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Stirling numbers of the first kind

In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one).

The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the first kind. Identities linking the two kinds appear in the article on Stirling numbers.

Definitions
Stirling numbers of the first kind are the coefficients $$s(n,k)$$ in the expansion of the falling factorial

 * $$(x)_n = x(x-1)(x-2)\cdots(x-n+1)$$

into powers of the variable $$x$$:

 * $$(x)_n = \sum_{k=0}^n s(n,k) x^k,$$

For example, $$(x)_3 = x(x-1)(x - 2) = x^3 - 3x^2 + 2x$$, leading to the values $$s(3, 3) = 1$$, $$s(3, 2) = -3$$, and $$s(3, 1) = 2$$.

Subsequently, it was discovered that the absolute values $$|s(n,k)|$$ of these numbers are equal to the number of permutations of certain kinds. These absolute values, which are known as unsigned Stirling numbers of the first kind, are often denoted $$c(n,k)$$ or $$\left[{n\atop k}\right]$$. They may be defined directly to be the number of permutations of $$n$$ elements with $$k$$ disjoint cycles. For example, of the $$3! = 6$$ permutations of three elements, there is one permutation with three cycles (the identity permutation, given in one-line notation by $$123$$ or in cycle notation by $$(1)(2)(3)$$), three permutations with two cycles ($$132 = (1)(23)$$, $$213 = (12)(3)$$, and $$321 = (13)(2)$$) and two permutations with one cycle ($$312 = (132)$$ and $$231 = (123)$$). Thus, $$\left[{3\atop 3}\right] = 1$$, $$\left[{3\atop 2}\right] = 3$$ and $$\left[{3\atop 1}\right] = 2$$. These can be seen to agree with the previous calculation of $$s(n, k)$$ for $$ n = 3$$. It was observed by Alfréd Rényi that the unsigned Stirling number $$\left[{n\atop k}\right]$$ also count the number of permutations of size $$n$$ with $$k$$ left-to-right maxima.

The unsigned Stirling numbers may also be defined algebraically, as the coefficients of the rising factorial:
 * $$ x^{\overline{n}} = x(x+1)\cdots(x+n-1)=\sum_{k=0}^n \left[{n\atop k}\right] x^k$$.

The notations used on this page for Stirling numbers are not universal, and may conflict with notations in other sources. (The square bracket notation $$\left[{n\atop k}\right]$$ is also common notation for the Gaussian coefficients.)

Definition by permutation
$$\left[{n\atop k}\right]$$ can be defined as the number of permutations on $$n$$ elements with $$k$$ cycles. The image at right shows that $$\left[{4\atop 2}\right] = 11$$: the symmetric group on 4 objects has 3 permutations of the form

 * $$ (\bullet \bullet)(\bullet \bullet)$$ (having 2 orbits, each of size 2),

and 8 permutations of the form

 * $$ (\bullet\bullet\bullet)(\bullet)$$ (having 1 orbit of size 3 and 1 orbit of size 1).

These numbers can be calculated by considering the orbits as conjugancy classes (last bullet point).

Signs
The signs of the (signed) Stirling numbers of the first kind are predictable and depend on the parity of $n − k$. In particular,
 * $$s(n,k) = (-1)^{n-k} \left[{n \atop k}\right] .$$

Recurrence relation
The unsigned Stirling numbers of the first kind can be calculated by the recurrence relation

 * $$ \left[{n+1\atop k}\right] = n \left[{n\atop k}\right] + \left[{n\atop k-1}\right]$$

for $$k > 0$$, with the initial conditions

 * $$\left[{0\atop 0}\right] = 1 \quad\mbox{and}\quad \left[{n\atop 0}\right]=\left[{0\atop n}\right]=0$$

for $$n>0$$.

It follows immediately that the (signed) Stirling numbers of the first kind satisfy the recurrence

 * $$ s(n + 1, k) = -n \cdot s(n, k) + s(n, k - 1)$$.

$$

$$

Table of values
Below is a triangular array of unsigned values for the Stirling numbers of the first kind, similar in form to Pascal's triangle. These values are easy to generate using the recurrence relation in the previous section.

Simple identities
Using the Kronecker delta one has,

 * $$\left[{n \atop 0}\right] = \delta_n$$

and

 * $$\left[{0\atop k}\right] = 0$$ if $$k > 0$$, or more generally $$\left[{n\atop k}\right] = 0$$ if k > n.

Also

 * $$\left[{n \atop 1}\right] = (n-1)!,

\quad \left[{n\atop n}\right] = 1, \quad \left[{n\atop n-1}\right] = {n \choose 2},$$

and

 * $$\left[{n\atop n-2}\right] = \frac{3n-1}{4} {n \choose 3}\quad\mbox{ and }\quad\left[{n\atop n-3}\right] = {n \choose 2} {n \choose 4}.$$

Similar relationships involving the Stirling numbers hold for the Bernoulli polynomials. Many relations for the Stirling numbers shadow similar relations on the binomial coefficients. The study of these 'shadow relationships' is termed umbral calculus and culminates in the theory of Sheffer sequences. Generalizations of the Stirling numbers of both kinds to arbitrary complex-valued inputs may be extended through the relations of these triangles to the Stirling convolution polynomials.

$$

Note that all the combinatorial proofs above use either binomials or multinomials of $$n$$.

Therefore if $$p$$ is prime, then:

$$p\ |\left[{p\atop k}\right]$$ for $$1<k<p$$.

Expansions for fixed k
Since the Stirling numbers are the coefficients of a polynomial with roots 0, 1, ..., $n − 1$, one has by Vieta's formulas that

$$ \left[ \begin{matrix} n \\ n - k \end{matrix} \right] = \sum_{0 \leq i_1 < \ldots < i_k < n} i_1 i_2 \cdots i_k.$$

In other words, the Stirling numbers of the first kind are given by elementary symmetric polynomials evaluated at 0, 1, ..., $n − 1$. In this form, the simple identities given above take the form

$$ \left[ \begin{matrix} n \\ n - 1 \end{matrix} \right] = \sum_{i = 0}^{n - 1} i = \binom{n}{2}, $$$$ \left[ \begin{matrix} n \\ n - 2 \end{matrix} \right] = \sum_{i = 0}^{n - 1}\sum_{j = 0}^{i - 1} ij = \frac{3n - 1}{4} \binom{n}{3}, $$$$ \left[ \begin{matrix} n \\ n - 3 \end{matrix} \right] = \sum_{i = 0}^{n - 1}\sum_{j = 0}^{i - 1}\sum_{k = 0}^{j - 1} ijk = \binom{n}{2} \binom{n}{4}, $$ and so on.

One may produce alternative forms for the Stirling numbers of the first kind with a similar approach preceded by some algebraic manipulation: since

 * $$(x+1)(x+2) \cdots (x+n-1)

= (n-1)! \cdot (x+1) \left(\frac{x}{2}+1\right) \cdots \left(\frac{x}{n-1}+1\right), $$

it follows from Newton's formulas that one can expand the Stirling numbers of the first kind in terms of generalized harmonic numbers. This yields identities like

 * $$\left[{n\atop 2}\right] = (n-1)!\; H_{n-1},$$

 * $$\left[{n\atop 3}\right] = \frac{1}{2} (n-1)! \left[ (H_{n-1})^2 - H_{n-1}^{(2)} \right]$$

 * $$\left[{n\atop 4}\right] = \frac{1}{3!}(n-1)! \left[ (H_{n-1})^3 - 3H_{n-1}H_{n-1}^{(2)}+2H_{n-1}^{(3)} \right], $$

where Hn is the harmonic number $$ H_n = \frac{1}{1} + \frac{1}{2} + \ldots + \frac{1}{n}$$ and Hn(m) is the generalized harmonic number $$ H_n^{(m)} = \frac{1}{1^m} + \frac{1}{2^m} + \ldots + \frac{1}{n^m}.$$

These relations can be generalized to give
 * $$\frac{1}{(n-1)!} \left[\begin{matrix} n \\ k+1 \end{matrix} \right] = \sum_{i_1=1}^{n-1} \sum_{i_2=i_1+1}^{n-1} \cdots \sum_{i_k=i_{k-1}+1}^{n-1} \frac{1}{i_1 i_2 \cdots i_k} = \frac{w(n, k)}{k!} $$

where w(n, m) is defined recursively in terms of the generalized harmonic numbers by
 * $$w(n, m) = \delta_{m, 0} + \sum_{k=0}^{m-1} (1-m)_k H_{n-1}^{(k+1)} w(n, m-1-k).$$

(Here δ is the Kronecker delta function and $$(m)_k$$ is the Pochhammer symbol.)

For fixed $$n \geq 0$$ these weighted harmonic number expansions are generated by the generating function

 * $$\frac{1}{n!} \left[\begin{matrix} n+1 \\ k \end{matrix} \right] = [x^k]\exp\left(\sum_{m \geq 1} \frac{(-1)^{m-1} H_n^{(m)}}{m} x^m\right), $$

where the notation $$[x^k]$$ means extraction of the coefficient of $$x^k$$ from the following formal power series (see the non-exponential Bell polynomials and section 3 of ).

More generally, sums related to these weighted harmonic number expansions of the Stirling numbers of the first kind can be defined through generalized zeta series transforms of generating functions.

One can also "invert" the relations for these Stirling numbers given in terms of the $$k$$-order harmonic numbers to write the integer-order generalized harmonic numbers in terms of weighted sums of terms involving the Stirling numbers of the first kind. For example, when $$k = 2, 3$$ the second-order and third-order harmonic numbers are given by

 * $$(n!)^2 \cdot H_n^{(2)} = \left[ \begin{matrix} n+1 \\ 2 \end{matrix} \right]^2 - 2 \left[ \begin{matrix} n+1 \\ 1 \end{matrix} \right] \left[ \begin{matrix} n+1 \\ 3 \end{matrix} \right]$$

 * $$(n!)^3 \cdot H_n^{(3)} = \left[ \begin{matrix} n+1 \\ 2 \end{matrix} \right]^3 - 3 \left[ \begin{matrix} n+1 \\ 1 \end{matrix} \right] \left[ \begin{matrix} n+1 \\ 2 \end{matrix} \right] \left[ \begin{matrix} n+1 \\ 3 \end{matrix} \right] + 3 \left[ \begin{matrix} n+1 \\ 1 \end{matrix} \right]^2 \left[ \begin{matrix} n+1 \\ 4 \end{matrix} \right].$$

More generally, one can invert the Bell polynomial generating function for the Stirling numbers expanded in terms of the $$m$$-order harmonic numbers to obtain that for integers $$m \geq 2$$

 * $$H_n^{(m)} = -m \times [x^m]\log\left(1+\sum_{k \geq 1} \left[ \begin{matrix} n+1 \\ k+1 \end{matrix} \right] \frac{(-x)^k}{n!}\right). $$

Finite sums
Since permutations are partitioned by number of cycles, one has
 * $$\sum_{k=0}^n \left[{n\atop k}\right] = n!$$

The identity
 * $$\sum_{p=k}^{n} {\left[{n\atop p}\right]\binom{p}{k}} = \left[{n+1\atop k+1}\right]$$

can be proved by the techniques on the page Stirling numbers and exponential generating functions.

