instruction stringlengths 12 30k |
|---|
Walking on a lattice. The number of various paths from (0,0) to (m,n) using north and east steps is binomial coefficient
C(m+n,m)
if he needs to go back (0,0) using south and west steps, and doesn't pass by the passed points. Then what is the number of various loops walking from (0,0) to (m,n) then returning t... |
the number of loops on lattice? |
A continuous function does not always map open sets to open sets, but a continuous function will map compact sets to compact sets. One could make list of such preservations of topological
properties by a continuous function $f$:
$$ f( \mathrm{open} ) \neq \mathrm{open} \;,$$
$$ f( \mathrm{closed} ) \neq \mathrm{cl... |
I was given this question (there were prior questions defining the equivelence relation ~)
let E be the set of equivelence classes of ~
define f from E to Reals
f:={([(x,y)],r)|x=yr,\: x,y,r in Reals}
I'm trying to understand what it means; is the following correct?
f:[(x,y)] mapsto r
... |
Am I understanding this question right? (function defined with set notation) |
Let $f : [0, 1] \to \mathbb{R}$ be a 'well behaved' function and define $c_n$ by the Riemann sums
$$
c_n := \frac{1}{n} \sum_{k = 1}^n f(\frac{k - \theta}{n})
$$
for some given $\theta \in [0, 1]$. Then obviously $c_n$ will converge to the integral of $f$. Under which conditions on $f$ will this convergence be mono... |
In the Portuguese book *Análise Matemática* (Mathematical Analysis) by C. Sarrico, it is proved there exist
$\displaystyle\lim_{n\rightarrow +\infty}f_{n}(x)=\lim_{n\rightarrow +\infty }e^{\log f_{n}(x)}$ (1),
where
$f_{n}(x)=\dfrac{n!n^{x}}{x(x+1)(x+2)\cdots (x+n)}$. (2)
And it is stated there that
$\d... |
Walking on a lattice. The number of various paths from (0,0) to (m,n) using north and east steps is binomial coefficient
C(m+n,m)
if he needs to go back (0,0) using south and west steps, and doesn't pass by the passed points. Then what is the number of various loops walking from (0,0) to (m,n) then returning t... |
Yes; the function is defined to take equivalence classes as inputs, and return reals as values. If you put in the class of $(x,y)$, which is $[(x,y)] = \\{(a,b)\in \mathbb{R}\times\mathbb{R}^{\star}\,|\, (x,y)\sim (a,b)\\}$, it returns a real number $r$. Which real number? The real number $r$ such that $x=yr$. (I'm usi... |
When does the product of two polynomials = $x^{k}$ ? |
I am not sure if it would be considered a classic or not, but I absolutely love [Differential Equations with Boundary-Value Problems][1] by Dennis G. Zill and Michael R. Cullen. I feel as though the definitions are written and explained in such a way, that if one was not a mathematician, they could still grasp a basic... |
Two ways of defining the gamma function $\Gamma (x)$. How to show they are equivalent? |
A couple of years ago I found the following continued fraction for $\frac{1}{e-2}$:
$\frac{1}{e-2} = 1+\frac{1}{2 + \frac{2}{3 + \frac{3}{4 + \frac{4}{5 + \frac{5}{6 + \frac{6}{7 + \frac{7}{\cdots}}}}}}}$
from fooling around with the well-known continued fraction for $\phi$. Can anyone here help me figure out why... |
A couple of years ago I found the following continued fraction for $\frac{1}{e-2}$:
$\frac{1}{e-2} = 1+\frac{1}{2 + \frac{2}{3 + \frac{3}{4 + \frac{4}{5 + \frac{5}{6 + \frac{6}{7 + \frac{7}{\cdots}}}}}}}$
from fooling around with the well-known continued fraction for $\phi$. Can anyone here help me figure out why... |
I asked this question a while back on MO : http://mathoverflow.net/questions/32689/how-should-a-homotopy-theorist-think-about-sheaf-cohomology
One thing that really helped in learning the Serre SS was doing particular computations (like $H^*(CP^{\infty})$)
I am curious, as a sort of followup if anyone can suggest... |
I have a problem I tried to solve, but couldn't because I don't know the method to solve it and I've never come across such problem.
Here's the problem.
