine warfare and special warfare (SEAL) groups; Mobile Inshore Underwater Warfare (MIUW) groups; and construction (SeaBee) regiments. Although not flag officers, modern day commodores in the U.S. Navy rate a blue and white command pennant, also known as a broad pennant, that is normally flown at their headquarters facilities ashore or from ships that they are embarked aboard when they are the senior officer present afloat (SOPA).

Argentina
In the Argentine Navy, the position of commodore was created in the late 1990s, and is usually, but not always, issued to senior captains holding rear-admirals' positions. It is not a rank but a distinction and, as such, can be issued by the chief of staff without congressional approval. Its equivalents are colonel-major in the Army and commodore-major in the Air Force. It is usually—but incorrectly—referred to as "navy commodore", to avoid confusion with the "air force commodore", which is equivalent to the navy's captain and army's colonel. The sleeve lace is identical to that of the Royal Navy, and wears one star on the epaulette.

Naval rank

 * Commodore (Canada)
 * Commodore (Finland)
 * Commodore (India)
 * Commodore (Sri Lanka)
 * Commodore (Royal Navy)
 * Commodore (United States)
 * Commodore (Pakistan)
 * Kommodore

Air force ranks
Commodore, in Spanish comodoro, is a rank in the Argentine Air Force. This rank is the equivalent of a colonel in the Argentine Army, and a colonel or group captain in other air forces of the world. The Argentine rank below commodore is the rank of vice-commodore (Spanish vicecomodoro) equivalent to a lieutenant-colonel in the Argentine Army, and a lieutenant-colonel or wing commander in other air forces.

Commodore is a rank in the Royal Netherlands Air Force. It is a one-star rank and has essentially the same rank insignia as the British air commodore.

Many air forces use the rank of air commodore. This rank was first used by the Royal Air Force and is now used in many countries such as Australia, Bangladesh, Greece, India, New Zealand, Nigeria, Pakistan, Thailand and Zimbabwe. It is the equivalent rank to the navy rank of "commodore", and the army ranks of brigadier and brigadier general.

The German air force used the concept of a unit commodore for the commander of a wing, usually in the rank of colonel (OF-5).

Merchant Service (Merchant Marine) rank and Yacht Club chief directors
Commodore is also a title held by many captains as recognition of exceptional navigation ability and seagoing seniority in the Merchant Service, and by the directors of a few yacht clubs and boating associations. Commodores 'in command' as Master aboard Merchant Marine ships wear distinctive rank and cap insignia denoting their honorific high rank position. In a few country the honorific high position of commodore it is indicated with the high rank denomination of senior captain. Traditionally, commodore is the title of the president of a yacht club.

Convoy commodore
During wartime, a shipping convoy will have a ranking officer—sometimes an active-duty naval officer, at other times a civilian master or retired naval officer—designated as the convoy commodore. This title is not related to the individual's military rank (if any), but instead is the title of the senior individual responsible for the overall operation of the merchant ships and naval auxiliary ships that make up the convoy. The convoy commodore does not command the convoy' escort forces (if any), which are commanded by a naval officer who serves as escort commander.

Commodore in Yachting Leadership
Civilian yacht clubs, yachting associations and fellowships with formal hierarchical structures, began to use the title "commodore" in countries around the world       for their presidents in the early twentieth century along with "vice commodore" in the same manner as "vice president,"and "rear-commodore" and "port captain' or "international bridge member" in the same manner as board members.

Commodores, vice-commodores and rear-commodores are also known as civilian flag officers because they have an epaulettes, regalia and maritime flags with designated symbols and number of stars for their ranks. Many of the clubs that are more than a century old, such as the Los Angeles Yacht Club have formal ceremonies, where commodores from more than 100 surrounding yacht clubs, flag officers of the US Navy and Coast Guard attend a ceremony at the beginning of the year. The ceremony includes a bagpipe entrance, a presentation of the country flag by commissioned officers of the country's navy and a cannon shot upon the raising of each individual officer's flags on a flag staff, (also known as flagpoles) for each flag officer (commodore, vice commodore, rear commodore) as their term of office officially begins. Sometimes a trumpet fanfare is also include for special occasions like ribbon cutting in 2019 for the 50th Transpacific Yacht Race. Salutes are given to commodores for special ceremonies, including opening days of the racing season.

Other uses
The U.S. Coast Guard Auxiliary also employs variants of the title of commodore. Members of the Auxiliary serve in the Coast Guard's uniformed auxiliary service and they do not have military rank, but who do wear modified U.S. Coast Guard uniforms and U.S. military-style officer rank insignia to indicate office. Auxiliary members who have been elected or appointed to positions in the highest levels of the organization, similar in nature to active and reserve rear admirals and vice admirals use the term commodore (e.g., district commodore, assistant national commodore, deputy national commodore, national commodore, etc.). These Coast Guard auxiliarists may permanently append the title commodore, sometimes abbreviated COMO, to their names (e.g., Commodore James A. Smith, National Commodore; or COMO Jim Smith, (NACO)).

In the Philippine Coast Guard Auxiliary—PCGA—each of the directors in command of the ten Coast Guard Auxiliary districts are commodores, as well as most of the Deputy National Directors (some may be rear admirals). Commodore is abbreviated to COMMO in the PCGA.

Vanderbilt University's intercollegiate athletics teams are nicknamed the "Commodores", a reference to Cornelius Vanderbilt's self-appointed title (he was the master of a large shipping fleet).

In the U.S. Sea Scouting program (which is part of the Boy Scouts of America), all National, Regional, Area, and Council committee chairs are titled as commodore, while senior committee members are addressed as vice commodore. Ship committee chairs do not hold this recognition.

title: Commodore (rank)


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Cauchy–Riemann equations

In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which form a necessary and sufficient condition for a complex function of a complex variable to be complex differentiable.

These equations are

and

where $u(x, y)$ and $v(x, y)$ are real differentiable bivariate functions.

Typically, $u$ and $v$ are respectively the real and imaginary parts of a complex-valued function $f(x + iy) = f(x, y) = u(x, y) + iv(x, y)$ of a single complex variable $z = x + iy$ where $x$ and $y$ are real variables; $u$ and $v$ are real differentiable functions of the real variables. Then $f$ is complex differentiable at a complex point if and only if the partial derivatives of $u$ and $v$ satisfy the Cauchy–Riemann equations at that point.

A holomorphic function is a complex function that is differentiable at every point of some open subset of the complex plane $C$. It has been proved that holomorphic functions are analytic and analytic complex functions are complex-differentiable. In particular, holomorphic functions are infinitely complex-differentiable.

This equivalence between differentiability and analyticity is the starting point of all complex analysis.

History
The Cauchy–Riemann equations first appeared in the work of Jean le Rond d'Alembert. Later, Leonhard Euler connected this system to the analytic functions. Cauchy then used these equations to construct his theory of functions. Riemann's dissertation on the theory of functions appeared in 1851.

Simple example
Suppose that $$z = x + iy$$. The complex-valued function $$f(z) = z^2$$ is differentiable at any point $$ in the complex plane. $$f(z) = (x + iy)^2 = x^2 - y^2 + 2ixy$$ The real part $$u(x,y)$$ and the imaginary part $$v(x, y)$$ are $$\begin{align} u(x, y) &= x^2 - y^2 \\ v(x, y) &= 2xy \end{align}$$ and their partial derivatives are $$u_x = 2x;\quad u_y = -2y;\quad v_x = 2y;\quad v_y = 2x$$

We see that indeed the Cauchy–Riemann equations are satisfied, $$u_x = v_y$$ and $$u_y = -v_x$$.

Interpretation and reformulation
The Cauchy-Riemann equations are one way of looking at the condition for a function to be differentiable in the sense of complex analysis: in other words, they encapsulate the notion of function of a complex variable by means of conventional differential calculus. In the theory there are several other major ways of looking at this notion, and the translation of the condition into other language is often needed.

Conformal mappings
First, the Cauchy–Riemann equations may be written in complex form

In this form, the equations correspond structurally to the condition that the Jacobian matrix is of the form $$\begin{pmatrix} a & -b \\ b & a  \end{pmatrix},$$ where $$ a = \partial u/\partial x = \partial v/\partial y$$ and $$ b = \partial v/\partial x = -\partial u/\partial y$$. A matrix of this form is the matrix representation of a complex number. Geometrically, such a matrix is always the composition of a rotation with a scaling, and in particular preserves angles. The Jacobian of a function $f(z)$ takes infinitesimal line segments at the intersection of two curves in z and rotates them to the corresponding segments in $f(z)$. Consequently, a function satisfying the Cauchy–Riemann equations, with a nonzero derivative, preserves the angle between curves in the plane. That is, the Cauchy–Riemann equations are the conditions for a function to be conformal.

Moreover, because the composition of a conformal transformation with another conformal transformation is also conformal, the composition of a solution of the Cauchy–Riemann equations with a conformal map must itself solve the Cauchy–Riemann equations. Thus the Cauchy–Riemann equations are conformally invariant.

Complex differentiability
Let $$ f(z) = u(z) + i \cdot v(z) $$ where $u$ and $$v$$ are real-valued functions, be a complex-valued function of a complex variable $ z = x + i y$  where $ x$  and $ y$  are real variables. $ f(z) = f(x + iy) = f(x,y)$ so the function can also be regarded as a function of real variables $ x$  and $ y$. Then, the complex-derivative of $ f $ at a point $ z_0=x_0+iy_0 $  is defined by $$ f'(z_0) =\lim_{\underset{h\in\Complex}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} $$ provided this limit exists (that is, the limit exists along every path approaching $ z_{0} $, and does not depend on the chosen path).