The table in section 6.1 of Concrete Mathematics provides a plethora of generalized forms of finite sums involving the Stirling numbers. Several particular finite sums relevant to this article include

\begin{align} \left[{n \atop m} \right] & = \sum_{k=m}^n \left[{n+1 \atop k+1 } \right] \binom{k}{m} (-1)^{m-k} \\ \left[{n+1 \atop m+1 } \right] &= \sum_{k=m}^n \left[{k \atop m } \right] \frac{n!}{k!} \\ \left[{m+n+1 \atop m } \right] & = \sum_{k=0}^m (n+k) \left[{n+k \atop k}\right] \\ \left[{n \atop l+m } \right] \binom{l+m}{l} & = \sum_k \left[{k \atop l} \right] \left[{n-k \atop m } \right] \binom{n}{k}. \end{align} $$

Additionally, if we define the second-order Eulerian numbers by the triangular recurrence relation

 * $$\left\langle \!\! \left\langle {n \atop k} \right\rangle \!\! \right\rangle = (k+1) \left\langle \!\! \left\langle {n-1 \atop k} \right\rangle \!\! \right\rangle + (2n-1-k) \left\langle \!\! \left\langle {n-1 \atop k-1} \right\rangle \!\! \right\rangle, $$

we arrive at the following identity related to the form of the Stirling convolution polynomials which can be employed to generalize both Stirling number triangles to arbitrary real, or complex-valued, values of the input $$x$$:

 * $$\left[{x \atop x-n}\right] = \sum_{k=0}^n \left\langle \!\! \left\langle {n \atop k} \right\rangle \!\! \right\rangle \binom{x+k}{2n}. $$

Particular expansions of the previous identity lead to the following identities expanding the Stirling numbers of the first kind for the first few small values of $$n := 1, 2, 3$$:

\begin{align} \left[\begin{matrix} x \\ x-1 \end{matrix} \right] & = \binom{x}{2} \\ \left[\begin{matrix} x \\ x-2 \end{matrix} \right] & = \binom{x}{4} + 2 \binom{x+1}{4} \\ \left[\begin{matrix} x \\ x-3 \end{matrix} \right] & = \binom{x}{6} + 8 \binom{x+1}{6} + 6 \binom{x+2}{6}. \end{align} $$

Software tools for working with finite sums involving Stirling numbers and Eulerian numbers are provided by the RISC Stirling.m package utilities in Mathematica. Other software packages for guessing formulas for sequences (and polynomial sequence sums) involving Stirling numbers and other special triangles is available for both Mathematica and Sage here and here, respectively.

Congruences
The following congruence identity may be proved via a generating function-based approach:

\begin{align} \left[\begin{matrix} n \\ m \end{matrix} \right] & \equiv \binom{\lfloor n/2 \rfloor}{m - \lceil n/2 \rceil} = [x^m] \left(     x^{\lceil n/2 \rceil} (x+1)^{\lfloor n/2 \rfloor}      \right) && \pmod{2}, \end{align} $$

More recent results providing Jacobi-type J-fractions that generate the single factorial function and generalized factorial-related products lead to other new congruence results for the Stirling numbers of the first kind. For example, working modulo $$2$$ we can prove that

\begin{align} \left[\begin{matrix} n \\ 1 \end{matrix} \right] & \equiv \frac{2^{n}}{4} [n \geq 2] + [n = 1] && \pmod{2} \\ \left[\begin{matrix} n \\ 2 \end{matrix} \right] & \equiv \frac{3 \cdot 2^{n}}{16} (n-1) [n \geq 3] + [n = 2] && \pmod{2} \\ \left[\begin{matrix} n \\ 3 \end{matrix} \right] & \equiv 2^{n-7} (9n-20) (n-1) [n \geq 4] + [n = 3] && \pmod{2} \\ \left[\begin{matrix} n \\ 4 \end{matrix} \right] & \equiv 2^{n-9} (3n-10) (3n-7) (n-1) [n \geq 5] + [n = 4] && \pmod{2} \end{align} $$

Where $$[b]$$ is the Iverson bracket.

and working modulo $$3$$ we can similarly prove that

\begin{align} \left[\begin{matrix} n \\ m \end{matrix} \right] & \equiv [x^m] (x^{\lceil n/3 \rceil} (x+1)^{\lceil (n-1)/3 \rceil} (x+2)^{\lfloor n/3 \rfloor} && \pmod{3} \\ & \equiv \sum_{k=0}^m \binom{\lceil (n-1)/3 \rceil}{k} \binom{\lfloor n/3 \rfloor}{m-k-\lfloor n/3 \rfloor} 2^{\lceil n/3 \rceil + \lfloor n/3 \rfloor - m + k} && \pmod{3} \end{align} $$

More generally, for fixed integers $$h \geq 3$$ if we define the ordered roots

 * $$\left(\omega_{h,i}\right)_{i=1}^{h-1} :=

\left\{ \omega_j : \sum_{i=0}^{h-1} \binom{h-1}{i} \frac{h!}{(i+1)!} (-\omega_j)^{i} = 0,\ 1 \leq j < h \right\}, $$

then we may expand congruences for these Stirling numbers defined as the coefficients

 * $$\left[\begin{matrix} n \\ m \end{matrix} \right] = [R^m] R(R+1) \cdots (R+n-1), $$

in the following form where the functions, $$p_{h,i}^{[m]}(n)$$, denote fixed polynomials of degree $$m$$ in $$n$$ for each $$h$$, $$m$$, and $$i$$:

 * $$\left[\begin{matrix} n \\ m \end{matrix} \right] =

\left(\sum_{i=0}^{h-1} p_{h,i}^{[m]}(n) \times \omega_{h,i}^{n}     \right) [n > m] + [n = m] \qquad \pmod{h}, $$

Section 6.2 of the reference cited above provides more explicit expansions related to these congruences for the $$r$$-order harmonic numbers and for the generalized factorial products, $$p_n(\alpha, R) := R(R+\alpha)\cdots(R+(n-1)\alpha)$$.

Generating functions
A variety of identities may be derived by manipulating the generating function (see change of basis):

 * $$H(z,u)= (1+z)^u = \sum_{n=0}^\infty {u \choose n} z^n = \sum_{n=0}^\infty \frac{z^n}{n!} \sum_{k=0}^n s(n,k) u^k

= \sum_{k=0}^\infty u^k \sum_{n=k}^\infty \frac {z^n}{n!} s(n,k).$$

Using the equality
 * $$(1+z)^u = e^{u\log(1+z)} = \sum_{k = 0}^\infty (\log(1 + z))^k \frac{u^k}{k!},$$

it follows that
 * $$\sum_{n=k}^\infty s(n,k) \frac{z^n}{n!} = \frac{(\log (1+z))^k}{k!}$$

and
 * $$\sum_{n=k}^\infty \left[ {n \atop k} \right] \frac{z^n}{n!} = \frac{(-\log(1-z))^k}{k!}.$$

(This identity is valid for formal power series, and the sum converges in the complex plane for |z| < 1.) Other identities arise by exchanging the order of summation, taking derivatives, making substitutions for z or u, etc. For example, we may derive:
 * $$\frac{\log^m (1+z)}{1+z}

= m!\sum_{k=0}^\infty \frac { s(k+1,m+1)\,z^k}{k!} ,\qquad m=1,2,3,\ldots\quad |z|<1$$ or
 * $$\sum_{n=i}^\infty \frac{\left[{n\atop i}\right]}{n \, (n!)} = \zeta(i+1),\qquad i=1,2,3,\ldots$$

and
 * $$\sum_{n=i}^\infty \frac{\left[{n\atop i}\right]}{n \, (v)_n} = \zeta(i+1,v),\qquad i=1,2,3,\ldots\quad \Re(v)>0 $$

where $$\zeta(k)$$ and $$\zeta(k,v)$$ are the Riemann zeta function and the Hurwitz zeta function respectively, and even evaluate this integral
 * $$\int_0^1 \frac{\log^z(1-x)}{x^k} \, dx = \frac{(-1)^z \Gamma(z+1)}{(k-1)!}\sum_{r=1}^{k-1} s(k-1,r)\sum_{m=0}^r \binom{r}{m}(k-2)^{r-m} \zeta(z+1-m) ,\qquad \Re(z) > k-1 ,\quad k=3,4,5,\ldots $$

where $$\Gamma(z)$$ is the gamma function. There also exist more complicated expressions for the zeta-functions involving the Stirling numbers. One, for example, has
 * $$\zeta(s,v) = \frac{k!}{(s-k)_k}\sum_{n=0}^\infty \frac{1}{(n+k)!}\left[{n+k \atop n}\right]\sum_{l=0}^{n+k-1}\!(-1)^l \binom{n+k-1}{l} (l+v)^{k-s},\quad k=1, 2, 3,\ldots$$

This series generalizes Hasse's series for the Hurwitz zeta-function (we obtain Hasse's series by setting k=1).

Asymptotics
The next estimate given in terms of the Euler gamma constant applies:

 * $$\left[ \begin{matrix} n+1 \\ k+1 \end{matrix} \right] \underset{n \to \infty}{\sim} \frac{n!}{k!} \left(\gamma+\ln n\right)^k,\ \text{ uniformly for } k = o(\ln n). $$

For fixed $$n$$ we have the following estimate :

 * $$\left[ \begin{matrix} n+k \\ k \end{matrix} \right] \underset{k \to \infty}{\sim} \frac{k^{2n}}{2^n n!}. $$

Explicit formula
It is well-known that we don't know any one-sum formula for Stirling numbers of the first kind. A two-sum formula can be obtained using one of the symmetric formulae for Stirling numbers in conjunction with the explicit formula for Stirling numbers of the second kind.
 * $$ \left[{ n \atop k } \right] = \sum_{j=n}^{2n-k} \binom{j-1}{k-1} \binom{2n-k}{j} \sum_{m=0}^{j-n} \frac{(-1)^{m+n-k} m^{j-k}}{m! (j-n-m)!} $$

As discussed earlier, by Vieta's formulas, one get$$ \left[ \begin{matrix} n \\ k \end{matrix} \right] = \sum_{0 \leq i_1 < \ldots < i_{n-k} < n} i_1 i_2 \cdots i_{n-k}.$$The Stirling number s(n,n-p) can be found from the formula

\begin{align} s(n,n-p) &= \frac{1}{(n-p-1)!} \sum_{0 \leq k_1, \ldots, k_p : \sum_1^p mk_m = p}  (-1)^K \frac{(n+K-1)!}{k_1! k_2! \cdots k_p! ~ 2!^{k_1} 3!^{k_2} \cdots (p+1)!^{k_p}} , \end{align} $$ where $$K =k_1 + \cdots + k_p.$$ The sum is a sum over all partitions of p.

Another exact nested sum expansion for these Stirling numbers is computed by elementary symmetric polynomials corresponding to the coefficients in $$x$$ of a product of the form $$(1+c_1 x) \cdots (1+c_{n-1}x)$$. In particular, we see that

\begin{align} \left[ {n \atop k+1} \right] & = [x^k] (x+1)(x+2) \cdots (x+n-1) = (n-1)! \cdot [x^k] (x+1) \left(\frac{x}{2}+1\right) \cdots \left(\frac{x}{n-1}+1\right) \\ & = \sum_{1 \leq i_1 < \cdots < i_k < n} \frac{(n-1)!}{i_1 \cdots i_k}. \end{align} $$

Newton's identities combined with the above expansions may be used to give an alternate proof of the weighted expansions involving the generalized harmonic numbers already noted above.

Relations to natural logarithm function
The nth derivative of the μth power of the natural logarithm involves the signed Stirling numbers of the first kind:

$${\operatorname{d}^n\!(\ln x)^\mu\over\operatorname{d}\!x^n}=x^{-n}\sum_{k=1}^{n}\mu^{\underline{k}}s(n,n-k+1)(\ln x)^{\mu-k},$$

where $$\mu^\underline{i}$$is the falling factorial, and $$s(n,n-k+1)$$is the signed Stirling number.

It can be proved by using mathematical induction.

