Challenge Math Challenge - September 2019

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fresh_42

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Summary
ring of functions
arithmetic-geometric mean
geometry
Hilbert space
quaternions
prime
convergence
curve
gauge transformation
Questions

1.
Consider the ring ##R= C([0,1], \mathbb{R})## of continuous functions ##[0,1]\to \mathbb{R}## with pointwise addition and multiplication of functions. Set ##M_c:=\{f \in R\mid f(c)=0\}##.

(a) (solved by @mathwonk ) Show that the map ##c \mapsto M_c, c \in [0,1]## is a bijection between the interval ##[0,1]## and the maximal ideals of ##R##.

(b) Show that the ideal ##M_c## is not finitely generated.

Hint: Assume not and take a generating set ##\{f_1, \dots, f_n\}##. Consider the function ##f = |f_1| + \dots + |f_n|## and play with the function ##\sqrt{f}## to obtain a contradiction.


2. We all know that the geometric mean is less than the arithmetic mean. I memorize it with ##3\cdot 5 < 4\cdot 4##. Now we consider the arithmetic-geometric mean ##M(a,b)## between the two others. Let ##a,b## be two nonnegative real numbers. We set ##a_0=a\, , \,b_0=b## and define the sequences ##(a_k)\, , \,(b_k)## by
$$
a_{k+1} :=\dfrac{a_k+b_k}{2}\, , \,b_{k+1}=\sqrt{a_kb_k}\quad k=0,1,\ldots
$$
Then the arithmetic-geometric mean ##M(a,b)## is the common limit
$$
\lim_{n \to \infty}a_n = M(a,b) = \lim_{n \to \infty}b_n
$$
It is not hard to show that both sequences converge and that their limit is the same by using the known inequality and the monotony of the sequences.

Prove that for positive ##a,b\in \mathbb{R}## holds
$$
T(a,b):=\dfrac{2}{\pi} \int_0^{\pi/2}\dfrac{d\varphi}{\sqrt{a^2\cos^2 \varphi +b^2 \sin^2 \varphi }} = \dfrac{1}{M(a,b)}
$$


3. (solved by @StoneTemplePython ) If ##A,B,C,D## are four points in the plane, show that
$$
\operatorname{det} \begin{bmatrix}
0&1&1&1&1 \\ 1&0&|AB|^2&|AC|^2&|AD|^2 \\ 1&|AB|^2&0&|BC|^2&|BD|^2 \\ 1&|AC|^2&|BC|^2&0&|CD|^2 \\ 1&|AD|^2&|BD|^2&|CD|^2&0
\end{bmatrix} = 0
$$


4. Let ##T \in \mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)## a linear, continuous (= bounded) operator on Hilbert spaces.
Prove that the following are equivalent:
(a) ##T## is invertible.
(b) There exists a constant ##\alpha > 0##, such that ##T^*T\geq \alpha I_{\mathcal{H}_1}## and ##TT^*\geq \alpha I_{\mathcal{H}_2}\,.## ##A \geq B## means ##\langle (A-B)\xi\, , \,\xi \rangle \geq 0## for all ##\xi\,.##


5. Let ##a,b \in \mathbb{F}## be non-zero elements in a field of characteristic not two. Let ##A## be the four dimensional ##\mathbb{F}-##space with basis ##\{\,1,\mathbf{i},\mathbf{j},\mathbf{k}\,\}## and the bilinear and associative multiplication defined by the conditions that ##1## is a unity element and
$$
\mathbf{i}^2=a\, , \,\mathbf{j}^2=b\, , \,\mathbf{ij}=-\mathbf{ji}=\mathbf{k}\,.
$$
Then ##A=\left( \dfrac{a,b}{\mathbb{F}} \right)## is called a (generalized) quaternion algebra over ##\mathbb{F}##.
Show that ##A## is a simple algebra whose center is ##\mathbb{F}##.


6. Prove that the quaternion algebra ##\left( \dfrac{a,1}{\mathbb{F}} \right)\cong \mathbb{M}(2,\mathbb{F})## is isomorphic to the matrix algebra of ##2\times 2## matrices for every ##a\in \mathbb{F}-\{\,0\,\}\,.##


7. (solved by @Not anonymous ) Show that there are infinitely many primes of the form ##4k+3\, , \,k\in \mathbb{N}_0##.


8. Do ##\displaystyle{\sum_{n=0}^\infty}\, \dfrac{(-1)^n}{\sqrt{n+1}}## and the Cauchy product ##\left( \displaystyle{\sum_{n=0}^\infty}\, \dfrac{(-1)^n}{\sqrt{n+1}} \right)^2## converge or diverge?


9. (solved by @DEvens ) Consider the curve ##\gamma\, : \,\mathbb{R}\longmapsto \mathbb{C}\, , \,\gamma(t)=\cos(\pi t)\cdot e^{\pi i t}\,.##

(a) Find the minimal period of ##\gamma##,

(b) prove that ##\gamma(\mathbb{R})\equiv\{\,(x,y)\in \mathbb{R}^2\,|\,x^2+y^2-x=0\,\}##,

(c) show that ##\gamma(\mathbb{R})## is symmetric to the ##x-##axis,

(d) parameterize ##\gamma## with respect to its arc length.


10. (solved by @DEvens ) Let ##\gamma\, : \,I \longrightarrow \mathbb{R}^n## be a regular curve with unit tangent vector ##T=\dfrac{d}{dt} \gamma\,.## A (orthonormal) frame is a (smooth) ##C^\infty-## transformation ##F\, : \,I \longrightarrow \operatorname{SO}(n)## with ##F(t)e_1=T(t)## where ##\{\,e_i\,\}## is the standard basis of ##\mathbb{R}^n\,.## The pair ##(\gamma,F)## is called a framed curve, and the matrix ##A## given by ##\dfrac{d}{dt}F=F'=FA## is called derivation matrix of ##F##.

Let ##F_0\, : \,\mathbb{R}\longrightarrow \operatorname{SO}(n)## be a frame of a regular curve ##\gamma\, : \,\mathbb{R}\longrightarrow \mathbb{R}^n##. Show that

(a) If ##F\, : \,\mathbb{R}\longrightarrow \operatorname{SO}(n)## is another frame of ##\gamma##, then there exists a transformation ##\Phi\, : \,\mathbb{R}\longrightarrow \operatorname{SO}(n)## with ##\Phi(t)e_1=e_1## for all ##t\in \mathbb{R}## and ##F=F_0\Phi\,.##

(b) If on the other hand ##\Phi\, : \,\mathbb{R}\longrightarrow \operatorname{SO}(n)## is a smooth transformation with ##\Phi(t)e_1=e_1\,,## then ##F:=F_0\cdot\Phi## defines a new frame of ##\gamma##.

(c) If ##A_0## is the derivation matrix of ##F_0##, and ##A## the derivation matrix of the transformed frame ##F:=F_0\Phi## with ##\Phi## as above, then
$$
A=\Phi^{-1}A_0\Phi +\Phi^{-1}\Phi'
$$


1567301112926.png




11. (solved by @Not anonymous ) Show that the number of ways to express a positive integer ##n## as the sum of consecutive positive integers is equal to the number of odd factors of ##n##.


