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Talk:Characterizations of the exponential function

Revamp
Give me a sec, I think the "intuitive" proof actually links defs 1 and 3, not 1 and 2, and might actually be able to replace the more clunky Havil proof I have here. But, I'm pooped and will return later. Revolver 01:21, 24 Jul 2004 (UTC)

Okay, I'm going to revamp this article a bit. In particular, I think that 2 of the arguments (the "technical" and "intuitive" ones) can be used to show the equivalent ways of defining the exponential function, not just the number e. Then, the equivalent definitions of e follow trivially by taking x = 1. Hopefully, this works! So, I'm going to change the title of the article to "definitions of the exponential function". Revolver 07:12, 24 Jul 2004 (UTC)

What would people think of moving this to characterizations of the exponential function? Michael Hardy 20:08, 24 Jul 2004 (UTC)

PS: see characterization (mathematics).

 * I don't really care, to be honest. Call it what you want. Revolver 22:42, 24 Jul 2004 (UTC)

Howcome self-derivative isn't included? It's always been the characteristic that jerked my knee... Kwantus 19:44, 4 Sep 2004 (UTC)
 * The main problem that pops into my mind is that it's not immediately obvious why such a function exists, i.e. why there is y s.t. y' = y and y(0) = 1. Without falling back on one of the others defs, I mean. If anyone knows of a nice way to prove this existence independent of the other 3 defs, I'd be happy to include it. Revolver 08:35, 20 Sep 2004 (UTC)

This is really a beautiful article. I like it very much. :) MathKnight 01:20, 1 Oct 2004 (UTC)

Characterization 4
Sorry, we have this problem on the german Exponentialfunktion. Maybe someone knows the exavt proof. I have only a german account. So Greetings Roomsixhu --83.176.135.233 02:50, 31 May 2005 (UTC)

Got it! How do I proove this? roomsixhu --83.176.135.227 19:02, 31 May 2005 (UTC)

Take characterization 4, which is:
 * $$y'=y,\quad y(0)=1.$$

Divide both sides by y and integrate. Get characterization 3, which is:
 * $$\int_{1}^{y} \frac{dt}{t} = x.$$

''is an increasing sequence which is bounded above. Since every bounded, increasing sequence of real numbers converges to a unique real number, this characterization makes sense.''
 * Do you call that a proof? It's not much more than a tautology. It should be explained why it is increasing and why it is bounded.--Army1987 20:31, 19 July 2005 (UTC)

What did I miss?
The proof of the equivelance of charactaristics 1 and 3 isn't actually a proof, it's a circular argument and I've no idea why it's there. It claims that:

$$\lim_{n\to\infty}\ln\left(1+\frac{x}{n}\right)^n=\lim_{n\to\infty}n\ln \left(1+\frac{x}{n}\right).$$

which should only hold surely, if we were dealing with a logarithm, but that's exactly what we're trying to proove, that ln x is the same thing as loge(x)

 * That's easy to prove; integral of dt/t from 1 to x^n, by the substitution u^n=t, becomes the integral of du n u^(n-1)/u^n = n u/du for u = 1 to x. Any good calculus book should have this proof. User:Ben Standeven as 70.249.214.16 03:05, 14 October 2005 (UTC)

Continuity required for characterization 5?
Is there an example of a discontinuous function $$f(x)$$ ($$\mathbb{R} \to \mathbb{R}$$) satisfying $$f(x+y)=f(x)f(y)$$ and $$f(0)=1$$, but for which $$f(x)\neq e^{cx}$$ for all $$c$$?

Of course $$f(x) = e^{cx}$$ if $$f(x)$$ is continuous, as proved in the article (I believe this is a homework assignment in Rudin), but I'm curious to see a counterexample showing that continuity is necessary. (And if it's not a necessary assumption, it would be good to note this in the article and to give a reference.)

—Steven G. Johnson 04:57, 22 November 2006 (UTC)

 * I answered my own question, finally: continuity is required, as one can prove the existence of a discontinuous counter-example. I'm not aware of any constructive counter-example; the existence proof seems to require the axiom of choice.

 * I came up with the proof of a discontinuous $$f(x)$$ satisfying $$f(x+y)=f(x)f(y)$$ on my own after quite a bit of puzzling. Then a friend of mine pointed out that this is also shown in Hewitt and Stromberg, Real and Abstract Analysis (exercise 18.46)...it turns out to be easy to do once you have proved the existence of a Hamel basis for R/Q.

 * —Steven G. Johnson 00:30, 23 February 2007 (UTC)

 * Monotonicity implies continuity: Use monotonicity to show that $$\sup\left\{f(q):q\leq x, q\in\mathbb{Q}\right\}\leq f(x)\leq \inf\left\{f(q):q\geq x, q\in\mathbb{Q}\right\}$$. Then suppose that $$\sup\left\{f(q):q\leq x, q\in\mathbb{Q}\right\}< \inf\left\{f(q):q\geq x, q\in\mathbb{Q}\right\}$$ and show you can find a rational $$q_0$$ such that $$\sup\left\{f(q):q\leq x, q\in\mathbb{Q}\right\}< f(q_0) < \inf\left\{f(q):q\geq x, q\in\mathbb{Q}\right\}$$ which is contradictory. You can then define for $$x\in\mathbb{R}\backslash\mathbb{Q}$$, $$f(x)=f(1)^x:=\sup\left\{f(q):q\leq x, q\in\mathbb{Q}\right\}= \inf\left\{f(q):q\geq x, q\in\mathbb{Q}\right\}$$.

 * To conclude, suppose $$f$$ is not continuous in $$x_0\in\mathbb{R}$$. Then $$\exists\varepsilon_0 > 0,\forall y_1\in f(]-\infty, x_0]), \forall y_2\in f(]x_0,+\infty[), |y_1-y_2|\geq \varepsilon_0$$. This implies that $$\inf\left\{f(q):q\geq x_0, q\in\mathbb{Q}\right\}\neq\sup\left\{f(q):q\leq x_0, q\in\mathbb{Q}\right\}$$, which is not true. Therefore $$f$$ is continuous everywhere.
 * —Deimos 28 06:23, 11 September 2007 (UTC)
 * Hello. I checked the exercise 18.46 of Hewitt and Stromberg, Real and Abstract Analysis, but it concludes that f is exp( i alpha x) for some alpha, with an imaginary unit in the exponent. I think that we should remove Lebesgue integrability from the list of alternative conditions because the exercise is different. What do you think? Caph1993 (talk) 18:06, 3 April 2024 (UTC)

Equivalence of 1 and 3
Hi,

The point raised on the discussion page about the equivalence of 1 and 3 is worthy of being addressed in the main article, I believe, as a perceptive reader will see that one is using a result whose prior establishment is not clear. (Indeed, I thought there was a problem and, having satisfied myself that there wasn't, inserted an edit to clear it up, before I had read the point on the discussion page. [Sorry for not checking first]) I hope the edit is still deemed worthy of retention.

Hugh McManus 14:56, 26 February 2007 (UTC)Hugh McManus

Equivalence of characterizations 1 and 2
The proof given only works for $$x \geq 0$$. In line 6, you are arguing $$(1-1/n)(1-2/n)\leq 1$$ implies that $$(x^3/3!)(1-1/n)(1-2/n)\leq x^3/3!$$ and this requires non-negative $$x$$.

Indeed, the statement in Characterization 1 that $$a_n=(1+x/n)^n$$ is an increasing sequence may not be valid if $$x<0$$. Fathead99 (talk) 14:24, 30 January 2008 (UTC)

Taylor series.
From the limit (char-1) follows the differential equation (char-4) by differentiation, and then the taylor series (char-2).

It is unclear whether the article talks about the real or complex exponential function. The reference to positive values of x indicate real exponentials, but the results are important for complex exponentials.

The formulation "in several cases can be extended to any Banach algebra" is confusing; in which cases can it be extended, and in which cases can it not be extended? Is differentiation defined for any Banach algebra? The Banach algebra article didn't tell.

Bo Jacoby 07:29, 23 July 2007 (UTC).

Real versus complex
For some of the characterizations given here (e.g. power series) one may take the domain to be the whole complex plane, but for others (monotonicity plus the functional equation) one must assume the domain is the real line. (There do exist continuous functions on the complex plane that satisfy the functional equation and the initial condition but are not equal to the natural exponential function; they are nowhere (complex-)differentiable. For example, observe that
 * $$ g(x + iy) = e^x(\cos(2y) + i\sin(2y))\, $$
 * $$ g(x + iy) = e^x(\cos(2y) + i\sin(2y))\, $$

for x and y real, is continuous everywhere in the complex plane and satisfies the functional equation
 * $$ g(z + w) = g(z)g(w)\, $$
 * $$ g(z + w) = g(z)g(w)\, $$

for z and w complex, and the initial condition
 * $$ g(1) = e\,$$
 * $$ g(1) = e\,$$

but not the Cauchy-Riemann equations, and is therefore nowhere complex-differentiable.) Michael Hardy 13:16, 26 August 2007 (UTC)

Motivating the definitions?
I think at least definitions 1., 2., and 5. may seem to a beginning student like rabbits pulled out of a hat. Among all possible limits, why study $$\lim_{n\to\infty} \left(1+\frac{x}{n}\right)^n$$? Among all possible power series, why study $$\sum_{n=0}^\infty {x^n \over n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$$? Among all measurable functions f satisfing f(x+y)=f(x)f(y), why study the one satisfying f(1)=e? I will try to answer these quations in what follows.

Even definitions 3. and 4. (which are really two sides of the same coin, the relation dy=ydx first when x is considered a function of y and then when y is considered a function of x) may need some motivation. In 3., why start the integral at 1? In 4., why chose y(0)=1? The answer is that otherwise we do not get the properties log(xy)=log(x)+log(y) and $$e^{(x+y)}=e^xe^y$$.

The differential equation y'=y is sufficiently well-behaved to have a power series solution. From 4. we thus get 2. Of course, 3. also leads to a power series: $$log(1+x)=x-x^2/2+x^3/3-x^4/4+...$$. Perhaps this power series should also be included among the definitions.

Although it is commonplace, I find the choice of the variable name "n" in 1. unfortunate. It suggests an integer variable when any complex number is perfectly permissible. I also prefer to use a variable that approaches 0 to one that approaches infinity, so rather than motivating 1., I will motivate: $$e^x = \lim_{h \to 0} {(1+xh)}^{1/h}$$.

If you try to differentiate $$a^x$$ with respect to x in the most straightforward of ways, you get $$\lim_{h\to0}\frac{a^h-1}{h} a^x$$. Since we know that the derivative should be $$log(a)a^x$$, and since a can be any positive real number, we must have come across an expression for the function log. Since the exponential function is just the inverse of log, it is natural to wonder whether you can turn the expression for log into one for the exponential function. You can, and what you get is exactly $$e^x = \lim_{h \to 0} (1+xh)^{1/h}$$. (And once we have this expression it is a good idea to use the binomial series to expand it; this constitutes an alternative path to the power series 2.)