Other formulas
Stirling numbers of the first kind appear in the formula for Gregory coefficients and in a finite sum identity involving Bell number

$$n! G_n = \sum_{l=0}^n \frac{s(n,l)}{l+1}$$

$$\sum_{j=0}^n \binom{n}{j} B_j k^{n-j} = \sum_{i=0}^k \left[{k \atop i}\right] B_{n+i} (-1)^{k-i}$$

Infinite series involving the finite sums with the Stirling numbers often lead to the special functions. For example

$$ \ln\Gamma(z) = \left(z-\frac{1}{2}\right)\!\ln z -z +\frac{1}{2}\ln2\pi + \frac{1}{\pi} \sum_{n=1}^\infty\frac{ 1}{n\cdot n!}\!\sum_{l=0}^{\lfloor n/2\rfloor}\! \frac{(-1)^{l} (2l)!}{(2\pi z)^{2l+1}} \left[{n\atop 2l+1}\right] $$

and

$$ \Psi(z) = \ln z - \frac{1}{2z} - \frac{1}{\pi z} \sum_{n=1}^\infty\frac{ 1}{n\cdot n!}\!\sum_{l=0}^{\lfloor n/2 \rfloor}\! \frac{(-1)^{l}(2l+1)!}{(2\pi z)^{2l+1}} \left[{n\atop 2l+1}\right] $$

or even

$$ \gamma_m=\frac{1}{2}\delta_{m,0}+\frac{(-1)^m m!}{\pi} \!\sum_{n=1}^\infty\frac{1}{n\cdot n!} \! \sum_{k=0}^{\lfloor n/2\rfloor}\frac{(-1)^{k}}{(2\pi)^{2k+1}} \left[{2k+2\atop m+1}\right] \left[{n\atop 2k+1}\right] \, $$

where γm are the Stieltjes constants and δm,0 represents the Kronecker delta function.

Notice that this last identity immediately implies relations between the polylogarithm functions, the Stirling number exponential generating functions given above, and the Stirling-number-based power series for the generalized Nielsen polylogarithm functions.

Generalizations
There are many notions of generalized Stirling numbers that may be defined (depending on application) in a number of differing combinatorial contexts. In so much as the Stirling numbers of the first kind correspond to the coefficients of the distinct polynomial expansions of the single factorial function, $$n! = n (n-1) (n-2) \cdots 2 \cdot 1$$, we may extend this notion to define triangular recurrence relations for more general classes of products.

In particular, for any fixed arithmetic function $$f: \mathbb{N} \rightarrow \mathbb{C}$$ and symbolic parameters $$x, t$$, related generalized factorial products of the form

 * $$(x)_{n,f,t} := \prod_{k=1}^{n-1} \left(x+\frac{f(k)}{t^k}\right)$$

may be studied from the point of view of the classes of generalized Stirling numbers of the first kind defined by the following coefficients of the powers of $$x$$ in the expansions of $$(x)_{n,f,t}$$ and then by the next corresponding triangular recurrence relation:

\begin{align} \left[\begin{matrix} n \\ k \end{matrix} \right]_{f,t} & = [x^{k-1}] (x)_{n,f,t} \\ & = f(n-1) t^{1-n} \left[\begin{matrix} n-1 \\ k \end{matrix} \right]_{f,t} + \left[\begin{matrix} n-1 \\ k-1 \end{matrix} \right]_{f,t} + \delta_{n,0} \delta_{k,0}. \end{align} $$

These coefficients satisfy a number of analogous properties to those for the Stirling numbers of the first kind as well as recurrence relations and functional equations related to the f-harmonic numbers, $$F_n^{(r)}(t) := \sum_{k \leq n} t^k / f(k)^r$$.

One special case of these bracketed coefficients corresponding to $$t \equiv 1$$ allows us to expand the multiple factorial, or multifactorial functions as polynomials in $$n$$.

The Stirling numbers of both kinds, the binomial coefficients, and the first and second-order Eulerian numbers are all defined by special cases of a triangular super-recurrence of the form

 * $$\left|\begin{matrix} n \\ k \end{matrix} \right| = (\alpha n + \beta k + \gamma) \left|\begin{matrix} n-1 \\ k \end{matrix} \right| + (\alpha^{\prime} n + \beta^{\prime} k + \gamma^{\prime}) \left|\begin{matrix} n-1 \\ k-1 \end{matrix} \right| + \delta_{n,0} \delta_{k,0}, $$

for integers $$n, k \geq 0$$ and where $$\left|\begin{matrix} n \\ k \end{matrix} \right| \equiv 0$$ whenever $$n < 0$$ or $$k < 0$$. In this sense, the form of the Stirling numbers of the first kind may also be generalized by this parameterized super-recurrence for fixed scalars $$\alpha, \beta, \gamma, \alpha^{\prime}, \beta^{\prime}, \gamma^{\prime}$$ (not all zero).

title: Stirling numbers of the first kind


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text:
Union Grove Township, Iredell County, North Carolina

Union Grove Township is a rural, non-functioning county subdivision established in 1868 in Iredell County, North Carolina, United States. Two towns have existed in Union Grove township: 1) the unincorporated community Union Grove, established in 1857 and 2) the former town of Williamsburg established in 1812. The Union Grove Post Office was established in 1857. In 2019, within the unincorporated town of Union Grove is the Union Grove Milling Company, Union Grove School (formerly Union Grove elementary through high school), Union Grove Volunteer Fire House, and two Methodist churches. In the later half of the 20th century, Union Grove was famous for the Fiddler's Convention which was held in Union Grove from the 1920s to 1970s. The Fiddler's Convention was originally established to benefit the Union Grove school.

Geography
Union Grove Township is located on a scenic spot with a view of the Blue Ridge, Brushy Mountains, and the pinnacle of Pilot Mountain some 60 miles away. The center of the town of Union Grove is located at 36.02444°N, -80.86639°W (36.024400, -80.86639). The center of Union Grove township is located at 36.00867730, -80.85436970. Union Grove Township is in the north-central area of Iredell County. It is bordered by Eagle Mills Township to the east, New Hope Township to the west, Olin Township and Turnersburg Township to the south, and Yadkin County and Wilkes County to the north.

Interstate 77 runs north-south and North Carolina Highway 901 runs east–west through the township.

Bodies of water flowing through Union Grove Township include Hunting Creek, Rocky Creek, Owens Branch, Camel Branch, and Brushy Creek.

Demographics
The area that is now Union Grove Township was first settled in the late 1700s. Early settlers included Campbell, Mullis, Morgan, Howard, and Tutterow families. The first town in the area was Williamsburg, which had a post office from 18121827 and from 18321905 with the name Williamsburgh.

Post offices were used to designate persons in the 1860 U.S. Federal Census. The area that became Union Grove Township in 1868 included the census districts north and south of Hunting Creek in Iredell County. This area included the following post offices: Williamsburg, Union Grove, Houstonville, and Eagle Mills post offices. Houstonville and Eagle Mills would become part of Eagle Mills Township. Before the U.S. Civil War, Union Grove was the largest town in this area. The Union Grove post office was called Craters Mills until December 21, 1857. John Templeton was postmaster of Craters Mills from September 17, 1851 until December 21, 1857. The following table shows the breakout of the population of these towns in June 1860:

The inhabitants of the area that became Union Grove in the early 1800s and Union Grove Township in 1868 were farmers and some also ran mills. By 1850, some farms grew larger, as evidenced by the number of slaves and size of the families. In 1850, Perciphull Campbell owned 21 slaves. In 1860, Hosea Redman on Rocky Creek owned 30 slaves. After the Civil War, the 1870 Census shows that there were only 70 "colored" men and women out of a total population of 1,034 in Union Grove Township. There was only one foreign born resident in 1870, a farmer's wife from England named Sophia Low. There were 281 inhabitants in the U.S. Census of 1860, including 242 in district North of Rocky Creek-Union Grove post office area and 39 in the Williamsburg district-Williamsburg post office area.

Demographic makeup in 2000
As of the census of 2000, there were 2,069 people, 775 households, and 579 families residing in the Union Grove township. There were 849 housing units at an average density of 57/sq mi (147.6/km$2$). The racial makeup of the town was 92.2% White, 3.2% Black or African American, 0.1% Native American, 0.5% Asian, <1% Pacific Islander, 3.4% from other races, and 0.5% from two or more races. 5.5% of the population were Hispanic or Latino of any race.

There were 775 households, out of which 74.7% had children under the age of 18 living with them, 61.8% were married couples living together, 8.4% had a female householder with no husband present, and 25.3% were non-families. 36.1% of all households were made up of individuals, and 8.0% had someone living alone who was 65 years of age or older. The average household size was 2.61 and the average family size was 3.02.

In the township the population was spread out, with 24.2% under the age of 18, 7.6% from 18 to 24, 30% from 25 to 44, 24.6% from 45 to 64, and 13.6% who were 65 years of age or older. The median age was 38.0 years. For every 100 females, there were 104 males. For every 100 females age 18 and over, there were 101 males.

The median income for a household in the township was $43,284, and the median income for a family was $46,134. Males had a median income of $31,906 versus $24,340 for females. The per capita income for the town was $18,321. About 4.8% of families and 8.0% of the population were below the poverty line, including 5.2% of those under age 18 and 22.2% of those age 65 or over.

History
The area that is now Union Grove Township was a favorite spot with the Native Americans before early settlers. Many Native American artifacts have been found in the area, especially a site near Jennings mill known as Indian Hill.

Several mills were built on the Hunting Creek in the south and Rocky Creek in the north, the two main streams in Union Grove Township. The mills were used as both a grist mill and saw mill. Mills in the area included the following:
 * Campbell Mill on Rocky Creek built by Perciphull Campbell in 1820, see also Perciphull Campbell House, added to the National Register of Historic Places in 1980.
 * Old Brothers Mill built before 1848, later the site of the Weisner mill;
 * Diffy Mill built before 1848, later the site of the Robertson mill.
 * Rev. Quentin Holten Mill built about 1850. Jennings mill was built on the site of the Holten mill. This mill, originally built by Sandy Hall and Sampson Ball, included a corn grist mill first, and later a sash mill was added.
 * About 1880, John R. Huey built a mill several miles down Rocky Creek from the Jennings mill to grind wheat and corn, as well to saw lumber.
 * Union Grove mill, an electric-powered mill in Union Grove built in the 1900s

Severe flooding in the area in 1848 left only three mills in operation in North Iredell County, the Campbell mill on Rocky Creek, the Jennings mill, and the Diffy mill.

The community churches grew up around the mills, two towns, and concentration of inhabitants. The following churches are within Union Grove township:
 * Barker's Grove Baptist Church
 * Cavalry Baptist Church
 * Friendship Baptist Church
 * Grassy Knob Baptist Church, founded in 1794, within New Hope Township, Union Grove Post Office
 * Macedonia United Methodist Church, in Williamsburg in 1829
 * Moss Chapel was named after Rev. William Moss, who died in 1826. Rev. Moss was a Circuit Riding Methodist preacher from Brunswick County, Virginia, founded in 1799
 * New Friendship Church, Baptist, founded in 1896
 * Smith Chapel
 * Sweet Home Baptist Church, founded in 1886, located nearby in Osborneville in Wilkes County
 * Union Grove Christian Church
 * Union Grove United Methodist, founded on September 23, 1846, Tradition says that Union Grove referred to the church meeting place in the grove where the church members met.
 * Wesley Chapel, formerly Bush Hill Methodist Church, founded on August 8, 1868
 * Winthrop Friends Meeting, Quaker, established in the 1870s
 * Zion Baptist Church, founded on June 4, 1825, organized in the Douthit School House, which stood within 75 feet of the present Zion Church building—just north of the Campbell Mill site on Rocky Creek. The Douthit School, established in the 1830s in a log cabin, was a subscription school.

The earliest known store in the area was operated on the John Howard farm, two miles north of Union Grove, by Jacob Fraley. Prior to the Civil War, Mr Fraley moved his store to Union Grove. Jesse E. Fraley and John A. Fraley operated a store in Union Grove around 1844. Jacob Fraley was Postmaster of the Union Grove Post Office on December 21, 1857.