12. (solved by @Not anonymous ) How many solutions in non-negative integers are there to the equation:
$$
x_1+x_2+x_3+x_4+x_5+x_6=32
$$


13. (solved by @etotheipi ) Let ##A, B, C## and ##D## be four points on a circle such that the lines ##AC## and ##BD## are perpendicular. Denote the intersection of ##AC## and ##BD## by ##M##. Drop the perpendicular from ##M## to the line ##BC##, calling the intersection ##E##. Let ##F## be the intersection of the line ##EM## and the edge ##AD##. Then ##F## is the midpoint of ##AD##.

Brahmagupta_2.png




14. (solved by @timetraveller123 ) Prove that every non negative natural number ##n\in \mathbb{N}_0## can be written as
$$
n=\dfrac{(x+y)^2+3x+y}{2}
$$
with uniquely determined non negative natural numbers ##x,y\in \mathbb{N}_0\,.##


15. Calculate
$$
S = \int_{\frac{1}{2}}^{3} \dfrac{1}{\sqrt{x^2+1}}\,\dfrac{\log(x)}{\sqrt{x}}\,dx \, + \, \int_{\frac{1}{3}}^{2} \dfrac{1}{\sqrt{x^2+1}}\,\dfrac{\log(x)}{\sqrt{x}}\,dx
$$
 
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8
I think the question was asking for a proof that F is the midpoint:

If we label up the angles in corresponding segments w, x, y and z as shown in orange below

Brahmagupta_2.png


We can see that triangles BEM and MEC are similar to AMD (by AAA). We label the angles AMF and FMD x and z respectively by the property of vertically opposite angles being equal.

Then we have two isosceles triangles, FAM and FMD, which share a common side FM. The lengths AF, FM and FD are equal. Since AF = FD, F is the midpoint of AD.
 

fresh_42

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I think the question was asking for a proof that F is the midpoint:

If we label up the angles in corresponding segments w, x, y and z as shown in orange below

View attachment 249011

We can see that triangles BEM and MEC are similar to AMD (by AAA).
You mean BMC and AMD are similar (by the first theorem of chords in an inscribed quadrilateral)?
We label the angles AMF and FMD x and z respectively by the property of vertically opposite angles being equal.

Then we have two isosceles triangles, FAM and FMD, which share a common side FM. The lengths AF, FM and FD are equal. Since AF = FD, F is the midpoint of AD.
Yes, that was the idea. It would have been a bit more work to do if done step by step (I think). It is Brahmagupta's Theorem, named after the Indian mathematician and astronomer (598–668).
 
(7. Show that there are infinitely many primes of the form ##4k+3\, , \,k\in \mathbb{N}_0##.)

Proof by contradiction. Suppose there are only a finite number of primes of the form ##4k+3\, , \,k\in \mathbb{N}_0##. Let these ##m## primes be denoted by ##q_1##, ##q_2##, .., ##q_m##, in increasing order of their value. Therefore, all other primes except 2 (the only even prime) must be of the form ##4k+1\, , \,k\in \mathbb{N}_0## (since 4k+0 and 4k+2 are even numbers and so cannot be odd primes).

Now consider the values ##x = 4(\prod_{i=2}^m q_i) + 3## and #### (note that the product of ##q_i##s starts with ##q_2##, so excludes ##q_1 = 3##). We observe the following:
1. Since ##4(\prod_{i=2}^m q_i)## is not a multiple of 3, ##4(\prod_{i=2}^m q_i) + 3## too cannot be a multiple of 3.
2. Since ##4(\prod_{i=2}^m q_i)## is even, ##4(\prod_{i=2}^m q_i) + 3##, i.e. ##x## must be odd.
3. ##x## cannot be a multiple of the other ##q_i## primes (i.e. ##q_2, q_3, .., q_m##) too because ##4(\prod_{i=2}^m q_i)## would be a multiple of any such ##q_i## and next smallest multiple divisible by ##q_i## would be ##4(\prod_{i=2}^m q_i) + q_i## which is greater than ##x = 4(\prod_{i=2}^m q_i) + 3## (since ##q_m \gt q_{m-1} \gt q_{m-2} \gt .. \gt q_2 \gt 3##).
4. Since ##x## is of the form ##4k+3## and by its definition must be greater than ##q_m## which was assumed to be the largest prime of the form ##4k+3##, ##x## cannot be prime
4. Therefore, ##2, q_1 = 3, q_2, .., q_m## have all been excluded as possible prime factors of ##x##. By initial assumption, all other primes must be of the form ##4k + 1##. So all prime factors of ##x## must be of the form ##4k + 1, k \in \mathbb{N}_0##, i.e. for any prime factor ##p_j##of ##x##, it must be that ##p_j \equiv 1 \ (\mod 4)##. But it is easy to show using modular arithmetic that the product of any number of such ##p_j## factors (including possible repeated values) will also be of the form ##4k + 1, k \in \mathbb{N}_0##.

Putting together the above observations, it follows that ##x## cannot have any prime factor less than itself, implying that ##x## itself is prime. Since ##x \gt q_m##, this contradicts that the initial assumption that ##q_m## is the largest prime of the form ##4k+3, k \in \mathbb{N}_0## which in turn was a result of assuming that there are only a finite number of primes of the form ##4k+3, k \in \mathbb{N}_0##. Therefore by contradiction, it follows that the number of primes of the form ##4k+3, k \in \mathbb{N}_0## cannot be finite; so there must be infinite primes of that form.
 

fresh_42

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Yes, well done! The trick with ##3## was nice. My solution didn't make that difference but was therefore a bit more complicated.

(7. Show that there are infinitely many primes of the form ##4k+3\, , \,k\in \mathbb{N}_0##.)

Proof by contradiction. Suppose there are only a finite number of primes of the form ##4k+3\, , \,k\in \mathbb{N}_0##. Let these ##m## primes be denoted by ##q_1##, ##q_2##, .., ##q_m##, in increasing order of their value. Therefore, all other primes except 2 (the only even prime) must be of the form ##4k+1\, , \,k\in \mathbb{N}_0## (since 4k+0 and 4k+2 are even numbers and so cannot be odd primes).