A fundamental result of complex analysis is that $$f$$ is complex differentiable at $$z_0$$ (that is, it has a complex-derivative), if and only if the bivariate real functions $$u(x+iy)$$ and $$v(x+iy)$$ are differentiable at $$(x_0,y_0),$$ and satisfy the Cauchy–Riemann equations at this point.

In fact, if the complex derivative exists at $ z_0$, then it may be computed by taking the limit at $ z_0$ along the real axis and the imaginary axis, and the two limits must be equal. Along the real axis, the limit is $$\lim_{\underset{h\in\Reals}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} = \left. \frac{\partial f}{\partial x} \right \vert_{z_0}$$ and along the imaginary axis, the limit is $$\lim_{\underset{h\in \Reals}{h\to 0}} \frac{f(z_0+ih)-f(z_0)}{ih} = \left. \frac{1}{i}\frac{\partial f}{\partial y} \right \vert _{z_0}.$$

So, the equality of the derivatives implies $$i \left. \frac{\partial f}{\partial x} \right \vert _{z_0} = \left. \frac{\partial f}{\partial y} \right \vert _{z_0}$$ which is the complex form of Cauchy–Riemann equations at $ z_0$.

(Note that if $$f$$ is complex differentiable at $$z_0$$, it is also real differentiable and the Jacobian of $$f$$ at $$z_0$$ is the complex scalar $$f'(z_0)$$, regarded as a real-linear map of $$\mathbb C$$, since the limit $$|f(z)-f(z_0)-f'(z_0)(z-z_0)|/|z-z_0|\to 0$$ as $$z\to z_0$$.)

Conversely, if $$ is differentiable at $ z_{0} $  (in the real sense) and satisfies the Cauchy-Riemann equations there, then it is complex-differentiable at this point. Assume that $z$ as a function of two real variables $$ and $f$ is differentiable at $z_{0}$ (real differentiable). This is equivalent to the existence of the following linear approximation $$ \Delta f(z_0) = f(z_0 + \Delta z) - f(z_0) = f_x \,\Delta x + f_y \,\Delta y + \eta(\Delta z)$$where $ f_x = \left. \frac{\partial f}{\partial x}\right \vert _{z_0} $, $ f_y = \left. \frac{\partial f}{\partial y} \right \vert _{z_0} $, $z = x + iy$, and $\eta(\Delta z) / |\Delta z| \to 0$ as $Δz → 0$.

Since $ \Delta z + \Delta \bar{z}= 2 \, \Delta x $ and $ \Delta z - \Delta \bar{z}=2i \, \Delta y $, the above can be re-written as

$$ \Delta f(z_0) = \frac{f_x - if_y}{2} \, \Delta z + \frac{f_x + if_y}{2} \, \Delta \bar{z} + \eta(\Delta z)\, $$$$\frac{\Delta f}{\Delta z} = \frac{f_x -i f_y}{2}+ \frac{f_x + i f_y}{2}\cdot \frac{\Delta\bar{z}}{\Delta z} + \frac{\eta(\Delta z)}{\Delta z}, \;\;\;\;(\Delta z \neq 0). $$

Now, if $\Delta z$ is real, $\Delta\bar z/\Delta z = 1$, while if it is imaginary, then $\Delta\bar z/\Delta z=-1$. Therefore, the second term is independent of the path of the limit $\Delta z\to 0$ when (and only when) it vanishes identically: $f_x + i f_y=0$, which is precisely the Cauchy–Riemann equations in the complex form. This proof also shows that, in that case, $$\left.\frac{df}{dz}\right|_{z_0} = \lim_{\Delta z\to 0}\frac{\Delta f}{\Delta z} = \frac{f_x - i f_y}{2}.$$

Note that the hypothesis of real differentiability at the point $$z_0$$ is essential and cannot be dispensed with. For example, the function $$f(x,y) = \sqrt{|xy|}$$, regarded as a complex function with imaginary part identically zero, has both partial derivatives at $$(x_0,y_0)=(0,0)$$, and it moreover satisfies the Cauchy–Riemann equations at that point, but it is not differentiable in the sense of real functions (of several variables), and so the first condition, that of real differentiability, is not met. Therefore, this function is not complex differentiable.

Some sources state a sufficient condition for the complex differentiability at a point $$z_0$$ as, in addition to the Cauchy–Riemann equations, the partial derivatives of $$u$$ and $$v$$ be continuous at the point because this continuity condition ensures the existence of the aforementioned linear approximation. Note that it is not a necessary condition for the complex differentiability. For example, the function $$f(z) = z^2e^{i/|z|}$$ is complex differentiable at 0, but its real and imaginary parts have discontinuous partial derivatives there. Since complex differentiability is usually considered in an open set, where it in fact implies continuity of all partial derivatives (see below), this distinction is often elided in the literature.

Independence of the complex conjugate
The above proof suggests another interpretation of the Cauchy–Riemann equations. The complex conjugate of $$z$$, denoted $\bar{z}$, is defined by $$\overline{x + iy} := x - iy$$ for real variables $$x$$ and $$y$$. Defining the two Wirtinger derivatives as$$ \frac{\partial}{\partial z} = \frac{1}{2} \left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right), \;\;\; \frac{\partial}{\partial\bar{z}} = \frac{1}{2} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right), $$ the Cauchy–Riemann equations can then be written as a single equation $$\frac{\partial f}{\partial\bar{z}} = 0,$$ and the complex derivative of $f$  in that case is $\frac{df}{dz}=\frac{\partial f}{\partial z}.$ In this form, the Cauchy–Riemann equations can be interpreted as the statement that a complex function $f$  of a complex variable $z$  is independent of the variable $\bar{z}$. As such, we can view analytic functions as true functions of one complex variable ($z$ ) instead of complex functions of two real variables ($x$  and $y$ ).

Physical interpretation
A standard physical interpretation of the Cauchy–Riemann equations going back to Riemann's work on function theory is that u represents a velocity potential of an incompressible steady fluid flow in the plane, and v is its stream function. Suppose that the pair of (twice continuously differentiable) functions u and v satisfies the Cauchy–Riemann equations. We will take u to be a velocity potential, meaning that we imagine a flow of fluid in the plane such that the velocity vector of the fluid at each point of the plane is equal to the gradient of u, defined by $$\nabla u = \frac{\partial u}{\partial x}\mathbf i + \frac{\partial u}{\partial y}\mathbf j.$$

By differentiating the Cauchy–Riemann equations for the functions u and v, with the symmetry of second derivatives, one shows that u solves Laplace's equation: $$\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0.$$ That is, u is a harmonic function. This means that the divergence of the gradient is zero, and so the fluid is incompressible.

The function v also satisfies the Laplace equation, by a similar analysis. Also, the Cauchy–Riemann equations imply that the dot product $\nabla u\cdot\nabla v = 0$ ($\nabla u\cdot\nabla v = \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \cdot \frac{\partial v}{\partial y} = \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial x} - \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial x}  = 0$ ), i.e., the direction of the maximum slope of u and that of v are orthogonal to each other. This implies that the gradient of u must point along the $v = \text{const}$ curves; so these are the streamlines of the flow. The $u = \text{const}$ curves are the equipotential curves of the flow.

A holomorphic function can therefore be visualized by plotting the two families of level curves $u=\text{const}$ and $v=\text{const}$. Near points where the gradient of u (or, equivalently, v) is not zero, these families form an orthogonal family of curves. At the points where $\nabla u=0$, the stationary points of the flow, the equipotential curves of $u=\text{const}$ intersect. The streamlines also intersect at the same point, bisecting the angles formed by the equipotential curves.

Harmonic vector field
Another interpretation of the Cauchy–Riemann equations can be found in Pólya & Szegő. Suppose that u and v satisfy the Cauchy–Riemann equations in an open subset of R2, and consider the vector field $$\bar{f} = \begin{bmatrix} u\\ -v \end{bmatrix}$$ regarded as a (real) two-component vector. Then the second Cauchy–Riemann equation ($f$) asserts that $$\bar{f}$$ is irrotational (its curl is 0): $$\frac{\partial (-v)}{\partial x} - \frac{\partial u}{\partial y} = 0.$$

The first Cauchy–Riemann equation ($x$) asserts that the vector field is solenoidal (or divergence-free): $$\frac{\partial u}{\partial x} + \frac{\partial (-v)}{\partial y}=0.$$

Owing respectively to Green's theorem and the divergence theorem, such a field is necessarily a conservative one, and it is free from sources or sinks, having net flux equal to zero through any open domain without holes. (These two observations combine as real and imaginary parts in Cauchy's integral theorem.) In fluid dynamics, such a vector field is a potential flow. In magnetostatics, such vector fields model static magnetic fields on a region of the plane containing no current. In electrostatics, they model static electric fields in a region of the plane containing no electric charge.

This interpretation can equivalently be restated in the language of differential forms. The pair u and v satisfy the Cauchy–Riemann equations if and only if the one-form $$v\,dx + u\, dy$$ is both closed and coclosed (a harmonic differential form).

Preservation of complex structure
Another formulation of the Cauchy–Riemann equations involves the complex structure in the plane, given by $$J = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}.$$ This is a complex structure in the sense that the square of J is the negative of the 2×2 identity matrix: $$J^2 = -I$$. As above, if u(x,y) and v(x,y) are two functions in the plane, put

$$f(x,y) = \begin{bmatrix}u(x,y)\\v(x,y)\end{bmatrix}.$$

The Jacobian matrix of f is the matrix of partial derivatives $$Df(x,y) = \begin{bmatrix} \dfrac{\partial u}{\partial x} & \dfrac{\partial u}{\partial y} \\[5pt] \dfrac{\partial v}{\partial x} & \dfrac{\partial v}{\partial y} \end{bmatrix}$$

Then the pair of functions u, v satisfies the Cauchy–Riemann equations if and only if the 2×2 matrix Df commutes with J.