By the requirements of the North Carolina Constitution of 1868, the county was divided into townships. Previous to that time, the subdivisions were Captain's Districts. While the Captain's Districts referred primarily to the militia, it served also for the election precinct, the tax listing and tax collecting district. Union Grove Township was named after the Union Grove postoffice in the center of that township. Union Grove Township has been used since 1870 as a census district.

The longest operating commercial activity in the area is the Jenning store, which is located one and one-half miles south of Union Grove. This store was originally operated by Rev. Quentin Holton about 1850. In 1886, Mr. John Myers and the Rev. T.L. Jennings founded a mercantile business on this site. Rev. Jennings son, J.T. Jennings, purchased the store from his father.

In the later 1800s or early 1900s, Union Grove citizens founded the Union Grove School. The earliest records show that in 1905 the average attendance was 50. There was only one teacher and the salary for one month was thirty dollars. The first schoolhouse was a one-room frame building. For the first time, the Union Grove school opened on September 14, 1925 with 67 high school students. The high school continued until 1966 when high schools in Iredell County where consolidated. The elementary and junior high students remained at Union Grove School, until 1991, when a consolidated middle school opened.

Fiddlers' convention
The Union Grove Old Time Fiddlers' Convention founded in 1924 by H. P. VanHoy and friends, as a fund raiser for the school. Held annually on the school grounds every Easter weekend, the Fiddlers' Convention was a popular community event that brought many area old-time fiddlers into the small unincorporated town of Union Grove to compete. With the folk music revival in the 1950s and 1960s, however, this small-town community event became increasingly popular with young people who would travel great distances to attend. The unconventional lifestyles of some of these young people disturbed many residents of the conservative farming community. The "Woodstock generation" generated huge crowds to Union Grove, thus overwhelming the community's resources. After the 1969 Convention (and after Woodstock) the county school board asked that the event be moved off school grounds. Thus the decision on where to relocate the Fiddlers' Convention came down to the two sons of H. P. VanHoy. Pierce VanHoy was the oldest son and program director of the Convention. Harper VanHoy (younger brother) was the Secretary and Treasurer. They disagreed on how to proceed and thus, went their separate ways creating two distinct "Fiddlers' Conventions."

Fiddler's Grove, an old-time-music and family-oriented campground, was developed on land purchased near the school by Harper Van Hoy and his wife, Wanona, in 1970. From the beginning, Fiddler's Grove was envisioned as the site for the annual fiddling competition. Central entertainment events at Fiddler's Grove have been the Old Time Fiddler's & Bluegrass Festival, a spring fiddling competition held annually since 1970, and the Square-Up, a fall clogging dance competition held until 1982.

Pierce Van Hoy was the program director and the main promoter of the Convention. He felt that it should be promoted as their father, the founder, intended. He held his event on his farm and retained the original name of "World's Championship Old-Time Fiddlers' Convention," with the blessings and support of their father. Financially committed to Fiddler's Grove, Harper Van Hoy felt it was important to focus on the music and the community by creating a family atmosphere and enforcing a ban on alcohol and illegal drugs. He called his event the Old Time Fiddler's & Bluegrass Festival and received early support from Doc Watson, Lee and Jean Shilling, Alan Jabbour, and Allein Stanley, JP Fraley, and Fred Coon, among others. While the World's Championship Old-Time Fiddlers Convention attracted large crowds and was immediately financially successful, Harper Van Hoy's Ole Time Fiddler's & Bluegrass Festival deliberately stayed small, despite early financial losses, by implementing an invitation-only attendance policy in 1973. Both fiddling events were held simultaneously in Union Grove over Easter weekends. After newspaper accounts erroneously reported drug-related incidents and other problems occurring at the Old Time Fiddler's & Bluegrass Festival instead of the World's Championship Old-Time Fiddlers Convention, Harper Van Hoy moved his festival to Memorial Day weekend in 1974 to distance it from his brother's festival.

Fiddler's Grove continued to host the Old Time Fiddler's & Bluegrass Festival on Memorial Day weekend along with a number of other events. The Festival's commitment to the preservation and perpetuation of old-time music is particularly reflected in its special competition categories of Certified Old-Time Fiddler (fiddlers 55 years and older who play strictly by ear with no formal violin training) and Heritage Tune (in which the fiddler must not only perform but tell the origin and story of an old tune). The Square-Up, an annual dance competition, held over Labor Day weekend was an institution at Fiddler's Grove until 1982. This dance competition featured traditional mountain, precision, and smooth clogging teams as well as individual buck and flatfoot dancers. Fiddler's Grove and the Old Time Fiddler's & Bluegrass Festival were nominated in the Local Legacy category by Senator Jesse Helms as part of the Library of Congress Bicentennial in 2000.

The Pierce VanHoy farm was acquired by Casey VanHoy (the late Pierce VanHoy's youngest son) and converted into an entertainment facility and RV Park - The VanHoy Farms Family Campground. In 2012, Casey decided to restart the World's Championship Old-Time Fiddlers' Convention, under the name Union Grove Fiddlers' Convention. The convention was a success - thus returning to its pre - "Woodstock generation" days, with a much smaller family oriented setting.


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Wikipedia:Articles for deletion/Calvary Christian High School

 This page is an archive of the proposed deletion of the article below. Further comments should be made on the appropriate discussion page (such as the article's talk page or on a Votes for Undeletion nomination). No further edits should be made to this page.

The result of the debate was No consensus. A majority wanted deletion, quite a few wanted BJAODN (which suggests they consider the content unencyclopedic but of some amusement value). While I'm tempted to interpret the BJAODN's as delete votes, I think there's the germ of an encyclopedia article here, and BJAODN does not necessarily imply deletion; it's a comment on content. There is some concern that the existing content may be a copyright violation, and this should be addressed in the appropriate place. If no attempt is made to turn this into a neutral encyclopedia article, relisting on VfD after 28 days may be in order. --Tony Sidaway|Talk 10:54, 30 July 2005 (UTC)
 * See talk page for further discussion.

Calvary Christian High School
This well-written parody article deserves a spot in BJAODN (reminds me of the Landover Church website), but it doesn't belong in the regular article space. Too bad the scrofulous school articles we have to put up with don't rise to this standard of excellence. Denni ☯ 00:33, 2005 July 14 (UTC)

DELETE: This is the page's original author. I started working on this page planning for it to be a true school website. When a friend came over to visit one night though, we decided to have a little fun with it. We would like it deleted now though, all evidence destroyed so-to-speak, to prevent any offense in the future. Thanks. -Hermes- DELETE: Co-creator would rather have the page deleted. We wish to cause no controversy. -Apollo-
 * I don't do this a lot, but delete unless basically rewritten from scratch before end of VfD. JYolkowski // talk 01:20, 14 July 2005 (UTC)
 * BJAODN --Carnildo 02:45, 14 July 2005 (UTC)
 * Creator would like it deleted, not moved to BJAODN. -Hermes-
 * 'keep this please but move it to the bjadon like carnildo suggested a real one can be made later by the original authors Yuckfoo 21:03, 14 July 2005 (UTC)
 * BJAODN! will accomodate it... the article space will not. GarrettTalk 09:01, 15 July 2005 (UTC)
 * Delete per author request - BJAODN might be a good choice - Tεx  τ  urε  14:29, 15 July 2005 (UTC)
 * BJAODN. Just because author regrets he made the article doesn't mean that we have to listen to him- when he hit the submit button, he released it into GFDL.  Besides, it's funny, and deserves its own spot in BJAODN.   ral  315  23:21, July 15, 2005 (UTC)
 * First of all, I don't regret making the article. Secondly, it's not a matter of what you don't have to do, but one of respect and kindness. -Hermes-

Delete First of all, "Willmcw," accurate scrutinization will reveal that everything below goals is not a copyright violation. Read the text carefully. Such prudence is crucial when broadcasting such bold claims to such a large constituency. -Apollo-
 * Delete It's not nice to keep an article just because the creator says to delete it. 64.109.253.204 04:27, 19 July 2005 (UTC)
 * Speedy delete as Wikipedia is not a webspace provider, and the original author has requested its deletion. --Idont Havaname 22:43, 19 July 2005 (UTC)
 * Delete. Yes, part of it is funny, but there will be more funny articles. Everything from the "Goals" section down is a copyright violation from here: . -Willmcw 04:51, July 21, 2005 (UTC)
 * 'Keep Article has been edited well beyond the author's right to request deletion M0rd0r 21:03, 14 July 2005 (UTC)
 * Still a copyvio. The changes you've made don't seem extensive enough to make it a protected parody. --Carnildo 17:49, 21 July 2005 (UTC)

Keep the article m0rd0r? It's just a big joke. -Hermes-
 * Keep - I cant see any reason to delete this page! - The Time Killer

Does this need to be moved to Votes for deletion/Old?
 * The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in an undeletion request). No further edits should be made to this page.


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Reginald Hall

Admiral Sir William Reginald Hall (28 June 1870 – 22 October 1943), known as Blinker Hall, was the British Director of Naval Intelligence (DNI) from 1914 to 1919. Together with Sir Alfred Ewing he was responsible for the establishment of the Royal Navy's codebreaking operation, Room 40, which decoded the Zimmermann telegram, a major factor in the entry of the United States into World War I.

Royal Navy career
Reginald Hall was born in Salisbury, Wiltshire, eldest son of Captain William Henry Hall, the first head of Naval Intelligence, who married the daughter of the Reverend George Armfield from Armley, Leeds. Hall decided on a naval career for himself when taken on a cruise on board HMS Flamingo (1876) by his father. He joined the training ship HMS Britannia in 1884 and two years later was appointed to the armoured cruiser Northampton. After a year he was transferred to the ironclad battleship Bellerophon which was part of the North American Station. In 1889 he became acting sub-lieutenant before attending courses for his Lieutenant's examinations at Greenwich, the gunnery school and torpedo school, where he achieved first class grades in all five of the subjects. Now as Lieutenant he was posted to the China Station where he served on the flagship, the armoured cruiser Imperieuse. In 1892 he was recommended to train as a gunnery officer, which involved a course in mathematics at the Royal Naval College, Greenwich, followed by a year's course at the gunnery school at HMS Excellent, a shore establishment at Whale Island, Portsmouth. Having qualified, he remained for a year on the staff.

In 1894 Hall married Ethel Wootton de Wiveslie Abney, daughter of Sir William. The engagement had begun when he was aged 19, five years before, but at the time this was still considered a young age for a naval officer to get married. In 1895 he was appointed as gunnery lieutenant on the cruiser Australia where he served for two years before being appointed to the senior staff at Whale Island. He was promoted to commander 1 January 1901, in recognition of services with naval forces in South Africa, and later served on the battleship Magnificent, flagship of the second in command of the Channel Fleet.

In 1904 he became commander of the pre-dreadnought battleship Cornwallis which was commissioning with a new crew to operate in the Mediterranean. Hall had a reputation as a strict disciplinarian with an ability to get the best out of a crew, so he was given some of the worst sailors to bring up to scratch. A trick of disaffected sailors was to remove gunsights and throw them overboard, and indeed this happened. Hall instructed the master at arms to look out for two sailors who had previously been friends but who now avoided each other. He took one, told him that the other had confessed they had removed the sights, and the sailor duly confessed. Behaviour on board improved. Although Hall imposed discipline strictly, he also showed concern for the men he commanded and for their welfare, which was unusual for that time. He took particular care for the boys and junior seamen.