Now consider the values ##x = 4(\prod_{i=2}^m q_i) + 3## and #### (note that the product of ##q_i##s starts with ##q_2##, so excludes ##q_1 = 3##). We observe the following:
1. Since ##4(\prod_{i=2}^m q_i)## is not a multiple of 3, ##4(\prod_{i=2}^m q_i) + 3## too cannot be a multiple of 3.
2. Since ##4(\prod_{i=2}^m q_i)## is even, ##4(\prod_{i=2}^m q_i) + 3##, i.e. ##x## must be odd.
3. ##x## cannot be a multiple of the other ##q_i## primes (i.e. ##q_2, q_3, .., q_m##) too ...
Or simply: ##q_i\,|\,x \Longrightarrow q_i\,|\,3 ## which is impossible as you excluded ##3##.
.... because ##4(\prod_{i=2}^m q_i)## would be a multiple of any such ##q_i## and next smallest multiple divisible by ##q_i## would be ##4(\prod_{i=2}^m q_i) + q_i## which is greater than ##x = 4(\prod_{i=2}^m q_i) + 3## (since ##q_m \gt q_{m-1} \gt q_{m-2} \gt .. \gt q_2 \gt 3##).
4. Since ##x## is of the form ##4k+3## and by its definition must be greater than ##q_m## which was assumed to be the largest prime of the form ##4k+3##, ##x## cannot be prime
4. Therefore, ##2, q_1 = 3, q_2, .., q_m## have all been excluded as possible prime factors of ##x##. By initial assumption, all other primes must be of the form ##4k + 1##. So all prime factors of ##x## must be of the form ##4k + 1, k \in \mathbb{N}_0##, i.e. for any prime factor ##p_j##of ##x##, it must be that ##p_j \equiv 1 \ (\mod 4)##. But it is easy to show using modular arithmetic that the product of any number of such ##p_j## factors (including possible repeated values) will also be of the form ##4k + 1, k \in \mathbb{N}_0##.
Maybe it's a personal attitude and I should excuse myself, but I really don't like these platitudes like "easy to show". In 9 of 10 cases they cover a pit. The shortest way to you conclude is:
$$
x \equiv \prod (4k_i+1) \equiv \prod 1 \equiv 1 \equiv 4 \prod q_i +3 \equiv 3 \mod 4 \; ; \;\text{a contradiction }
$$
because modulo is a ring homomorphism. Alternatively you could name the two rules which holds:
$$
\rho(a\cdot b) \equiv \rho(a)\cdot \rho(b) \mod n \text{ and } \rho(a+ b) \equiv \rho(a)+ \rho(b) \mod n
$$
where ##\rho ## is "take the remainder of the division by ##n##.
Putting together the above observations, it follows that ##x## cannot have any prime factor less than itself, implying that ##x## itself is prime. Since ##x \gt q_m##, this contradicts that the initial assumption that ##q_m## is the largest prime of the form ##4k+3, k \in \mathbb{N}_0## which in turn was a result of assuming that there are only a finite number of primes of the form ##4k+3, k \in \mathbb{N}_0##. Therefore by contradiction, it follows that the number of primes of the form ##4k+3, k \in \mathbb{N}_0## cannot be finite; so there must be infinite primes of that form.
 
(12. How many solutions in non-negative integers are there to the equation:
##x_1+x_2+x_3+x_4+x_5+x_6=32##​

)

I assume that the order of values matters, i.e. for example, ##(x_1=x_2=x_3=x_4=x_5=0, x_6 = 32)## and ##(x_1=32, x_2=x_3=x_4=x_5=x_6 = 0)## are to be counted as 2 different solutions.

The number of possible solutions for that equation can be found using combinatorics. Each possible solution to that equation is logically equivalent to a partitioning of 32 objects into 6 bins, with some bins also allowed to be empty. Therefore, I consider the example of 32 marbles placed in a row and try to find the number of different ways in which 5 sticks can be placed between the balls or at the start or end of the row so that they partition the row of marbles into 6 groups, with some groups possibly being empty (which is equivalent to some ##x_i = 0##). With 32 marbles and 5 sticks, we have 37 positions and we need to select 5 positions for placing the sticks. The number of such choices is given by ##{_{37}C_5} = 435897##. Therefore, the number of possible whole-numbers-only solution for the equation ##x_1+x_2+x_3+x_4+x_5+x_6=32## is 435897.
 

DEvens

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a) By Euler's formula, the curve is ##\cos^2 t \pi + i \sin t \pi \cos t \pi## and so by the half-angle formula, the period for each of these is t from 0 to 1.

b) Let me use shorthand of ##C## for ##\cos(t \pi)## and similarly ##S## for ##\sin(t \pi )##. The real part, x, is ##C^2## and the imaginary, y, is ##C S##. So ##x^2+y^2-x = C^4+C^2 S^2 - C^2 = C^2 (C^2+S^2-1) = 0##. Looking at the construction in part d) shows that each point on the circle so defined can be associated with a single value of t in the range 0 to 1. And so the entire circle is included.

c) From b), the function is even in y.

d) Examine the figure attached. Add the shorthand ##C2## for ##\cos (2 t \pi)##, and similar for ##S2##. The center of the circle in b) is the point (0.5,0), the radius is 0.5. Beginning from t=0, the point (1,0), find the arc length to a point specified by t. The x and y values are given in b). We again need the half-angle formulas. From the center of the circle to this point the y interval is just the y value of the point, or ## C S = S2 / 2##. The x interval is ##C^2 - 1/2 = C2/2##. So the tangent of the angle required is ## S2 / C2##. That is, the angle required is ##\theta = 2 t \pi##. But the radius of this circle is 0.5, so the arc length is ##t \pi##. The curve is originally parameterized in its arc length divided by ##\pi##, or ##\gamma(t) = \gamma(\frac{s}{\pi})##.


gamma.png
 
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fresh_42

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a) By Euler's formula, the curve is ##\cos^2 t \pi + i \sin t \pi \cos t \pi## and so by the half-angle formula, the period for each of these is t from 0 to 1.

b) Let me use shorthand of ##C## for ##\cos(t \pi)## and similarly ##S## for ##\sin(t \pi )##. The real part, x, is ##C^2## and the imaginary, y, is ##C S##. So ##x^2+y^2-x = C^4+C^2 S^2 - C^2 = C^2 (C^2+S^2-1) = 0##.
This shows ##\gamma(\mathbb{R}) \subseteq \{\,(x,y)\in \mathbb{R}^2\,|\,x^2+y^2-x=0\,\}##, but why do those points lie on the curve?
c) From b), the function is even in y.

d) Add the shorthand ##C2## for ##\cos (2 t \pi)##, and similar for ##S2##. The center of the circle in b) is the point (0.5,0), the radius is 0.5. Beginning from t=0, the point (1,0), find the arc length to a point specified by t. The x and y values are given in b). We again need the half-angle formulas. From the center of the circle to this point the y interval is just the y value of the point, or ## C S = S2 / 2##. The x interval is ##C^2 - 1/2 = C2/2##. So the tangent of the angle required is ## S2 / C2##. That is, the angle required is ##2 t \pi##. But the radius of this circle is 0.5, so the arc length is ##t \pi##. The curve is originally parameterized in its arc length.
I don't fully understand your proof, but the conclusion is more or less correct, depending which version of your answer we concentrate on. Could you write the formula in dependence of the arc length, say ##s##?
 

mathwonk

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For question 1,a, since for every point c, the ideal Mc of functions vanishing on c is the kernel of the evaluation map to R, a field, hence Mc is a maximal ideal, one wants to show conversely that every maximal ideal J is contained in, hence equal to, an ideal of form Mc, i.e. that J has a common zero point c. Contrapositively one wants to show that an ideal J with no common zero, is not a proper ideal, i.e. contains a unit. So assume that for every point, there is a function in the given ideal J that does not vanish at that point. Then try to use the definition of continuity, and compactness to conclude.
 