This interpretation is useful in symplectic geometry, where it is the starting point for the study of pseudoholomorphic curves.

Other representations
Other representations of the Cauchy–Riemann equations occasionally arise in other coordinate systems. If ($y$) and ($$) hold for a differentiable pair of functions u and v, then so do $$ \frac{\partial u}{\partial n} = \frac{\partial v}{\partial s},\quad \frac{\partial v}{\partial n} = -\frac{\partial u}{\partial s} $$

for any coordinate system $(n(x, y), s(x, y))$ such that the pair $(\nabla n,\nabla s)$ is orthonormal and positively oriented. As a consequence, in particular, in the system of coordinates given by the polar representation $$z = r e^{i\theta}$$, the equations then take the form $$ {\partial u \over \partial r} =  {1 \over r}{\partial v \over \partial\theta},\quad {\partial v \over \partial r} = -{1 \over r}{\partial u \over \partial\theta}. $$

Combining these into one equation for $f$ gives $${\partial f \over \partial r} = {1 \over ir}{\partial f \over \partial\theta}.$$

The inhomogeneous Cauchy–Riemann equations consist of the two equations for a pair of unknown functions $u(x, y)$ and $v(x, y)$ of two real variables $$\begin{align} \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} &= \alpha(x, y) \\[4pt] \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} &= \beta(x, y) \end{align}$$

for some given functions $α(x, y)$ and $β(x, y)$ defined in an open subset of R2. These equations are usually combined into a single equation $$\frac{\partial f}{\partial\bar{z}} = \varphi(z,\bar{z})$$ where f = u + iv and 𝜑 = (α + iβ)/2.

If 𝜑 is Ck, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided 𝜑 is continuous on the closure of D. Indeed, by the Cauchy integral formula, $$f\left(\zeta, \bar{\zeta}\right) = \frac{1}{2\pi i} \iint_D \varphi\left(z, \bar{z}\right) \, \frac{dz\wedge d\bar{z}}{z - \zeta}$$ for all ζ ∈ D.

Goursat's theorem and its generalizations
Suppose that $f = u + iv$ is a complex-valued function which is differentiable as a function $f : R^{2} → R^{2}$. Then Goursat's theorem asserts that f is analytic in an open complex domain Ω if and only if it satisfies the Cauchy–Riemann equation in the domain. In particular, continuous differentiability of f need not be assumed.

The hypotheses of Goursat's theorem can be weakened significantly. If $f = u + iv$ is continuous in an open set Ω and the partial derivatives of f with respect to x and y exist in Ω, and satisfy the Cauchy–Riemann equations throughout Ω, then f is holomorphic (and thus analytic). This result is the Looman–Menchoff theorem.

The hypothesis that f obey the Cauchy–Riemann equations throughout the domain Ω is essential. It is possible to construct a continuous function satisfying the Cauchy–Riemann equations at a point, but which is not analytic at the point (e.g., $f(z) = z^{5}/|z|^{4})$. Similarly, some additional assumption is needed besides the Cauchy–Riemann equations (such as continuity), as the following example illustrates

$$f(z) = \begin{cases} \exp\left(-z^{-4}\right) & \text{if }z \not= 0\\ 0 & \text{if }z = 0 \end{cases}$$

which satisfies the Cauchy–Riemann equations everywhere, but fails to be continuous at z = 0.

Nevertheless, if a function satisfies the Cauchy–Riemann equations in an open set in a weak sense, then the function is analytic. More precisely:
 * If f(z) is locally integrable in an open domain Ω ⊂ C, and satisfies the Cauchy–Riemann equations weakly, then f agrees almost everywhere with an analytic function in Ω.

This is in fact a special case of a more general result on the regularity of solutions of hypoelliptic partial differential equations.

Several variables
There are Cauchy–Riemann equations, appropriately generalized, in the theory of several complex variables. They form a significant overdetermined system of PDEs. This is done using a straightforward generalization of the Wirtinger derivative, where the function in question is required to have the (partial) Wirtinger derivative with respect to each complex variable vanish.

Complex differential forms
As often formulated, the d-bar operator $$\bar{\partial}$$ annihilates holomorphic functions. This generalizes most directly the formulation $${\partial f \over \partial \bar z} = 0,$$ where $${\partial f \over \partial \bar z} = {1 \over 2}\left({\partial f \over \partial x} + i{\partial f \over \partial y}\right).$$

Bäcklund transform
Viewed as conjugate harmonic functions, the Cauchy–Riemann equations are a simple example of a Bäcklund transform. More complicated, generally non-linear Bäcklund transforms, such as in the sine-Gordon equation, are of great interest in the theory of solitons and integrable systems.

Definition in Clifford algebra
In the Clifford algebra $$C\ell(2)$$, the complex number $$z = x+iy $$ is represented as $$z \equiv x + J y$$ where $$J \equiv \sigma_1 \sigma_2$$, ($$\sigma_1^2=\sigma_2^2=1, \sigma_1 \sigma_2 + \sigma_2 \sigma_1 = 0$$, so $$J^2=-1$$). The Dirac operator in this Clifford algebra is defined as $$\nabla \equiv \sigma_1 \partial_x + \sigma_2\partial_y$$. The function $$f=u + J v$$ is considered analytic if and only if $$\nabla f = 0$$, which can be calculated in the following way:

$$ \begin{align} 0 & =\nabla f= ( \sigma_1 \partial_x + \sigma_2 \partial_y )(u + \sigma_1 \sigma_2 v) \\[4pt] & =\sigma_1 \partial_x u + \underbrace{\sigma_1 \sigma_1 \sigma_2}_{=\sigma_2} \partial_x v + \sigma_2 \partial_y u + \underbrace{\sigma_2 \sigma_1 \sigma_2}_{=-\sigma_1} \partial_y v =0 \end{align} $$

Grouping by $$\sigma_1$$ and $$\sigma_2$$:

$$\nabla f = \sigma_1 ( \partial_x u - \partial_y v) + \sigma_2 ( \partial_x v + \partial_y u) = 0 \Leftrightarrow \begin{cases} \partial_x u - \partial_y v = 0\\[4pt] \partial_x v + \partial_y u = 0 \end{cases}$$

Hence, in traditional notation:

$$\begin{cases} \dfrac{ \partial u }{ \partial x } = \dfrac{ \partial v }{ \partial y }\\[12pt] \dfrac{ \partial u }{ \partial y } = -\dfrac{ \partial v }{ \partial x } \end{cases}$$

Conformal mappings in higher dimensions
Let Ω be an open set in the Euclidean space Rn. The equation for an orientation-preserving mapping $$f:\Omega\to\mathbb{R}^n$$ to be a conformal mapping (that is, angle-preserving) is that $$Df^\mathsf{T} Df = (\det(Df))^{2/n}I$$

where Df is the Jacobian matrix, with transpose $$Df^\mathsf{T}$$, and I denotes the identity matrix. For $n = 2$, this system is equivalent to the standard Cauchy–Riemann equations of complex variables, and the solutions are holomorphic functions. In dimension $n > 2$, this is still sometimes called the Cauchy–Riemann system, and Liouville's theorem implies, under suitable smoothness assumptions, that any such mapping is a Möbius transformation.

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

title: Cauchy–Riemann equations


text:
Talk:Cuisine

Best season to visit
Best season to visit 103.240.234.141 (talk) 14:38, 28 October 2023 (UTC)

title: Talk:Cuisine


text:
Chaim Topol

Chaim Topol (חיים טופול; September 9, 1935 – March 8, 2023), mononymously known as Topol, was an Israeli actor. He is best known for his portrayal of Tevye, the lead role in the stage musical Fiddler on the Roof and the 1971 film adaptation, performing this role more than 3,500 times from 1967 through 2009.

Topol began acting during his Israeli army service as a member of the Nahal entertainment troupe. He later toured Israel with kibbutz theatre and satirical theatre companies. He was a co-founder of the Haifa Theatre. His breakthrough film role came in 1964 as the title character in Sallah Shabati, by Israeli writer Ephraim Kishon, for which he won a Golden Globe for Most Promising Newcomer—Male. Topol went on to appear in more than 30 films in Israel and the United States, including Galileo (1975), Flash Gordon (1980), and For Your Eyes Only (1981). He was described as Israel's only internationally recognized entertainer from the 1960s through the 1980s. He won a Golden Globe for Best Actor and was nominated for an Academy Award for Best Actor for his 1971 film portrayal of Tevye, and was nominated for a Tony Award for Best Actor for a 1991 Broadway revival of Fiddler on the Roof.

Topol was a founder of Variety Israel, an organization serving children with special needs, and Jordan River Village, a year-round camp for Arab and Jewish children with life-threatening illnesses, for which he served as chairman of the board. In 2015 he was awarded the Israel Prize for lifetime achievement.

Biography
Chaim Topol was born on September 9, 1935, in Tel Aviv, in what was then Mandatory Palestine. His father Jacob Topol was born in Russia and in the early 1930s immigrated to Mandatory Palestine, where he worked as a plasterer; he also served in the Haganah paramilitary organization. His mother Imrela "Rel" (née Goldman) Topol was a seamstress.