In December 1905 he was promoted to captain and appointed by the First Sea Lord, Sir John Fisher, as Inspecting Captain of the new Mechanical Training Establishments, which Fisher had established to give engineering training to ordinary sailors (1906–7). He was next appointed captain of the cadet training ship HMS Cornwall. Although not a conventional warship, this now involved Hall in intelligence work. The ship visited foreign ports, particularly in Germany which was now seen as the navy's greatest potential enemy, and Hall started the tour with a long list of places to investigate. In Kiel he was tasked with discovering how many slips had been constructed for building large vessels. To do this he devised a ruse with the help of the Duke of Westminster, who was visiting the port and agreed to lend his motor boat for the task. Hall and a couple of officers dressed down as sailors and took the yacht on a full speed circuit of the harbour, pretending to break down by the naval dockyard. A concealed camera was then used to take photographs of the installations. More information was discovered by careful questions ashore.

In 1910 two officers, Lieutenant Brandon and Captain Trench, who had been part of Hall's crew gathering intelligence, were sent on a 'holiday' in Germany to collect information about coastal defences by Captain Regnart of the intelligence division. The two were captured and served two-and-a-half years of a four-year sentence, before being pardoned in May 1913 as part of a visit by King George V to Germany. The admiralty then denied any responsibility for what had happened to the pair and their considerable financial loss as a result. When appointed director of intelligence, Hall arranged for them to be compensated.

Hall was appointed to the armoured cruiser Natal following the death of her captain, F.C.A. Ogilvy. Ogilvy had obtained a high reputation for his ship which had beaten all records at gunnery but Hall managed to step into Ogilvy's place, retaining the confidence of the crew and bettering the gunnery record the following year. His reputation for unorthodox treatment of his men continued to grow, making it his business to reform recalcitrant sailors rather than simply punishing them. He had the knack of threatening men with punishments he could not deliver (such as dismissal from the service) if they did not reform, and succeeded. Natal was called upon to assist a sailing ship, the Celtic Race, which had lost most of her sails in a storm and was in danger of sinking. Despite the risk to his own ship, he escorted Celtic Race into Milford Haven. He was rewarded by the owners and underwriters, being presented with a silver table centrepiece by the Lord Mayor of Liverpool on their behalf.

From 1911 to 1913 he served as assistant to the Controller of the Royal Navy.

In 1913 he became captain of the battle cruiser Queen Mary. When he took over he was asked to take part in an experiment by the Admiralty to dispense with ships' police, transferring their duties to petty officers. He also introduced an innovation of his own, to divide the ship's crew into three watches rather than the customary two. Hall believed that in the war with Germany which he expected, it would be impossible to run a ship continuously with just two watches. The change attracted much ridicule but when war came it was adopted on all large ships. A religious man, he arranged that a chapel was provided on board the ship, and this innovation too was adopted generally. He felt it important that the authority of petty officers should be boosted, and he did this by improving their quarters. He arranged for washing machines on board, for the convenience of the officers who no longer had to pay for laundry ashore, and to the benefit of sailors who were 'tipped' to carry out the washing. He introduced a bookshop on board, and the navy's first shipboard cinema. Inadequacies in the water supply meant that it took an hour for all the stokers to finish washing after duty: Hall insisted that his engineers find a solution to provide enough hot water that they could all wash in 15 minutes, giving them more free time. These changes too became standard. Hall was criticised for being too soft with his crews, but his requirements for discipline were not relaxed. Rather, he believed that reward was necessary as well as punishment. Living conditions in society were improving generally and he believed the Navy must keep pace with standards ashore.

Queen Mary took part in the battlecruiser squadron's visit to Cronstadt, where the fleet officers and sailors were entertained by the Russian royal family, and a lavish ball was held on board ship as the culmination of the visit. It was the last such visit before World War I commenced and the battlecruisers transferred to their initial wartime base at Scapa Flow. Ships spent long periods at sea, before the strategic situation became clearer and the fear of immediate German raids diminished. Queen Mary took part in the Battle of Heligoland Bight, where the battlecruisers were called upon to support a raid by destroyers and cruisers against German patrol ships operating off Heligoland. The raid was hailed as a British success, although behind the scenes it was marred by poor communications between the British forces involved.

Hall's health had deteriorated under the stress of continuous sea service, and after three months matters had reached the point where he was obliged to request posting away from the ship.

Director of the Intelligence Divisions
His seagoing career cut short by ill-health, Hall was appointed Director of the Intelligence Division (DID) by the Admiralty in October 1914, replacing Captain Henry Oliver. According to Oliver, Hall's wife wrote to him on behalf of her husband requesting that he replace Oliver in the Intelligence Division. Hall served as DID (the title eventually reverted to the pre-1911 "DNI") until January 1919, when he retired from active duty. It turned out to be a fortunate appointment, for he was responsible for building up the naval intelligence organization during the war, encouraged codebreaking and radio-intercept efforts, and provided the fleet with good intelligence, making the division the pre-eminent British intelligence agency during the war. He also encouraged cooperation with other British intelligence organizations, such as MI5 (under Vernon Kell), MI6 (under Mansfield Smith-Cumming) and the Special Branch of Scotland Yard (under Basil Thomson).

Easter Rising
Thanks to intercepts from Room 40, Hall was instrumental in the interception of the steamer Aud on 21 April 1916 by HMS Bluebell, which was carrying German arms to Ireland. That morning the man who had organised the arms shipment, Sir Roger Casement, was arrested in Tralee Bay after disembarking from a German U-Boat. Hall was aware of the upcoming Easter Rising in Dublin, but refused to reveal his sources, so that when information of the rising did reach the government, its authenticity was questioned. Hall interrogated Casement and allegedly refused Casement the opportunity of making a public demand for the cancellation of the uprising.

'German Plot'
When the United States broke off diplomatic relations with Germany, intercepted traffic between the US German legation and Berlin dried up as a key source of British intelligence. Less reliable information was used by intelligence chiefs including Admiral Hall, such as 'spy-obsessed loyalist residents in Ireland'. The mass arrests of known Sinn Féin activists, following the discovery of a spurious 'German Plot' in Ireland has been interpreted as: a striking illustration of the apparent manipulation of intelligence in order to prod the Irish authorities into more forceful action...when the British Government was unable to provide convincing evidence of a 'German Plot', nationalist Ireland concluded that it had been invented as retribution for the defeat of conscription.' In that analysis, Irish public opinion was wrong. Based on the faulty intelligence information made available to them, 'British ministers sincerely believed the threat was real'.

German counter-intelligence
Room 40's decryptions also led to the capture of Captain Franz von Rintelen, a veteran field agent in the intelligence wing of the German Imperial Navy, who had operated covertly in the still neutral United States and, among many other things, had financed and encouraged strikes by anti-war labor unions, attempted a hostile takeover of the Du Pont corporation, and firebombed munitions ships and armaments factories.

In 1917, Hall was awarded the Japanese Order of the Rising Sun, Gold and Silver Star, which represents the second highest of eight classes associated with the award. In that same year, he was promoted to rear-admiral. He was knighted in 1918 and was promoted to vice-admiral in 1922 and to full admiral in 1926, both on the retired list.

Political career
Upon retirement Hall served as a Conservative Member of Parliament (MP) for Liverpool West Derby from 1919 to 1923, then for Eastbourne 1925–1929. As an MP, in 1919 he and a group of industrialists founded a group to counter subversive actions against free enterprise known as National Propaganda, which was later renamed the Economic League. Even in the House of Commons he was still said to be involved in the Zinoviev letter affair in 1924, which led to the victory of the Conservatives in the general election of that year. In the 1920s and 1930s he travelled extensively in the United States to give lectures on intelligence gathering matters. Too old to return to active service on the outbreak of World War II, Hall nevertheless served in the British Home Guard until his death.

Despite his retirement from military and political life, Hall by the late 1930s had been identified as an important target person by the National Socialist police apparatus: in early 1940 the Reichssicherheitshauptamt in Berlin, the headquarters of the intelligence service of the SS, added his name to the Sonderfahndungsliste G.B., a list of people residing in the UK, whom the Nazi leadership and/or its intelligence service regarded as particularly important or (from their point of view) dangerous and who for that reason were slated to be tracked down and apprehended with heightened priority by special task forces of the SS, that were to follow on the heels of the occupying forces in case of a successful invasion of the British islands by the Wehrmacht.

Hall was described by the U.S. Ambassador to the Court of St. James's Walter Page as a "clear case of genius", while American attaché Edward Bell described him as "a perfectly marvelous person but the coldest-hearted proposition that ever was – he'd eat a man's heart and hand it back to him."

He was known as "Blinker" on account of a chronic facial twitch, which caused one of his eyes to "flash like a Navy signal lamp". His daughter attributed this to childhood malnutrition. (He had attended a military boarding school in which the boys had to fill their bellies by stealing turnips from neighbouring farms). Today, such a twitch is also thought to be symptomatic of a mild form of Dyspraxia.

Honours and awards

 * Knight Commander of the Order of St Michael and St George
 * Companion of the Order of the Bath
 * Order of the Rising Sun, Gold and Silver Star (Japan)
 * Commander of the Order of St. Maurice and St. Lazarus (Italy)
 * Commander of the Legion of Honour (France)
 * Navy Distinguished Service Medal
 * Grand Officer of the Order of the Crown of Italy

title: Reginald Hall


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Talk:Disentanglement puzzle

VfD results
This article was nominated for deletion. The result was no consensus, but some merging/redirecting might be in order. For details, please see Votes for deletion/MoreKarlScherer. -- BD2412 talk 00:35, July 14, 2005 (UTC)

Burr puzzles
Any thoughts on whether burr puzzles and ball-in-cage puzzles actually belong in the list of disentanglement puzzles? Dalgety & Hordern's classification places them in "Interlocking Puzzles" which is distinct from their "Tanglement Puzzles," although I don't know how well their classification is accepted.

In my mind (as a casual collector of disentanglement puzzles), there are two features which distinguish burr puzzles from disentanglement puzzles. First, disentanglement puzzles are not rigid in the assembled state; burr puzzles are. Second, all relevant features of disentanglement puzzles are visible in the assembled state; this is not so in burr puzzles, where the shape of each piece is not fully visible in the assembled state. However, I admit I currently have no basis for these two features as attributes of disentanglement puzzles other than my own intuition and the classification I linked to above. Chuck 19:44, July 22, 2005 (UTC)

 * I think you have hit on the main reason that -Ril- has been putting these articles up on VfD. The apparent fact is that the contributor has largely invented his own classification scheme which is, at least in some places, at odds with other, more authoritative schemes. Since classification is just about the only content of the article, you can perhaps understand why there has been some desire to delete them or merge the descriptive sections into a larger article like mechanical puzzles. — HorsePunchKid → 龜 20:45, July 22, 2005 (UTC)

 * Having heard no objection, I will be bold and remove them. If anyone disagrees, please cite a source. Chuck 17:59, August 1, 2005 (UTC)

 * Looks good to me. The less cruft we have here, the easier it will be to merge this into a more general, more useful article when the time comes. :) — HorsePunchKid → 龜  22:02, 1 August 2005 (UTC)

Vfd
Keep vote on vfd per Votes_for_deletion/KarlSchererRevisited2. Please wait a bit before putting this up again. --Woohookitty 02:33, 27 July 2005 (UTC)

Can't you vacuum
before taking a picture of your puzzle to post on wikipedia?--345Kai (talk) 00:52, 11 May 2009 (UTC)

 * I've got pictures of a few of mine at against white backgrounds.  Let me know if you think a couple of these would be better for the article.  I haven't released them under a Wikipedia-compatible license yet, but would be willing to do so. Chuck (talk) 04:59, 11 May 2009 (UTC)

horseshoe Disentanglement puzzle
horseshoe-ring puzzles sold in gift shops are a fond childhood memory. This specific puzzle isn't mentioned on any of the other pages, so it might make sense to note them here. --Zqrrld (talk) 01:00, 9 September 2012 (UTC)

Impossible?
Is a "puzzle" with no solution really a puzzle, or is it just a piece of sculpture? — Preceding unsigned comment added by 70.89.176.249 (talk) 22:05, 24 September 2018 (UTC)
 * I have to agree. If there is no solution, then it's not a puzzle.  — Preceding unsigned comment added by 74.95.43.253 (talk) 19:54, 5 December 2022 (UTC)

title: Talk:Disentanglement puzzle


text:
Talk:Packing problems

VfD results
This article was nominated for deletion. The result was no consensus, but some merging/redirecting might be in order. For details, please see Votes for deletion/MoreKarlScherer. -- BD2412 talk 00:36, July 14, 2005 (UTC)

This article was nominated for deletion a second time, this time kept outright, See Votes for deletion/KarlSchererRevisited1. Sjakkalle (Check!)  09:16, 27 July 2005 (UTC)

Dubious-ness
Could someone with a large (by number of edits, not by time) edit history (so that it is obviously not Karl Scherer or friends) verify the maths examples in this article. They strike me as inaccurate and/or non-noteworthy and/or presented with Weasel words

21:15, 20 July 2005 (UTC)

 * Why are you disputing these? Weren't they there a long time ago? Charles Matthews 10:05, 27 July 2005 (UTC)

Correct me if I'm wrong,

 * Maybe I did not find the best packing but the first box I could put an extra disc is 2x238 after that each extra 238 gives me extra two balls...