As for 5., it is really incomplete as a definition since it presupposes that you already have a definition of e. It seems to me that what we do have is an interesting characterisation of the function $$f(x)=a^x$$. It is determined once you know: I. that $$f(x+y)={f(x)}{f(y)}$$, II. that f is measurable at one point, and III. the value of f(x) for any value of x except 0. Mattias Wikstrom (talk) 18:02, 8 February 2008 (UTC)

Why characterization 1 makes sense: not valid explanation
I came here looking for exactly why characterization 1 is true. This site generally is good about having formal proofs, rather than intuitive explanations. In this "explanation," the only thing seen is that the sequence is convergent, no information is given about what the limit is. There is not sufficient information to prove that the sequence in characterization 1 converges to the exponential function. —Preceding unsigned comment added by 71.154.242.50 (talk) 16:28, 28 June 2008 (UTC)

The idea is that any of these characterizations could be taken as a definition of the exponential function. It is later proved that all the characterizations are equivalent, but you can't prove anything about the exponential function without first defining it. Argoreham (talk) 19:50, 18 July 2008 (UTC)

There's a reason that we don't have sufficient information to prove that the sequence in characterization 1 converges. The text claims, "It can be shown that the sequence [removed] is an increasing sequence which is bounded above." My calculator, on the other hand, claims that the sequence that approaches exp(-5) is 1, -4, 2.25, -0.296296, 0.003906, 0, etc, which clearly does not increase monotonically. We don't prove our proposition because it isn't true. 209.216.175.60 (talk) 01:29, 27 September 2008 (UTC)

 * I've changed what it says.
 * Do you really need a calculator to see that? Or was that just rhetoric? Michael Hardy (talk) 22:04, 27 September 2008 (UTC)
 * Do you really need a calculator to see that? Or was that just rhetoric? Michael Hardy (talk) 22:04, 27 September 2008 (UTC)

Missing Characterization?
Maybe I'm missing something here, but how come none of the characterizations are the very simple elementary/middle school definition of an exponent? exp(2) = e*e, exp(3) = e*e*e, etc. This can then be extended to all numbers 1/n using integer roots, and from there to all rational numbers, and from there to all real numbers using limits. It seems to me that this should be the very first characterization. Maybe it isn't the most analytically useful, but it is I believe the first definition historically, and certainly the first thing that comes to mind when one thinks "exponential function." 75.83.25.40 (talk) 01:18, 13 November 2008 (UTC)LFStokols

 * This is essentially characterization 5 (which may - like much of the material on this page - be too technical to be accessible to beginners).
 * One could argue, though, that this type of characterisation is incomplete, since it requires one to already have a definition of the number e (and I am afraid that the most natural way of defining e is to say that e=exp(1), where exp has been defined without mentioning e). I think the best way of making them complete is to replace the requirement that exp(1)=e with the requirement that exp'(0)=1, where exp' is the derivative of exp. Such a characterisation may not be one of the "most common", but I do not see why this page should deal only with the "most common" characterizations. Mattias Wikstrom (talk) 10:58, 15 January 2009 (UTC)

Equiv. of 1 & 2
Hi, As someone with training in science and engineering, but not a mathematician, I might lack the sophistication to understand why the laborious proof of the equivalence of characterizations 1 and 2 is needed. To my lesser-trained eye, it would seem that something like the following is sufficient:

$$ \begin{align} R & \equiv (1+\frac{x}{n})^n \\ & = \sum_{k=0}^{n}{n \choose k} \left( \frac{x}{n} \right) ^k \\ & = \sum_{k=0}^{n}\frac{n!}{k! (n-k)!} \left( \frac{x}{n} \right) ^k \\ \end{align} $$

where we used The Binomial Theorem to arrived at the sum, and the the definition of "n choose k".

Now, note that there are as many factors (viz. k of them) in $$ \frac{n!}{(n-k)!} $$ as there are in $$x^k$$, so we'll group them, and then group the other factors as $$\frac{x^k}{k!}$$

\begin{align} & = \sum_{k=0}^{n}  \left[  \frac{n}{n}\frac{n-1}{n} \cdot\cdot\cdot \frac{n-k+1}{n} \right]  \frac{x^k}{k!} \\ & = \sum_{k=0}^{n}  \left[  \prod_{j=1}^{k-1} \left( \frac{n-j}{n} \right)  \right]  \frac{x^k}{k!} \\ & = \sum_{k=0}^{n}  \left[  \prod_{j=1}^{k-1} \left( 1-\frac{j}{n} \right)  \right]  \frac{x^k}{k!} \\ \end{align} $$

Now, take the limit $$n\to\infty$$. In that case, $$\frac{j}{n}\to 0$$, and the product thusly approaches 1, giving

\begin{align} \lim_{n\to\infty} R & = \sum_{k=0}^{\infty}   \frac{x^k}{k!} \\ \end{align} $$

and we have arrived at characterization 2.

What are the nuances that make this proof fail?

Thanks! --Filiusfob (talk) 13:25, 16 July 2010 (UTC)

 * The final bit, taking the limit as n->infinity and asserting that the products
 * $$\prod_{j=1}^{k-1} \left( 1-\frac{j}{n} \right)$$
 * go to 1, isn't strictly rigorous. More important, though, is that we need to figure out how far off the partial sums are from the usual Maclaurin series. Each term in the sum contributes some error--do they add up to a finite amount? That is, does each product above go to 1 quickly enough so that the total error actually decreases as more terms are added? It should be possible to construct a counterexample, replacing (j/n) with a more slowly decreasing function--maybe (j/(n^(1/j))--which satisfies all the requirements of your proof without satisfying the conclusion.
 * That said, your proof is pretty intuitive and would be what I'd expect to see in non-math books discussing this topic. Physics texts are especially bad about using "as x->A, f(x)->B" arguments without really justifying them, in my experience. I think they fundamentally rely on mathematicians pointing out errors over the years if their proofs can't actually be made rigorous. That, or physical intuition to confirm their conclusion and, circularly, their reasoning. 67.158.43.41 (talk) 09:43, 4 November 2010 (UTC)

You don't proof anything
"Equivalence of characterizations 1 and 3

Here, we define the natural logarithm function in terms of a definite integral as above. By the fundamental theorem of calculus,

 * $$\frac{d}{dx}\left( \ln x \right) = \frac{1}{x}.$$

Now, let x be any fixed real number, and let

 * $$y=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n.$$

We will show that ln(y) = x, which implies that y = ex, where ex is in the sense of definition 3. We have

 * $$\ln y=\ln\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n=\lim_{n\to\infty}\ln\left(1+\frac{x}{n}\right)^n.$$

Here, we have used the continuity of ln(y), which follows from the continuity of 1/t:

 * $$\ln y=\lim_{n\to\infty}n\ln \left(1+\frac{x}{n}\right)=\lim_{n\to\infty}\frac{x\ln\left(1+(x/n)\right)}{(x/n)}.$$

Here, we have used the result lnan = nlna. This result can be established for n a natural number by induction, or using integration by substitution. (The extension to real powers must wait until ln and exp have been established as inverses of each other, so that ab can be defined for real b as eb lna.)

 * $$=x\cdot\lim_{h\to 0}\frac{\ln\left(1+h\right)}{h} \quad \mbox{ where }h=\frac{x}{n}$$

 * $$=x\cdot\frac{d}{dt}\left( \ln t\right) \quad \mbox{ at }t=1$$

 * $$=x\cdot\frac{1}{t} \quad \mbox{ at }t=1$$

 * $$\!\, = x.$$"

This should proof why $$\int \frac{1}{x} dx= \ln x+C,$$ but it don't proving anything. It only proves, that at t=1, $$ln y=x$$:
 * $$\ln y=\ln (e^x)=x\cdot\frac{1}{t} \quad \mbox{ at }t=1,$$
 * $$\ln y=\ln e^x= x.$$

And I don't see any relation in this equation:
 * $$ x\cdot\lim_{h\to 0}\frac{\ln\left(1+h\right)}{h}=x\cdot\frac{d}{dt}\left( \ln t\right).$$  —Preceding unsigned comment added by 84.240.9.58 (talk) 10:10, 24 December 2010 (UTC)

"Alternative characterizations

Other characterizations of e are also possible: one is as the limit of a sequence, another is as the sum of an infinite series, and still others rely on integral calculus. So far, the following two (equivalent) properties have been introduced:

1. The number e is the unique positive real number such that
 * $$\frac{d}{dt}e^t = e^t.$$

2. The number e is the unique positive real number such that
 * $$\frac{d}{dt} \log_e t = \frac{1}{t}.$$

The following three characterizations can be proven equivalent:

3. The number e is the limit
 * $$e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n$$

Similarly:
 * $$e = \lim_{x\to 0} \left( 1 + x \right)^{1/x}$$

4. The number e is the sum of the infinite series
 * $$e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots$$

where n! is the factorial of n.

5. The number e is the unique positive real number such that
 * $$\int_{1}^{e} \frac{1}{t} \, dt = {1}$$."

It should prove, why $$e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n,$$ but it don't proof anything.

 * Could you summarize please what your point is thanks. Dmcq (talk) 12:05, 24 December 2010 (UTC)

One more thing, $$\lim_{h\to 0}\ln(1+h)^{1\over h}=\lim_{h\to 0}\frac{\ln(1+h)}{h}=1.$$ And $$\lim_{h\to 0}(1+h)^{1\over h}=e.$$ So $$\ln e=1$$. —Preceding unsigned comment added by 84.240.9.58 (talk) 09:44, 27 December 2010 (UTC)

 * Could you summarize please what your point is thanks. Rather than just writing more equations say which section you think should be changed and why. Dmcq (talk) 09:59, 27 December 2010 (UTC)

To summarize, let's prove, that $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n=e.$$ So $$\ln(e)$$ and $$\ln\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n $$, should give the same answer 1, because $$\ln(e)=1$$. So need to prove, that $$\ln\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n =1.$$ So $$y=e$$ and $$y=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n.$$ So let's begin proving:
 * $$y=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n,$$
 * $$\ln y=\ln\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n=\lim_{n\to\infty}\ln\left(1+\frac{1}{n}\right)^n=\lim_{n\to\infty}n\ln \left(1+\frac{1}{n}\right)=\lim_{n\to\infty}\frac{1\cdot\ln\left(1+(1/n)\right)}{(1/n)}=$$
 * $$=1\cdot\lim_{h\to 0}\frac{\ln\left(1+h\right)}{h}=1\cdot \ln(\lim_{h\to 0}(1+h)^{1\over h})=1\cdot \ln e =1\cdot 1=1 \quad \mbox{ where }h=\frac{1}{n}.$$ —Preceding unsigned comment added by 84.240.9.58 (talk) 10:13, 27 December 2010 (UTC)

To summarize, let's prove, that $$\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n=e^x.$$ So $$\ln(e^x)$$ and $$\ln\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n $$, should give the same answer x, because $$\ln(e^x)=x$$. So need to prove, that $$\ln\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n =x.$$ So $$y=e^x$$ and $$y=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n.$$ So let's begin proving:
 * $$y=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n,$$
 * $$\ln y=\ln\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n=\lim_{n\to\infty}\ln\left(1+\frac{x}{n}\right)^n=\lim_{n\to\infty}n\ln \left(1+\frac{x}{n}\right)=\lim_{n\to\infty}\frac{x\ln\left(1+(x/n)\right)}{(x/n)}=$$
 * $$=x\cdot\lim_{h\to 0}\frac{\ln\left(1+h\right)}{h}=x\cdot \ln(\lim_{h\to 0}(1+h)^{1\over h})=x\cdot \ln e =x\cdot 1=x \quad \mbox{ where }h=\frac{x}{n}.$$

 * You obviously have a problem just pointing out where you think there is a problem. I think reading through all this the problem you have is going between the integral and the logarithm. In fact the integral is used as a definition of the logarithm and the logarithm rule can be proved as follows:

 * $$ \ln(tu) = \int_1^{tu} \frac{1}{x} \, dx \ \stackrel {(1)} = \int_1^{t} \frac{1}{x} \, dx + \int_t^{tu} \frac{1}{x} \, dx \ \stackrel {(2)} = \ln(t) + \int_1^u \frac{1}{w} \, dw = \ln(t) + \ln(u). $$