Topol's parents had been members of the Betar Zionist youth movement in Warsaw. His father had Hasidic roots, with a mother coming from a family of Gerrer Hasidim and a father from Aleksander Hasidim.

Topol and his two younger sisters grew up in the South Tel Aviv working-class neighborhood of Florentin. As a young child, although he wanted to become a commercial artist, his elementary school teacher, the writer Yemima Avidar-Tchernovitz, saw a theatrical side to him, and encouraged him to act in school plays and read stories to the class.

At age 14 he began working as a printer at Davar newspaper while pursuing his high school studies at night. He graduated at age 17 and moved to Kibbutz Geva. A year later, he enlisted in the Israeli army and became a member of the Nahal entertainment troupe, singing and acting in traveling shows. He rose in rank to troupe commander.

Twenty-three days after being discharged from military service on October 2, 1956, and two days after marrying Galia Finkelstein, a fellow Nahal troupe member, Topol was called up for reserve duty in the Sinai Campaign. He performed for soldiers stationed in the desert.

After the war, he and his wife settled in Kibbutz Mishmar David, where Topol worked as a garage mechanic. Topol assembled a kibbutz theatre company made up of friends from his Nahal troupe; the group toured four days a week, worked on their respective kibbutzim for two days a week, and had one day off. The theatre company was in existence from early 1957 to the mid-1960s. Topol both sang and acted with the group, doing both "loudly".

Topol and his wife Galia Finkelstein had three children: a son and two daughters. The couple resided in Galia's childhood home in Tel Aviv. Topol's hobbies included sketching and sculpting.

In June 2022, Topol's son, Omer, revealed that his father was suffering from Alzheimer's disease.

On March 8, 2023, Topol's family notified the press that he was near death and "living his final hours", and asked the public to respect the family's privacy. He died overnight at the age of 87. The day before his burial at Kvutzat Shiller on March 10, a memorial was held at the Cameri Theater in Tel Aviv.

Singing and acting career
Between 1960 and 1964, Topol performed with the Batzal Yarok ("Green Onion") satirical theatre company, which also toured Israel. Other members of the group included Uri Zohar, Nechama Hendel, Zaharira Harifai, Arik Einstein, and Oded Kotler. In 1960, Topol co-founded the Haifa Municipal Theatre with Yosef Milo, serving as assistant to the director and acting in plays by Shakespeare, Ionesco, and Brecht. In 1965 he performed in the Cameri Theatre in Tel Aviv.

Topol's first film appearance was in the 1961 film I Like Mike, followed by the 1963 Israeli film El Dorado. His breakthrough role came as the lead character in the 1964 film Sallah Shabati. Adapted for the screen by Ephraim Kishon from his original play, the social satire depicts the hardships of a Sephardic immigrant family in the rough conditions of ma'abarot, immigrant absorption camps in Israel in the 1950s, satirizing "just about every pillar of Israeli society: the Ashkenazi establishment, the pedantic bureaucracy, corrupt political parties, rigid kibbutz ideologues and ... the Jewish National Fund's tree-planting program". Topol, who was 29 during the filming, was familiar playing the role of the family patriarch, having performed skits from the play with his Nahal entertainment troupe during his army years. He contributed his ideas to the part, playing the character as a more universal Mizrahi Jew instead of specifically a Yemenite, Iraqi, or Moroccan Jew, and asking Kishon to change the character's first name from Saadia (a recognizably Yemenite name) to Sallah (a more general Mizrahi name).

The film won the Golden Globe Award for Best Foreign Language Film, and Topol won the 1964 Golden Gate Award for Best Actor at the San Francisco International Film Festival and the 1965 Golden Globe for Most Promising Newcomer—Male,  alongside Harve Presnell and George Segal. Sallah Shabati was nominated for the Academy Award for Best Foreign Language Film, losing to the Italian-language Yesterday, Today and Tomorrow.

In 1966, Topol made his English-language film debut as Abou Ibn Kaqden in the Mickey Marcus biopic Cast a Giant Shadow.

Tevye the Dairyman
Topol came to greatest prominence in his portrayal of Tevye the Dairyman on stage and screen. He first played the lead role in the Israeli production of the musical Fiddler on the Roof in 1966, replacing Shmuel Rodensky for 10 weeks when Rodensky fell ill. Harold Prince, producer of the original Fiddler on the Roof that opened on Broadway in 1964, had seen Topol in Sallah Shabati and called him to audition for the role of the fifty-something Tevye in a new production scheduled to open at Her Majesty's Theatre in London on February 16, 1967. Not yet fluent in English, Topol memorized the score from listening to the original Broadway cast album and practiced the lyrics with a British native.

When Topol arrived at the audition, Prince was surprised that this 30-year-old man had played Shabati, a character in his sixties. Topol explained, "A good actor can play an old man, a sad face, a happy man. Makeup is not an obstacle". Topol also surprised the producers with his familiarity with the staging, since he had already acted in the Israeli production, and was hired. He spent six months in London learning his part phonetically with vocal coach Cicely Berry. Jerome Robbins, director and choreographer of the 1964 Broadway show who came over to direct the London production, "re-directed" the character of Tevye for Topol and helped the actor deliver a less caricatured performance. Topol's performance received positive reviews.

A few months after the opening, Topol was called up for reserve duty in the Six-Day War and returned to Israel. He was assigned to an army entertainment troupe on the Golan Heights. Afterward he returned to the London production, appearing in a total of 430 performances.

It was during the London run that he began being known by his last name only, as the English producers were unable to pronounce the voiceless uvular fricative consonant Ḥet at the beginning of his first name, Chaim, instead calling him "Shame".

In casting the 1971 film version of Fiddler on the Roof, director Norman Jewison and his production team sought an actor other than Zero Mostel for the lead role. This decision was a controversial one, as Mostel had made the role famous in the long-running Broadway musical and wanted to star in the film. But Jewison and his team felt Mostel would eclipse the character with his larger-than-life personality. Jewison flew to London in February 1968 to see Topol perform as Tevye during his last week with the London production, and chose him over Danny Kaye, Herschel Bernardi, Rod Steiger, Danny Thomas, Walter Matthau, Richard Burton, and Frank Sinatra, who had also expressed interest in the part.

Then 36 years old, Topol was made to look 20 years older and 30 lb heavier with makeup and costuming. As in his role as Shabati, Topol used the technique of "locking his muscles" to convincingly play an older character. He later explained:

"As a young man, I had to make sure that I didn't break the illusion for the audience. You have to tame yourself. I'm now someone who is supposed to be 50, 60 years old. I cannot jump. I cannot suddenly be young. You produce a certain sound [in your voice] that is not young."

For his performance, Topol won the Golden Globe Award for Best Actor in a Motion Picture – Musical or Comedy, the Sant Jordi Award for Best Performance in a Foreign Film, and the 1972 David di Donatello for Best Foreign Actor, sharing the latter with Elizabeth Taylor. He was also nominated for the 1971 Academy Award for Best Actor, losing to Gene Hackman in The French Connection.

In 1983 Topol reprised the role of Tevye in a revival of Fiddler on the Roof on the West End in London. In 1989, he played the role in a 30-city U.S. touring production. As he was by then the approximate age of the character, he commented, "I didn't have to spend the energy playing the age". In 1990–1991, he again starred as Tevye in a Broadway revival of Fiddler at the Gershwin Theatre. In that production Rosalind Harris, who had played eldest daughter Tzeitel in the film, played Tevye's wife Golde opposite Topol. In 1991, he was nominated for a Tony Award for Best Performance by a Leading Actor in a Musical, losing to Jonathan Pryce in Miss Saigon. Topol again played Tevye in a 1994 London revival, which became a touring production. In that production, the role of one of his daughters was played by his daughter, Adi Topol Margalith.

Topol reprised the role of Tevye for a 1997–1998 touring production in Israel, as well as a 1998 show at the Regent Theatre in Melbourne. In September 2005 he returned to Australia for a Fiddler on the Roof revival at the Capitol Theatre in Sydney, followed by an April 2006 production at the Lyric Theatre in Brisbane, and a June 2006 production at Her Majesty's Theatre in Melbourne. In May 2007, he starred in a production at the Auckland Civic Theatre.

In 2009, Topol began a farewell tour of Fiddler on the Roof as Tevye, opening in Wilmington, Delaware. He was forced to withdraw from the tour in Boston owing to a shoulder injury, and was replaced by Theodore Bikel and Harvey Fierstein, both of whom had portrayed Tevye on Broadway. Topol estimated that he performed the role more than 3,500 times.

In 2014, he appeared in Raising the Roof, a 50th-anniversary tribute to Fiddler at New York City's Town Hall produced by National Yiddish Theatre. The evening featured Chita Rivera, Joshua Bell, Sheldon Harnick, Andrea Martin, Jerry Zaks, and more, and was co-directed by Gary John La Rosa and Erik Liberman.

Other stage and film roles

In 1976, Topol played the lead role of the baker, Amiable, in the new musical The Baker's Wife, but was fired after eight months by producer David Merrick. In her autobiography, Patti LuPone, his co-star in the production, claimed that Topol had behaved unprofessionally on stage and had a strained relationship with her off-stage. The show's composer, Stephen Schwartz, claimed that Topol's behavior greatly disturbed the cast and directors and resulted in the production not reaching Broadway as planned. In 1988, Topol starred in the title role in Ziegfeld at the London Palladium. He returned to the London stage in 2008 in the role of Honoré, played by Maurice Chevalier in the 1958 film Gigi.

Topol appeared in more than 30 films in Israel and abroad. Among his notable English-language appearances are the title role in Galileo (1975), Dr. Hans Zarkov in Flash Gordon (1980), and Milos Columbo in the James Bond film For Your Eyes Only (1981). He was said to be Israel's "only internationally-recognized entertainer" from the 1960s through to the 1980s.