This is very different from numbers in the article, could anyone provide a ref? Tosha 23:24, 18 October 2005 (UTC)

Fiction? in article
I am a Maths Graduate and teacher, and yet I cannot see any possible way to pack more than 2n circles in an n X 2cm strip. To do so, essentially you would have either fit more circles (or part circles) vertically (there's no room) or horizontally. To benefit from hexagonal packing you would have to have a wider strip, and hexagonal packing is the most efficient way of packing circles. Furthermore, the article talks about a classic problem, and yet that problem is not mentioned on this article in Math World about circle packing.

As citations were asked for a long time ago, and have not been provided, I will now change the article and remove contentious references (they can be restored if citations are found).

Captainj 17:29, 22 May 2006 (UTC)

 * OK, I think I can posted the answer to your problem. Stand by... Michael Hardy 20:15, 22 May 2006 (UTC)

Maybe I'll return to this later. For now I'll just suggest thinking about how many circles in the figure above should be "centered" and how many should not be. Michael Hardy 20:38, 22 May 2006 (UTC)

I still can't make this work. If we use the packing method in the picture, and have 2 circles (in a vertical line) followed by a centered circle followed by another two circles, you get a total area of 3.92 for the five circles. The length of the strip that they occupy is $$1 + \sqrt(5) \approx 6.47$$. This means the area of the strip they occupy is approximately 8.47, giving an efficiency of 60.7%%. But the "standard efficiency" (all the circles two "neat" lines) is 78.5%. Therefore every time a centered circle is but between two standard packed circles, you lose area. So there doesn't seem to be any benefit in doing it that way.

If the circles were only put at then end, this wouldn't help either, because, in any case, the length of the strip is an integer.

Sorry I haven't drawn all the diagrams etc, but I don't have the software to do it easily. (I did all the working out back of the envelope). I hope my above Maths is correct, its clear I am going rusty in places.Captainj 21:28, 22 May 2006 (UTC)

 * You can do better than this. First lay a layer on the bottom.  Then lay a layer above it, touching it, but offset by half a ball.  Now notice that there is a gap at the top.  Therefore this whole structure can be zigzagged a bit to compact it.99of9 (talk) 02:01, 1 September 2009 (UTC)

 * So you can pack two rows of circles whose diameter is a bit greater than half the width of the strip; that's an answer to a question other than what was asked here, I think. —Tamfang (talk) 02:49, 1 September 2009 (UTC)

 * Well I meant you could zigzag the bottom row a bit (and the top row would still fit), increasing the lineal density, rather than expanding the circles. --99of9 (talk) 05:52, 10 May 2010 (UTC)

 * How does that increase the density? —Tamfang (talk) 16:37, 10 May 2010 (UTC)

 * Easier to explain with an image. I'm impressed that you're still watching the page - I almost didn't bother replying when I saw how long it had been! --99of9 (talk) 00:26, 11 May 2010 (UTC)

 * [[Image:Circle-strip.svg|frameless|500px|Packing on a strip with density 1.006046496.]]

 * Ah! —Tamfang (talk) 02:46, 11 May 2010 (UTC)
 * Well, you've convinced me. But it should really still be sourced.  — Preceding unsigned comment added by Captainj (talk • contribs) 23:16, 4 January 2012 (UTC)
 * That's easy. See Fig. 1.2 (c) on page 4 of the PDF Leob (talk) 21:05, 11 March 2016 (UTC)

Walter Stromquist
Where did the fact about Walter Stromquist proving something come from? I can't find it in the sources given. Leon math 23:44, 14 December 2006 (UTC)

Other known results
Not a single reference or proof for any of these. —Preceding unsigned comment added by 84.92.32.151 (talk) 21:09, 26 November 2007 (UTC)

balls inside a ball?
After noticing some interest at the RD/M I have added a short description about the case of packing few balls into a ball (of whom I don't have references)... I thought it could be nice to have a complete description of an easy packing problem. NOw I wonder if maybe it should better be shortened...--pma (talk) 09:12, 30 April 2009 (UTC)

de Bruijn's theorem: neccessary condition?
De Brujin's theorem, as stated, de Bruijn's theorem: A box can be packed with a harmonic brick a × a b × a b c if the box has dimensions a p × a b q × a b c r for some natural numbers p, q, r (i.e., the box is a multiple of the brick.) is obvious. Perhaps what is really meant is "if and only if?" —Preceding unsigned comment added by 92.193.118.17 (talk) 09:59, 4 October 2009 (UTC)

"Klarner's" theorem
The article says:

 * Klarner's theorem: An a × b rectangle can be packed with 1 × n strips iff n | a or n | b.

No cite is given for the name "Klarner's theorem". Is "Klarner" David A. Klarner?

I added a citation of this theorem:. But Wagon does not refer to the theorem as Klarner's; he cites it as a special case of a theorem of N. G. de Bruijn which may be the de Bruijn's theorem mentioned in this wikipedia article just after "Klarner's".

cites to a book by Ross Honsberger. Can someone confirm that this book really calls the theorem "Klarner's"? Or is this just another Mathworld oddity?

—Dominus (talk) 16:19, 6 January 2010 (UTC)

 * The cited source does not bear out the attribution.  Honsberger credits Klarner, but does not refer to the result as "Klarner's theorem". I think this is a Mathworld neologism.  Klarner is indeed David A. Klarner.  Honsberger's cite is:  —Mark Dominus (talk) 07:16, 5 January 2012 (UTC)

Same problem in 3D
Does wikipedia have any references to this same problem in 3d, and for arbitrary shapes?

Is there any algorithm to solve these problems, other than just "try all possibilities"? 87.194.84.113 (talk) 23:35, 9 May 2010 (UTC)
 * Tetrahedron packing is one article I know of. I know a lot about binary sphere packing (in Euclidean space), but haven't written anything for wiki.  What shapes are you interested in? --99of9 (talk) 05:54, 10 May 2010 (UTC)

Rectangle Packing
This section is particularly sparse, and only talks about one special case. Could it possibly be fleshed out?Samineru (talk) 22:01, 1 March 2011 (UTC)

No techniques, no proofs
I believe the topic is important to have a wiki article on. I also believe the implementation is unsatisfactory in part because: 1) no word in it of how the packings presented were found;2) how they were proven (if at all). Both items, especially, 2 are hard to present fully and they may be connected. (The article has some names of people who prove optimalitry of some packings). I would not expect any complete description on these, but no word at all... that is too much to not have (or too little to have) in a good article. As to how some of the packings were found, I may suggest an item on which I've just written a wiki article Lubachevsky-Stillinger algorithm. This is certainly not the only one in existence, so go ahead and put a list of other techniques. But, as a matter of fact, I know that many of the optimal (proven or conjectured) packings, e.g., of 91 congruent circles in a circle, were found first with this very algorithm (and then confirmed with other algorithms, which is usually much easier, than to discover; some recent proof techniques are based on knowing the packing with high precision, so discovering the packing numerically becomes, essentially, a first step of the proof). See Curved Hexagonal Packings of Equal Disks in a Circle http://www.math.ucsd.edu/~ronspubs/95_02_dense_packings.pdf It was also fun to read a comment and its resolution above on packing more than 2n unit diam circles in 2 times n rectangles. That problem was one of many by Erdos. Does this article mention that name? Yes, but not emphatically enough for a math genius of the 20th century. (Remember notability requirement of wiki). The mentioned above algorithm produces the required packings (for some smaller n of course). If I remember it correctly, for n=1000 you can exceed 2n by 7.

Lsalgo (talk) 04:18, 23 May 2011 (UTC)

Finite uniform 3D sphere packing
There does not seem to be any discussion of uniform 3D sphere packing on the English Wikipedia; as "sited" here it is only discussed on the German Wikipedia here, even though there is an English summary of research as of 1999 by Conway and Sloane here. Mark Hurd (talk) 07:54, 28 August 2012 (UTC)

 * Sphere packing? Close-packing of equal spheres? —Tamfang (talk) 08:01, 28 August 2012 (UTC)
 * Yeah "not ... any" is a little strong. Specifically nothing seems to mention the "sausage collapse" after 55 spheres. Mark Hurd (talk) 13:00, 28 August 2012 (UTC)

unequal-sphere dimmers
What is unequal-sphere dimmer? Where I can read about it? Jumpow (talk) 13:11, 8 June 2016 (UTC)

May be it is misspelling and must be dimer? Jumpow (talk) 14:45, 8 June 2016 (UTC)

Again, what means woodpulp stowage in context of packing of a single rectangle? Woodpulp refers to article Pulp (paper): Pulp is a lignocellulosic fibrous material prepared by chemically or mechanically separating cellulose fibres from wood, fiber crops or waste paper. Jumpow (talk) 16:31, 8 June 2016 (UTC)

What means pack one of each n-omino into a rectangle? Pack set of n-omino into a single rectangle (each only ones)? Into given single rectangle? Into minimal single rectangle? Not all n-omino? Jumpow (talk) 17:12, 8 June 2016 (UTC)

Is the puzzle template appropriate?
The first line of the article reads

"Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers."

Given the technical nature of the article, is it appropriate to have a template with links such as Optical illusion and Thinking outside the box?