Delta2

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I have a sketch of the proof for 2 but I need a hint on how to fully implement it.
We take the Riemann sum ##S_n## of the integral $$\int_0^{\pi/2}\frac{d\phi}{\sqrt{a^2\cos^2\phi+b^2\sin^2\phi}}=\lim_{n\to\infty}S_n$$ and prove by induction on ##n## that $$\frac{1}{a_n}\leq \frac{2}{\pi}S_n\leq\frac{1}{b_n}$$ and then the result follows by taking the limits and using the sandwich theorem for limits.

But I have problem with the induction... Any hints and of course am I on the right track or there is a completely different approach??
 

fresh_42

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I have a sketch of the proof for 2 but I need a hint on how to fully implement it.
We take the Riemann sum ##S_n## of the integral $$\int_0^{\pi/2}\frac{d\phi}{\sqrt{a^2\cos^2\phi+b^2\sin^2\phi}}=\lim_{n\to\infty}S_n$$ and prove by induction on ##n## that $$\frac{1}{a_n}\leq \frac{2}{\pi}S_n\leq\frac{1}{b_n}$$ and then the result follows by taking the limits and using the sandwich theorem for limits.

But I have problem with the induction... Any hints and of course am I on the right track or there is a completely different approach??
This would only show, that the limits exist. It follows from ##b_1 \leq b_2 \leq \ldots b_n \leq \ldots \leq a_n \leq \ldots \leq a_2 \leq a_1##.

The proof is a bit tricky. It uses a recursion ##T(a,b)=T(f(a,b),g(a,b))## with suitable ##f## and ##g##.
 
q14:

##
let f(x,y) =\frac{{(x+y)}^2+3x+y}{2} x,y \geq 0\\
##then if ##n = f(x,y)## then ## n+1 = f(x+1,y-1) ## and ## n-1=f(x-1,y+1)##

consider what values of n can be represented by x=0

##
f(0,y) = n
## using quadratic formula we can get that n must be of the form
##
n = k(2k+1)
##
for positive integer k with the corresponding y being
## y = 2k##
we can then raise x by 1 and decrease y by one each time to get subsequent value of n and since y must be at least zero we have covered
##
k(2k+1) \leq n \leq k(2k+1) + 2k = k (2k+3)
##

similarly
##
f(x,0) = n
## implies
##
n = \frac{(m-1)(m+2)}{2}
##
with corresponding x being
##x = m-1##
if m is odd then writing m as 2k+1 we get
##
n = k(2k+3)
##
this is the case previously derived
if m is even then we get
##
n= (2k-1)(k+1)
##
we also have that
##
(2k-1)(k+1) + 1= k(2k+1)
##

this is value is covered by the x=0 case
then for the case m is even we can decrease x by 1 and increase y by 1 to cover lower values of n and since x =m-1 = 2k-1 we can cover until

##
(2k-1)(k+1) - (2k-1) = 2k^2-k
##
we note that
##
2k^2-k -1 = (k-1)(2k+1) = (k')(2k'+3)
##

which has been covered before
thus all the values in between
##
k(2k+1)
##
and
##
(k+1)(2(k+1) +1)
##
is covered then i think we are done
 

fresh_42

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Maybe it will seem a bit nitpicking, but I'll comment it as I would comment an exercise given to me for correction.
q14:

##
let f(x,y) =\frac{{(x+y)}^2+3x+y}{2} x,y \geq 0\\
##
Let ## f(x,y) =\frac{{(x+y)}^2+3x+y}{2} \quad ; \quad x,y \geq 0##
then if ##n = f(x,y)## then ## n+1 = f(x+1,y-1) ## and ## n-1=f(x-1,y+1)##
The reader automatically associates a natural number with ##n##. But why is ##f(x,y)\in \mathbb{N}_0\,\text{?}## You could first tell us why, but you do not need it if you write ##f(x+1,y-1)=f(x,y)+1\, , \,f(x-1,y+1)=f(x,y)-1## or the lazy way: ##f(x \pm 1, y \mp 1)=f(x,y) \pm 1\;\;(*)##
consider what values of n can be represented by x=0

##
f(0,y) = n
## using quadratic formula we can get that n must be of the form
##
n = k(2k+1)
##
for positive integer k with the corresponding y being
## y = 2k##
Same as before. What is ##n##? It always looks as if you used at the beginning what you want to prove at the end. Do you assume a given ##n##? Is it short for ##f(0,y)##? Does your proof go: ##n=f(x,y) \Longrightarrow \ldots \Longrightarrow n=\dfrac{x^2+2xy+y^2+3x+y}{2}## Yipeeh, I wrote ##n## as required?
Short: Do not use unqualified variables!

What you considerer was: ##f(0,y)=\dfrac{y^2+y}{2}=\dfrac{y(y+1)}{2}##. Now either ##y## or ##y+1## is even, so ##f(0,y)\in \mathbb{N}_0##. Without loss of generality we set ##y=2k##...
we can then raise x by 1 and decrease y by one each time to get subsequent value of n and since y must be at least zero we have covered
##
k(2k+1) \leq n \leq k(2k+1) + 2k = k (2k+3)
##
... and ##k(2k+1)=f(0,y)##. With ##(*)## we get all predecessors and successors of ##k(2k+1)##, hence all integers.

The existence part of the problem is finished here. You have written all natural numbers in the required way. I don't understand why you need the rest of your considerations, especially as you did not really said which role ##k,n,m## have.
similarly
##
f(x,0) = n
## implies
##
n = \frac{(m-1)(m+2)}{2}
##
with corresponding x being
##x = m-1##
if m is odd then writing m as 2k+1 we get
##
n = k(2k+3)
##
this is the case previously derived
if m is even then we get
##
n= (2k-1)(k+1)
##
we also have that
##
(2k-1)(k+1) + 1= k(2k+1)
##

this is value is covered by the x=0 case
then for the case m is even we can decrease x by 1 and increase y by 1 to cover lower values of n and since x =m-1 = 2k-1 we can cover until

##
(2k-1)(k+1) - (2k-1) = 2k^2-k
##
we note that
##
2k^2-k -1 = (k-1)(2k+1) = (k')(2k'+3)
##

which has been covered before
thus all the values in between
##
k(2k+1)
##
and
##
(k+1)(2(k+1) +1)
##
is covered then i think we are done
I do not see, why this presentation of a given natural number ##n## as ##n=f(x,y)## is unique?
You are close and on the right track.

Hint: Instead of solving ##f(x,y)=f(x',y') \Longrightarrow \ldots \Longrightarrow x=x'\, , \,y=y'## which looks awful to do, consider which numbers you can cover with ##s=x+y##. Observe that ##(*)## doesn't change the value of ##s##. So the question is: Which interval can we cover with one specific ##s## by your method, and can there be a point in an interval for ##s## which is also in an interval for ##t\neq s##?