In Israel, Topol acted in and produced dozens of films and television series. As a voice artist, he dubbed the voice of Bagheera in the Hebrew-language versions of The Jungle Book and the 2003 sequel as well as Rubeus Hagrid in the first two films of the Harry Potter film series. He was also a playwright and screenwriter. Topol was featured on two BBC One programmes, the six-part series Topol's Israel (1985) and earlier It's Topol (1968). A Hebrew-language documentary of his life, Chaim Topol – Life as a Film, aired on Israel's Channel 1 in 2011, featuring interviews with his longtime actor friends in Israel and abroad.

Musical recordings
A baritone, Topol recorded several singles and albums, including film soundtracks, children's songs, and Israeli war songs. His albums include Topol With Roger Webb And His Orchestra - Topol '68 (1967), Topol Sings Israeli Freedom Songs (1967), War Songs By Topol (1968), and Topol's Israel (1984). He appeared on the soundtrack album for the film production of Fiddler on the Roof (1971) and the London cast album (1967).

Mossad missions
After Topol's death, the family revealed that he had been involved in Mossad missions in the 1960s and 1970s. They said he went on unexplained trips abroad while equipped with a miniature state-of-the-art camera and tape recorder, and that he was in regular contact with Mossad officer Peter Malkin, who came on visits to the family home through the backyard in disguise. On several occasions, Topol carried out wiretapping and other operations with Malkin, using his international acclaim to divert attention from Malkin.

Literary and art career

His autobiography, Topol by Topol, was published in London by Weindenfel and Nicholson (1981). He also authored To Life! (1994) and Topol's Treasury of Jewish Humor, Wit and Wisdom (1995).

Topol illustrated approximately 25 books in both Hebrew and English. He also produced drawings of Israeli national figures. His sketches of Israeli presidents were reproduced in a 2013 stamp series issued by the Israel Philatelic Federation, as was his self-portrait as Tevye for 2014 commemorative stamp marking the 50th anniversary of the Broadway debut of Fiddler on the Roof.

Philanthropy
In 1967, Topol founded Variety Israel, an organization serving children with special needs. He was also a co-founder and chairman of the board of Jordan River Village, a vacation village for Arab and Jewish children with life-threatening illnesses, which opened in 2012. It was inspired by Paul Newman's Hole in the Wall Gang Camp. The village is operated almost entirely by volunteers. Topol described it as the project he was "most connected to."

Awards and recognition

Topol was a recipient of Israel's Kinor David award in arts and entertainment in 1964. He received a Best Actor award from the San Sebastián International Film Festival for his performance in the 1972 film Follow Me! In 2008, he was named an Outstanding Member of the Israel Festival for his contribution to Israeli culture.

In 2014, the University of Haifa conferred upon Topol an honorary degree in recognition of his 50 years of activity in Israel's cultural and public life. In 2015, he received the Israel Prize for lifetime achievement.

In 2015, Chaim Topol was honoured by the Chief Rabbi of Ukraine, Rabbi Moshe Reuven Azman and the Ukrainian Jewish Community. Topol's portrayal of Tevye in Fiddler on the Roof led to the inspiration for the Anatevka Refugee Village which was named in commemoration of the fictional village.

Legacy
Shortly after Topol's death, President Isaac Herzog issued a statement honouring "one of the most prominent Israeli stage artists, a gifted actor who conquered many stages in Israel and overseas, filled the cinema screens with his presence and, above all, deeply entered our hearts". Prime minister Benjamin Netanyahu stated "his wide smile, warm voice, and unique sense of humour made him a folk hero who won the hearts of the people" and former prime minister Yair Lapid remarked "He and his smile will continue to accompany Israeli culture, his rich legacy will forever remain a part of Israel".

title: Chaim Topol


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Christadelphians

The Christadelphians are a restorationist and nontrinitarian Christian denomination. The name means 'brothers and sisters in Christ', from the Greek words for Christ (Christos) and brothers (adelphoi).

Christadelphians believe in the inspiration of the Bible, the Virgin Birth, the status of Jesus Christ as the son of God, believer's baptism, the resurrection of the dead, the second coming of Christ, and the future kingdom of God on earth. However, they reject a number of mainstream Christian doctrines, for example the Trinity and the immortality of the soul, believing these to be corruptions of original Christian teaching.

The movement developed in the United Kingdom and North America in the 19th century around the teachings of John Thomas and they were initially found predominantly in the developed English-speaking world, expanding in developing countries after the Second World War. There are approximately 50,000 Christadelphians in around 120 countries. Congregations are traditionally referred to as "ecclesias".

Background
Many of the beliefs later associated with Christadelphians emerged in the Radical Reformation. For example, rejection of the doctrine of the Trinity and the pre-existence of Christ was a feature of Socinianism. However, restrictions on religious freedom in Britain meant that those with unorthodox views generally stayed within mainstream churches. With the passing of the Doctrine of the Trinity Act of 1813, penalties for denying the Trinity which had been in place since the Blasphemy Act of 1697 were removed. Meanwhile in the United States, the Second Great Awakening gave birth to Adventism which stressed the Second Coming of Christ, and Restorationism which sought to restore Christianity to its 'primitive' origins.

19th century
The Christadelphian movement traces its origins to John Thomas (1805–1871). He initially associated with emerging Restoration Movement in the United States but later separated from them. The Christadelphian community in the United Kingdom effectively dates from Thomas's first lecturing tour of Britain (May 1848 – October 1850). During this period, he wrote Elpis Israel in which he laid out his understanding of the main doctrines of the Bible. Since his medium for bringing change was print and debate, it was natural for the origins of the Christadelphian body to be associated with books and journals, such as Thomas's Herald of the Kingdom. His message was particularly welcomed in Scotland, and Campbellite, Unitarian and Adventist friends separated to form groups of "Baptised Believers".

In his desire to seek to establish Biblical truth and test orthodox Christian beliefs through independent scriptural study he was not alone. Among other churches, he had links with the Adventist movement and with Benjamin Wilson (who later set up the Church of God of the Abrahamic Faith in the 1860s). Although the Christadelphian movement originated through the activities of John Thomas, he never saw himself as making his own disciples. He believed rather that he had rediscovered 1st century beliefs from the Bible alone, and sought to prove that through a process of challenge and debate and writing journals. Through that process a number of people became convinced and set up various fellowships that had sympathy with that position. Groups associated with John Thomas met under various names, including Believers, Baptised Believers, the Royal Association of Believers, Baptised Believers in the Kingdom of God, Nazarines (or Nazarenes), and The Antipas until the time of the American Civil War (1861–1865). At that time, church affiliation was required in the United States and in the Confederate States of America in order to register for conscientious objector status, and in 1864 Thomas chose for registration purposes the name Christadelphian.

Through the teaching of John Thomas and the need in the American Civil War for a name, the Christadelphians emerged as a denomination, but they were formed into a lasting structure through a passionate follower of Thomas's interpretation of the Bible, Robert Roberts. In 1864, he began to publish The Ambassador of the Coming Age magazine. John Thomas, out of concern that someone else might start a publication and call it The Christadelphian, urged Robert Roberts to change the name of his magazine to The Christadelphian, which he did in 1869. His editorship of the magazine continued with some assistance until his death in 1898. In church matters, Roberts was prominent in the period following the death of John Thomas in 1871, and helped craft the structures of the Christadelphian body.

Initially, the denomination grew in the English-speaking world, particularly in the English Midlands and in parts of North America.. Two thirds of ecclesias, and members, in Britain before 1864 were in Scotland. In the early days after the death of John Thomas, the group could have moved in a number of directions. Doctrinal issues arose, debates took place, and statements of faith were created and amended as other issues arose. These attempts were felt necessary by many to both settle and define a doctrinal stance for the newly emerging denomination and to keep out error. As a result of these debates, several groups separated from the main body of Christadelphians, most notably the Suffolk Street fellowship in 1885 (with members believing that the whole of the Bible was not inspired), and the Unamended fellowship.

20th century
The Christadelphian position on conscientious objection came to the fore with the introduction of conscription during the First World War. Varying degrees of exemption from military service were granted to Christadelphians in the United Kingdom, Canada, Australia, New Zealand, and the United States. In the Second World War, this frequently required the person seeking exemption to undertake civilian work under the direction of the authorities.

During the Second World War, the Christadelphians in Britain assisted in the Kindertransport, helping to relocate several hundred Jewish children away from Nazi persecution by founding a hostel, Elpis Lodge, for that purpose. In Germany, the small Christadelphian community founded by Albert Maier went underground from 1940 to 1945, and a leading brother, Albert Merz, was imprisoned as a conscientious objector and later executed.

After the Second World War, moves were taken to try to reunite various of the earlier divisions. By the end of the 1950s, most Christadelphians had united into one community, but there are still a number of small groups of Christadelphians who remain separate.

Today
The post-war and post-reunions periods saw an increase in co-operation and interaction between ecclesias, resulting in the establishment of a number of week-long Bible schools and the formation of national and international organisations such as the Christadelphian Bible Mission (for preaching and pastoral support overseas), the Christadelphian Support Network (for counselling), and the Christadelphian Meal-A-Day Fund (for charity and humanitarian work).

The period following the reunions was accompanied by expansion in the developing world, which now accounts for around 40% of Christadelphians.