104.228.101.152 (talk) 05:04, 16 March 2017 (UTC)

Untitled
O(a7/11) is not consistent with Wikipedia's other pages. It changed several decades ago. Thanks Paul Erdös — Preceding unsigned comment added by 1.136.109.183 (talk) 09:22, 20 October 2021 (UTC)

title: Talk:Packing problems


text: File:Grant Weaver user photo.jpg

title: File:Grant Weaver user photo.jpg


text:
Wikipedia:Peer review/Military history of Canada/archive1

Military history of Canada
An article I started back in 2003 that has since been much improved by a number of editors. In the last few weeks I've worked to cover the last omissions with the hope of elevating it to featured article status. At 40kb this article is substantial, but the subject is a large one. One option to somewhat reduce the length would be to move the WWII section to its own article and replace it with a shorter summary of similar size to the WWI section. I also would not be at all surprised if there are still some typos and grammar problems and would be much obliged for any copyediting. - SimonP 00:37, July 14, 2005 (UTC)
 * It looks pretty good; very comprehensive. A minor point is that there is no mention of Canadian actions following the battle of the Falaise Gap. I know they were involved in clearing the Scheldt estuary so that the port of Antwerp could be opened. They also had a nasty little scrap at Hochwald Forest in 1945. — RJH 15:52, 14 July 2005 (UTC)
 * I like it also. However, I think there should be some kind of picture in the lead; either a modern Canadian soldier or perhaps some historical scene to serve as a "hook". --Scimitar 15:55, 14 July 2005 (UTC)

Demote
I have moved that this article be demoted Labattblueboy (talk) 17:00, 4 July 2008 (UTC)


text:
Talk:Rattle puzzle

VfD results
This article was nominated for deletion. The result was no consensus, but some merging/redirecting might be in order. For details, please see Votes for deletion/MoreKarlScherer. -- BD2412 talk 00:37, July 14, 2005 (UTC)

It was nominated again on the same day the previous listing was closed. The discussion is at Votes for deletion/KarlSchererRevisited3. The result again was no consensus. This doesn't stop people being bold and merging on their own account. --Tony Sidaway|Talk 04:27, 31 July 2005 (UTC)

title: Talk:Rattle puzzle


text:
Alphonsa of the Immaculate Conception

Alphonsa of the Immaculate Conception, FCC, born as Anna Muttathupadathu); 19 August 1910 – 28 July 1946), was a religious sister in the Franciscan Clarist Congregation. She was also known for being a victim soul, visionary and prophetess in the Kottayam pergunna of the erstwhile Travancore province of British India, in the present-day Kerala, India.

She is the first woman of Indian origin to be canonised as a saint after decades of enquiry by the Sacred Congregation for the Causes of Saints, she is also the first saint of the Syro-Malabar Church, an Eastern Catholic community of Eastern Christianity.

In the 1990s, a postal stamp commemorating Alphonsa was published by Kerala state's former minister, K Karunakaran. She is the first woman keralite to appear in the Indian Stamp.

Early life
Alphonsa was born as Annakutty Muttathupadathu in a Malankara Nasrani family to Joseph Muttathupadathil and Mary Puthukari in Kudamalloor, near Kottayam, on 19 August 1910. She was baptised on 27 August. Alphonsamma, as she was locally known, was born in Arpookara, a village in the princely state of Travancore, what would become Kerala state. The village lies within the Archeparchy of Changanassery.

Her parents nicknamed her Annakutty (little Anna). She had a difficult childhood characterised by the loss of a parent. Her mother died early, so her maternal aunt helped to raise her. Hagiographies describe her early life as one of hardships at the hands of her stern aunt and foster mother and one of teasing by schoolmates. Anna was educated by her great-uncle, Joseph Muttathupadathu. When Anna was three-years-old, she contracted eczema and suffered for over a year.

In 1916, Anna started school in Arpookara. She received her First Communion on 27 November 1917. When the first school cycle ended in 1920, the time had come to transfer to Muttuchira, to the house of her aunt Anna Murickan, to whom her mother had entrusted her, before she died. She was raised in the Murickan family, which was an age-old and prominent family in the area. As a child her grandmother would tell her stories of saints, teach prayers and Christian songs. As an adolescent Anna received many marriage proposals from reputed families. However, she had a feeling in her mind that she wants to be a bride of Christ and do good things for people. In these years, Thérèse of Lisieux appeared and told her that she will one day become a saint. There are many similarities in the lives of Anna and Thérèse, and she was Anna's all-time favorite patron saint and role model.

Anna sacrificed her material fortune and wanted to dedicate her life to Jesus Christ. In 1923, Anna's feet were burnt when she fell into a pit of burning chaff; local hagiographies describe this as a self-inflicted injury in order to avoid her foster mother's attempt to arrange a marriage for her and thereby to fulfill her desire for becoming a religious sister instead. This accident left her partially disabled for the rest of her life.

Sister Alphonsa
When it became possible, Anna joined the Franciscan Clarist Congregation, a religious congregation of the Third Order of St. Francis, and through them, completed her schooling.

Anna arrived at the Clarist convent at Bharananganam, Kottayam district, on Pentecost Sunday 1927. She received the postulant's veil on 2 August 1928 and took the name Alphonsa of the Immaculate Conception in honor of Alphonsus Liguori, whose feast day it was. In May 1929 Alphonsa was assigned to teach at Malayalam High School at Vazhappally. Her foster mother died in 1930. She resumed her studies at Changanacherry, while working as a temporary teacher at a school at Vakakkad. On 19 May 1930 Alphonsa entered the novitiate of the congregation at Bharananganam. On 11 August 1931 she completed the novitiate and took her first vows. And became a religious in the Catholic Church.

Health decline
The period 1930–1935 was characterised by grave illness. Alphonsa took her permanent vows on 12 August 1936. Two days later she returned to Bharananganam from Changanacherry. She then taught high school at St. Alphonsa Girl's High School, but was often sick and unable to teach. For most of her years as a Clarist sister she endured serious illness.

In December 1936, it is claimed that she was cured from her ailments through the intervention of the Kuriakose Elias Chavara (who was beatified at the same ceremony as she), but on 14 June 1939 she was struck by a severe attack of pneumonia which left her weakened. On 18 October 1940, a thief entered her room in the middle of the night. This traumatic event caused her to suffer amnesia and weakened her again.

Her health continued to deteriorate over a period of months. She received extreme unction on 29 September 1941. The next day it is believed that she regained her memory, though not complete health. Her health improved over the next few years, until in July 1945 she developed gastroenteritis and liver problems that caused violent convulsions and vomiting. During the last year of her life she came to know Sebastian Valopilly (later Bishop of Kerala) who frequently brought her communion. This bishop became famous in Kerala for championing the cause of poor people from all religious backgrounds who had come to live in Thalassery as a result of shortages elsewhere.

Death
After a series of serious health problems, she died on 28 July 1946, at a young age of 35. She was buried at St. Mary's Catholic Church, Bharananganam, Travancore (present day Kottayam district) in the Diocese of Palai. Her death news made her loved ones sad. Many people came to see Alphonsa's funeral.

Veneration
Claims of her miraculous intervention began almost immediately upon her death and often involved the children of the convent school where she used to teach. On 2 December 1953, Cardinal Eugène Tisserant inaugurated the diocesan process for her beatification and Alphonsa was declared a Servant of God.

In 1985, Pope John Paul II formally approved a miracle attributed to her intercession and on 9 July she became "Venerable Sister Alphonsa".

Beatification
Alphonsa was beatified along with Kuriakose Elias Chavara at Kottayam, on 8 February 1986 by Pope John Paul II during his apostolic pilgrimage to India.

During his speech at Nehru Stadium, the Pope said: "From early in her life, Sister Alphonsa experienced great suffering. With the passing of the years, the heavenly Father gave her an ever fuller share in the Passion of his beloved Son. We recall how she experienced not only physical pain of great intensity, but also the spiritual suffering of being misunderstood and misjudged by others. But she constantly accepted all her sufferings with serenity and trust in God. ... She wrote to her spiritual director: 'Dear Father, as my good Lord Jesus loves me so very much, I sincerely desire to remain on this sick bed and suffer not only this, but anything else besides, even to the end of the world. I feel now that God has intended my life to be an oblation, a sacrifice of suffering' (20 November 1944). She came to love suffering because she loved the suffering Christ. She learned to love the Cross through her love of the crucified Lord."

Miracles
Hundreds of miraculous cures are claimed from her intervention, many of them involving straightening of clubbed feet, possibly because of her having lived with deformed feet herself. Two of these cases were submitted to the Congregation for the Causes of Saints as proof of her miraculous intervention. The continuing cures are chronicled in the magazine PassionFlower.

Bishop Sebastian reported: "About ten years ago, when I was in a small village in Wayanad outside Manatavady, I saw a boy walking with some difficulty, using a stick. As he approached me I noted that both of his feet were turned upside down. I had a stack of holy cards in my pocket with Alphonsa's picture on them, so I pulled one of them out and gave it to the boy. When I told the boy that he should pray to this woman for the cure of his feet, the boy – he was quite smart for a ten-year-old boy – replied: 'But I'm a Muslim, and, besides, I was born this way.' I replied that God is very powerful, so let's pray. A few months later, a boy and a gentleman appeared at the house here. I didn't recognize them at first but soon learned that it was the Muslim boy with his father, here to tell me that his feet had been cured through their prayers to Sister Alphonsa. They showed me the calluses on the tops of his feet, and you could see the marks which had been made from the years of his walking with his feet turned under. Before they left, the three of us had our pictures taken."

The boy had reportedly taken Alphonsa's picture card and asked Alphonsa to help fix his feet. Several days afterwards one of his feet supposedly turned around. He and the other members of his family then prayed for the cure of the second foot, which also supposedly turned around later.

Canonization
On Sunday, 12 October 2008, Pope Benedict XVI announced her canonization at a ceremony at Saint Peter's Square. Indians from across the world, especially people from Kerala, gathered at the ceremony in Rome. Among them was a 10-year-old Kerala boy Jinil Joseph whose clubfoot – a birth defect – was, in the judgment of Vatican officials, miraculously healed after prayers to Alphonsa in 1999.

The final ceremony for the canonization began with the holy relics of Alphonsa being presented to the Pope by Sister Celia, Mother General of the Franciscan Clarist Congregation, the congregation to which Alphonsa belonged. Celia was accompanied by Vice Postulator Francis Vadakkel and former Kerala minister K. M. Mani, all holding lit candles. Speaking in English, the Pope declared Alphonsa a saint, after reading excerpts from the Bible. The Pope himself read out the biography of Alphonsa after the ceremony.

In the homily, Pope Benedict XVI recalled Alphonsa's life as one of "extreme physical and spiritual suffering".

"This exceptional woman ... was convinced that her cross was the very means of reaching the heavenly banquet prepared for her by the Father,, By accepting the invitation to the wedding feast, and by adorning herself with the garment of God's grace through prayer and penance, she conformed her life to Christ's and now delights in the 'rich fare and choice wines' of the heavenly kingdom."

"(Her) heroic virtues of patience, fortitude and perseverance in the midst of deep suffering remind us that God always provides the strength we need to overcome every trial," the pope stated before the ceremony ended.

The canonization was greeted with the bursting of firecrackers and the toll of church bells. St Mary's Forane church at Kudmaloor, her home parish, also celebrated a special Mass. The grave at St. Mary's Forane Church, Bharananganam where the Franciscan Clarist Sister was buried had a chapel built there, which houses her mortal remains.

Reserve bank of India Issued a 5 Rupees Commemorative Coin in 2009 marking the birth centenary of Alphonsa.

Shrine
Her tomb at St. Mary's Syro-Malabar Catholic Church, Bharananganam, has become a pilgrimage site as miracles have been reported by some of the faithful.

St. Alphonsa Church Valiyakolly is known as the "Bharanaganam of North Kerala" (unofficial). This church is under the diocese of Thamarassery. Valiyakolly church is the first church named after Alphonsa in the diocese. And hence this church is one of the main pilgrimage centers of Saint Alphonsa in North Kerala. Novena on the name of Saint Alphonsa is being conducted here on every Friday evening

Feast
Thousands of people converge on the small town of Bharananganam when they celebrate the feast of Saint Alphonsa from 19 to 28 July each year; her tomb has been designated as a pilgrimage site with numerous miracles being reported by pious devotees.

Documentary video
In 2008, Sibi Yogiaveedan directed a documentary video of Alphonsa, titled Vishudha (saint) Alphonsama- The Passion Flower. In this video the director gathers the perspectives of Alphonsa's childhood friends and relatives, through which a glimpse into the real life of Alphonsa can be understood. In addition, Alphonsa's character was played by Nikhila Vimal. The video was made in Malayalam language. For this reason, English subtitles have been included.

title: Alphonsa of the Immaculate Conception


text:
Thunder Bay (film)

Thunder Bay is a 1953 American adventure film distributed by Universal International, produced by Aaron Rosenberg, directed by Anthony Mann, and starring James Stewart, Joanne Dru, Gilbert Roland, and Dan Duryea. It was shot in Technicolor and was released on May 20, 1953. This film tells the story of two engineers drilling for oil in the Louisiana gulf while dealing with hostility of the local shrimp fishermen fearing for their livelihood, and features the first non-western collaboration between Stewart and Mann.