You already had the crucial idea with ##(*)## and I think uniqueness is hidden somewhere in your argumentation, but I was unable to extract it.
 

DEvens

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For question 10, are there some typos? Show that (a) Is F: etc. What? Maybe show that (a) If F is etc. then Phi as specified exists?

Part b includes the word "new." If ##\Phi## is just the identity, which is a smooth transformation that satisfies the requirement, then it's not a new frame but the same frame.
 

fresh_42

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For question 10, are there some typos? Show that (a) Is F: etc. What? Maybe show that (a) If F is etc. then Phi as specified exists?

Part b includes the word "new." If ##\Phi## is just the identity, which is a smooth transformation that satisfies the requirement, then it's not a new frame but the same frame.
Yes. Lost in translation. It's corrected now.
 

Math_QED

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For question 1,a, since for every point c, the ideal Mc of functions vanishing on c is the kernel of the evaluation map to R, a field, hence Mc is a maximal ideal, one wants to show conversely that every maximal ideal J is contained in, hence equal to, an ideal of form Mc, i.e. that J has a common zero point c. Contrapositively one wants to show that an ideal J with no common zero, is not a proper ideal, i.e. contains a unit. So assume that for every point, there is a function in the given ideal J that does not vanish at that point. Then try to use the definition of continuity, and compactness to conclude.
Hi Mathwonk. That's certainly the idea for the (a) part of the question! Are you willing to fill in the details so other readers can benefit from your knowledge?
 

mathwonk

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well sure. choose for each point t a function ft that does not vanish at t. Then there is, by the definition of continuity, an open interval It on which ft does not vanish. These intervals cover the interval J and by compactness of J, a finite number It1,...,Itn already cover it. Then the sum of the squares of the functions ft1,...,ftn, is a continuous function on J that vanishes nowhere, hence is a unit, since its inverse is again continuous.
 
I don't understand why you need the rest of your considerations, especially as you did not really said which role k,n,mk,n,mk,n,m have.
i have made a mistake in my part 1 there is a another case i forgot to consider here is new proof
let n be some non negative natural number thus if
f(0,y) = n is to have a solution with y being a non negative natural then
n must be of the form either k(2k+1) or (k+1)(2k+1) with the corresponding y being 2k and 2k+1 respectively
i forgot the second case this makes the whole of second part of the previous solution irrelevant
(k+1)(2k+1) - k(2k+1) - 1 = 2k thus we can cover all numbers from (k)(2k+1) to (k+1)(2k+1) by decreasing y by 1 and increasing x by 1 each time
and since
k(2k+1) < (k+1)(2k+1) < (k+1)(2(k+1) + 1)
we have covered all the numbers in the first inequality for the second inequality we can also cover all the numbers since
(k+1)(2(k+1) + 1) - (k+1)(2k+1) -1 = 2k+1 and since this is the y value for n = (k+1)(2k+1)
we can also cover the second inequality so we have covered all integers between and including
k(2k+1) to (k+1)(2(k+1) + 1) . thus that covers all non negative integers

in my first attempt i forgot to consider the second case that made things weird.
 

fresh_42

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Let me summarize, @timetraveller123.

Say we have a number ##n=k(2k+1) \geq 0##. Then ##n=f(0,2k)##. With your equations ##f(x\pm 1,y\mp 1)\stackrel{(*)}{=}f(x,y)\pm 1## we get all numbers up to ##f(2k,0)= k(2k+3)=n+2k##.
On the other hand, we have ##f(2k,0)=2k^2+3k=n+2k## and by ##(*)## all numbers down to ##f(0,2k)=n##, which doesn't give us new numbers.

So we look at ##f(0,2k-1)=2k^2-k=k(2k-1)=n-2k## and by ##(*)## all numbers up to ##f(2k-1,0)=2k^2+k-1=n-1##

Both "machines" give us therefore all numbers form ##n-2k, n-2k+1,\ldots,n-1,n,n+1,n+2k##.

##k=0## yields ##n=0##.
##k=1## yields ##n=1\cdot (2\cdot 1 +1)=3## and the "machine" outputs ##3-2,3-1,\ldots,3+2=1,2,3,4,5##.
##k=2## yields ##n=2\cdot (2\cdot 2 +1)=10## and the "machine" outputs ##10-2\cdot 2=6,\ldots ,14##.

And so on is o.k., an induction exaggerated, but we can write:

After the numbers ##n-2k,\ldots ,n+2k## we increase ##k## by one and get as new values ##k'=k+1## and ##n'=k'(2k'+1)=(k+1)(2(k+1)+1)= 2k^2+5k+3##. Then our "machine" yields all numbers from ##n'-2k'=2k^2+5k+3-2k-2=2k^2+3k+1=k(2k+1)+2k+1=n+2k+1## which is the next number our "machine" hasn't produced with ##k##, but does so with ##k+1##.

Sorry, I had to sort out your strategy for myself to understand what you have. At this point we have all numbers covered, AND the intervals produced by the "machine" for one specific value ##k## do not intersect with those of other values, i.e. uniqueness for ##x,y## in ##n=f(x,y)## is also given.
 
Last edited:
yes that is what i meant
 

StoneTemplePython

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No one ever does the matrix problems... :frown:
This one is quite nice and relates to a favorite topic of mine: Gram matrices, as well as rudimentary geometry.
There may be a more expedient route but I used a rank based argument. This ended up being a bit longer than I had hoped....

Problem 3
##\mathbf W = \begin{bmatrix}
0&1&1&1&1 \\ 1&0&|AB|^2&|AC|^2&|AD|^2 \\ 1&|AB|^2&0&|BC|^2&|BD|^2 \\ 1&|AC|^2&|BC|^2&0&|CD|^2 \\ 1&|AD|^2&|BD|^2&|CD|^2&0
\end{bmatrix} = \begin{bmatrix} 0 &\mathbf 1^T \\\mathbf 1 & \mathbf W_p\end{bmatrix}##

where ##\mathbf W_p## is the principal submatrix of main interest
we need to prove ##\operatorname{det}\big(\mathbf W\big) = 0##

lemma
The sum of ##d## rank one updates has rank at most ##d##.

