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Here's what I recall of the question from CNML Grade 11, 2010/2011 Contest #3, Question 7:
> **There are 2010 points on a circle,
> evenly spaced.** Ford Prefect will*
> randomly choose three points on the
> circle. He will* connect these points
> to form a shape. What is the
> probability that the resulting sh... |
Hint: You can choose $3$ points in ${2010}\choose {3}$ ways. To form a right angled triangle, we need to choose a pair of diametrically opposite points (how many ways to do this?) and then any third point will form a right angled triangle with the chosen pair. |
I am using math as a tool and for entertainment, I am not a pro, I do not understand most of the posts here but I am glad such site exists.
Anyway, a friend asked me a question the other day, and I was able to redefine it in terms of a directed graph where all nodes in and out degree equal to 1. I know that If if ea... |
You should take a look at the paper: [Some New Formulas for ][1] by Gert Almkvist, Christian Krattenthaler, and Joakim Petersson, Experiment. Math. Volume 12, Number 4 (2003), 441-456.
[1]: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.em/1087568020 |
What is the $d\sigma/\sigma$ rule? |
1. Can a CFG over large alphabet describe a language, where each terminal appears even number of times?
2. If yes, would the Chomsky Normal Form be polynomial in |Σ| ?
Find a context-free grammar (CFG) for the language L of all
words such that each terminal in a word occurs even number of times over a possibly l... |
Can a polynomial size CFG over large alphabet describe a language, where each terminal appears even number of times? |
I was watching this [TED talk][1], which suggested that on average your friends tend to individually have more friends than you do. To define this more formally, we are comparing the average number of friends with:
average over each person p of:
friend popularity, defined as:
average over each ... |
Do your friends on average have more friends than you do? |
I was watching this [TED talk][1], which suggested that on average your friends tend to individually have more friends than you do. To define this more formally, we are comparing the average number of friends with:
average over each person p of:
friend popularity, defined as:
average over each ... |
I was watching this [TED talk][1], which suggested that on average your friends tend to individually have more friends than you do. To define this more formally, we are comparing the average number of friends with:
average over each person p of:
friend popularity, defined as:
average over each ... |
It's possible, that count of unique algebras (up to isomorphism) with one unary operation on set with $n$ elements is $2^n-1$?
For $n=1,2,3,4$ is this hypothesis true (I still have not verified it on the computer). Is there any counterexample, or idea for proof?
Thanks. |
I was trying to expand $\displaystyle \left(1+x\right)^{y}$ as a power series in terms of $y$ using the Generalized Binomial Theorem,
$\displaystyle \left(1+x\right)^{y}=\sum_{k=0}^{\infty}\binom{y}{k}x^{k}$
Assume $x,y \in \mathbb{R}$ and $\left | x \right | < 1$. With this in mind I would get something like
... |
It is well known that any conformal mapping of the unit disk onto itself extends to the unit cirle.
However, is there an homeomorphism of the unit disk onto itself which does not extend to a continuous function on the closed unit disk?
If yes, can you give an explicit one?
Thank you,
Malik |
Let $\Omega$ be a domain of the complex plane. The Hardy space $H^p(\Omega)$ is defined, for $1 \leq p<\infty$, as the class of functions $f$ that are holomorphic on $\Omega$ such that $|f|^p$ has a harmonic majorant on $\Omega$, i.e. there is a function $u$ harmonic on $\Omega$ such that
$$|f(z)|^p \leq u(z) $$
for ... |
Can this sum be simplified: $\sum_{k=0}^{n} \binom{n}{k} 2^{\sqrt{k}}$
Or at least is there a simple fairly tight upperbound? |
My motivation for asking this question is that a classmate of mine asked me some kind of question that made me think of this one. I can't recall his exact question because he is kind of messy (both when talking about math and when thinking about math).
I'm kind of stuck though. I feel like the set $A^{\mathbb{N}} =... |
What's the cardinality of all sequences with coefficients in an infinite set? |
1. Can a CFG over large alphabet describe a language, where each terminal appears even number of times?
2. If yes, would the Chomsky Normal Form be polynomial in |Σ| ?
Find a context-free grammar (CFG) for the language L of all
words such that each terminal in a word occurs even number of times over a possibly l... |
My motivation for asking this question is that a classmate of mine asked me some kind of question that made me think of this one. I can't recall his exact question because he is kind of messy (both when talking about math and when thinking about math).