 * Is this what was worrying you and you'd like in or what? 12:33, 27 December 2010 (UTC)
 * You forget to say, that at t=1, because this:
 * $$ \ln(tu) = \int_1^{tu} \frac{1}{x} \, dx \ \stackrel {(1)} = \int_1^{t} \frac{1}{x} \, dx + \int_t^{tu} \frac{1}{x} \, dx \ \stackrel {(2)} = \ln(t) + \int_1^u \frac{1}{w} \, dw = \ln(t) + \ln(u), $$
 * can be rewritten as this:
 * $$ \ln(tu) = \int_1^{tu} \frac{1}{x} \, dx \ \stackrel {(1)} = \int_1^{1} \frac{1}{x} \, dx + \int_t^{tu} \frac{1}{x} \, dx \ \stackrel {(2)} = (\ln(1)-\ln(1)) + \int_1^u \frac{1}{w} \, dw = 0 + \ln(u), $$
 * so you still moving in circles and don't proving anything. Proof should be about why $$\frac{d}{dx}\left( \ln x \right) = \frac{1}{x}$$ and not about why $$\ln (c\cdot d)=\ln(c)+\ln(d).$$ 16:37, 6 January 2011 (UTC)
 * The integral is the definition of log. Therefore the deriviative is 1/x by the fundamental theorem of calculus. One doesn't nmeed to prove it is 1/x - that's the definition. I was showing the definition worked to give a log type function. The log is not defined as the inverse function of exponentiation. Dmcq (talk) 17:09, 6 January 2011 (UTC)
 * See Logarithm where it says 'the right hand side can serve as a definition of the natural logarithm'. That is exactly what is done here. Dmcq (talk) 17:12, 6 January 2011 (UTC)
 * So it's just definition like rest of integrals and guessing, that $$\int\frac{1}{x}\; dx=\log_e x.$$ But one thing you still can not do by math rules in this equation:
 * $$ \ln(tu) = \int_1^{tu} \frac{1}{x} \, dx \ \stackrel {(1)} = \int_1^{t} \frac{1}{x} \, dx + \int_t^{tu} \frac{1}{x} \, dx \ \stackrel {(2)} = \ln(t) + \int_1^u \frac{1}{w} \, dw = \ln(t) + \ln(u), $$
 * you can't say, that $$\int_t^{tu} \frac{1}{x} \, dx=\int_1^u \frac{1}{w} \, dw.$$ You only can claim, that $$\int_t^{tu} \frac{1}{x} \, dx=\ln(tu)-\ln(t)=\ln(t)+\ln(u)-\ln(t)=\ln(u).$$
 * And this means moving in circle, repeating, that $$\ln(t\cdot u)=\ln(t)+\ln(u).$$ See for yourself, if w=x/t, then assume this equation is correct:
 * $$ \ln(tu) = \int_1^{tu} \frac{1}{x} \, dx \ \stackrel {(1)} = \int_1^{t} \frac{1}{x} \, dx + \int_t^{tu} \frac{1}{x} \, dx \ \stackrel {(2)}, $$

but I think you can't do this (sum up two parts of integrals) by math rules. But let's assume, then:
 * $$ \ln(tu) = \int_1^{tu} \frac{1}{x} \, dx \ \stackrel {(1)} = \int_1^{t} \frac{1}{x} \, dx + \int_t^{tu} \frac{1}{x} \, dx \ \stackrel {(2)} = \ln(t)-\ln(1) + \int_t^{tu} \frac{t}{x} \, d(x/t) = \ln(t)-0 + t\ln(tu)-t\ln(t)=\ln(t)+t\ln(u)+t\ln(t)-t\ln(t)=\ln(t)+t\ln(u), $$

where dw=d(x/t)=dx/t; dx=t*dw.

 * If you looked at that section I pointed you at you'd see that the equality of those two integrals was shown by the change of variables substitution w = x/t. Dmcq (talk) 01:27, 7 January 2011 (UTC)

Missing proof
Looking through this, I notice that there is a proof that 5 implies 1, but not that 1 (or one of the others) implies 5. Should be easy to fix (a proof should be in Rudin or any other standard textbook). — Steven G. Johnson (talk) 18:09, 11 December 2011 (UTC)

Characterization 4 again
Am I missing something or does the article never link characterization 4 to anything else? Dmcq (talk) 04:49, 8 February 2012 (UTC)

 * Plugging the Taylor series (characterization 2) into characterization 4 quickly shows the relation. About characterization 5, even though it is a nice characterization, it is far less common than the other four characterizations. It should probably be not placed on the same footing.  Perhaps the main list should have only 4 characterizations (we can keep the 5'th, writing something like:  Besides these, there are other characterizations as well, for example: .. #5).  71.229.28.197 (talk) 17:38, 22 February 2014 (UTC)MvH

 * Perhaps Dmcq's point is that we should have a subsection Equivalence of characterizations 2 and 4, rather than leaving it to the reader? —Quondum 18:13, 22 February 2014 (UTC)

Error in series for exponent
This article contains the errorneous series for exponent
 * $$e^x = \sum_{n=0}^\infty {x^n \over n!}$$

But it's clear that for $$x=0$$ the first item $${0^0 \over 0!}$$ is nonsense because $$0^0$$ isn't correct mathematical expression, see Indeterminate form.

I modified the text to correct syntax:
 * $$e^x = 1 + \sum_{n=1}^\infty {x^n \over n!}$$

but undid my modification with comment: «Unsourced claim of error in edit summary, and atypical presentation». No source is needed for the obvious failure. LGB (talk) 10:40, 14 February 2014 (UTC)

 * You're confusing the indeterminate form 00 with the expression 00. Talk:Exponentiation and several archives on the topic should make it clear that it is not so simple. —Quondum 06:29, 15 February 2014 (UTC)

 * (1) The section «the expression 00» has no reference to sources. (2) The same Exponentiation section warns honestly: «Not all sources define $$0^0$$ to be 1». (3) You have no basis to suppose that all Wikipedia readers know and support such strange convention — even without any commentary and reference to Exponentiation section. My modified formula is clearly better for a reader and I hope you'll not insist upon equivocal and confusing formula. Otherwise let Wikipedia talk:WikiProject Mathematics will discuss the subject. LGB (talk) 11:22, 15 February 2014 (UTC)

 * You're welcome to do discuss the matter there, but be aware that this topic has been discussed at length on Wikipedia, at times quite heatedly. It would make sense to first acquaint yourself with the existing consensus (or lack thereof!) by finding and examining the extensive past discussions on the topic, such as the example to which I linked, before reopening the discussion, as many will have tired of it already. If the rationale behind your edit were to be accepted, it would imply a similar revision to many other articles, including Exponentiation, Exponential function, Taylor series, Laurent series, Binomial theorem, Derivative and at a guess no less than hundreds of others. There is no shortage of sources that use the expression $x^{0}$ to mean 1 in exactly the way that you object to. To find a source that carefully avoids its use with this meaning might be a challenge, though. —Quondum 16:29, 15 February 2014 (UTC)

 * I agree with LGB that it is wrong to say in one page that 0^0 is undefined, while assuming in another page that 0^0 will be evaluated to 1. Unfortunately, that is precisely what many calculus textbooks do.  The mathematical solution to this problem is very easy (accept the empty product rule, which makes 0! and 0^0 equal to 1).  The goal in wikipedia is to say what the textbooks say, so if the textbooks contradict themselves, then wikipedia will do likewise.  This means that the only way to fix this contradiction on wikipedia is to convince authors of calculus books that 0^0 is 1.  MvH (talk) 18:51, 21 February 2014 (UTC)MvH

 * This article says nothing of the sort. Perhaps you're confusing an expression with a limiting form? The shorthand 00 for the limit $$\lim_{x\to 0^+,y\to 0}x^y$$ looking like the expression might be confusing, but that is all it is: a shorthand. —Quondum 02:38, 22 February 2014 (UTC)

 * Quondum, yes, I agree that one can do this correctly (0^0 = 1, 0/0 = undefined, while "0^0" and "0/0" refer to limits that require more than direct substitution). Widespread confusion arises because there are many calculus textbooks that leave 0^0 undefined, even though the same textbooks expect the reader to evaluate 0^0 to 1 in many formulas.  It's not the readers fault when they are confused. In the past, 1 was considered a prime number, until it was found that this was inconvenient in an increasingly large number of situations.  The same will happen with the view that 0^0 is undefined, and for the same reason.  The option "undefined" is inconvenient in an increasingly large number of places, so eventually people will settle on "1". In research papers, 0^0=1 is already the status quo. Calculus textbooks will likely be the last ones to adopt this.  71.229.28.197 (talk) 16:03, 22 February 2014 (UTC)MvH
 * I agree with you about problems relating to any (lack of) definition of 00 and the consequent confusion, but that was not my point, so I guess I was unclear. I'm saying that since "00" and "the indeterminate form 00" refers to distinct concepts, one must take care to use the full phrase in the latter. Conflating the two distinct concepts seems to have been underlying LGB's assertion. The eventual resolution may occur differently from your prediction. I would like to see notation arising that distinguishes between the two fundamentally different types of exponentiation, which will allow 00=1 without those in analysis objecting (other than to the change of notation). —Quondum 17:50, 22 February 2014 (UTC)

As LGB knows that$$\sum_{n=0}^\infty {x^n \over n!}= 1 + \sum_{n=1}^\infty {x^n \over n!}$$ he obviously knows that $${x^0 \over 0!}= 1$$ for all values of x. People who deny that 00=1 are spreading confusion. Bo Jacoby (talk) 12:24, 17 February 2014 (UTC).

Missing characterization
One characterization I've seen used in calc textbooks (eg Briggs, Cochran, Gillett): let $e$ be the number such that $\lim e^h-1/h=1.$ Then $e^x$ is the exponential function with that base. This should also be shown equivalent. -lethe talk [ +] 04:29, 21 October 2015 (UTC)

More general proof that 1 implies 2 using the Dominated Convergence Theorem
The generalisation to negative values of $$x$$ is done in an ad-hoc way. It raises the question of how to do the same for the complex numbers.

I think that using the Dominated convergence theorem allows for a generalisation to all unital Banach algebras.

--Svennik (talk) 21:50, 30 September 2019 (UTC)

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


text:
Job (biblical figure)

Job (אִיּוֹב 'Īyyōv; Iṓb) is the central figure of the Book of Job in the Bible. In Islam, Job (أيوب) is also considered a prophet.

Job is presented as a good and prosperous family man who is suddenly beset with horrendous disasters that take away all he holds dear—a scenario intended to test Job's faith in God. Struggling mightily to understand this situation, Job reflects on his despair but consistently remains devout.

In the Hebrew Book of Job

The Hebrew Book of Job is part of Ketuvim ("Writings") of the Hebrew Bible. Not much is known about Job based on the Masoretic Text.

The characters in the Book of Job consist of Job, his wife, his three friends (Bildad, Eliphaz, and Zophar), a man named Elihu, God, and angels.

It begins with an introduction to Job's character—he is described as a blessed man who lives righteously in the Land of Uz. The Lord's praise of Job prompts an angel with the title of "satan" ("Adversary") to suggest that Job served God simply because God protected him. God removes Job's protection and gives permission to the angel to take his wealth, his children, and his physical health (but not his life). Despite his difficult circumstances, he does not curse God, but rather curses the day of his birth. And although he anguishes over his plight, he stops short of accusing God of injustice. Job's miserable earthly condition is simply God's will.

In the following, Job debates with three friends concerning his condition. They argue whether it was justified, and they debate solutions to his problems. Job ultimately condemns all their counsel, beliefs, and critiques of him as false. God then appears to Job and his friends out of a whirlwind. God rebukes the three friends and gives them instruction for the remission of sin, followed by Job being restored to an even better condition than his former wealthy state. Job is blessed to have seven sons, and three daughters named Jemimah (which means "dove"), Keziah ("cinnamon"), and Keren-happuch ("horn of eye-makeup"). His daughters were said to be the most beautiful women in the land.

In the Greek Old Testament Book of Job
The Septuagint, an ancient Greek translation of the Hebrew Old Testament, has a revised and updated final verse that claims Job's genealogy, asserting him to be a grandson of Esau and a ruler of Edom. "This man is described in the Syriac book as living in the land of Ausis, on the borders of Idumea and Arabia: and his name before was Jobab; and having taken an Arabian wife, he begot a son whose name was Ennon. And he himself was the son of his father Zare, one of the sons of Esau, and of his mother Bosorrha, so that he was the fifth from Abraam. And these were the kings who reigned in Edom, which country he also ruled over: first, Balac, the son of Beor, and the name of his city was Dennaba: but after Balac, Jobab, who is called Job, and after him Asom, who was governor out of the country of Thaeman: and after him Adad, the son of Barad, who destroyed Madiam in the plain of Moab; and the name of his city was Gethaim. And his friends who came to him were Eliphaz, of the children of Esau, king of the Thaemanites, Baldad sovereign of the Sauchaeans, Sophar king of the Minaeans."