(PQ)²=RSP
Where P, Q, R, S are distinct single digit natural numbers, then R=?
We need to find out the value of R, and given option are
a) 1
b) 2
c) 3... |
(PQ)²=RSP, where P,Q,R,S are distinct single digit natural numbers, then R=? |
Let $X = \mathbb{R}^n$ and let $Z$ be a closed subspace of $X$. The inclusion map $i: Z \hookrightarrow X$ induces a map $C_0(X) \to C_0(Z)$ (via pullback). My question is simple: what is the kernel? The answer should either be $C_0(X - Z)$ or $C(X - Z)$ and I think I know which one is correct. But I need a sanity ... |
I have a problem I tried to solve, but couldn't because I don't know the method to solve it and I've never come across such problem.
Here's the problem.
(PQ)²=RSP
Where P, Q, R, S are distinct single digit natural numbers, then R=?
We need to find out the value of R, and given option are
a) 1
b) 2
... |
Here's one more question I need help with.
Samir has some amount of rupees with him, the difference between 45.45% and 36.36% of which is Rs. 789435689. Find the amount.
a) 8683792579
b) 8673792579
c) 7573781479
d) 88667292579
e) None of these
Now, Samir is the name of Indian male, rupess is India... |
**HINT** $\;\quad\rm mod \; 97: \:\quad 100 \;\equiv\; 3 $
Hence $\; 3007 \;\equiv\; 30 \cdot 100 + 7$
$\quad\quad\quad\quad\quad\quad\;\; \;\equiv\; 30 \:\;\cdot\;\: 3 \;\: + \; 7 \;$
$\quad\quad\quad\quad\quad\quad\;\; \;\equiv\; 0$
I.e. **cast out 97's** in analogy to **cast out nines**. See also [he... |
In the Portuguese book *Análise Matemática* (Mathematical Analysis) by C. Sarrico, it is proved there exist
$\displaystyle\lim_{n\rightarrow +\infty}f_{n}(x)=\lim_{n\rightarrow +\infty }e^{\log f_{n}(x)}$ (1),
where
$f_{n}(x)=\dfrac{n!n^{x}}{x(x+1)(x+2)\cdots (x+n)}$. (2)
And it is stated there that
$\d... |
Why does "convex function" mean "concave *up*"? |
(20:30)
First we notice that 43, 47 and 97 are all prime numbers and 93=3x31. Clearly 3 is not a factor of 3007, let's see about the rest using the good ol' Euclidean algorithm.
- 43 is clearly not (as 43x7=280+21=301 therefore 300=43x6+42, and 427 is clearly not divisible by 43)
- in 47 case, 300=6x(47+3) there... |
In the Portuguese book *Análise Matemática* (Mathematical Analysis) by C. Sarrico, it is proved there exist
$\displaystyle\lim_{n\rightarrow +\infty}f_{n}(x)=\lim_{n\rightarrow +\infty }e^{\log f_{n}(x)}$ (1),
where
$f_{n}(x)=\dfrac{n!n^{x}}{x(x+1)(x+2)\cdots (x+n)}$. (2)
And it is stated there that
$\d... |
When do Riemann sums converge monotonically? |
Some years ago I was interested in the following Markov chain
whose state space is the positive integers. The chain begins at state "1",
and from state "n" the chain next jumps to a state uniformly
selected from {n+1,n+2,...,2n}.
As time goes on, this chain goes to infinity, with occasional
large jumps. In any ... |
Is there a caracterization of the entire functions $f$ that satisfy $|f(z)|=1$ for all $|z|=1$?
Clearly, $f(z)=z^n$ works for all $n$. Also, it's not difficult to show that if $f$ is such an entire function, then $f$ must vanish somewhere inside the unit disk. What else can be said about those functions?
Thank yo... |
I read this fascinating article:<br>
http://www.newscientist.com/article/mg20627550.200-enter-the-matrix-the-deep-law-that-shapes-our-reality.html
Unfortunately all the other papers I find googling are just not tangible to me :-(
Could anyone please point me to some material that bridges the gap from this popula... |
Gives a nice concise overview:<br>
http://www.wolframalpha.com/input/?i=Sqrt(i) |
Try this:<br>
http://www.wolframalpha.com/input/?i=sin
Click on all the "more"s - works of course for "cos" too! |
This is a follow-up to the MathStackexchange question 3305 and MathOverflow question 23229.