Beliefs
The Christadelphian body has no central authority or co-ordinating organisation to establish and maintain a standardised set of beliefs, but there are core doctrines accepted by most Christadelphians. In the formal statements of faith a more complete list is found; for instance, the Birmingham Amended Statement of Faith has 30 doctrines to be accepted and 35 to be rejected.

The Bible
Christadelphians state that their beliefs are based wholly on the Bible, and they do not see other works as inspired by God. They regard the Bible as inspired by God and, therefore, believe that in its original form, it is error-free apart from errors in later copies due to errors of transcription or translation.

God
Christadelphians believe that God, Jehovah, is the creator of all things and the father of true believers, that he is a separate being from his son, Jesus (who is subordinate to him). They reject the doctrine of the Trinity.

Jesus
Christadelphians believe that Jesus is the promised Jewish Messiah, in whom the prophecies and promises of the Old Testament find their fulfilment. They believe he is the Son of Man, in that he inherited human nature (with its inclination to sin) from his mother, and the Son of God by virtue of his miraculous conception by the power of God. Christadelphians reject the doctrine of Christ's pre-existence. They teach that he was part of God's plans from the beginning and was foreshadowed in the Old Testament, but was no independent creature prior to his earthly birth. Although he was tempted, Jesus committed no sin, and was therefore a perfect representative sacrifice to bring salvation to sinful humankind. They believe that God raised Jesus from death and gave him immortality, and he has ascended to Heaven, God's dwelling place, until he returns to set up the Kingdom of God on earth.

The Holy Spirit
Christadelphians believe that the Holy Spirit is the power of God used in creation and for salvation. They also believe that the phrase Holy Spirit sometimes refers to God's character/mind, depending on the context in which the phrase appears, but reject the view that people need strength, guidance and power from the Holy Spirit to live the Christian life, believing instead that the spirit a believer needs within themselves is the mind/character of God, which is developed in a believer by their reading of the Bible (which, they believe, contains words God gave by his Spirit) and trying to live by what it says during the events of their lives which God uses to help shape their character. Christadelphians deny the personhood of the Holy Spirit,   and the present-day possession of the Holy Spirit (both "gift of" and "gifts of") (see cessationism).

The Kingdom of God
Christadelphians believe that Jesus Christ will return to the Earth in person to set up the Kingdom of God in fulfilment of the promises made to Abraham and David. This includes the belief that the coming Kingdom will be the restoration of God's first Kingdom of Israel, which was under David and Solomon. For Christadelphians, this is the focal point of the gospel taught by Jesus and the apostles. They believe that the Kingdom will be centred upon Israel, but Jesus Christ will also reign over all the other nations on the Earth. Old Paths Christadelphians continue to believe that the Kingdom of God is to be restored to the land of Israel promised to Abraham and ruled over in the past by David, with a worldwide empire.

The Devil
Christadelphians believe that the word devil is a reference in the scriptures to sin and human nature in opposition to God, while the word satan is merely a reference to an adversary or opponent (be it good or bad) and is frequently applied to human beings. According to Christadelphians, these terms are used in reference to specific political systems or individuals in opposition or conflict and not to an independent spiritual being or fallen angel. Accordingly, they do not define Hell as a place of eternal torment for sinners, but as a state of eternal death and non-existence due to annihilation of body and mind.

Salvation
Christadelphians believe that people are separated from God because of their sins but that humankind can be reconciled to him by becoming disciples of Jesus Christ. This is by belief in the gospel, through repentance, and through baptism by total immersion in water. They reject assurance of salvation, believing instead that salvation comes as a result of remaining "in Christ". After death, believers are in a state of non-existence, knowing nothing until the Resurrection at the return of Christ. Following the judgement at that time, the accepted receive the gift of immortality, and live with Christ on a restored Earth, assisting him to establish the Kingdom of God and to rule over the mortal population for a thousand years (the Millennium). Christadelphians deny the immortality of the soul.

Life in Christ
The Commandments of Christ demonstrates the community's recognition of the importance of biblical teaching on morality. Marriage and family life are important. Most Christadelphians believe that sexual relationships should be limited to heterosexual marriage, ideally between baptised believers.

General organisation
In the absence of centralised organisation, some differences exist amongst Christadelphians on matters of belief and practice. This is because each congregation (commonly styled 'ecclesias') is organised autonomously, typically following common practices which have altered little since the 19th century. Many avoid the word "church" due to its association with mainstream Christianity, and its focus on the building as opposed to the congregation. Most ecclesias have a constitution, which includes a 'Statement of Faith', a list of 'Doctrines to be Rejected' and a formalised list of 'The Commandments of Christ'. With no central authority, individual congregations are responsible for maintaining orthodoxy in belief and practice, and the statement of faith is seen by many as useful to this end. The statement of faith acts as the official standard of most ecclesias to determine fellowship within and between ecclesias, and as the basis for co-operation between ecclesias. Congregational discipline and conflict resolution are applied using various forms of consultation, mediation, and discussion, with disfellowship (similar to excommunication) being the final response to those with unorthodox practices or beliefs.

The relative uniformity of organisation and practice is undoubtedly due to the influence of a booklet, written early in Christadelphian history by Robert Roberts, called A Guide to the Formation and Conduct of Christadelphian Ecclesias. It recommends a basically democratic arrangement by which congregational members elect 'brothers' to do arranging and serving duties, and includes guidelines for the organisation of committees, as well as conflict resolution between congregational members and between congregations. Christadelphians do not have paid ministers. Male members (and increasingly female in some places) are assessed by the congregation for their eligibility to teach and perform other duties, which are usually assigned on a rotation basis, as opposed to having a permanently appointed preacher. Congregational polity typically follows a democratic model, with an elected arranging committee for each individual ecclesia. This unpaid committee is responsible for the day-to-day running of the ecclesia and is answerable to the rest of the ecclesia's members.

Inter-ecclesial organisations co-ordinate the running of, among other things, Christadelphian schools and elderly care homes, the Christadelphian Isolation League (which cares for those prevented by distance or infirmity from attending an ecclesia regularly) and the publication of Christadelphian magazines.

Adherents
No official membership figures are published, but the Columbia Encyclopaedia gives an estimated figure of 50,000 Christadelphians, spread across approximately 120 countries. Estimates for the main centers of Christadelphian population are as follows: Mozambique (17,800), Australia (9,734), the United Kingdom (8,200), Malawi (7,000), United States (6,500), Canada (3,000), Kenya (2,700), India (2,300) and New Zealand (1,785),. Figures from Christadelphian mission organisations are as follows: Africa (32,500), Asia (4,000), the Caribbean (400), Europe (including Russia) (700), Latin America (275), and the Pacific (200).

Fellowships
The Christadelphian body consists of a number of fellowships – groups of ecclesias which associate with one another, often to the exclusion of ecclesias outside their group. They are to some degree localised. The Unamended Fellowship, for example, exists only in North America. Christadelphian fellowships have often been named after ecclesias or magazines who took a lead in developing a particular stance.

The majority of Christadelphians today belong to what is commonly known as the Central Fellowship. The term "Central" came into use around 1933 to identify ecclesias worldwide who were in fellowship with the Birmingham (Central) Ecclesia. These were previously known as the "Temperance Hall Fellowship". The "Suffolk Street Fellowship" arose in 1885 over disagreements surrounding the inspiration of the Bible. Meanwhile, in Australia, division concerning the nature of Jesus Christ resulted in the formation of the "Shield Fellowship". Discussions in 1957–1958 resulted in a worldwide reunion between the Central, Suffolk Street and Shield Fellowships.

The Unamended Fellowship, consisting of around 1,850 members, is found in the East Coast and Midwest USA and Ontario, Canada. This group separated in 1898 as a result of differing views on who would be raised to judgement at the return of Christ. The majority of Christadelphians believe that the judgement will include anyone who had sufficient knowledge of the gospel message, and is not limited to baptised believers. The majority in England, Australia and North America amended their statement of faith accordingly. Those who opposed the amendment became known as the "Unamended Fellowship" and allowed the teaching that God either could not or would not raise those who had no covenant relationship with him. Opinions vary as to what the established position was on this subject prior to the controversy. Prominent in the formation of the Unamended Fellowship was Thomas Williams, editor of the Christadelphian Advocate magazine. The majority of the Unamended Fellowship outside North America joined the Suffolk Street Fellowship before its eventual incorporation into Central Fellowship. There is also some co-operation between the Central (Amended) and Unamended Fellowships in North America – most recently in the Great Lakes region, where numerous Amended and Unamended ecclesias are working together to unify their ecclesias. The Central Fellowship in North America is still often referred to today as the Amended Fellowship.

The Berean Fellowship was formed in 1923 as a result of varying views on military service in England, and on the atonement in North America. The majority of the North American Bereans re-joined the main body of Christadelphians in 1952. A number continue as a separate community, numbering around 200 in Texas, 100 in Kenya and 30 in Wales. Most of the divisions still in existence within the Christadelphian community today stem from further divisions of the Berean Fellowship.

The Dawn Fellowship are the result of an issue which arose in 1942 among the Berean Fellowship regarding divorce and remarriage. The stricter party formed the Dawn Fellowship who, following re-union on the basis of unity of belief with the Lightstand Fellowship in Australia in 2007 increased in number. There are now thought to be around 800 members in England, Australia, Canada, India, Jamaica, Poland, the Philippines, Russia and Kenya.

The Old Paths Fellowship was formed in 1957 in response to the reunion of the Central and Suffolk Street Fellowships. A minority from the Central Fellowship held that the reasons for separation remained and that full unity of belief on all fundamental principles of Bible teaching was necessary; thus reunion was only possible with the full agreement and understanding of all members rather than a decision by majority vote. Ecclesias forming the Old Paths Fellowship arose in England, Australia, New Zealand and Canada numbering around 500 members in total. They now number around 250 members in total, with members in Australia, England, Mexico and New Zealand. They maintain that they hold to the original Central Fellowship position held prior to the 1957 Reunion.