Judaism

 * He is mentioned in the Book of Ezekiel.
 * He is cited as someone "who held fast to all the ways of justice" in the deuterocanonical Book of Sirach.

Christianity

 * He is praised for his perseverance in the Christian Epistle of James.
 * He is the protagonist of a pseudepigraphal book called the Testament of Job.

Mormonism

 * He is also mentioned in the Doctrine and Covenants, one of the four sacred texts of the Church of Jesus Christ of Latter-day Saints (LDS Church).

Islam
The Quran describes Job as a righteous servant of Allah, who was afflicted by suffering for a lengthy period of time. However, it clearly states that Job never lost faith in God and forever called to God in prayer, asking Him to remove his affliction:
 * He is discussed as a prophet in the Quran by the name of Ayūb.

And [mention] Job, when he called to his Lord, "Indeed, adversity has touched me, and you are the Most Merciful of the merciful."

— Quran, sura 21 (The Prophets), ayah 83

Baháʼí

 * In the writings of the Baháʼí Faith: A lengthy tablet was written by Bahá'u'lláh, the first part of which is focused on Job. The Tablet is often referred to as the Tablet of Patience or the Tablet of Job.

Job in Judaism
A clear majority of rabbis saw Job as having in fact existed as a historically factual figure.

According to a minority view, Job never existed. In this view, Job was a literary creation by a prophet who used this form of writing to convey a divine message. On the other hand, the Talmud (in Tractate Baba Batra 15a–16b) goes to great lengths trying to ascertain when Job actually lived, citing many opinions and interpretations by the leading sages.

Job is further mentioned in the Talmud as follows:

 * Job's resignation to his fate.
 * When Job was prosperous, anyone who associated with him even to buy from him or sell to him, was blessed.
 * Job's reward for being generous.
 * David, Job and Ezekiel described the Torah's length without putting a number to it.
 * Job was in fact one of three advisors that Pharaoh consulted, prior to taking action against the increasingly multiplying Israelites in the Book of Exodus. As described in the Talmud: Balaam urged Pharaoh to kill the Hebrew new-born boys; Jethro opposed this decree; and Job, though personally opposed to the decree, kept silent and did not protest it. It is for Job's silence that God subsequently punishes him with his bitter afflictions. However, the Book of Job itself contains no indication of this, and to the prophet Ezekiel, Yahweh refers to Job as a righteous man of the same calibre as Noah and Daniel.

Christian views
Christianity accepts the Book of Job as canon in its Old Testament. In addition, Job is mentioned in the New Testament of the Christian Bible: the Epistle of James paraphrases Job as an example of patience in suffering.

Job's declaration, "I know that my redeemer liveth", is considered by some Christians to be a proto-Christian reference to Christ as the Redeemer, and is the basis of several Christian hymns, as well as the opening scene of Part III of Handel's Messiah. However, Jewish bible commentators and scholars point out that Job "insists on a divine hearing in his lifetime" (cf. Job 16:19–22).

Feast Days
He is commemorated by the Lutheran Church–Missouri Synod in their Calendar of Saints on May 9, by the Roman Catholic Church on May 10, and by the Eastern Orthodox and Byzantine Catholic churches on May 6.

He is also commemorated by the Armenian Apostolic Church on May 6 and December 26, and by the Coptic Orthodox Church on April 27 and August 29. The Armenian Apostolic Church commemorates Job along with John the Baptist on the Thursday after the third Sunday of the Feast of the Assumption.

Islamic views and Quranic account
In the Qur'an, Job (أيّوب) is considered a prophet in Islam. The narrative frame of Job's story in Islam is similar to the Hebrew Bible story but, in Islam, the emphasis is paid to Job remaining steadfast to God and there is no record of his despair, or mention of discussions with friends. Some Muslim commentators also spoke of Job as being the ancestor of the Romans. Muslim literature also comments on Job's time and place of prophecy, saying that he came after Joseph in the prophetic series and that he preached to his own people rather than being sent to a specified community. Tradition further recounts that Job will be the leader of the group of "those who patiently endured" in Heaven.

Local traditions regarding Job
There are at least two locations that claim to be the place of Job's ordeal, and at least three that claim to have his tomb.

The Eyüp Sultan Mosque in Istanbul, Turkey, holds the tomb of Abu Ayyub al-Ansari, a companion of Muhammad, not the biblical/Qur'anic Job (Ayyub in Arabic, Eyüp in Turkish), though some locals tend to conflate the two.

Modern Israel
In Palestinian folk tradition, Job's place of trial is Al-Jura, or Al-Joura, a village outside the town of Al-Majdal (today's Ashkelon, Israel). It was there God rewarded him with a fountain of youth that removed whatever illnesses he had, and gave him back his youth.

To the northwest of the depopulated Palestinian village of Dayr Ayyub is an area which, according to the village belief, contained the tomb of the prophet Ayyub, the biblical Job.

In the area of Tabgha (Greek: Heptapegon), on the shore of the Sea of Galilee, a few sites are associated by local tradition with the life of Ayyub. A small grotto near the base of what is known to Christians as the Mount of Beatitudes, or Mount Eremos, is known as Mghraret Ayub ("Job's Cave"). Two of the towers built in the Byzantine period to collect the water of the Heptapegon springs are named in Arabic Tannur Ayub ("Job's Kiln") and Hammam Ayyub ("Job's Bath"). Hammam Ayyub was initially called "the Leper's Bath", but the leper was later identified with Job; the nearby spring, now a waterfall, is known as Ain Ayub, "Job's Spring".

Hauran, Syria
The town of al-Shaykh Saad in the Hauran region in Syria has been associated with Job since at least the 4th-century AD. Karnein was mentioned in Eusebius' Onomasticon as a town of Bashan that was said to be the location of the house of Job. Egeria the pilgrim relates that a church was built over the place in March or February 384 AD, and that the place was known as the "town of Job", or "civitas Job". According to Egeria's account the body of Job was laid in a stone coffin below the altar. According to tradition, Hammam Ayyub is a fountain in the town where Job washed himself when he was sick, and is reputed to have healing powers. Another holy artifact in the town is the "Rock of Job", known in local folklore as the place where he sat when he was afflicted with the disease.

Adma', Upper Mesopotamia
The city of Urfa (ancient Adma', later Edessa) in the Şanlıurfa Province, or Harran region of southeastern Turkey, also claims to be the location at which Job underwent his ordeal in a cave. The location boasts an Ottoman-style mosque and madrasa that runs as shops today. A well exists within the complex, said to be the one formed when he struck the ground with his foot as described in the Quran. The water is considered to be miraculously curing. The whole complex underwent recent restoration. The tomb of Job is located outside the city of Urfa.

Oman
The Tomb of Job is also said to be situated in Jabal Qarah outside the city of Salalah in southern Oman.

El-Chouf mountains, Lebanon
Additionally, the Druze community also maintains a shrine for the Prophet Job in Niha village in the Chouf mountains of Lebanon. This shrine is said to be the place where Job was healed from his ailments after his wife carried his frail body up the steep mountain in a basket so he dies up there. Instead, he was healed and given an even larger wealth.

title: Job (biblical figure)


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Category:Academia

Academia is a general term for the whole of higher education and academic research. The word comes from the Greek referring to the larger body of knowledge, its development and transmission across generations.

 * See also: List of academic disciplines and Category:Academic disciplines.

title: Category:Academia


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Enoggera railway station

Enoggera railway station is located on the Ferny Grove line in Queensland, Australia. It serves the Brisbane suburb of Enoggera.

History
Enogerra was the original terminus when the line opened in 1899. The line was progressively extended to Samford in 1918.

Services
Enogerra station is served by all stops Ferny Grove line services from Ferny Grove to Roma Street, Park Road, Coopers Plains and Beenleigh.

title: Enoggera railway station


text:
Ingo

Ingo is a masculine given name in contemporary Scandinavia and Germany, and a historical name in France. It is the male version of the name Inga, used in the same region.

It means "protected by Yngvi", who is the main god for the Ingvaeones, and is probably a different name for the Germanic god Freyr.

Persons with the name Ingo

 * Ingemar Johansson (1932-2009), Swedish boxer, nicknamed "Ingo"
 * Ingo Anderbrügge (born 1964), German footballer
 * Ingo Appelt (born 1961), German former bobsledder
 * Ingo Appelt (born 1967), German comedian
 * Ingo Brigandt, Canadian philosopher
 * Ingo Bodtke (born 1965), German politician
 * Ingo Buding (1942-2003), German tennis player
 * Ingo Freyer (born 1971), German professional basketball coach and former national team player
 * Ingo Gerhartz (born 1965), German Air Force general
 * Ingo Giezendanner (born 1975), Swiss artist
 * Ingo Hoffmann (born 1953), Brazilian former race car driver
 * Ingo Maurer (1932–2019), German industrial designer
 * Ingo Metzmacher (born 1957), German music conductor
 * Ingo Molnár, Hungarian computer programmer
 * Ingo Mörth (born 1949), Austrian sociologist
 * Ingo Nugel (1976-2007), German composer of video game music
 * Ingo Preminger (1911-2006), American movie producer, brother of director Otto Preminger
 * Ingo Rademacher (born 1971), German-born Australian actor
 * Ingo Rechenberg (1934–2021), German researcher and professor currently in the field of bionics
 * Ingo Renner (1940–2022), Australian glider pilot, four-time world champion
 * Ingo Schmitt (fl. 1995-present), German politician
 * Ingo Schulze (born 1962), German author
 * Ingo Schwichtenberg (1965-1995), German rock drummer
 * Ingo Simon (1875–1964), British singer and archer
 * Ingo Steinhöfel (born 1967), German former weightlifter
 * Ingo Steuer (born 1966), German figure skater and coach
 * Ingo Swann (1933-2013), American psychic
 * Ingo Titze, American vocal scientist
 * Ingo Wegener (1950-2008), German computer scientist
 * Ingo Wellenreuther (born 1959), German politician
 * Ingo Zamperoni (born 1974), German news presenter

Fictional characters

 * Ingo, an associate of Talon in The Legend of Zelda series
 * Ingo, the missing Subway Boss in Pokémon Black and White and Pokémon Legends: Arceus
 * Ingo Manfred, a main character in the Broken Sky trilogy by L. A. Weatherly.

Literature

 * Ingo (novel), a 2005 children's novel by Helen Dunmore

title: Ingo


text:
File:Sazaesanvolume11.jpg

Summary
Sazae-san, Volume 11. Selected comics from 1968 to 1971. Fair use book cover.

title: File:Sazaesanvolume11.jpg


text:
Gaythorne railway station

Gaythorne railway station is located on the Ferny Grove line in Queensland, Australia. It serves the Brisbane suburb of Gaythorne.

History
The station was built in 1916 and originally named Rifle Range Station. It was built as an extension of the Ferny Grove line from Enoggera to service the new military camps established in the area during World War 1, such as Bells Paddock. After the war, in 1923, a petition was circulated calling for the station's named to be changed to Gaythorne after the closure of the wartime camps.