I'm kind of stuck though. I feel like the set $A^{\mathbb{N}} =... |
We can let $G$ act on the set of left cosets of $H$, by $g(xH) = gxH$; this is not a group, but a group action. We can also, separtely, put group structures on the set of cosets of $H$ in $G$, since it will be a set, and every nonempty set can be given a group structure (assuming the Axiom of Choice; in fact, equivalen... |
Can this sum be simplified: $\sum_{k=0}^{n} \binom{n}{k} 2^{\sqrt{k}}$
Or at least is there a simple fairly tight upperbound?
**EDIT**
So I think this sum is more easily bounded than I previously thought:
Clearly, $\sum_{k=0}^{n} \binom{n}{k} 2^{\sqrt{k}} \leq 2^{\sqrt{n}} \sum_{k=0}^{n} \binom{n}{k} = 2^{n+\sq... |
It's pretty easy to construct an (R-S) bi-module M and a left S-module N such that neither M_S nor N is a projective S-module, but the tensor product is a non-zero projective R-module.
However, taking R=S=Z defies my zoo of examples. Here projective=free is closed under direct sums and summands, and so it seems lik... |
Can the tensor product of two non-free abelian groups be non-zero free? |
this question may be shameful, but nevertheless I can't help myself.
Let $U \subset \mathbb R^n$ be arbitrary, in particular not the whole of the space itself. I wonder about the dual of the space $W^{1,p}(U)$, for $p < \infty$.
For $U = \mathbb R^n$, we have $(W^{1,p})' = (W^{1,p'})$ with $p' = \frac{p}{p-1}$. H... |
Prove that if k is an inaccessible cardinal, then V(sub k) satisfies all the axioms of ZFC.
How is this done for the axiom of choice and for regularity? |
Prove that if $\kappa$ is an inaccessible cardinal, then $V_{\kappa}$ satisfies all the axioms of ZFC.
How is this done for the axiom of choice and for regularity? |
Why $(x-c)^p \equiv (x-c)$ mod $x^p-x$? |
Given four coordinates that define the corners of an irregular quadrilateral and a point defined by its coordinates, what is the simplest way to determine if the point is within or outside of the quadrilateral? |
Why is $(x-c)^p \equiv x-c \pmod{x^p-x}$? |
While I tagged this question homework, it's not really homework. I'm really studying for a test and don't understand how to do something. I have the answer to the question, so the important part is for my to understand how to get there.
X has distribution `Gamma(2,4)`
`X = Y-3` Find: `P(Y>11|Y>7)` Which I know is e... |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ strictly monotonically increasing.
(i) Is $f$ not continuous at $p \in \mathbb{R}$, there exists a non-empty, open interval $(a_p, b_p) \subset \mathbb{R}$ such that $f(x)\leq a_p$ for all $x < p$ and $f(x) \geq b_p$ for all $x > p$.
(ii) The set of discontinuity points ... |
Is there a closed form expression for the number of terms in a [monomial symmetric polynomial][1] on a given number of variables for a particular partition of exponents, in terms of which/how many exponents are repeated?
I feel the answer should be straightforward, and it's probably just a contrived statement of a m... |
Is there a closed form expression for the number of terms in a [monomial symmetric polynomial][1] in a given number of variables for a particular partition of exponents, in terms of which/how many exponents are distinct?
I feel the answer should be straightforward, and it's probably just a contrived statement of a m... |
Is there a closed form expression for the number of terms in a [monomial symmetric polynomial][1] in a given number of variables for a particular partition of exponents, in terms of which/how many exponents are distinct?
I feel the answer should be straightforward, and it's probably just a contrived statement of a m... |
In Hatcher's "Algebraic Topology" in the Poincare Duality section he introduces the subject by doing orientable surfaces. He shows that there is a dual cell structure to each cell structure and it's easy to see that the first structure gives the cellular chain complex, while the other gives the cellular cochain comple... |
Let R be a ring and every left R-module RM has a projective cover, then M is projective.
Is it possible help me: why J(R)=0? |
Update: See the last section for a proof that $a_n < n$ for all $n \geq 18$.
<HR>
Here are some bounds on $E(n,m)$ that turn out to be useful.