Other fellowships (ranging in numbers from as few as 10 to over 200 members) include the Watchman Fellowship, the Companion Fellowship and the Pioneer Fellowship.

According to Bryan Wilson, functionally the definition of a "fellowship" within Christadelphian history has been mutual or unilateral exclusion of groupings of ecclesias from the breaking of bread. This functional definition still holds true in North America, where the Unamended Fellowship and the Church of God of the Abrahamic Faith are not received by most North American Amended ecclesias. But outside North America this functional definition no longer holds. Many articles and books on the doctrine and practice of fellowship now reject the notion itself of separate "fellowships" among those who recognise the same baptism, viewing such separations as schismatic. Many ecclesias in the Central fellowship would not refuse a baptised Christadelphian from a minority fellowship from breaking bread; the exclusion is more usually the other way.

They tend to operate organisationally fairly similarly, although there are different emphases. Despite their differences, the Central, Old Paths, Dawn and Berean fellowships generally subscribe to the Birmingham Amended Statement of Faith (BASF), though the latter two have additional clauses or supporting documents to explain their position. Most Unamended ecclesias use the Birmingham Unamended Statement of Faith (BUSF) with one clause being different. Within the Central fellowship individual ecclesias also may have their own statement of faith, whilst still accepting the statement of faith of the larger community. Some ecclesias have statements around their positions, especially on divorce and re-marriage, making clear that offence would be caused by anyone in that position seeking to join them at the 'Breaking of Bread' service. Others tolerate a degree of divergence from commonly held Christadelphian views. While some communities of Christadelphian origin have viewed previous statements of faith as set in stone, others have felt it necessary to revise them in order to meet contemporary issues, update language or add supporting Biblical quotations.

For each fellowship, anyone who publicly assents to the doctrines described in the statement and is in good standing in their "home ecclesia" is generally welcome to participate in the activities of any other ecclesia.

Related groups
There are a number of groups who, while sharing a common heritage and many Christadelphian teachings, have adopted alternative names in order to dissociate themselves from what they believe to be false teachings and/or practice within the main Christadelphian body. Ranging in size from two or three members in size to around 50, each group restricts fellowship to its own members. These include the Nazarene Fellowship, the Ecclesia of Christ, the Remnant of Christ's Ecclesia, the Apostolic Fellowship of Christ and the Apostolic Ecclesia.

The Church of God of the Abrahamic Faith (CGAF) also has common origins with Christadelphians and shares Christadelphian beliefs. Numbering around 400 (primarily Ohio and Florida, USA), they are welcomed into fellowship by some "Central" Christadelphians and are currently involved in unity talks.

Historical antecedents
One criticism of the Christadelphian movement has been over the claim of John Thomas and Robert Roberts to have "re-discovered" scriptural truth. However one might argue that all Protestant groups make the same claims to some extent. Although both men believed that they had "recovered" the true doctrines for themselves and contemporaries, they also believed there had always existed a group of true believers throughout the ages, albeit marred by the apostasy.

The most notable Christadelphian attempts to find a continuity of those with doctrinal similarities since that point have been geographer Alan Eyre's two books The Protesters (1975) and Brethren in Christ (1982) in which he shows that many individual Christadelphian doctrines had been previously believed. Eyre focused in particular on the Radical Reformation, and also among the Socinians and other early Unitarians and the English Dissenters. In this way, Eyre was able to demonstrate substantial historical precedents for individual Christadelphian teachings and practices, and believed that the Christadelphian community was the 'inheritor of a noble tradition, by which elements of the Truth were from century to century hammered out on the anvil of controversy, affliction and even anguish'. Although noting in the introduction to 'The Protestors' that 'Some recorded herein perhaps did not have "all the truth" — so the writer has been reminded', Eyre nevertheless claimed that the purpose of the work was to 'tell how a number of little-known individuals, groups and religious communities strove to preserve or revive the original Christianity of apostolic times', and that 'In faith and outlook they were far closer to the early springing shoots of first-century Christianity and the penetrating spiritual challenge of Jesus himself than much that has passed for the religion of the Nazarene in the last nineteen centuries'.

Eyre's research has been criticized by some of his Christadelphian peers, and as a result Christadelphian commentary on the subject has subsequently been more cautious and circumspect, with caveats being issued concerning Eyre's claims, and the two books less used and publicised than in previous years.

Nevertheless, even with most source writings of those later considered heretics destroyed, evidence can be provided that since the first century BC there have been various groups and individuals who have held certain individual Christadelphian beliefs or similar ones. For example, all the distinctive Christadelphian doctrines (with the exception of the non-literal devil), down to interpretations of specific verses, can be found particularly among sixteenth century Socinian writers (e.g. the rejection of the doctrines of the trinity, pre-existence of Christ, immortal souls, a literal hell of fire, original sin). Early English Unitarian writings also correspond closely to those of Christadelphians. Also, recent discoveries and research have shown a large similarity between Christadelphian beliefs and those held by Isaac Newton who, among other things, rejected the doctrines of the trinity, immortal souls, a personal devil and literal demons. Further examples are as follows:
 * The typical Old Testament belief in unconsciousness until resurrection, The Christadelphian understanding of Daniel 12:2, etc.
 * 'Barr is surely right to stress that the Genesis story as it now stands indicates that humans were not created immortal, but had (and lost) the chance to gain unending life.', Wright, 'The Resurrection of the Son of God', p. 92 (2003); Wright himself actually interprets some passages of Scripture as indicating alternative beliefs, 'The Bible offers a spectrum of belief about life after death', Wright, 'The Resurrection of the Son of God', p. 129 (2003)
 * 'In contrast to the two enigmatic references to Enoch and Elijah, there are ample references to the fact that death is the ultimate destiny for all human beings, that God has no contact with or power over the dead, and that the dead do not have any relationship with God (see, inter alia, Ps. 6:6, 30:9–10, 39:13–14, 49:6–13, 115:16–18, 146:2–4). If there is a conceivable setting for the introduction of a doctrine of the afterlife, it would be in Job, since Job, although righteous, is harmed by God in the present life. But Job 10:20–22 and 14:1–10 affirm the opposite.', Gillman, 'Death and Afterlife, Judaic Doctrines Of', in Neusner, 'The Encyclopedia of Judaism', volume 1, p. 176 (2000)
 * ' "Who knows whether the breath of human beings rises up and the breath of an animal sinks down to the earth?" (Eccles 3:21). In Qohelet's day there were perhaps people who were speculating that human beings would enjoy a positive afterlife, as animals would not. Qohelet points out that there is no evidence for this.', Goldingay, 'Old Testament Theology', volume 2, p. 644 (2006)
 * 'The life of a human being came more directly from God, and it is also evident that when someone dies, the breath (rûaḥ, e.g., Ps 104:29) or the life (nepeš, e.g., Gen 35:18) disappears and returns to the God who is rûaḥ. And whereas the living may hope that the absence of God may give way again to God's presence, the dead are forever cut off from God's presence.241 Death means an end to fellowship with God and to fellowship with other people. It means an end to the activity of God and the activity of other people. Even more obviously, it means an end to my own activity. It means an end to awareness.', Goldingay, 'Old Testament Theology', volume 2, p. 640 (2006) instead of the immortality of the soul, has been held marginally throughout the history of both Judaism and Christianity; such sources include certain Jewish pseudepigraphal works, rabbinical works, Clement of Rome, Arnobius in the third to fourth century, a succession of Arabic and Syrian Christians from the third to the eighth century 'others arose in Arabia, putting forward a doctrine foreign to the truth. They said that during the present time the human soul dies and perishes with the body, but that at the time of the resurrection they will be renewed together.', Eusebius (a contemporary), 'Ecclesiastical History' (6.37.1), NPNF2 1:297
 * 'It is unclear if Arabian thnetopsychism ['soul death'] is related to the Syriac tradition of the soul's dormition [sleep] espoused by writers like Aphrahat (d. ca. 345), Ephrem (d. 373), and Narsai (d. 502), according to whom the souls of the dead are largely inert, having lapsed into a state of sleep, in which they can only dream of their future reward or punishments.', Constas, '"To Sleep, Perchance to Dream": The Middle State of Souls in Patristic and Byzantine Literature', in Talbot (ed.), 'Dunbarton Oaks Papers', No. 55, p. 110 (2001)
 * 'Gouillard notes that variations of thnetopsychism ['soul death'] and hypnopsychism ['soul sleep'] existed alongside the views of the official church until the 6th century when they were resoundingly denounced by Eustratios.', Constas, '"To Sleep, Perchance to Dream": The Middle State of Souls in Patristic and Byzantine Literature', in Talbot (ed.), 'Dunbarton Oaks Papers', No. 55, p. 111 (2001).
 * 'Thnetopsychism ['soul death'] continued to challenge the patience and ingenuity of church officials, as evidenced by writers such as John the Deacon, Niketas Stethatos, Philip Monotropos (Dioptra, pp. 210, 220), and Michael Glykas, all of whom are keenly interested in the survival of consciousness and memory among the souls of the departed saints. John the Deacon, for example, attacks those who "dare to say that praying to the saints is like shouting in the ears of the deaf, as if they had drunk from the mythical waters of Oblivion" (line 174).', Murray, 'Symbols of church and kingdom: a study in early Syriac tradition', p. 111 (2006)
 * 'The Syriac tradition of the soul's "sleep in the dust" (Job 21:26), with its links to the Old Testament and Jewish apocalyptic, stands as a corrective to overly Hellenized views of the afterlife, and was canonized at a Nestorian synod in the 8th century (786–787) presided over by Timothy I (d. 823), who rejected anything else as blatant Origenism.', Murray, 'Symbols of church and kingdom: a study in early Syriac tradition', p. 111 (2006).
 * 'In virtually every period of Byzantine history, critical voices denied that the souls of the dead could involve themselves in the affairs of the living or intercede on their behalf in heaven. Based on a more unitive, materialist notion of the self as irreducibly embodied, some thinkers argued that the souls of the dead (sainted or otherwise) were largely inert, having lapsed into a state of cognitive oblivion and psychomotor lethargy, a condition sometimes described as a state of "sleep" in which the soul could only "dream" of its future punishment or heavenly reward. Still others argued for the outright death of the soul, which, they claimed, was mortal and perished with the body, and which would be recreated together with the body only on the day of resurrection.', Constas, '"To Sleep, Perchance to Dream": The Middle State of Souls in Patristic and Byzantine Literature', in Talbot (ed.), 'Dunbarton Oaks Papers', No. 55, p. 94 (2001)
 * 'Till the end of the sixth century and beyond, Christians in Nisibis and Constantinople, Syria and Arabia adduced Leviticus 17:11 which states that "The soul of the whole flesh is the blood" to argue that the soul after death sank into non-existence, that it lost its sensibility and stayed inert in the grave together with the body.', Samellas, 'Death in the eastern Mediterranean (50–600 AD.): the Christianization of the East: An Interpretation', Studien Und Text Zu Antike Und Christentum, pp. 55–56 (2002) including Aphrahat, Ephrem, Narsai, Isaac of Nineveh (d.700), and Jacob of Sarug, Jewish commentators such as Abraham Ibn Ezra (1092–1167), Maimonides (1135–1204), and Joseph Albo (1380–1444), and later Christians such as John Wycliffe, Michael Sattler, and many Anabaptists, long before Martin Luther challenged Roman Catholic views on heaven and hell with his teaching of "soul sleep".
 * The Christadelphian denial of the pre-existence of Christ, and interpretation of verses such as "I came down from heaven" (John 6:38) as relating to the virgin birth and Christ's mission only, are found in the teachings of: the early Jewish Christians, the Ebionites, the Nazoreans (or Nazarenes),  the Theodotians of Theodotus the Cobbler (who believed Jesus was supernaturally begotten but a man nonetheless), Artemon, Paul of Samosata, the Pseudo-Clementines, and Photinus (d.376); naturally however, given that non-Trinitarian beliefs were punishable with death from the fourth century to the seventeenth, it would be foolish to expect to discover any consistent line of people or groups holding such beliefs. Such attempts become possible only after the Protestant Reformation. Christadelphian Christology is found from the publication of Lelio Sozzini's commentary on John (1561) through to the increasing resistance to the miraculous among English Unitarians after 1800.
 * Affinities with the Christadelphian concept of the devil and/or demons are found in a range of early Jewish and later Christian sources such as: Jonathan ben Uzziel (100s AD); Joshua Ben Karha (135–160); Levi ben Gershon (d. 1344); David Kimhi (1160); Saadia ben Joseph (892–942); Shimon ben Lakish (230–270), David Joris (1501-1556), Lelio Sozzini (1525-1562), Fausto Sozzini (1539-1604), Gerrard Winstanley (1609-1676), Joseph Mede (1640), Jacob Bauthumley (1650), Thomas Hobbes (1651), Lodowick Muggleton (1669), Dr. Anthonie van Dale (1685), Emanuel Swedenborg (1688-1772), Balthasar Bekker (1695), Isaac Newton; Christian Thomasius (1704), Arthur Ashley Sykes (1737), Nathaniel Lardner (1742), The New-Light Quakers of Lynn and New Bedford (1800s), Elias Hicks (1748-1830), Dr. Richard Mead (1755), Hugh Farmer (at least in the account of Christ's temptation; 1761), William Ashdowne (1791), John Simpson (1804), John Epps (1842) and Primitive Baptist Universalists also known No-Hellers (1907 to present)