Services
Gaythorne station is served by all stops Ferny Grove line services from Ferny Grove to Roma Street, Park Road, Coopers Plains and Beenleigh.

title: Gaythorne railway station


text:
Bob

Bob, BOB, or B.O.B. may refer to:

People, fictional characters, and named animals

 * Bob (given name), a list of people and fictional characters
 * Bob (surname)
 * Bob (dog), a dog that received the Dickin Medal for bravery in World War II
 * Bob the Railway Dog, a part of South Australian Railways folklore

Places

 * Mount Bob, New York, United States
 * Bob Island, Palmer Archipelago, Antarctica

Television, games, and radio

 * Bob (TV series), an American comedy series starring Bob Newhart
 * B.O.B. (video game), a side-scrolling shooter
 * Bob FM, on-air brand of a number of FM radio stations in North America

Musicians and groups

 * B.o.B (born 1988), American rapper and record producer
 * Bob (band), a British indie pop band
 * The Bobs, an American a cappella group
 * Boyz on Block, a British pop supergroup

Songs

 * "B.O.B" (song), by OutKast
 * "Bob" ("Weird Al" Yankovic song), from the 2003 album Poodle Hat by "Weird Al" Yankovic
 * "Bob", a song from the album Brighter Than Creation's Dark by the Drive-By Truckers
 * "Bob", a song from the album White Trash, Two Heebs and a Bean by NOFX
 * "Bob", a song from the album Pork Soda by Primus
 * "Bob", a song from the album One Hot Minute by the Red Hot Chili Peppers
 * "The Bob (Medley)", a song from Roxy Music's self-titled album

Science and technology

 * Bob (physics), the mass at the end of a pendulum
 * BOB (psychedelic), a drug
 * Alice and Bob, placeholder characters in cryptography and physics problems
 * Blitter object, a graphics construct
 * Hurricane Bob (disambiguation)
 * Microsoft Bob, a software product

Acronyms

 * Bank of Baroda, in India
 * Bank of Botswana
 * Bank One Ballpark, now Chase Field
 * Bayerische Oberlandbahn, a private German railway company
 * Beijing Olympic Broadcasting
 * Bend Over Boyfriend, slang term for a sexual practice
 * Berlin Operating Base, the CIA station in Berlin during the Cold War
 * Berner Oberland Bahn, a mountain railway in Switzerland
 * Best of Biotech, a business plan competition organized by Austria Wirtschaftsservice
 * Boys of Bangladesh, a network of gay men in Bangladesh
 * Breakout board, a minimal printed circuit board with a single component
 * Bug-out bag, an evacuation kit
 * Buy on board, food or beverages purchased in-flight

Codes

 * Aweer language ISO 639-3 code
 * Bolivian boliviano ISO 4217 currency code
 * Bora Bora Airport IATA code

Other uses

 * Bob, a slang term in Great Britain for the pre-decimal coin, the shilling
 * Bob, short for bobsleigh or bobsled
 * Bob cut, a hairstyle

title: Bob


text:
School of Fish

School of Fish was an alternative rock band which formed in 1989 and disbanded in 1994. The core members were Josh Clayton-Felt (lead vocals and guitar) and Michael Ward (guitar). School of Fish released two albums and are remembered for the hit single "3 Strange Days" (1991).

History
Clayton-Felt and Ward started as a duo who would play club dates in Los Angeles, California, accompanied by programmed drums and bass.

Recruiting drummer Michael Petrak and bassist David Lipson the band signed with Capitol Records in 1990, and released their first single in that same year on 7-inch vinyl. Their self-titled debut was released in March 1991, followed by their first single, "3 Strange Days." Also during the same year, they appeared on a KROQ-FM concert disc called the Acoustic Christmas.

After the release of the first album, the rhythm section changed from Michael Petrak and Dominic Nardini to Chad Fischer and Chris MacDonald respectively. However the new members did not play on the band's second album, which featured session musicians Josh Freese and John Pierce. Nevertheless, Chad Fischer stayed with the band to play live.

School of Fish's second album, Human Cannonball, was released in 1993. The album did not sell well and was not as successful as the self-titled album. Shortly thereafter, the band broke up.

Members careers
Singer Josh Clayton-Felt embarked on a solo career and released two studio albums and one live album before he was diagnosed with cancer in 1999. He died on January 19, 2000, at the age of 32 but his estate has since overseen three posthumous releases.

Guitarist Michael Ward went on to play with John Hiatt and eventually became a member of The Wallflowers. After The Wallflowers split, Ward spent many years as a member of Ben Harper's band, The Innocent Criminals. He was also a sought-after session guitarist in LA, and the author of the children's books 'Mike and the Bike' and 'Mike and the Bike Meet Lucille The Wheel.' Ward died on April 1, 2024.

Chad Fischer went on to form the band Lazlo Bane, known for original theme song of the TV show Scrubs. He also established himself as a record producer and composer.

Christopher MacDonald, who played bass with the band on tour, but did not play on any School of Fish recordings, records synthesizer space music under the name Telomere.

Releases
Two previously unreleased songs were released as downloads circa 2007. "Broken Arm" (recorded before the Human Cannonball sessions) and "Goodbye Green World" (recorded before the self-titled album sessions).

The idea of releasing a rarities compilation was in talks with surviving past members but was aborted completely.

Two songs, "Goodbye Green World" and "Who Am I Today", performed by Josh-Clayton Felt and Michael Ward were issued on Josh's third posthumous release, the EP The Spirit Shines Through.

In 2020 Chad Fischer reunited with the original members of School of Fish to record a new version of the song "3 Strange Days" which was released as a video. The track was later included on the Lazlo Bane album Someday We'll Be Together.

Members

 * Josh Clayton-Felt – vocals, guitars (1989-1994: died 2000)
 * Michael Ward – guitars, vocals (1989-1994, 2020: died 2024)
 * David Lipson – bass (1989)
 * Craig Aaronson – drums (1989)
 * Dominic Nardini – bass, tambourine (1990-1991, 2020)
 * Michael Petrak (aka M.P.) – drums (1990-1991, 2020)
 * John Pierce – bass, cello (1992-1994)
 * Josh Freese – drums (1992-1993, appears on Human Cannonball only)
 * Chad Fischer – live drums (1991-1994) vocals, guitar (2020)
 * Chris MacDonald – live bass (1991-1992)

Singles

 * Promotional Singles

title: School of Fish


text:
Oxford Park railway station

Oxford Park railway station is located on the Ferny Grove line in Brisbane, Queensland, Australia. It is one of two railway stations serving the Brisbane suburb of Mitchelton, the other being Mitchelton railway station.

History
The station was upgraded in 2008 as part of the Mitchelton to Keperra duplication project.

Services
Oxford Park station is served by all stops Ferny Grove line services from Ferny Grove to Roma Street, Park Road, Coopers Plains and Beenleigh.

title: Oxford Park railway station


text:
American Bantam

The American Bantam Car Company was an American automobile manufacturing company incorporated in the state of Pennsylvania. American Bantam is credited with inventing the original World War II 1⁄4ton jeep in 1940. The company's founders, Roy Evans and William A. Ward Jr., combined resources to purchase the assets of the bankrupt American Austin Car Company in August of 1935 during liquidation.

In 1935 the new company produced vehicles based on the American Austin tooling, operating as Evans Operations Inc. The new company was incorporated as American Bantam Car Company in June 1936. The new company launched a public fundraising campaign and redesigned their entire vehicle line to launch a completely refreshed selection of American Bantam roadsters and delivery vans in 1937. The company continued to make vehicles into 1943, until all of its production – like all the U.S. automotive industry – served the World War II efforts, with vehicles and weapons manufacturing, including (amphibious) cargo trailers, aircraft controls and other parts, and engines and tail gearing for torpedoes.

History
On June 2, 1936, Roy Evans and his co-founders incorporated American Bantam Car Company in Pennsylvania and transferred the assets purchased from the American Austin Car Company, along with $500 in cash to the new company. In early 1937, Evans phoned Count Alexis de Sakhnoffsky, the original Austin body designer, and asked him to design a new line of bodies improvised around the existing tooling. No funds were available for a costly styling program, much less to engineer new body dies, so the new design had to use as much of the original tooling as possible.

In early 1937 Alexis de Sakhnoffsky visited the American Bantam factory in Butler, Pennsylvania, and designed a completely new front grille, new front fenders and new rear fenders. The Austin engine was also redesigned with a new aluminum induction system and cylinder head, retaining the 45.6 cubic inch (747 cc) displacement, but while adopting a fully pressurized oil system, increasing the compression ratio by 40% to a 7:1 ratio and implementing plain babbitt crankshaft bearings to produce 20 horsepower at 4,000 rpm, a 50% improvement over the Austin engine. American Bantam's 1938 model was the inspiration for Donald Duck's car which was first seen in Don Donald (1937). Despite a wide range of Bantam body styles, ranging from light trucks to woodie station wagons, only about 6,000 Bantams of all types were produced. American Bantam continued to build cars until August 18, 1943.

Invention of the Original Jeep

American Bantam is credited with the invention of the original jeep and the first serial production military jeeps ordered by the U.S. Army in 1940.

The idea of a small, durable automobile to replace the horse was championed by American Bantam salesman Navy Commander Charles 'Harry' Payne (retired), working closely with Robert Brown, a civilian consultant working for the U.S. Army Quartermaster Corps (QMC), American Bantam chief engineer Harold Crist and American Bantam president Frank Fenn, who together during the spring of 1940 laid out the specifications for the first Bantam Reconnaissance Car. Harold Crist did the lion's share of specifying, conceiving, designing and building the car with Karl Probst drafting and formalizing pre-existing layout and specifications set out by Crist.

American Bantam delivered the first jeep to the QMC on September 23, 1940, at Camp Holabird, a U.S. Army base to the east of Baltimore, Maryland. Engineers from Ford and Willys-Overland on-hand at Camp Holabird during testing to learn more about the new vehicle. The original jeep designs were handed over to Willys-Overland and Ford and became the basis for the design of the World War II jeep. After the delivery of the first jeep, American Bantam kicked off serial production of the Mark II (also called the BRC-60) jeeps with improvements suggested by the QMC. American Bantam was the sole manufacturer of jeeps put into service by the U.S. Army during 1940.

All together American Bantam built 2,675 jeeps from 1940 through 1943, with the bulk of those vehicles being delivered during 1941. More than half of the initial production went to the British Army, and some to the Soviet Union. Some of the motors and chassis were imported from Toledo, Ohio; the original bodies were made at the American Bantam factory in Butler, Pennsylvania.

The Bantam company produced the most fuel-efficient engine and first prototype under the original U.S. Army tender specifications and was awarded the first contract. However, because elements favorable to Ford within the Quartermaster Corps claimed that Bantam lacked production capacity to produce the vehicle on the scale needed by the Army, the awarding of ongoing contracts was reopened. Eventually the Army gave the BRC (Bantam Reconnaissance Car) 40 designs to Willys-Overland and awarded the bulk of orders to Willys and Ford, while Bantam went on to produce jeep trailers (T-3).

After World War II American Bantam continued to make trailers for the consumer market. In 1943 American Bantam launched an Advertising campaign boasting that "Ivan got his first Jeep from Bantam" in response to an application by Willys-Overland to the United States Patent and Trademark Office to trademark "JEEP" filed on February 13, 1943.

Trailer production continued until the company was taken over by American Rolling Mills in 1956.

title: American Bantam


text:
Grovely railway station

Grovely railway station is located on the Ferny Grove line in Brisbane, Queensland, Australia. It serves the Brisbane suburb of Keperra.

History
The station was upgraded in 2008 as part of the Mitchelton to Keperra duplication project. The upgrade included a new platform & footbridge.

Services
Grovely station is served by all stops Ferny Grove line services from Ferny Grove to Roma Street, Park Road, Coopers Plains and Beenleigh.

title: Grovely railway station


text:
NRBQ

NRBQ is an American rock band founded by Terry Adams (piano), Steve Ferguson (guitar) and Joey Spampinato (bass). Originally the "New Rhythm and Blues Quintet", the group was formed circa 1965. Adams disbanded it for a time, and the group re-formed in 1967. The quartet is known for its live performances, containing a high degree of spontaneity and levity, and blending rock, pop, jazz, blues and Tin Pan Alley styles. Its membership comprises the quartet of pianist Adams, bassist Casey McDonough, guitarist Scott Ligon, and drummer John Perrin. Some of the members in the band's long history are singer, writer and bassist Joey Spampinato, guitarists Al Anderson and Johnny Spampinato; drummers Tom Staley and Tom Ardolino; and vocalist Frank Gadler.

History
NRBQ began in late 1965 as a rehearsal band in the Shively, Kentucky, home of brothers Terry and Donn Adams, and they appeared on stage for the first time in 1966. Along with drummer Charlie Craig, they made home tapes of their experiments. The first known reference to the band's name can be heard on one of these home tapes, with Donn announcing, "Here they are, the New Rhythm and Blues Quintet!" as though presenting them to a live audience.