The Bohr-Mollerup theorem states that the Gamma function is the unique function that satisfies:
$(1)\ f(x+1) = xf(x), \ (2)\ f(1) = 1, \ (3)\ \ln \circ f \ $ is convex.
Assume a function $f\colon \mathbb{R^{+}} \to \ma... |
Is it possible to represent every huge number in abreviated form? |
Length, in Euclidean geometry, is a relation, not a number. We say that two segments are of the same length if they are congruent to each other, and congruence is one of the undefined notions of Euclidean geometry. The Pythagorean theorem comes about as soon as you decide to map lengths to positive real numbers, and it... |
Consider the following expression.
16313107343153908912074032799466965289077771751767944648966669091376847859711382649033004075188224
This is a 98 decimal digit number.
This can be represented as 424^37 which has just 5 digits.
or consider this number:
1690735149233357049107817709433863585132662626950821... |
Is there a characterization of the nonconstant entire functions $f$ that satisfy $|f(z)|=1$ for all $|z|=1$?
Clearly, $f(z)=z^n$ works for all $n$. Also, it's not difficult to show that if $f$ is such an entire function, then $f$ must vanish somewhere inside the unit disk. What else can be said about those functions... |
The following is an attempt at an interpolation between (among?) the answers of Grigory M., Pierre-Yves Gaillard and Bill Dubuque (the last of which I confess that I do not completely understand, but this is one possible interpretation of it).
Let $J$ be an open interval in $\mathbb{R}$, and let $x_0 \in J$. Follow... |
I had asked this question sometime ago here: http://math.stackexchange.com/questions/2536/finding-f-such-that-int-f-sum-f
Now i have a question which i think is more or less related to it.
Let $f$ be an increasing function with $f(1)=0$. Consider the sequence $s_{n}= ( \sum\limits_{k=1}^{n} f(k) - \int\limits_{... |
I had asked this question sometime ago here: http://math.stackexchange.com/questions/2536/finding-f-such-that-int-f-sum-f
Now i have a question which i think is more or less related to it.
Let $f$ be an increasing function (continuous of course!) with $f(1)=0$. Consider the sequence $s_{n}= ( \sum\limits_{k=1}^... |
Some years ago I was interested in the following Markov chain
whose state space is the positive integers. The chain begins at state "1",
and from state "n" the chain next jumps to a state uniformly
selected from {n+1,n+2,...,2n}.
As time goes on, this chain goes to infinity, with occasional
large jumps. In any ... |
Does this minimal surface also minimize maximum distance? |
I’m interested in some sequences which have no square factor.
\[{{s}_{i}}=\left\{ \begin{array}{*{35}{l}}
4 & \text{if }i=0; \\
s_{i-1}^{2}-2 & \text{otherwise}\text{.} \\
\end{array} \right.\]
This is Lucas–Lehmer primality test sequence. A003010 in OEIS
when ${{s}_{i}}$ has a square factor? I had ch... |
I’m interested in some sequences which have no square factor.
\[{{s}_{i}}=\left\{ \begin{array}{*{35}{l}}
4 & \text{if }i=0; \\
s_{i-1}^{2}-2 & \text{otherwise}\text{.} \\
\end{array} \right.\]
This is Lucas–Lehmer primality test sequence. A003010 in OEIS
when ${s_i}$ has a square factor? I had checke... |
**Definition** A subgroup of a group is termed strongly potentially characteristic if there is an embedding of the bigger group in some group such that, in that embedding both the group and the subgroup become characteristic.
One can refer this link: http://groupprops.wiki-site.com/index.php/Strongly_potentially... |
If I create a 10 digit password with the following requirements:
- At least one uppercase letter A-Z - 26
- At least one lowercase letter a-z - 26
- At least one digits 0-9 - 10
- At least one common symbol (#,$,%,etc) - 32
By inclusion-exclusion, I can calculate I have ~ 3.2333E+19 possible combinations
... |
How can I calculate the number of potential combinations in a password? |
In particular, $\lfloor\frac{10^{20000}}{3+10^{100}}\rfloor$. (From a putnam exam many years ago.)