Using $E_{n,m}$ for $E(n,m)$ in Ross Millikan's notation, let $R_{n,m} = \lceil E_{n,m} \rceil - E_{n,m}$.
Simplifying the expression for $E_{n,m}$ in Ross's first a... |
I have already taken up to Multivariable Calculus, Linear Algebra and Diff Eq.
I want to learn Calculus on Manifolds by myself, could you recommend a good introductory book on this subject?
Should I look at other things first, like topology, to get a better background? |
Update: See the last section for a possible proof (**) that $a_n < n$ for all $n \geq 18$.
<HR>
Here are some bounds on $E(n,m)$ that turn out to be useful.
Using $E_{n,m}$ for $E(n,m)$ in Ross Millikan's notation, let $R_{n,m} = \lceil E_{n,m} \rceil - E_{n,m}$.
Simplifying the expression for $E_{n,m}$ in ... |
Let $R$ be a ring and assume that for every left $R$-module ${}_RM$, if $M$ has a projective cover, then $M$ is projective.
Can someone help me prove that in that case, $J(R)=0$? |
Any ideas about how to prove
$$!n = n!- \sum_{i=1}^{n} {{n} \choose {i}} \quad!(n-i)$$
from <a href="http://en.wikipedia.org/wiki/Subfactorial">
Wikipedia's article on derangements</a>?
Here, $!n$ is the number of derangements of a set with $n$ elements.
|
Any hints about how to prove
$$!n = n!- \sum_{i=1}^{n} {{n} \choose {i}} \quad!(n-i)$$
from <a href="http://en.wikipedia.org/wiki/Subfactorial">
Wikipedia's article on derangements</a>?
Here, $!n$ is the number of derangements of a set with $n$ elements.
I am not looking for proofs, just nudges in the right... |
I have a question about Hatcher's proof that the fundamental group of a circle is Z. Specifically, halfway through, (http://www.math.cornell.edu/~hatcher/AT/ATch1.pdf, page 30), he proves an important a lemma stated rather generally:
Given a map $Y\times I\rightarrow S^1$ (Y is any space, and I is an interval), ther... |
I have a question about Hatcher's proof that the fundamental group of a circle is Z. Specifically, halfway through, ( http://www.math.cornell.edu/~hatcher/AT/ATch1.pdf , page 30), he proves an important a lemma stated rather generally:
Given a map $Y\times I\rightarrow S^1$ (Y is any space, and I is an interval), th... |
Sum of 1/n*n for n = 1 ,2 ,3, ... ? |
we are a group of three, and we've got this question in an assignment (the question is originally in french, so bare with me) :
> Let X<sub>1</sub>...X<sub>5</sub> a random sample drawn from a population of average μ and variance σ<sup>2</sup>. We define the following estimators :
>
> ![alt text][1]
>
> 1. Sh... |
This is from one of the exercises in "[Concrete mathematics][1]", and is something I'm doing privately, not homework.
This is a variant on the classic tower of hanoi, where all moves must be made clockwise only. The recursive solution is to consider Qn to be the number of moves to move a tower size n one step across... |
This is from one of the exercises in "[Concrete mathematics][1]", and is something I'm doing privately, not homework.
This is a variant on the classic tower of hanoi, where all moves must be made clockwise only. The recursive solution is to consider Qn to be the number of moves to move a tower size n one step across... |
This is from one of the exercises in "[Concrete Mathematics][1]", and is something I'm doing privately, not homework.
This is a variant on the classic tower of hanoi, where all moves must be made clockwise only. The recursive solution is to consider Qn to be the number of moves to move a tower size n one step across... |
I have the following overdetermined linear equation system:
$$Ax=b$$
where $A$ is a matrix of $n \times k$, $x$ and $b$ are vectors of $k \times 1$, where $n>k$.
We all know this is an overdetermined linear equation system.
The question is how to check whether the solve for $x$, and check that the vector is... |
we are a group of three, and we've got this question in an assignment (the question is originally in French, so bear with me) :
> Let $X_1 \dots X_4$ be a random sample drawn from a population of average $\mu$ and variance $\sigma^2$. We define the following estimators:
>
> $$\hat{\mu}_1=\frac{X_1+2X_2-2X_3+5X_4}... |
I have the following overdetermined linear equation system:
$$Ax=b$$
where $A$ is a matrix of $n \times k$, $x$ is of $k \times 1$,$b$ is of $n \times 1$, where $n>k$.