In late spring of 1966, guitarist Steve Ferguson was invited to join NRBQ after he quit Mersey-Beats USA (no relation to the Liverpool group who recorded "I Stand Accused"). He met Adams when the latter briefly joined Mersey-Beats USA to fill in for the regular keyboard player. After playing a few live dates in Louisville, Jimmy Orten (Soul Inc) was recruited on bass and vocals and the band left for Florida in late 1966.

In Miami, early January 1967, NRBQ played six nights at The Cheetah. Soon after, Orten and Ferguson returned to Louisville. Adams stayed behind and joined The Seven of Us, a band that was playing the same club. By August, the NRBQ was Adams (keyboards), Ferguson (guitar), Spampinato (bass) and Frank Gadler (vocals) from The Seven of Us, and Tom Staley (drums).

In December 1968, they began recording with Eddie Kramer at the Record Plant and by early 1969 were signed to a two-record deal with Columbia Records. Their self-titled debut album was released that year, with songs by both Eddie Cochran and Sun Ra, and a number of similarly wide-ranging original songs. The following year, the group collaborated with rockabilly legend Carl Perkins on an album titled Boppin' the Blues.

Over the next three years, the band experienced personnel shifts, with the departure of Ferguson (replaced for one year by Ken Sheehan), Gadler, and Staley, and the arrival of two new members: guitarist/singer Al Anderson formerly of The Wildweeds, known for the Connecticut and Massachusetts regional hit "No Good To Cry", and drummer Tom Ardolino. The Adams / Spampinato / Anderson / Ardolino quartet stayed together longer than any other incarnation of the band (20 years, from 1974 until 1994), and was often augmented by the Whole Wheat Horns (Donn Adams, Keith Spring and others).

Starting in September 1989, NRBQ opened for R.E.M. on the "Acronym Tour" with the first concert taking place on the 27th at the University of Dayton Arena. The last stop on the tour was at the Shoreline Amphitheater at Mountain View in Mountain View, California on October 21. Rolling Stone's Sam Freedman noted that the NRBQ played a compressed 30-minute club show. He also reported that both Mike Mills and Peter Buck of R.E.M. watched each of the NRBQ's opening sets during their portion of the tour from the edge of the stage "as if to endorse NRBQ for their fans".

In 1994 Anderson departed the group: he would become an award-winning Nashville songwriter for many country and western acts. He was replaced in NRBQ by Joey Spampinato's younger brother, Johnny Spampinato, who was (and still is) a member of power-pop band The Incredible Casuals.

On April 30 and May 1, 2004, the group celebrated its 35th anniversary with concerts at the Calvin Theater in Northampton, Massachusetts. The shows featured every former and current member of the band, as Ferguson, Gadler, Staley, Sheehan and Anderson came back for a NRBQ reunion.

Hiatus and return
After the Halloween engagement at Shank Hall in Milwaukee in 2004, NRBQ went on hiatus. Adams had developed stage 4 throat cancer. During this time, Ardolino and the Spampinato brothers started playing shows as a trio, under the name "Baby Macaroni" and as the "Spaminato Bros.". After a number of months, Adams recovered well enough to tour with former drummer Staley and Japanese rockabilly group the Hot Shots.

In June 2006, Adams and Ferguson released the album Louisville Sluggers (with Ardolino on drums, Pete Toigo on bass and other supporting musicians), and this album's line-up performed some live shows in the U.S. and Japan as "The Terry Adams – Steve Ferguson Quartet" and "Rock & Roll Summit Meeting."

Also in September 2006 came the release of a SpongeBob SquarePants album, The Best Day Ever, which included backing music by all four NRBQ members, as well as Anderson. The album, a collection of '60s-influenced pop/rock produced by Andy Paley, and co-written by Paley and the voice of SpongeBob, Tom Kenny, also included such musical luminaries as Brian Wilson, Tommy Ramone, James Burton, Flaco Jiménez and Philadelphia DJ Jerry "The Geator" Blavat.

In November 2007, Adams formed "The Terry Adams Rock & Roll Quartet" with Scott Ligon on guitar and vocals, Pete Donnelly (of The Figgs) on bass and Conrad Choucroun and Ardolino on drums.

In March 2011, Adams posted an open letter to fans announcing that with the release of the upcoming album, Keep This Love Goin', this line-up would take on the NRBQ name. He also explained that while he did have tendinitis, the real reason for the hiatus was his treatment for cancer. In May 2012, the group released a live album, We Travel the Spaceways, on Clang! records.

During September 2012, bassist Donnelly was replaced by Scott Ligon's friend Casey McDonough. (As of 2021, Ligon and McDonough are also members of the Flat Five, based in Chicago.)

In July 2013, NRBQ toured with drummer Joe Camarillo of The Waco Brothers and Hushdrops, where he recorded two songs on the Brass Tacks album: "Greetings from Delaware" and "I'm Not Here". Bobby Lloyd Hicks became the drummer from 2013 to 2015. He played on "Love In Outer Space" and "Let Go" from the 5-CD box set High Noon. John Perrin became the drummer in 2015.

Steve Ferguson died of cancer on October 7, 2009.

Tom Ardolino died on January 6, 2012, following a long illness.

Bobby Lloyd Hicks died of bronchiectasis on February 20, 2017.

Joe Camarillo died on January 24, 2021, of a stroke after contracting COVID-19.

Reunion shows
On April 27 and 28 of 2007, NRBQ gave a pair of "38th Anniversary" performances in Northampton, Massachusetts, the first public NRBQ shows since 2004. Both Anderson and Johnny Spampinato appeared in the line-up, along with "Whole Wheat Horns" Donn Adams and Jim Bob Hoke, and unannounced guest appearances by John Sebastian, original NRBQ drummer Staley and the band's former road manager Klem Klimek on saxophone. Gadler, former lead vocalist, also appeared.

Style and influence
The band's music is a blend of styles from rockabilly to Beatles-influenced pop to Thelonious Monk-inspired jazz. They have attracted fans as diverse as Bob Dylan, Paul McCartney, Elvis Costello, Keith Richards, The Replacements, John Sebastian, Dave Edmunds, Ira Kaplan and Penn & Teller. NRBQ songs have been performed by Bonnie Raitt, Los Lobos, and Dave Edmunds among others. Also, the group served as the unofficial "house band" for The Simpsons for the season 10–12 period in which NRBQ fan Mike Scully was head writer and executive producer. NRBQ allowed several of their songs to be used on The Simpsons, including "Mayonnaise and Marmalade", written specifically for the show. The band also appeared in animated form and on camera during the end credits to perform the show's theme song during the episode "Take My Wife, Sleaze" and to perform Edmunds's cover of "Me & The Boys". The band also recorded a song entitled "Birdman" for an episode of Space Ghost Coast to Coast entitled "Pilot". The group appeared in feature films, including Day of the Dead, Shakes the Clown, and 28 Days. Their cover version of "Down in My Heart" appeared in the series finale of Wilfred.

NRBQ has a following from years of live shows. The band has been known to perform without a set list which makes the band, in the words of AllMusic's Mark Deming, "a stellar and wildly unpredictable live act." In addition to its own compositions, the band performs a broad range of cover material and audience requests.

In their nearly 50-year history, NRBQ's records have been released by many record companies, including Columbia Records, Kama Sutra Records, Mercury Records, Virgin Records, Rhino Entertainment, Rounder Records, and more. Their song "Get That Gasoline Blues" (on Kama Sutra) reached No. 70 in 1974 on the Billboard Hot 100 chart. Over the years, the group played sets while wearing pajamas, hired professional wrestling manager "Captain" Lou Albano as the band's manager (for whom they penned a song in tribute), and exploded Cabbage Patch Dolls on stage.

Current line-up

 * Terry Adams – keyboards, vocals (1966–2004, 2011-present); occasional live drums (1974–2004, 2016-present)
 * Scott Ligon – guitar, vocals (2011–present)
 * Casey McDonough – bass, vocals (2012–present)
 * John Perrin – drums (2015–present)

Former members

 * Steve Ferguson – guitar, vocals (1966–1970, 1973–1974; guest 2004; died 2009)
 * Joey Spampinato – bass, vocals (1967–2004, 2007)
 * Tom Staley – drums (1967–1974; guest 2004, 2007)
 * Frank Gadler – lead vocals (1967–1972; guest 2004, 2007)
 * Joe Gallivan – drums (1968)
 * Ken Sheehan – guitar (1970; guest 2004)
 * Al Anderson – guitar, vocals (1971–1994, 2007; guest 2004)
 * Tom Ardolino – drums, vocals (1974–2004; died 2012)
 * Johnny Spampinato – guitar, vocals (1994–2004, 2007)
 * Conrad Choucroun – drums (2011–2013)
 * Pete Donnelly – bass, vocals (2011–2012)
 * Joe Camarillo – drums (2013, died 2021)
 * Bobby Lloyd Hicks – drums (2013–2015, died 2017)

Studio albums

 * NRBQ (1969) (Columbia) 1969 No. 162 US; rereleased (Omnivore) 2018
 * Boppin' the Blues (w/ Carl Perkins) (Columbia) 1970
 * Scraps (Kama Sutra) 1972
 * Workshop (Kama Sutra) 1973
 * All Hopped Up (Red Rooster) 1977; rereleased (Omnivore) 2018
 * At Yankee Stadium (Mercury) 1978
 * Kick Me Hard (Rounder/Red Rooster) 1979
 * Tiddlywinks (Rounder/Red Rooster) 1980
 * Grooves in Orbit (Bearsville) 1983
 * Tapdancin' Bats (Rounder/Red Rooster) 1983
 * She Sings, They Play (w/ Skeeter Davis) (Rounder/Red Rooster) 1985
 * Lou and the Q (w/ "Captain" Lou Albano) (Rounder/Red Rooster) 1986
 * Wild Weekend (Virgin) 1989 No. 198 US
 * Message for the Mess Age (Rhino) 1994
 * You're Nice People You Are (Rounder) 1997
 * Christmas Wish (Rounder) 1997
 * NRBQ (sometimes known as "The Yellow Album") (Rounder) 1999
 * Atsa My Band (Edisun) 2002
 * Dummy (Edisun) 2004
 * Keep This Love Goin' (Clang!) 2011
 * Brass Tacks (Clang!) 2014
 * Happy Talk [5 song EP] (Omnivore) 2017
 * April Showers [3 song EP] (Omnivore) 2018
 * Dragnet (Omnivore) 2021

Live albums

 * God Bless Us All (Rounder) 1987
 * Diggin' Uncle Q (Rounder) 1988
 * Honest Dollar (Rykodisc) 1992
 * Tokyo (Rounder) 1996
 * You Gotta Be Loose (Rounder) 1998
 * Live from Mountain Stage (Live performances from the Mountain Stage radio show) (Blue Plate) 2002
 * Live at the Wax Museum (previously unreleased concert from 1982, with guest John Sebastian) (Edisun) 2003
 * Froggy's Favorites Vol. 1 (compilation of unreleased live tracks 1979–1999) (Edisun) 2006
 * Ludlow Garage 1970 (previously unreleased concert from 1970) (Sundazed) 2006
 * We Travel the Spaceways (Clang!) 2012
 * Talk Thelonious (Euclid Records) 2015
 * Turn On, Tune In (Omnivore) 2019
 * NRBQ at the Ardmore Music Hall 2015 (arQive) 2020
 * NRBQ & the Whole Wheat Horns Park West 83 (arQive) 2022