Is there a general technique for attacking such problems? |
In particular, problems like $\lfloor\frac{10^{20000}}{3+10^{100}}\rfloor$. (From a putnam exam many years ago.)
Is there a general technique for attacking such problems? |
I was recently reminded of this by my answer to the recent <a href="http://math.stackexchange.com/questions/3320/does-the-number-pi-have-any-significance-besides-being-the-ratio-of-a-circles-di">question about $\pi$</a>. By using the <a href="http://terrytao.wordpress.com/2010/01/02/254a-notes-0a-stirlings-formula/">t... |
Stirling's formula via the central limit theorem? |
In <a href="http://terrytao.wordpress.com/2010/01/02/254a-notes-0a-stirlings-formula/">these notes by Terence Tao</a> is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity
$$n! = \int_{0}^{\infty} t^n e^{-t} dt$$
coming from the Gamma function. I have a... |
Intuition for the definition of the Gamma function? |
I’m interested in some sequences which have no square factor.
$$
s_i =\begin{cases}
4 & i=0; \\\\
s_{i-1}^2 - 2 & \text{otherwise}
\end{cases}$$
This is Lucas–Lehmer primality test sequence. [A003010 in OEIS](http://www.research.att.com/~njas/sequences/A003010).
when ${s_i}$ has a square factor? I had checked... |
Motivated by [this][1] question I would like to know whether there is an exact definition of modern mathematics. In which point in time (century, year) does one think, when speaking about modern mathematics. Does it refers to the Abstract Algebra?
[1]: http://math.stackexchange.com/questions/3250/ |
What is Modern Mathematics? Is this an exact concept with a clear meaning? |
I'm looking for clear mathematical rules on rounding a number to n decimal places.
Everything seems perfectly clear for positive numbers. Here is for example what I found on [math.about.com](http://math.about.com/od/arithmetic/a/Rounding.htm) :
>**Rule One** Determine what your rounding digit is and look to the r... |
I'm looking for clear mathematical rules on rounding a number to n decimal places.
Everything seems perfectly clear for positive numbers. Here is for example what I found on [math.about.com](http://math.about.com/od/arithmetic/a/Rounding.htm) :
>**Rule One** Determine what your rounding digit is and look to the r... |
Motivated by [this][1] question I would like to know whether there is an exact definition of modern mathematics. In which point in time (century, decade) does one think, when speaking about modern mathematics. Does it refers to the Abstract Algebra?
[1]: http://math.stackexchange.com/questions/3250/ |
Motivated by [this][1] question I would like to know whether there is an exact definition of modern mathematics. In which point in time (century, decade) does one think, when speaking about modern mathematics. Does it refers to the Abstract Algebra?
**Edit:** I got -1. If this is not a correct question, please stat... |
Motivated by [this][1] question I would like to know whether there is an exact definition of modern mathematics. In which point in time (century, decade) does one think, when speaking about modern mathematics. Does it refers to the Abstract Algebra?
**Edit:** I got -1. If this is not a correct question, please stat... |
Factorial is defined as
$n! = n(n-1)(n-2)...1$
But why mathematicians named this thing as **FACTORIAL**?
Has it got something to do with factors?
|
Why the name 'FACTORIAL'? |
Which one is bigger $2^{n!}$ or $(2^{n})!$? |
Which one is bigger $2^{n!}$ or $(2^{n})!$?
where n∈N. |
[Indefinite](http://www.wolframalpha.com/input/?i=Plot[sign[2^n!+-+%282^n%29!]%2C+{n%2C+0%2C+100}]):
for $n=0$: $2^{0!}=2^1=2,$ $(2^0)!=1!=1$, so $2^{0!}>(2^0)!$
for $n=2$: $2^{2!}=2^2=4,$ $(2^2)!=4!=12$, so $2^{2!}<(2^2)!$. |
Actually, $(2^n)!$ is larger if 1 < n < [4.97399743597](http://www.wolframalpha.com/input/?i=log_2%28%282%5en%29!%29%3Dn!). See [this graph](http://www.wolframalpha.com/input/?i=plot+log%28log%282%5e%28n!%29%29%29+%2B+i+log%28log%28%282%5en%29!%29%29+for+n+%3D+0.1+to+10) for detail. |
Let $f: D \to \mathbb{R}\ $ be a function on non-singular, convex domain $D \subseteq \mathbb{R}^d$ and let us assume the second-order derivatives of $f$ exist. It is well known that $f$ is convex if and only if its Hessian $\nabla^2 f(x)$ is positive semi-definite for all $x \in D$. It is also known that if $\nabla^2 ... |
A continuous function does not always map open sets to open sets, but a continuous function will map compact sets to compact sets. One could make list of such preservations of topological
properties by a continuous function $f$:
$$ f( \mathrm{open} ) \neq \mathrm{open} \;,$$
$$ f( \mathrm{closed} ) \neq \mathrm{cl... |
My textbook doesn't give any example of this kind of series. Could you provide some?