We all know this is an overdetermined linear equation system.
The question is how to check whether the solve for $x$, and check that the vec... |
I am trying to work through Boothby's An Introduction to Differentiable Manifolds on my own and, embarassingly, have got stuck at the very first chapter. At the end of section 4, chapter 1 (called: Further examples of manifolds: cutting and pasting), there's this question:
> Prove that adding a handle to a 2-manifold... |
Let's say I have a list of point cloud ( in 3D coordinates!) with the vertex $V$, how to find all the triangular faces $F$ ( which has three edges $E$) that covers **all** of the points?
The conditions are that the faces must not overlap or cut through one another. And it would help if all the $F$ has not-too-narrow... |
How can I construct a CW complex A with H_0(A) = Z, H_2(A) = Z/4Z, H_4(A) =
Z ⊕ Z and all other homology groups trivial?
Any idea?
Thanks! |
I am trying to work through Boothby's An Introduction to Differentiable Manifolds on my own and, embarassingly, have got stuck at the very first chapter. At the end of section 4, chapter 1 (called: Further examples of manifolds: cutting and pasting), there's this question:
> Prove that adding a handle to a 2-manifold... |
Can you tell me why the following is true?
> A ring $R$ is local if and only if every principal left $R$-module is indecomposable. |
A module $M_{R}$ is called semi-artinian if every nonzero image of $M$ contains a simple submodule. Given $m\in M$ and $a_1,a_2,...$ in J(R). Why $ma_1a_2...a_{n-1}a_n=0$ for some $n\geq 1$.
----
*(Edit by KennyTM: The above is OP's original question. The latest, completely changed question follows:)*
if $R$ i... |
Can you tell me why the following is true?
> A ring $R$ is local if and only if every principal left $R$-module is indecomposable.
----
*(Edit by KennyTM: The above is OP's original question. The latest, completely changed question follows:)*
what is the relation between regular ring and this property:
... |
**Definition**: n is said to be a highly composite number if and only if $\sigma(n)>\sigma(m)$ for all $m<n$, where $\sigma(n)$ denotes number of divisors of n.
**Questions**:
1) Are there any theorems about highest prime that divides n, assuming that n is highly composite?. It seems that the highest prime is of ... |
1. Can a CFG over large alphabet describe a language, where each terminal appears even number of times?
2. If yes, would the Chomsky Normal Form be polynomial in |Σ| ?
EDIT: Solving this may give interesting extension to <a href="http://cstheory.stackexchange.com/questions/3816/restricted-read-twice-ibdds-and-con... |
***You are given three non-negative integers $A, B$ and $C$,find a number $X$(say) satisfy
$X^A = B(\mod(2C + 1))$ and $0 \le X \le 2C$***
I am inquisitive about how to approach this one? |
Common form of system of linear equations is A*X = B, X is unknown. But how to find A, if X and B are known? |
Common form of system of linear equations is A*X = B, X is unknown. But how to find A, if X and B are known?
A is MxN matrix, X is column vector(N), B is column vector(M) |
Several papers give poly-time algorithms for constrained paths on labelled graphs, e.g. [1]
Quote:
----------
Given an alphabet Σ, a (directed) graph G whose edges are weighted and Σ-labeled,
and a formal language L ⊆ Σ∗ , the formal-language-constrained shortest/simple path problem con-
sists of finding a short... |
Let A be a 4 × 4 matrix with eigenvalues -5, -2, 1, 4. Which of the following is an
eigenvalue of
A I
I A
where I is the 4 × 4 identity matrix?
(A) -5 (B) -7 (C) 2 (D) 1
I am not able to find out relation between eigenvalues of matrix A and of the given block matrix.
|
My problem is:
What is the expression in $n$ that equals to $\sum_{i=1}^n \frac{1}{i^2}$?
Thank you very much~ |
What is the expression of $n$ that equals to $\sum_{i=1}^n \frac{1}{i^2}$? |
Let's say I have a list of point cloud ( in 3D coordinates!) with the vertex $V$, how to find all the triangular faces $F$ ( which has three edges $E$) that covers **all** of the points?