Compilations

 * RC Cola and a Moon Pie (Rounder/Red Rooster) 1986
 * Uncommon Denominators (Rounder-era compilation covering 1972 through 1984) (Rounder) 1987
 * Kick Me Hard — The Deluxe Edition (reissue, w/ 8 bonus tracks) (Rounder) 1989
 * Peek-A-Boo (multi-label compilation covering 1969 through 1989) (Rhino) 1990
 * Stay with We (compilation of Columbia years, w/ unreleased songs) (Columbia/Legacy) 1993
 * Tapdancin' Bats — The Anniversary Edition (reissue, w/ 4 bonus tracks) (Rounder) 1998
 * Ridin' in My Car (reissue of All Hopped Up, w/ unreleased songs) (Rounder) 1999
 * Scraps (reissue, remastered, w/ 3 bonus tracks) (Rounder) 2000
 * Scraps Companion (15 tracks from radio show from Memphis in 1972 and 6 outtakes from Scraps sessions) (Edisun) 2000
 * Transmissions (2-disc Japan-only compilation featuring about 40 percent unissued material) (Caraway) 2004
 * Christmas Wish — Deluxe Version (Clang!) 2007
 * High Noon: A 50-Year Retrospective [106-track 5-CD version] (Omnivore) 2016
 * High Noon: A 50-Year Retrospective [17-track digital version] (Omnivore) 2016
 * High Noon: A 50-Year Retrospective [26-track 2-LP version] (Omnivore) 2017
 * In • Frequencies (Omnivore) 2020

Singles

 * "Stomp / I Didn't Know Myself" (Columbia) 1969 No. 122 US
 * "C'mon Everybody / Rocket Number 9" (Columbia) 1969
 * "Down In My Heart / Sure To Fall (In Love With You)" (Columbia) 1969
 * "All Mama's Children / Step Aside" (with Carl Perkins) (Columbia) 1970
 * "Howard Johnson's Got His Ho-Jo Workin' / Only You" (Kama Sutra) 1972
 * "Magnet / Do You Feel It?" (Kama Sutra) 1972
 * "C'mon If You're Comin' / RC And A Moon Pie" (Kama Sutra) 1973
 * "Get That Gasoline Blues / Mona" (Kama Sutra) 1973 No. 70 US
 * "Rumors / Sourpuss" (Select-O-Hit) 1974
 * "Ridin' In My Car / Do The Bump" (Red Rooster) 1977
 * "I Got A Rocket In My Pocket / Tapdancin' Bats" (Red Rooster) 1977
 * "I Love Her, She Loves Me / Green Lights" (Mercury) 1978
 * "Hot Biscuits And Sweet Marie / Don't She Look Good" (Red Rooster/Rounder) 1979
 * "Get That Gasoline / Wacky Tobacky" (Red Rooster/Rounder) 1979 No. 105 US Cashbox
 * "Me And The Boys / People" (Red Rooster/Rounder) 1980
 * "Christmas Wish / Jolly Old St. Nicholas" (Red Rooster/Rounder) 1980
 * "Never Take The Place Of You / Radio Spot / TV Spot" (Red Rooster/Rounder) 1980
 * "Captain Lou / Boardin' House Pie" (W/ Captain Lou Albano) (Red Rooster) 1982
 * "Rain At The Drive-In / Smackaroo" (Instrumental) (Bearsville) 1983
 * "Wild Weekend / This Love Is True" (Virgin) 1989
 * "If I Don't Have You" (Virgin) 1989
 * "A Little Bit Of Bad / Tom Dooley" (Forward) 1994
 * "Over Your Head / Blues Stay Away From Me / Spampinato" (Forward) 1994
 * "Careful What You Ask For" (Rounder) 1999
 * "Everybody Say Yeah! / Hornin' In" (Euclid) 2012
 * "Never Cop Out / Scram! It's The Fuzz" (B-side By Los Straitjackets) (Spinout) 2016
 * "Do The Primal Thing" (Extended Version) (Omnivore) 2020
 * "I'm Not Here / All I Have To Do Is Dream" (Folc) 2021

title: NRBQ


text:
Keperra railway station

Keperra railway station is located on the Ferny Grove line in Queensland, Australia. It serves the Brisbane suburb of Keperra. The station was opened in 1932.

Services
Keperra station is served by all stops Ferny Grove line services from Ferny Grove to Roma Street, Park Road, Coopers Plains and Beenleigh.

title: Keperra railway station


text:
Under the Sun (Paul Kelly album)

Under the Sun is the second album by Australian rock group Paul Kelly & The Coloured Girls and was originally released in December 1987 by Mushroom Records. In the North American and European markets, it was released by A&M Records in 1988 with the band credited as Paul Kelly & The Messengers, with a different track order and listing.

On the Australian albums charts, it peaked at #19, with the single "To Her Door" peaking at #14. Another single, "Dumb Things," peaked at #36 in early 1989 on the Australian Recording Industry Association (ARIA) singles charts; it reached #16 on the Billboard Modern Rock chart. The song was included in the soundtrack for the 1988 Yahoo Serious film Young Einstein.

"To Her Door" won an ARIA Award in 1988 for 'Best Video' directed by Claudia Castle. In 2001, the Australasian Performing Right Association (APRA) listed the Top 30 Australian songs of all time, including "To Her Door," written by Kelly.

Background
After relocating from Melbourne to Sydney in 1985, Paul Kelly began to play and record with a full-time band, which included Michael Armiger on bass guitar, Michael Barclay on drums, Steve Connolly on guitar, eventually bassist Jon Schofield, and keyboardist Peter Bull joined. Through a joke based on Lou Reed's song "Walk on the Wild Side", the band became known as Paul Kelly and the Coloured Girls. In September 1986 the band released their debut double LP Gossip. Due to possible racist connotations the band changed its name, for international releases, to Paul Kelly and the Messengers. They made an American tour, initially supporting Crowded House and then head-lining, travelling across the United States by bus. Jon Schofield replaced Armiger on bass guitar, Chris Coyne on tenor saxophone and Chris Wilson on harmonica

On the Australian albums charts, it peaked at #19 with the single "To Her Door" peaking at #14. First single from the album, "Bradman," had been released in January 1987 as a double-A side with "Leaps and Bounds" from Gossip but had little chart success. The third and fourth singles, "Forty Miles to Saturday Night" and "Don't Stand So Close to the Window" also had little chart success. Another single, "Dumb Things," peaked at #36 in early 1989 on the Australian Recording Industry Association (ARIA) singles charts; it reached #16 on the Billboard Modern Rock chart. The song was included in the soundtrack for the 1988 Yahoo Serious film Young Einstein.

In 1988, "To Her Door" won an ARIA Award for 'Best Video' directed by Claudia Castle. In 2001, the Australasian Performing Right Association (APRA) listed the Top 30 Australian songs of all time, including "To Her Door," written by Kelly.

"Desdemona" was featured in an episode of the Australian TV show Packed To The Rafters.

Track listing
All tracks written by Paul Kelly unless otherwise indicated.

Original Australian LP/MC release
 * 1) "Dumb Things" (aka "I've Done all the Dumb Things) – 2:31
 * 2) "Same Old Walk" – 4:08
 * 3) "Big Heart" – 3:22
 * 4) "Don't Stand So Close to the Window" (Paul Kelly, A McGregor) – 2:35
 * 5) "Forty Miles to Saturday Night" – 3:11
 * 6) "I Don't Remember a Thing" – 2:04
 * 7) "Know Your Friends" – 3:37
 * 8) "To Her Door" – 3:18
 * 9) "Under the Sun" – 4:18
 * 10) "Untouchable" – 2:04
 * 11) "Desdemona" – 3:46
 * 12) "Happy Slave" – 2:30
 * 13) "Crosstown" – 2:23
 * 14) "Bicentennial" – 3:04

Bonus tracks for Australian CD release
 * 1)  "Bradman" – 7:26
 * 2) "Pastures of Plenty" (Woody Guthrie) – 2:26

North American/European release
 * 1) "Dumb Things" – 2:31 ^^
 * 2) "Same Old Walk" – 4:08
 * 3) "Big Heart" – 3:22
 * 4) "Don't Stand So Close to the Window" (Paul Kelly, A McGregor) – 2:35
 * 5) "Forty Miles to Saturday Night" – 3:11 ^^
 * 6) "Untouchable" – 2:04
 * 7) "Know Your Friends" – 3:37
 * 8) "To Her Door" – 3:18 ^^
 * 9) "Under the Sun" – 4:18
 * 10) "Desdemona" – 2:07
 * 11) "Happy Slave" – 3:46
 * 12) "Crosstown" – 2:30
 * 13) "Little Decisions" – 2:25
 * 14) "Bicentennial" – 3:04

(^^) The European release featured slightly different mixes of these tracks.

Personnel
Paul Kelly and the Coloured Girls
 * Paul Kelly — acoustic guitar, vocals
 * Michael Barclay — drums, vocals
 * Peter Bull — keyboards
 * Steve Connolly — guitar (electric), vocals
 * Chris Coyne — saxophone (tenor)
 * Jon Schofield — bass guitar
 * Chris Wilson — harmonica, vocals, saxophone (baritone)

Additional musicians
 * Jessica Kenny — vocals
 * Joe Camilleri — saxophone (tenor)
 * Steve Miller — whistle (human)
 * Lucky Oceans — guitar (steel)
 * Ian Simpson — banjo

Recording details
 * Producer — Alan Thorne and Paul Kelly except "Dumb Things", produced by Martin Armiger and Paul Kelly
 * Engineer — Alan Thorne
 * Assistant engineer — Kathy Naunton
 * Recording & mixing studio — Alberts and Trafalgar Studios

Art work
 * Design — Melanie Nissen
 * Photography — Francine McDougall (cover photo), Isabel Snyder

title: Under the Sun (Paul Kelly album)


text:
Sussex, New Brunswick

Sussex is a town in Kings County, New Brunswick, Canada. Sussex is located in south central New Brunswick, between the province's three largest cities, Saint John, Moncton, and Fredericton.

Sussex straddles the Kennebecasis River, 70 km northeast of Saint John, and is a major dairy product producer in the province. It is home to Atlantic Canada's largest hot air balloon festival.

On 1 January 2023, Sussex amalgamated with the village of Sussex Corner and part of the local service district of the parish of Sussex. Revised census figures have not been released, although combining the 2016 census nuvmbers the total population would be approximately 5,900.

History
In 1857 the European and North American Railway was opened, connecting the farming communities of the Kennebecasis River valley with Saint John and Moncton. Sussex was incorporated in accordance with Chapter 44 of the Town Incorporation Act of 1896 and was officially established as a Town on June 2, 1904. Prior to its incorporation, the area was known as Sussex Vale from 1810 – 1898 and was a farming community of 185 people. This grew to include the surrounding area of Lower Sussex.

The settlers were for the most part British Loyalists who had fled the American Revolution in 1776, with many Irish refugees of the Great Famine from the mid-19th century settling in the nearby farming communities.

In 1885, the Sussex Military Camp was established on the eastern edge of the town. The facility was closed following the Second World War and the town purchased the land to expand the municipal boundaries. Today the agricultural exhibition and some areas remain as open land on the former site of Camp Sussex.

Sussex underwent several changes in the post-war period. In the early 1960s, several local roads were upgraded as part of the Trans-Canada Highway project which saw Route 2 pass immediately north of the town between Fredericton and Moncton. At the same time, a series of local roads in the Kennebecasis River valley were designated as Route 1, running from an interchange with the Trans-Canada at Sussex, southwest to Saint John.

The creation of Sussex as a highway interchange in this post-war period led to some transportation planners in New Brunswick calling for the consolidation or closure of the Fredericton, Saint John and Moncton airports to be replaced by a single airport located in Sussex to serve all three population centres of southern New Brunswick; this being in the late 1950s/early 1960s before these facilities underwent considerable expansion.

Potash was subsequently discovered in large quantities in the area surrounding Sussex, with the deposit being the second largest in the world after an area in Saskatchewan. Three mines were built near the town, two at Penobsquis, 8 km to the east, and another at Cassidy Lake (no longer operational), 10 km to the southwest. CN Rail built track to serve both mines, which employed hundreds from the surrounding area. Since 2003, natural gas has been produced from the McCully field near Sussex. Price decreases of the potash market combined with underground structural issues caused the Potash Corporation of Saskatchewan to close its Picadilly mining operation in January 2016, and then its Penobsquis mine in November 2018.