Organised worship in England for those whose beliefs anticipated those of Christadelphians only truly became possible in 1779 when the Act of Toleration 1689 was amended to permit denial of the Trinity, and only fully when property penalties were removed in the Doctrine of the Trinity Act 1813. This is only 35 years before John Thomas' 1849 lecture tour in Britain which attracted significant support from an existing non-Trinitarian Adventist base, particularly, initially, in Scotland where Arian, Socinian, and unitarian (with a small 'u' as distinct from the Unitarian Church of Theophilus Lindsey) views were prevalent.

Practices and worship
Christadelphians are organised into local congregations, that commonly call themselves ecclesias, which is taken from usage in the New Testament and is Greek for gathering of those summoned. Congregational worship, which usually takes place on Sunday, centres on the remembrance of the death and celebration of the resurrection of Jesus Christ by the taking part in the "memorial service". Additional meetings are often organised for worship, prayer, preaching and Bible study.

Ecclesias are typically involved in preaching the gospel (evangelism) in the form of public lectures on Bible teaching, college-style seminars on reading the Bible, and Bible Reading Groups. Correspondence courses are also used widely, particularly in areas where there is no established Christadelphian presence. Some ecclesias, organisations or individuals also preach through other media like video, and internet forums. There are also a number of Bible Education/Learning Centres around the world.

Only baptised (by complete immersion in water) believers are considered members of the ecclesia. Ordinarily, baptism follows someone making a "good confession" (cf. 1 Tim. 6:12) of their faith before two or three nominated elders of the ecclesia they are seeking to join. The good confession has to demonstrate a basic understanding of the main elements – "first principles" – of the faith of the community. The children of members are encouraged to attend Christadelphian Sunday schools and youth groups. Interaction between youth from different ecclesias is encouraged through regional and national youth gatherings, conferences and camping holidays.

Christadelphians understand the Bible to teach that male and female believers are equal in God's sight, and also that there is a distinction between the roles of male and female members. Women are typically not eligible to teach in formal gatherings of the ecclesia when male believers are present, are expected to cover their heads (using hat or scarf, etc.) during formal services, and do not sit on the main ecclesial arranging (organising) committees. They do, however: participate in other ecclesial and inter-ecclesial committees; participate in discussions; teach children in Sunday schools as well as at home, teach other women and non-members; perform music; discuss and vote on business matters; and engage in the majority of other activities. Generally, at formal ecclesial and inter-ecclesial meetings the women wear head coverings when there are acts of worship and prayer.

There are ecclesially accountable committees for co-ordinated preaching, youth and Sunday school work, conscientious objection issues, care of the elderly, and humanitarian work. These do not have any legislative authority, and are wholly dependent upon ecclesial support. Ecclesias in an area may regularly hold joint activities combining youth groups, fellowship, preaching, and Bible study.

Christadelphians refuse to participate in any military or police force because they are conscientious objectors (not to be confused with pacifists).

There is a strong emphasis on personal Bible reading and study and many Christadelphians use the Bible Companion to help them systematically read the Bible each year.

Hymnody and music
Christadelphian hymnody makes considerable use of the hymns of the Anglican and English Protestant traditions (even in US ecclesias the hymnody is typically more English than American). In many Christadelphian hymn books a sizeable proportion of hymns are drawn from the Scottish Psalter and non-Christadelphian hymn-writers including Isaac Watts, Charles Wesley, William Cowper and John Newton. Despite incorporating non-Christadelphian hymns however, Christadelphian hymnody preserves the essential teachings of the community.

The earliest hymn book published was the "Sacred Melodist" which was published by Benjamin Wilson in Geneva, Illinois in 1860. The next was the hymn book published for the use of Baptised Believers in the Kingdom of God (an early name for Christadelphians) by George Dowie in Edinburgh in 1864. In 1865 Robert Roberts published a collection of Scottish psalms and hymns called The Golden Harp (which was subtitled "Psalms, Hymns, and Spiritual Songs, compiled for the use of Immersed Believers in 'The Things concerning the Kingdom of God and the Name of Jesus Christ). This was replaced only five years later by the first "Christadelphian Hymn Book" (1869), compiled by J. J. and A. Andrew, and this was revised and expanded in 1874, 1932 and 1964. A thorough revision by the Christadelphian Magazine and Publishing Association resulted in the latest (2002) edition which is almost universally used by English-speaking Christadelphian ecclesias. In addition some Christadelphian fellowships have published their own hymn books.

Some ecclesias use the Praise the Lord songbook. It was produced with the aim of making contemporary songs which are consistent with Christadelphian theology more widely available. Another publication, the "Worship" book is a compilation of songs and hymns that have been composed only by members of the Christadelphian community. This book was produced with the aim of providing extra music for non-congregational music items within services (e.g. voluntaries, meditations, et cetera) but has been adopted by congregations worldwide and is now used to supplement congregational repertoire.

In the English-speaking world, worship is typically accompanied by organ or piano, though in recent years a few ecclesias have promoted the use of other instruments (e.g. strings, wind and brass as mentioned in the Psalms). This trend has also seen the emergence of some Christadelphian bands and the establishment of the Christadelphian Art Trust to support performing, visual and dramatic arts within the Christadelphian community.

In other countries, hymn books have been produced in local languages, sometimes resulting in styles of worship which reflect the local culture. It has been noted that Christadelphian hymnody has historically been a consistent witness to Christadelphian beliefs, and that hymnody occupies a significant role in the community.

title: Christadelphians