The conditions are that the faces must not overlap or cut through one another. And it would help if all the $F$ has not-too-narrow... |
Let's say I have a list of point cloud ( in 3D coordinates!) with the vertex $V$, how to find all the triangular faces $F$ ( which has three edges $E$) that covers **all** of the points?
The conditions are that the faces must not overlap or cut through one another. And it would help if all the $F$ has not-too-narrow... |
This steps seems to be unnecessary. In what follows, I'll assume you have p. 30 in front of you (since it's available freely online).
Start with the part where he has constructed neighborhood $N$ of $y_0 \in Y$ and a partition $0 = t_0 < \dots t_n = 1$ such that $F(N \times [t_i,t_{i+1}]) \subset U_i$ for each $i... |
**Definition**: n is said to be a highly composite number if and only if $d(n)>d(m)$ for all $m<n$, where $d(n)$ denotes number of divisors of n.
**Questions**:
1) Are there any theorems about highest prime that divides n, assuming that n is highly composite?. It seems that the highest prime is of the order of ln... |
In our analysis course, the following question came up and could, up to now, not be solved:
> Let $a: \mathbb{N} \to \mathbb{C}$ be a sequence of complex numbers. What are necessary and sufficient conditions for the existence of a function $\mu: 2^\mathbb{N} \to \mathbb{C}$ satisfying the properties
> - $\mu(\{i\... |
My question is ; How can I solve the following integral question?
$ \int{{tanx}.{cos2x}dx} $
Thanks in advance,
|
My question is ; How can I solve the following integral question?
$$\int\tan\;x\;\cos\;2x\;\mathrm dx$$
Thanks in advance,
|
**HINT**
(1) $\cos 2x = \cos^2x-\sin^2x=2\cos^2x-1$.
(2) $2\sin x\cos x = \sin 2x$.
(3) $\frac{d}{dx}\log(f(x))=?$ |
Suppose I gave you an integral of the form
$\displaystyle \int \tan x \ \ f(\sin x) \ \text{dx}$
Can you think of a substitution to get rid of the $\tan x$ term?
For a concrete example, can you try evaluating
$\displaystyle \int \tan x \ \ (1 + \sin^5 x) \ \ \text{dx}$ ?
|
1. Can a CFG over large alphabet describe a language, where each terminal appears even number of times?
2. If yes, would the Chomsky Normal Form be polynomial in |Σ| ?
EDIT: What about a language where each terminal appears 0 or 2 times?
EDIT: Solving this may give interesting extension to <a href="http://csth... |
Such algorithms do exist. Which one you mean depends on what you mean by "no way to decrypt it". You refer to either:
- A function which takes an input and generates a unique output, with there being ideally no known way to derive the input from the output. These are called hashing algorithms; examples of which inc... |
Suppose I gave you an integral of the form
$\displaystyle \int \cot x \ \ f(\sin x) \ \text{dx}$
Can you think of a substitution to get rid of the $\cot x$ term?
For a concrete example, can you try evaluating
$\displaystyle \int \cot x \ \ (1 + \sin^5 x) \ \ \text{dx}$ ?
|
Looking on Internet, I mostly found this definition:
Given quantities on a cyclic domain D, first rescale the domain to [0;2$\pi$[, then, find $z_n$ the point on the unit circle corresponding to your value, and compute the average by:
$$z_m = \sum_{n=1}^N z_n$$
The average angle is then $\theta_m = \arg z_m$ a... |
How to average cyclic quantities? |
How can I find $\int\tan\;x\;\cos\;2x\;\mathrm dx$? |
Is it true that $O(M^3 + NM^2) \, = \, O(M^3 + N)$, where $M$ and $N$ are variables of the function? |
1. Can a CFG over large alphabet describe a language, where each terminal appears even number of times?
2. If yes, would the Chomsky Normal Form be polynomial in |Σ| ?
EDIT: What about a language where each terminal appears 0 or 2 times?