This is implied by familiar matrix multiplication rank properties.
##\mathbf S \in \mathbb R^{\text{d x n}}## and conforming ##\mathbf C##, so

##\mathbf C\mathbf S ##
##= \bigg[\begin{array}{c|c|c|c|c}
\mathbf c_1 & \mathbf c_2 &\cdots & \mathbf c_{d-1} & \mathbf c_{d}
\end{array}\bigg]\begin{bmatrix}
\tilde{ \mathbf s_1}^T \\
\tilde{ \mathbf s_2}^T \\
\vdots\\
\tilde{ \mathbf s}_{d-1}^T \\
\tilde{ \mathbf s}_d^T
\end{bmatrix} ##
##= \mathbf c_1 \tilde{ \mathbf s_1}^T + \mathbf c_2 \tilde{ \mathbf s_2}^T +... + \mathbf c_d \tilde{ \mathbf s_d}^T ##
##= \sum_{k=1}^d\mathbf c_k \tilde{ \mathbf s_k}^T##

so
##\text{rank}\Big(\sum_{k=1}^d\mathbf c_k \tilde{ \mathbf s_k}^T\Big)= \text{rank}\Big(\mathbf C\mathbf S\Big) \leq \text{rank}\Big(\mathbf S\Big) \leq \min\Big(d,n\Big) \leq d##

Geometrically pleasant special case:
suppose all of these points (A,B,C,D --> relabelled as 1,2,3,4 or ##\mathbf x_1, \mathbf x_2, \mathbf x_3, \mathbf x_4## ) are in the plane, and this mysterious column zero is also in the plane. In effect we can interpret point 0 as the origin.

It isn't needed, but the geometry associated with this very nice in this special case: the above squared distance matrix, suggests that in effect point zero is the origin and points 1,2,3,4 are all (squared) distance of 1 from it -- i.e. they are on the boundary of the unit circle.

In general, the idea is that we have ##m## points and we know the (squared) distance between them. And we know the distance was created by use of an inner product (and specifically via standard dot product of vectors ##\in \mathbb R^2##). In general whenever have have an inner product, I've suggested that its wise to try to look at the Gram Matrix ##\mathbf G##. So we have

##2\mathbf G = -\mathbf W + \mathbf 1\mathbf y^T + \mathbf y \mathbf 1^T##
where ##\mathbf y## is a vector containing the original squared 2-norm of points ##\mathbf x_k## --we have 5 in this case, but it is useful to think in general of ##m## points and ##\mathbf G \in \mathbb R^\text{m x m}##

examine component i,j of this matrix to verify that
##2\cdot g_{i,j} ##
##= -\big \Vert \mathbf x_i - \mathbf x_j\big \Vert_2^2 + \big\Vert \mathbf x_i\big\Vert_2^2 + \big\Vert \mathbf x_j \big\Vert_2^2##
##= -\Big\{\langle \mathbf x_i - \mathbf x_j, \mathbf x_i - \mathbf x_j\rangle \Big\}+ \langle \mathbf x_i, \mathbf x_i\rangle+ \langle \mathbf x_j, \mathbf x_j\rangle##
##=-\Big\{ \langle \mathbf x_i, \mathbf x_i\rangle + \langle \mathbf x_j, \mathbf x_j\rangle - 2\langle \mathbf x_i , \mathbf x_j\rangle \Big\}+ \langle \mathbf x_i, \mathbf x_i\rangle+ \langle \mathbf x_j, \mathbf x_j\rangle##
##= 2\langle \mathbf x_i , \mathbf x_j\rangle ##

which confirms that ##\mathbf G## is a Gram Matrix. Since we know these points are in ##\mathbb R^2##, we know that in fact

##\mathbf X^T\mathbf X = \mathbf G##
##\mathbf X = \bigg[\begin{array}{c|c|c|c}
\mathbf x_0 &\mathbf x_1 &\cdots & \mathbf x_{m-1}
\end{array}\bigg]##
where ##\mathbf X \in \mathbb R^{\text{2 x m}}##

hence
##\text{rank}\big(\mathbf G \big) = \text{rank}\big(\mathbf X^T\mathbf X \big) =\text{rank}\big(\mathbf X\big) \leq d = 2##
and in particular we can use the 'outer product/rank one update' interpretation of matrix multiplication to write ##\mathbf X^T \mathbf X## as a the sum of (at most) 2 rank one matrices. For avoidance of doubt revisit the lemma and let ##\mathbf C:= \mathbf X^T## and ##\mathbf S := \mathbf X## and of course ##d =2## and ##n=m##.

so
##\mathbf G = \sum_{k=1}^2\mathbf c_k \tilde{ \mathbf s_k}^T##

the fact that this holds up to rescaling suggests it would be nicer to write
##-2\mathbf G := \sum_{k=1}^2\mathbf c_k \tilde{ \mathbf s_k}^T##
So I use this going forward -- if its an issue I can introdue more notation I suppose

But we know
##\mathbf W= -2\mathbf G + \mathbf 1\mathbf y^T + \mathbf y \mathbf 1^T = \big(\sum_{k=1}^2\mathbf c_k \tilde{ \mathbf s_k}^T\big) +\mathbf 1\mathbf y^T + \mathbf y \mathbf 1^T = \big(\sum_{k=1}^4\mathbf c_k \tilde{ \mathbf s_k}^T\big) ##
(after re-labelling)

which allows us to conclude
##\text{rank}\big(\mathbf W\big) \leq 4 \lt 5 \longrightarrow \det\big(\mathbf W\big) = 0##

algebraic tweak for the more general case
The above was nice but technically we don't know anything about ##\mathbf x_0##, whether it actually exists in the plane, etc. The hack is to just zero out the thing we are unsure of re-run the above argument with just

##\mathbf X = \bigg[\begin{array}{c|c|c|c|c}
\mathbf 0 &\mathbf x_1 &\cdots & \mathbf x_{m-1}
\end{array}\bigg]##

i.e. to get
##\mathbf G = \begin{bmatrix} 0 &\mathbf 0^T \\\mathbf 0 & \mathbf G_p\end{bmatrix}##
where ##\mathbf G_p## is a principal submatrix of interest.
Equivalently we are re-running the above argument with ##m=4## not ##m=5## this time, and then zero padding the result.

and we consider
##\mathbf y := \begin{bmatrix} 0 \\ \mathbf y_p\end{bmatrix}##

where ##\mathbf y_p## is the 'real result' associated with those 4 points, i.e. associated with the principal submatrix. Putting this together gives

##\mathbf W_{\text{almost}}= -2\mathbf G + \mathbf 1\mathbf y^T + \mathbf y \mathbf 1^T = \begin{bmatrix} 0 &\mathbf 0^T \\\mathbf 0 & \mathbf W_p\end{bmatrix}##

our above argument says that
##\text{rank}\Big(\mathbf W_{\text{almost}}\Big)##
##= \text{rank}\Big(\big(\sum_{k=1}^2\mathbf c_k \tilde{ \mathbf s_k}^T\big)+ \mathbf 1\mathbf y^T + \mathbf y \mathbf 1^T\Big) ##
##= \text{rank}\Big(\sum_{k=1}^4\mathbf c_k \tilde{ \mathbf s_k}^T\Big)##
##\leq 4 ##
##\lt 5##

so
##\det\Big(\mathbf W_{\text{almost}}\Big)=0##

This appears to be a weak result since we can eyeball the column of all zeros and see that ##\mathbf W_{\text{almost}}## is singular. But if we notice the homogeneity in the components of the first row and column (all ones except for the diagonal) we can hack our way to the desired result:

##\mathbf D := \begin{bmatrix} -1 &\mathbf 0^T \\\mathbf 0 & \mathbf I_p\end{bmatrix}##
##\mathbf y_1 := \begin{bmatrix} 1 \\ \mathbf y_p\end{bmatrix}##
##\mathbf y_2 := \begin{bmatrix} -1 \\ \mathbf y_p\end{bmatrix}##
i.e. ##\mathbf y_2 = \mathbf D\mathbf y_1##
- - - -
so
##-2\mathbf G + \mathbf 1\mathbf y_1^T + \mathbf y_2 \mathbf 1^T = \begin{bmatrix} 0 &-\mathbf 1^T \\\mathbf 1 & \mathbf W_p\end{bmatrix}##
i.e.
##\mathbf D\mathbf W = -2\mathbf G + \mathbf 1\mathbf y_1^T + \mathbf y_2 \mathbf 1^T##

making use of the involution, ##\mathbf D^2 = \mathbf I##, and the fact that multiplication by invertible matrices doesn't change the rank, we have
##\text{rank}\Big(\mathbf W\Big) = \text{rank}\Big(\mathbf D^2\mathbf W\Big) = \text{rank}\Big(\mathbf D\mathbf W\Big) \leq 4 \lt 5 \longrightarrow \det\Big(\mathbf W\Big) =0##

which completes the proof.
 

fresh_42

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No one ever does the matrix problems... :frown:
This one is quite nice and relates to a favorite topic of mine: Gram matrices, as well as rudimentary geometry.
There may be a more expedient route but I used a rank based argument. This ended up being a bit longer than I had hoped....

Problem 3
##\mathbf W = \begin{bmatrix}
0&1&1&1&1 \\ 1&0&|AB|^2&|AC|^2&|AD|^2 \\ 1&|AB|^2&0&|BC|^2&|BD|^2 \\ 1&|AC|^2&|BC|^2&0&|CD|^2 \\ 1&|AD|^2&|BD|^2&|CD|^2&0
\end{bmatrix} = \begin{bmatrix} 0 &\mathbf 1^T \\\mathbf 1 & \mathbf W_p\end{bmatrix}##

where ##\mathbf W_p## is the principal submatrix of main interest
we need to prove ##\operatorname{det}\big(\mathbf W\big) = 0##

lemma
The sum of ##d## rank one updates has rank at most ##d##.

This is implied by familiar matrix multiplication rank properties.
##\mathbf S \in \mathbb R^{\text{d x n}}## and conforming ##\mathbf C##, so

##\mathbf C\mathbf S ##
##= \bigg[\begin{array}{c|c|c|c|c}
\mathbf c_1 & \mathbf c_2 &\cdots & \mathbf c_{d-1} & \mathbf c_{d}
\end{array}\bigg]\begin{bmatrix}
\tilde{ \mathbf s_1}^T \\
\tilde{ \mathbf s_2}^T \\
\vdots\\
\tilde{ \mathbf s}_{d-1}^T \\
\tilde{ \mathbf s}_d^T
\end{bmatrix} ##
##= \mathbf c_1 \tilde{ \mathbf s_1}^T + \mathbf c_2 \tilde{ \mathbf s_2}^T +... + \mathbf c_d \tilde{ \mathbf s_d}^T ##
##= \sum_{k=1}^d\mathbf c_k \tilde{ \mathbf s_k}^T##

so
##\text{rank}\Big(\sum_{k=1}^d\mathbf c_k \tilde{ \mathbf s_k}^T\Big)= \text{rank}\Big(\mathbf C\mathbf S\Big) \leq \text{rank}\Big(\mathbf S\Big) \leq \min\Big(d,n\Big) \leq d##

Geometrically pleasant special case:
suppose all of these points (A,B,C,D --> relabelled as 1,2,3,4 or ##\mathbf x_1, \mathbf x_2, \mathbf x_3, \mathbf x_4## ) are in the plane, and this mysterious column zero is also in the plane. In effect we can interpret point 0 as the origin.

It isn't needed, but the geometry associated with this very nice in this special case: the above squared distance matrix, suggests that in effect point zero is the origin and points 1,2,3,4 are all (squared) distance of 1 from it -- i.e. they are on the boundary of the unit circle.

In general, the idea is that we have ##m## points and we know the (squared) distance between them. And we know the distance was created by use of an inner product (and specifically via standard dot product of vectors ##\in \mathbb R^2##). In general whenever have have an inner product, I've suggested that its wise to try to look at the Gram Matrix ##\mathbf G##. So we have

##2\mathbf G = -\mathbf W + \mathbf 1\mathbf y^T + \mathbf y \mathbf 1^T##
where ##\mathbf y## is a vector containing the original squared 2-norm of points ##\mathbf x_k## --we have 5 in this case, but it is useful to think in general of ##m## points and ##\mathbf G \in \mathbb R^\text{m x m}##

examine component i,j of this matrix to verify that
##2\cdot g_{i,j} ##
##= -\big \Vert \mathbf x_i - \mathbf x_j\big \Vert_2^2 + \big\Vert \mathbf x_i\big\Vert_2^2 + \big\Vert \mathbf x_j \big\Vert_2^2##
##= -\Big\{\langle \mathbf x_i - \mathbf x_j, \mathbf x_i - \mathbf x_j\rangle \Big\}+ \langle \mathbf x_i, \mathbf x_i\rangle+ \langle \mathbf x_j, \mathbf x_j\rangle##
##=-\Big\{ \langle \mathbf x_i, \mathbf x_i\rangle + \langle \mathbf x_j, \mathbf x_j\rangle - 2\langle \mathbf x_i , \mathbf x_j\rangle \Big\}+ \langle \mathbf x_i, \mathbf x_i\rangle+ \langle \mathbf x_j, \mathbf x_j\rangle##
##= 2\langle \mathbf x_i , \mathbf x_j\rangle ##

which confirms that ##\mathbf G## is a Gram Matrix. Since we know these points are in ##\mathbb R^2##, we know that in fact

##\mathbf X^T\mathbf X = \mathbf G##
##\mathbf X = \bigg[\begin{array}{c|c|c|c}
\mathbf x_0 &\mathbf x_1 &\cdots & \mathbf x_{m-1}
\end{array}\bigg]##
where ##\mathbf X \in \mathbb R^{\text{2 x m}}##

hence
##\text{rank}\big(\mathbf G \big) = \text{rank}\big(\mathbf X^T\mathbf X \big) =\text{rank}\big(\mathbf X\big) \leq d = 2##
and in particular we can use the 'outer product/rank one update' interpretation of matrix multiplication to write ##\mathbf X^T \mathbf X## as a the sum of (at most) 2 rank one matrices. For avoidance of doubt revisit the lemma and let ##\mathbf C:= \mathbf X^T## and ##\mathbf S := \mathbf X## and of course ##d =2## and ##n=m##.