Sussex also began to see a growing tourism trade, with many flocking to see the collection of wood-constructed covered bridges throughout the central area of Kings County.

Today, Sussex is primarily a regional service centre for the surrounding agricultural communities of the upper Kennebecasis River valley, as well as a highway service centre on Route 1, the primary highway between Moncton and Saint John, as well as being the most heavily travelled route in the Maritimes to the United States. Her Majesty Queen Elizabeth II visited the town as part of her Golden Jubilee tour of Canada. On October 12, 2002, she attended the opening of a new wing of Sussex Elementary School, unveiling a commemorative plaque at the event. She then visited Princess Louise Park for an agricultural exhibition. While there, it was announced that the community hall would be renamed Jubilee Hall in her honour. Both appearances drew thousands of visitors from Sussex and beyond.

Amalgamation
On January 1, 2023, Sussex and the village of Sussex Corner were amalgamated. The village of Sussex Corner was incorporated into the Town of Sussex.

Economy
The town is home to Sussex Golden Ginger Ale, a Maritime beverage.

The town entered economic difficulty during the late 1990s after the Cassidy Lake potash mine flooded, resulting in hundreds of lay-offs. Another significant blow came in October 2002 with a realignment of the Trans-Canada Highway (Route 2) between Fredericton and Moncton which no longer passed through Sussex, instead carrying the province's east–west interprovincial traffic 30 km (19 mi) north of the town. At the same time, Route 1 was extended east on the old Trans-Canada alignment to an interchange at River Glade near Three Rivers and Route 10 was extended on the old Trans-Canada from an interchange with the new Route 2 alignment at Young's Cove Road south to Sussex.

In 2022 there were rumours that the regional potash mines might reopen, due to a substantial rise in the market price of the commodity, but Nutrien said no.

Media
Kings County Record has been in existence since 1887. It is a weekly paper which serves Sussex and the surrounding areas. The paper is published on Tuesdays.

The town plays host to a handful of radio stations:
 * CJCW AM 590, local radio station
 * CITA 107.3 FM, a Moncton Christian radio station rebroadcaster
 * CBC Radio One 106.7 FM

Tourism

 * As the heart of Kings County with its 16 covered bridges, Sussex is known as the Covered Bridge Capital of Atlantic Canada. Eight of these wooden structures are within a ten-minute drive of town hall.
 * 26 murals were created during the summers of 2006 and 2007, establishing its reputation as the Mural Capital of Atlantic Canada. Murals exist as an outdoor art gallery in which the town provides a free digital tour guide.
 * An international hot air balloon festival is held every September, and Canada's largest outdoor flea market each August.
 * An agricultural fair takes place every fall at the Agricultural Museum of New Brunswick. The first fair was held in 1841.
 * Sussex is home to the Sussex Drive-In, a single screen drive-in theatre, which has been in operation since 1967. The Sussex Drive-In has a 24m high screen and 300 car capacity. It is opened seasonally in the summer.
 * Southern New Brunswick's only alpine ski hill, Poley Mountain, in the Caledonia Highlands is located southeast of the town.

Atlantic International Balloon Fiesta
The largest hot air balloon festival in Atlantic Canada is held each summer in Sussex. On the weekend after Labour Day, Sussex hosts up to 50,000 - 80,000 visitors who come to watch 40 hot air balloons. Along with the twice-daily flights, the event includes a giant craft fair, a free outdoor concert and an amusement park. the festival has been held since 1985.

Princess Louise Park Show Centre
The Princess Louise Park Show Centre is Eastern Canada's premier Agricultural Exhibition Centre. Located in Sussex, this facility is the host of many shows that include equine, dogs, livestock, sales and exhibitions as well as 4H and youth group activities. Princess Louise Park Show Centre is a board owned and operated exhibition ground that was founded in 1985. It hosts shows attracting people from all over the county, province, and country. It doubles as a storage facility for RVs and trailers during the winter months. The centre has five barns with stalls, three riding rings, and electrical and sewage hookups.

Agricultural Museum of New Brunswick
Sussex has been home to the Agricultural Museum of New Brunswick since 1983. The museum was opened by community members and it is now maintained by a board of volunteers. The land that the museum is situated on once belonged to the 8th Hussars. The museum is closed every year from October to May.

Primary and Secondary Education
The Anglophone South School District manages 4 public schools within the town:
 * Sussex Elementary School
 * Sussex Corner Elementary School
 * Sussex Middle School
 * Sussex Regional High School

Sussex also has one private school for students from Junior Kindergarten through to Grade 12:
 * Sussex Christian School

Higher Education
Sussex is also home to Kingswood University (formerly Bethany Bible College). Kingswood University is an accredited Christian university, it is the only college in Canada which is owned by the Wesleyan Church.

Demographics
In the 2021 Census of Population conducted by Statistics Canada, Sussex had a population of $4,440$ living in $2,065$ of its $2,219$ total private dwellings, a change of 3.7% from its 2016 population of $4,282$. With a land area of 8.9 km2, it had a population density of in 2021.

Most people in Sussex have some Scottish, French, English, or Irish ancestry. There are smaller groups of German and Dutch lineage. The town is almost universally anglophone, being in the heart of English-speaking southern New Brunswick.

Notable people

 * Julian Austin, country music artist
 * Mike Eagles, NHL hockey player, 1983-2000
 * Christian Meier, cyclist
 * William Pugsley, lawyer and 11th Premier of New Brunswick
 * Melanie Robillard, curling World Champion 2010
 * Tim Steeves, comedian, 1964-2022
 * Ellen Watters, cyclist, 1988-2016. In 2017, community members created Ellen's Tour de Sussex, an annual gran fondo in memory of Watters.

title: Sussex, New Brunswick


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Ferny Grove railway station

Ferny Grove railway station is the terminus station of the Ferny Grove line in Queensland, Australia. It serves the Brisbane suburb of Ferny Grove.

The Brisbane Tramway Museum is a short walk south of the station.

History
The station opened in 1918 as part of the extension to Samford.

The line formerly extended to Dayboro, but was cut back to Ferny Grove in 1955, as part of a larger rationalisation by the Queensland Government of its railway lines. The line now terminates here with bus services running to Samford during peak times.

The station was upgraded in 2012 as part of the Keperra-Ferny Grove duplication project. The upgrade included relocating the platform, a new station building and car park.

Camp Mountain rail accident
The Camp Mountain rail accident occurred in 1947 west of Ferny Grove on the Dayboro stretch of the line. It is Queensland's worst railway accident. 16 people were killed when a picnic excursion train failed to negotiate a bend on the line between Camp Mountain and Samford. A small memorial marks the site of the accident.

Services
Ferny Grove is the terminus for all stops services to and from Roma Street, Park Road, Coopers Plains and Beenleigh.

Ferny Grove Central Development
Honeycombes Property Group are developing the station to be Transit Oriented Development (TOD). Ferny Grove Staition is currently being redeveloped to improve parking, retail, add apartments and a public plaza directly outside of the station.

As of November 2023, the commuter car park has been completed with stage 2 construction underway, and expected to be opened in 2025.

Transport links
Transport for Brisbane operate three bus routes from Ferny Grove station:
 * 367: to Upper Kedron
 * 397: to Mitchelton via Everton Hills
 * 398: to Mitchelton via Arana Hills

Brisbane Bus Lines operate one route from Ferny Grove station:
 * 399 to Samford Village

title: Ferny Grove railway station


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American Chocolate

The American Chocolate, later known as the Walter, was an American automobile manufactured from 1902 to 1906 by vending machine company American Chocolate Machinery Co. in Manhattan, New York. He decided to expand his business by assembling automobiles in his factory. The cars were exhibited at the New York automobile show.

The cars were built from imported components, and were 30, 40, and 50 hp models. Production was supervised by Swiss engineer William Walter, who began building cars in 1898. The company relocated to Trenton, New Jersey in 1906, where it began producing cars under the name Walter.

After 1910 Walter was building 4 X 4 trucks, which were used by the military and municipalities. After World War I Walter truck switched from the brand's own four-cylinder motor to the Waukesha engine, and the company moved to a new location on Long Island in 1923. The company would be finally absorbed by Kovatch Mobile Equipment Corp (KME) in the late 1970s.

title: American Chocolate


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Aurora Snow

Aurora Snow is an American retired pornographic actress, director, and columnist.

Early life
Snow was raised by a single mother and was the oldest of three siblings.

Adult films
Snow entered the pornographic entertainment industry when she was 18 years old, planning to work in porn for only a year. One of her first professionally made and distributed movies was More Dirty Debutantes 152, produced by Ed Powers.

Snow was a contract performer and director for Sin City between May 2003 and May 2005. She made her directorial debut for the company with the film Assploitations. Snow hosted Private Calls on Playboy TV.

In July 2010, she reported on the obscenity case against John Stagliano for Adult Video News.

Despite not using condoms during most of her scenes, she came out as one of the few performers in favor of the 2012 Measure B vote in Los Angeles, mandating the use of condoms for all pornographic movie scenes. Snow left the pornographic film industry after ten years in 2011 and moved from Los Angeles to Macon, Missouri.

Mainstream media appearances
Snow appeared in the 2002 feature film The Rules of Attraction as Masked Naked Girl at Party.

Snow appeared in a cameo in the Judd Apatow film Superbad. On January 6, 2010, Snow appeared in an episode of 1000 Ways to Die titled "Death On Arrival" to discuss edible underwear during the "Way to Die #444: Deadliest Munch" segment of the episode.

On March 12, 2013, Snow appeared on The Young Turks to discuss racism in pornography and the reasons why some Caucasian pornographic actresses refuse to do interracial sex scenes.

Snow has been a contributor for The Daily Beast and Glammonitor, writing articles about the porn industry and women's issues.

On March 6, 2014, Snow spoke at Harvard Law School on the topic of "Sex, Drugs, and Rolling Dice: The Regulation of Vice." On March 19, 2015, Snow spoke at the University of South Carolina School of Law on "Vice, Porn and American Culture."

Personal life
In September 2013, Snow wrote an article for The Daily Beast titled "A Porn Star's Letter to Her Unborn Son" in which she announced she was pregnant with her first child and stated how she would explain her choice to work in the porn industry to her son once he is older. The letter quickly went viral on the Internet. She gave birth to her son in December 2013. Her father stated that he was proud upon finding out about her occupation.

Awards

 * 2002 XRCO Award – Best Threeway Sex Scene (Up Your Ass 18) with Mr. Marcus & Lexington Steele
 * 2002 XRCO Award – Best Group Sex Scene (Gangbang Auditions 7)
 * 2002 XRCO Award – Cream Dream
 * 2003 XRCO Award – Best Threeway Sex Scene (Trained Teens) with Gauge & Jules Jordan
 * 2003 AVN Award – Female Performer of the Year
 * 2011 XRCO Hall of Fame inductee
 * 2017 AVN Hall of Fame inductee

title: Aurora Snow


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Mitchelton railway station

Mitchelton railway station is located on the Ferny Grove line in Queensland, Australia. It is one of two railway stations serving the Brisbane suburb of Mitchelton, the other being Oxford Park railway station.

History
The station opened in March 1918 due to public demand. To the north of the station lies a turnback siding.

Services
Mitchelton station is served by all stops Ferny Grove line services from Ferny Grove to Roma Street, Park Road, Coopers Plains and Beenleigh.

Transport links
Transport for Brisbane operate four bus routes from Mitchelton station:
 * 369: to Toombul via Stafford
 * 396: to Arana Hills
 * 397: to Ferny Grove station via Everton Hills
 * 398: to Ferny Grove station via Arana Hills

In addition to the above, Transport for Brisbane buses also call at the nearby Brookside bus station.

title: Mitchelton railway station


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Category:Amazon (company)

Categorization for articles about Amazon (company) and its divisions and services.

title: Category:Amazon (company)


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Sledging

Sledging can mean:

 * Sledding or sleighing
 * Sledging (cricket), verbal abuse or comments meant to intimidate or put off an opposing batsman in cricket.
 * Sledging (bodyboarding)

title: Sledging