EDIT: Solving this may give interesting extension to <a href="http://csth... |
The surface integral of a vector field, $F$ over a surface $S$ (or flux as it is called) is, as stated above, the net flow of fluid through the surface. The purpose of orientation is to simply indicate which side of the surface is the "positive" side. For instance, if I calculate flux to be +10, this means their is a... |
Find Fourier cosine transform of e^(-a²x²) and hense evaluate Fourier sine transform of x.e^(-a²x²).
i can solve this ques. only if there is x instead of x² in the exponential function e^(-a²x²). bcoz in this situation i can use integral formula :- ⌠e^(-ax)cos(sx)dx = e^(-ax)/a²+x²[bsin(sx)-acos(sx)] but what should... |
How to start creating Mathematical models? |
**Added.** So, what is the problem? Let's look at the simplest example of a non-normal subgroup. Take $G=S_3$, and $H=\{e,(1,2)\}$. If we compose permutations right to left, the left cosets of $H$ in $G$ are:
\begin{align*}
eH = (1,2)H &= \\{ e, (1,2)\\};\\\
(1,2,3)H = (1,3)H &= \\{(1,2,3), (1,3)\\};\\\
(1,3,2)H = ... |
Let $R$ be a ring, $A$ is an ideal of $R$, and assume that $A\subseteq J(R)$.
Let $r\longmapsto \overline{r}$ denote the coset map $R\rightarrow \overline{R}=R/A$. The following are equivalent:
(1) $A$ is lifting.
(2) If $\overline{R}=\overline{K_1}\oplus...\oplus \overline{K_n}$ where the $\overline{K_i}$ are ... |
**How does a technical and theoretical study of a project of implementing an algorithm on a specific hardware architecture?**
Function example:
%Expresion 1:
y1 = exp (- (const1 + x) ^ 2 / ((conts2 ^ 2)),
y2 = y1 * const3
Where x is the input variable,
y2 is the output and const1, const3 const2 ... |
Generally it's difficult to compute such $\rm\:A$'th roots in $\rm \mathbb Z/m\ $ except for simple special cases, e.g. when $\rm\ gcd(A,\phi(m)) = 1\:.\ $ Otherwise one generally has to resort to factoring $\rm\:m\:$ to get $\rm\: \phi(m)\: $ and hence the factorization of $\rm\ Z/m^*\ $ into a product of cyclic group... |
Looking on Internet, I mostly found this definition:
Given quantities on a cyclic domain D, first rescale the domain to [0;2$\pi$[, then, find $z_n$ the point on the unit circle corresponding to the $n$th value, and compute the average by:
$$z_m = \sum_{n=1}^N z_n$$
The average angle is then $\theta_m = \arg z... |
Consider the [Chernoff bound](http://en.wikipedia.org/wiki/Chernoff_bound): It assumes $X_1, \ldots, X_n$ are independent Bernoulli random variables, each having probability $p > 1/2$, and seeks the probability of simultaneous occurrence of more than $n/2$ of the events.
I once heard that, in the proof of the Cherno... |
You are given three non-negative integers $A$, $B$ and $C$, find a number $X$ (say) satisfy
$X^A \equiv B\pmod{2C + 1}$ and $0 \le X \le 2C$.
I am inquisitive about how to approach this one? |
The last time I taught mathematical modeling we used *[Introduction to Mathematical Modeling][1],* by Giordano, Fox, Horton, and Weir. The first part of Chapter 2 contains a nice introduction to the modeling process.
A [search on Google books for mathematical modeling texts][2] should yield some more good results.
... |
I am a calculus TA a bit stumped at a question a student has posed me.
What is the method for finding the sum of the series $$\sum_{i=0}^\infty\frac{1+n}{3^n}$$ If you split the numerator up, one becomes a geometric series summing to 3/2, and the other sums to 3/4 by <a href="http://www.wolframalpha.com/input/?i=sum... |
**Added.** So, what is the problem? Let's look at the simplest example of a non-normal subgroup. Take $G=S_3$, and $H=\{e,(1,2)\}$. If we compose permutations right to left, the left cosets of $H$ in $G$ are:
\begin{align*}
eH = (1,2)H &= \\{ e, (1,2)\\};\\\
(1,2,3)H = (1,3)H &= \\{(1,2,3), (1,3)\\};\\\
(1,3,2)H = ... |
Let $G$ be a group of permutations of $n$ objects. Let subsets $A$,$B$ of the objects be equivalent if $A=g(B)$ for some $g \in G$.
Is there a way to show that a bijection exists between the equivalence classes and the $2$-colorings of the objects that are invariant under those permutations in $G$? |
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