Plot
Penniless but full of ideas, Steve Martin and Johnny Gambi, engineers who served in the Navy during World War II, walk down a quiet road on the gulf coast of Louisiana. Teche Bossier, owner of the Port Felicity Fish Co., agrees to drive them into the shrimping town Port Felicity for five dollars. On reaching their destination, Gambi rents a shrimp boat from Dominique Rigaud, although the fisherman's daughter, Stella, distrusts them immediately.

Gambi and Steve use the boat to show potential investor Kermit MacDonough the location in which they plan to drill for offshore oil. Claiming that he has designed a drilling platform that can withstand any storm, Steve estimates that by investing one million dollars now they will soon tap an oil reserve worth two billion. His enthusiasm is so infectious that MacDonough agrees to fund the project, against the advice of his secretary, Rawlins. However, MacDonough warns Steve that he must discover oil within three months, or his company, due to huge investments made in an offshore oil lease, will put them both out of work.

Several weeks later, Gambi meets and falls for Stella's younger sister Francesca, but, according to custom, she has been betrothed since childhood to Philippe Bayard. After singing a love song in the local gathering place Bon Chance, Philippe is upset to see Francesca enter with Gambi. Teche, who good-naturedly calls the oilmen "foreigners," agrees to help Steve and Gambi, but Stella refuses to accept Steve's statement that oil will be good for the town, claiming that she learned about "their kind" during her stay in Chicago. Nevertheless, the outsiders hire a crew and begin their search for oil.

When Teche sees dynamite charges being dropped into the gulf, he begs them to stop, believing that the explosions will kill the shrimp and worsen an already dismal shrimping season. Steve maintains that the charges are safe, but Teche returns to town and incites the fishermen to form an angry mob. Steve manages to scare the mob away by exploding sticks of dynamite behind them, and he placates Stella by warning Gambi to stay away from Francesca. Steve gently advises Francesca to "go back to [her] people."

With one month gone, Steve drives the building crew relentlessly, and the platform and rig are completed on schedule. He immediately orders the drilling crew to get started, and the exhausted Gambi is relieved when a hurricane warning gives the men an excuse to take the night off. Gambi and his men enter the Bon Chance with Francesca, and Philippe furiously punches his rival and starts a brawl. The sheriff arrests the oilmen and Francesca angrily denounces all the men.

During the storm Stella visits Steve at the rig, determined to have Gambi fired so that he stops seeing her sister. Steve explains to Stella that, if he could pull up a resource that has been in the earth for millions of years, then he will truly have accomplished something. Stella finally abandons her suspicion and kisses Steve, but back in town Philippe persuades Teche to help him destroy the oil rig. With the hurricane winds rising, Philippe climbs onto the platform and lights a bundle of dynamite, but Steve sees him and the two men fight. Philippe trips and disappears under the waves, and Steve, horrified, assumes that Stella was involved in Philippe's plot.

The rig survives the storm, and in the morning drilling begins. However, eight days before the deadline, MacDonough visits Steve and sadly delivers the news that the board of his company has voted to stop the drilling operation the following day, fearing a penalty for non-payment on their lease. MacDonough has already spent all of his own money, and the crew is unable to work for no pay. Gambi soon returns from town, announcing that he has just married Francesca. Steve punches Gambi, who in reply loudly chastises Steve for having driven him and the men too hard. Steve orders them all to leave, intending to do the drilling himself, whereupon Gambi hesitates and then persuades the crew to remain. While the men are drilling, they discover that the troublesome shrimp that have been clogging the valves are actually the huge golden shrimp that have so long eluded the local fishermen.

Steve later takes Francesca to the rig, infuriating Dominique, who inflames the fishermen by declaring that the oilmen will steal their daughters and destroy the town. At Stella's request, Teche go to warn Steve about the impending mob on its way to the rig. There, Steve feigns ignorance about the golden shrimp and asks Teche if he can help him get rid of the creatures. He then addresses the furious mob to assure the men on their concerns: Francesca's marriage is a happy one and, moreover, oil will bring progress and prosperity to Port Felicity. Despite these words the mob decides to destroy the structure, but at that moment oil explodes through the rig and onto the platform. Later, the fishermen discover that the golden shrimp bed is huge, and consequently the conflict between the oilmen and the fishermen is resolved. Teche then convinces Steve that Stella was not involved in Philippe's plot and the lovers finally come together.

Cast

 * James Stewart as Steve Martin
 * Joanne Dru as Stella Rigaud
 * Gilbert Roland as Teche Bossier
 * Dan Duryea as Johnny Gambi
 * Jay C. Flippen as Kermit MacDonald
 * Marcia Henderson as Francesca Rigaud
 * Robert Monet as Phillipe Bayard
 * Antonio Moreno as Dominique Rigaud
 * Harry Morgan as Rawlings, MacDonald's assistant (as Henry Morgan)
 * Fortunio Bonanova as Sheriff Antoine Chighizola
 * Mario Siletti as Louis Chighizola

Production
Production for this film started from late September to mid-November 1952. It was filmed in 1.37:1 Academy ratio while it was released in 1.85:1 widescreen.

The film was Universal-International's first with stereophonic sound. It was also originally planned to be photographed in 3-D, but those plans were scrapped sometime during production.

The film was shot in Morgan City, Louisiana with some scenes shot in New Orleans and on an oil-drilling barge in the Gulf of Mexico. While filming the scenes on-location in Louisiana, Dan Duryea slipped and fell from the roof of a tugboat (which appears throughout the film). He suffered a broken rib, contusion, and bruises but was able to continue filming after a day or two of rest.

Mann later expressed dissatisfaction with the film. "Its story was weak and we never were really able to lick it. I think it was a little too commercial and it fell down on its basic plot. Some of the things we showed were effective and beautiful I think,  but I don’t think it was a very good script. Borden [Chase] worked on it, John Michael Hayes worked on it who was a good writer and has developed into a very good one. There were too many writers, and it became too  fabricated. They wanted a picture with Jimmy Stewart and we concocted one."

Release
The film premiered at Loew's State Theatre in New York City on Tuesday, May 19, 1953 demonstrating Universal-International's widescreen process and stereophonic sound. It began its release the following day.

Reception
The film received favourable reviews although some complained that the sound from the film's stereophonic presentation, with its use of three speakers, was loud and distracting.

It opened with a gross of $42,000, the best opening for a Universal film at Loew's State.

Home media
Universal first released this film on VHS on March 1, 1992. Then, on June 12, 2007, it was released on DVD as part of the James Stewart Screen Legend Collection, a 3-disc set featuring four other films (Next Time We Love, You Gotta Stay Happy, The Glenn Miller Story, and Shenandoah). This film was re-released on August 30, 2013 as a stand-alone DVD as part of the Universal Vault Series. (The DVD releases of the movie are not presented in the film's original 1.33:1 aspect ratio.)

title: Thunder Bay (film)


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User:Grant Weaver

Contributions

 * EarthBound
 * List of commercial failures in computer and video gaming

title: User:Grant Weaver


text: Talk:Wire puzzle

title: Talk:Wire puzzle


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Talk:Wire-and-string puzzle

VfD results
This article was nominated for deletion. The result was no consensus, but some merging/redirecting might be in order. For details, please see Votes for deletion/MoreKarlScherer. -- BD2412 talk 00:40, July 14, 2005 (UTC)

A second debate also ended with keep. See Votes for deletion/KarlSchererRevisited7. Sjakkalle (Check!)  09:31, 27 July 2005 (UTC)

title: Talk:Wire-and-string puzzle


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John G. Hanna

John Griffin Hanna (1889–1948) was a sailboat designer, famous for designing the Tahiti ketch.

Hanna was born in Galveston, Texas, on October 12, 1889. During his childhood he was afflicted with deafness following scarlet fever and lost a foot in a traffic accident. Around 1917, he settled in Dunedin, Florida, and was greatly influenced by the Greek double-ended sponge boats found in nearby Tarpon Springs, Florida. Shortly after his move to Dunedin, Hanna purchased a double-ended ketch-rigged sponge boat that had been built in Apalachicola, Florida by a Greek-American shipwright named Demo George. This vessel, and others that Hanna studied, would inspire the design of Hanna's famous Tahiti ketch. Hanna died in 1948.

Hanna originally wrote of his Tahiti ketch as a 30 foot deep-sea auxiliary cruiser. The design is described and illustrated in detail in the 1935 new edition of How to build 20 boats, pp. 118–133.

Hanna designed a 70 foot research vessel, Iorano, for amphibian tractor inventor Donald Roebling, which was constructed on Roebling's own property, and which took part in a Smithsonian-connected scientific expedition led by Paul Bartsch in 1937. His 1926 Story-built ketch Faith was later purchased by director John Ford and renamed Araner. Ford, a naval reservist, used the boat both for personal recreation and for naval intelligence. The boat was later taken into U.S. Navy service directly as USS Araner (IX-57) during World War II.

At least two boats of Hanna's design have circumnavigated the world twice: Jean Gau in the Atom; and Tom Steele in the Adios.

Hanna was a gifted small vessel designer, but perhaps his greatest strength -and weakness- was as a writer and critic of other's designs. He was let go by the Rudder after a drawn-out feud with L. Francis Herreshoff, who was also a columnist there, and debated acerbically with Thomas C. Gillmer over Tahiti's design antecedents, as well as with Philip Rhodes, and Howard Chapelle.

title: John G. Hanna


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Harvey Bailey

Harvey John Bailey (August 23, 1887 – March 1, 1979), called "The Dean of American Bank Robbers", was an American criminal who spanned a long career and was one of the most successful bank robbers during the 1920s, walking off with over $1 million.

His career
Born in West Virginia, Bailey robbed his first bank c. 1921 and his last in Kingfisher, Oklahoma, on September 9, 1933. He was incarcerated in Dallas on July 8, 1932, until he escaped on June 1, 1933, during a breakout in which the warden was kidnapped and used as a human shield. He was recaptured and found guilty of complicity in the Urschel kidnapping and was sentenced to life in prison on October 7, 1933.

According to Jay Nash, Bailey was holed up in a separate part of the ranch where Machine Gun Kelly was holding kidnap victim Charles F. Urschel, unaware of the kidnapping and unable to travel because of a bullet wound in the leg sustained during his prison escape.

As the Kelly gang was preparing to pull up stakes, one of the minor players in the gang visited Bailey and peeled off some bills from his ransom and gave them to Bailey for doctor bills.

The marked ransom money was found on Bailey when he was arrested, and he received extra years on his sentence as an "accomplice" in a kidnapping which ironically he had no part of.

Originally sent to Leavenworth, he was transferred to Alcatraz on September 1, 1934. He was returned to Leavenworth in 1946 and transferred in 1960 to Seagoville Federal Correctional Institution in Texas, where he remained until he was released on March 30, 1964.

One of the many possible suspects listed as one of the four assassins in the Saint Valentine's Day Massacre is Fred "Killer" Burke. In his 1973 autobiography, however, Bailey insisted that he and Burke were planning a bank robbery together in Calumet City, Illinois, about 20 miles south of the massacre site, at the time the massacre took place. In 1966, Bailey married the widow of Herbert Allen "Deafy" Farmer and found work as a woodworker in a furniture factory. Esther Farmer Bailey died in 1981.

Bailey died peacefully in Joplin, Missouri, on March 1, 1979, at the age of 91.

title: Harvey Bailey