thanks. |
Example of a trigonometric series that is not fourier series? |
Take a look at [this wikipedia stub](http://en.wikipedia.org/w/index.php?title=Trigonometric_series&oldid=370829487). A trigonometric series, which is not a Fourier series, must therefore be one that is defined by
$$ f(x) = A_{0}+\displaystyle\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx}) $$
for which the r... |
Take a look at [this wikipedia stub](http://en.wikipedia.org/w/index.php?title=Trigonometric_series&oldid=370829487). A trigonometric series, which is not a Fourier series, must therefore be one that is defined by
$$ f(x) = A_{0}+\displaystyle\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx}) $$
for which the r... |
My textbook doesn't give any example of this kind of series. Could you provide some?
Trigonometric series is defined in wikipedia as :
$A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$
When
$A_{n}=\frac{1}{\pi} \int^{2 \pi}_0 f(x) \cos{nx} dx\qquad (n=0,1,2,3 \dots)$
$B_{n}=\frac{1}{\pi} \int... |
Motivated by [this][1] question I would like to know whether there is an exact definition of modern mathematics. In which point in time (century, decade) does one think, when speaking about modern mathematics. Does it refer to the Abstract Algebra?
**Edit:** I got -1. If this is not a correct question, please state... |
My textbook doesn't give any example of this kind of series. Could you provide some?
Trigonometric series is defined in wikipedia as :
$A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$
When
$A_{n}=\frac{1}{\pi} \int^{2 \pi}_0 f(x) \cos{nx} dx\qquad (n=0,1,2,3 \dots)$
$B_{n}=\frac{1}{\pi} \int... |
Motivated by [this][1] question I would like to know whether there is an exact definition of modern mathematics. In which point in time (century, decade) does one think, when speaking about modern mathematics. Does it refer to the Abstract Algebra?
**Edit:** I got -1. If this is not a correct question, please state... |
I've been thinking about maps between sets. Injections, surjections and the rest. Often when thinking about some kind of map, it is interesting to say "what about the maps from a set to _itself_?" Call these maps endomaps. Permutations of elements of the set are a special case of endomaps: they are bijective endomaps. ... |
Do endomaps of sets have interesting properties? |
Motivated by [this][1] question I would like to know whether there is an exact definition of modern mathematics. In which point in time (century, decade) does one think, when speaking about modern mathematics. Does it refer to the Abstract Algebra?
**Edit:** I got -1. If this is not a correct question, please state... |
A standard example is
$$
f(t)= \sum_{n>1} \sin(nt)/\log(n)$$
The conjugate of $f$ is a Fourier series but $f\not\in L^1(\mathbb{T})$.
For further explanation see [Katznelson's book page 85.][1]
[1]: http://books.google.com/books?id=gkpUE_m5vvsC&printsec=frontcover&dq=katznelson&hl=sv&ei=R8l3TN_tA5DLOIGhgBA&... |
$ 2^{n!}=(2^n)^{(n-1)!} $, so we have to compare $(2^n)^{(n-1)!} $ and $ (2^n)! $. For $n\geq 6$, $(n-1)!>2^n$. I may be wrong, but doesn't the result follow immediately from this? It's still true for $n=5$ though, but not for $n=4$. |
A standard example is
$$
f(t)= \sum_{n>1} \sin(nt)/\log(n)$$
The conjugate of $f$ is a Fourier series but $f\not\in L^1(\mathbb{T})$ and hence is no Fourier series.
For further explanation see [Katznelson's book page 85.][1]
[1]: http://books.google.com/books?id=gkpUE_m5vvsC&printsec=frontcover&dq=katznelso... |
The generalization of cycle decomposition to endomaps is quite interesting; rather than just cycles, endomaps break up into cycles in which each vertex is the root of a tree. Counting endomaps (of which there are $n^n$) is therefore relevant to counting trees, and this is the basis of a beautiful proof due to Joyal (w... |
Motivated by [this][1] question I would like to know whether there is an exact definition of modern mathematics. In which point in time (century, decade) does one think, when speaking about modern mathematics. Does it refer to the Abstract Algebra?
**Edit:** I got -1. If this is not a correct question, please state... |
In a very interesting [blog discussion][1] at the $n$-category cafe, an anonymous poster made the following remark: "... using the dictionary between number fields and function fields, Weil suggested that G-with-adelic-entries is analogous to the group of gauge transformations of a principal G-bundle over a Riemann sur... |
How are gauge transformations of a $G$-bundle related to the adelic points of $G$? |
I don't know the answer, but in my opinion a lot of people are misinterpreting the question, so perhaps it is worth an answer to try to set this straight.
Here is an analogy: trigonometric series is to Fourier series as power series is to Taylor series. In other words, a trigonometric series is just any series of t... |
Generalizing from decimal to arbitrary radix, a quick computer search seems to indicate that there are only a handful of radices with such a unique solution. This leads to the following conjecture - which I haven't yet had a chance to try to prove. Is anyone game for a proof, or independent empirical verification?
... |
A standard example is
$$
f(t)= \sum_{n>1} \sin(nt)/\log(n)$$
The conjugate of $f$ is a Fourier series but $f\not\in L^1(\mathbb{T})$ and hence is no Fourier series.
For further explanation see [Katznelson's book page 85.][1]
(Edit: If $f$ is not in $L^1(\mathbb{T})$ it is hard to define Fourier coefficients.)
... |
In [an answer][1] to [another question I asked][2], Isaac suggested a book that is the standard "reform calculus" book. In a comment, I asked what the phrase "reform calculus" means, and Isaac provided a link to [this summary page][3]. It seems to be a new or revised teaching methodology. However, the page linked to do... |
What is "reform calculus"? |
I'm brushing up on my calculus, and geometry and trigonometry keep coming up. However, these are probably my weakest areas mathematically. Are there any good sites or free PDFs/ebooks that I can use to quickly (in a weekend, preferably) work through geometry and trigonometry and refresh myself in topics such as areas a... |
What are some good resources for brushing up on geometry and trigonometry? |
I know how to prove $\zeta (2)=\pi ^{2}/6$ by using the *trigonometric* Fourier series expansion of $x^{2}/4$. How can one prove the same result using the *complex* Fourier series of $f(x)=x$ for $0\leq x\leq 1$? Any suggestion? |
Here's an excerpt from [Reviewing Reformed Calculus][0], by Lisa Murphy, 2006.
>The very beginnings of mathematics reform started in the 1960's, but the big push for Calculus
>reform started in earnest in 1989 with the publication of the National Council of Teachers of Mathematics'
>(NCTM) Principles and Standards... |
100 boxes of fruits - pick 51 and get at least half of each type? |
Use the definition:
Say $f$ is defined on $[-\pi, \pi]$.
If $f(z) = \sum_{-\infty}^{\infty} {c_{n} e^{inz}}$
then
$c_{n} = \int_{-\pi}^{\pi}{f(z)e^{-inz}} dz$
If you put $f(z) = z$, can you work out what $c_{n}$ turns out to be?
To integrate, you can try integration by parts. |
What's the simplest formula to calculate the optimal sample size from a population size? I've been reading a little about it and they ask me stuff like confidence level or confidence interval which I have no idea about. I need to test some HTML elements in web pages to get a better idea about general properties of HTML... |
Optimal sample size? |
Some years ago I was interested in the following Markov chain
whose state space is the positive integers. The chain begins at state "1",
and from state "n" the chain next jumps to a state uniformly
selected from {n+1,n+2,...,2n}.
As time goes on, this chain goes to infinity, with occasional
large jumps. In any ... |
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