so
##\mathbf G = \sum_{k=1}^2\mathbf c_k \tilde{ \mathbf s_k}^T##

the fact that this holds up to rescaling suggests it would be nicer to write
##-2\mathbf G := \sum_{k=1}^2\mathbf c_k \tilde{ \mathbf s_k}^T##
So I use this going forward -- if its an issue I can introdue more notation I suppose

But we know
##\mathbf W= -2\mathbf G + \mathbf 1\mathbf y^T + \mathbf y \mathbf 1^T = \big(\sum_{k=1}^2\mathbf c_k \tilde{ \mathbf s_k}^T\big) +\mathbf 1\mathbf y^T + \mathbf y \mathbf 1^T = \big(\sum_{k=1}^4\mathbf c_k \tilde{ \mathbf s_k}^T\big) ##
(after re-labelling)

which allows us to conclude
##\text{rank}\big(\mathbf W\big) \leq 4 \lt 5 \longrightarrow \det\big(\mathbf W\big) = 0##

algebraic tweak for the more general case
The above was nice but technically we don't know anything about ##\mathbf x_0##, whether it actually exists in the plane, etc. The hack is to just zero out the thing we are unsure of re-run the above argument with just

##\mathbf X = \bigg[\begin{array}{c|c|c|c|c}
\mathbf 0 &\mathbf x_1 &\cdots & \mathbf x_{m-1}
\end{array}\bigg]##

i.e. to get
##\mathbf G = \begin{bmatrix} 0 &\mathbf 0^T \\\mathbf 0 & \mathbf G_p\end{bmatrix}##
where ##\mathbf G_p## is a principal submatrix of interest.
Equivalently we are re-running the above argument with ##m=4## not ##m=5## this time, and then zero padding the result.

and we consider
##\mathbf y := \begin{bmatrix} 0 \\ \mathbf y_p\end{bmatrix}##

where ##\mathbf y_p## is the 'real result' associated with those 4 points, i.e. associated with the principal submatrix. Putting this together gives

##\mathbf W_{\text{almost}}= -2\mathbf G + \mathbf 1\mathbf y^T + \mathbf y \mathbf 1^T = \begin{bmatrix} 0 &\mathbf 0^T \\\mathbf 0 & \mathbf W_p\end{bmatrix}##

our above argument says that
##\text{rank}\Big(\mathbf W_{\text{almost}}\Big)##
##= \text{rank}\Big(\big(\sum_{k=1}^2\mathbf c_k \tilde{ \mathbf s_k}^T\big)+ \mathbf 1\mathbf y^T + \mathbf y \mathbf 1^T\Big) ##
##= \text{rank}\Big(\sum_{k=1}^4\mathbf c_k \tilde{ \mathbf s_k}^T\Big)##
##\leq 4 ##
##\lt 5##

so
##\det\Big(\mathbf W_{\text{almost}}\Big)=0##

This appears to be a weak result since we can eyeball the column of all zeros and see that ##\mathbf W_{\text{almost}}## is singular. But if we notice the homogeneity in the components of the first row and column (all ones except for the diagonal) we can hack our way to the desired result:

##\mathbf D := \begin{bmatrix} -1 &\mathbf 0^T \\\mathbf 0 & \mathbf I_p\end{bmatrix}##
##\mathbf y_1 := \begin{bmatrix} 1 \\ \mathbf y_p\end{bmatrix}##
##\mathbf y_2 := \begin{bmatrix} -1 \\ \mathbf y_p\end{bmatrix}##
i.e. ##\mathbf y_2 = \mathbf D\mathbf y_1##
- - - -
so
##-2\mathbf G + \mathbf 1\mathbf y_1^T + \mathbf y_2 \mathbf 1^T = \begin{bmatrix} 0 &-\mathbf 1^T \\\mathbf 1 & \mathbf W_p\end{bmatrix}##
i.e.
##\mathbf D\mathbf W = -2\mathbf G + \mathbf 1\mathbf y_1^T + \mathbf y_2 \mathbf 1^T##

making use of the involution, ##\mathbf D^2 = \mathbf I##, and the fact that multiplication by invertible matrices doesn't change the rank, we have
##\text{rank}\Big(\mathbf W\Big) = \text{rank}\Big(\mathbf D^2\mathbf W\Big) = \text{rank}\Big(\mathbf D\mathbf W\Big) \leq 4 \lt 5 \longrightarrow \det\Big(\mathbf W\Big) =0##

which completes the proof.
I have found this problem in Tery's Newsletter to his Blog. It is the Cayley Menger Determinant. Your algebraic and his proof are along the same idea: Use ##|AB|^2=|A|^2\cdot|B|^2 -2A\cdot B## and write the matrix ##W=M+M^\tau-2G## where ##M## has the rows ##1,|A|^2,|B|^2,|C|^2,|D|^2## and is thus of rank ##1## and ##G## are all the inner products, surrounded by zeroes in the first column and row.

The coclusion then goes:
##\operatorname{rk}M = \operatorname{rk}M^\tau = 1 ## and ##G=S\cdot S^\tau## with the ##5 \times 2## matrix with rows ##0,A,B,C,D##, i.e. ##\operatorname{rk}G \leq 2##. Hence ##\operatorname{rk}\left( M+M^\tau -2G\right) \leq 1+1+2=4<5 ## and its determinant vanishes.
 

DEvens

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Oh man! I hit the cancel button after typing in my whole answer. Try again.

For the following I will be using that SO(n) is a group and that ##\{ e_i \}## is a basis for it. So if ##F e_1 = T## then ##F^{-1} T = e_1##.

a) Given ##F e_1 = F_0 e_1 = T## then by group properties the inverse of ##F## and ##F_0## must exist.

Try ##\Phi = F_0^{-1} F##.

Then ##F_0 \Phi = F_0 F_0^{-1} F = F##, so the first part is true. And ##\Phi e_1 = F_0^{-1} F e_1 = F_0^{-1} T = F_0^{-1} F_0 e_1 = e_1## so the second part is true.

b) Since ##\Phi e_1 = e_1## and ##F_0 e_1 = T## then ##F e_1 = F_0 \Phi e_1 = F_0 e_1 = T##. That is, ##F## gives a frame also. Whether it is distinct depends on ##\Phi##.

c) Start with ##F= F_0 \Phi## and take the time derivative.

##\frac{d F}{dt} = F A = \frac{d F_0} {dt} \Phi + F_0 \Phi^{\prime} = F_0 A_0 \Phi + F_0 \Phi^{\prime}##

Now use that ##(AB)^{-1} = B^{-1} A^{-1}## and multiply the whole thing by ##F^{-1}##.

## A = F^{-1} F_0 A_0 \Phi +F^{-1} F_0 \Phi^{\prime}##
## = (F_0^{-1} F)^{-1} A_0 \Phi +(F_0^{-1} F)^{-1}\Phi^{\prime}##
## = \Phi^{-1} A_0 \Phi +\Phi^{-1}\Phi^{\prime}##
 
Last edited:

DEvens

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I have updated my post in 7. Can you please have a look?
 

fresh_42

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The curve is originally parameterized in its arc length divided by ##\pi, or \gamma(t) = \gamma(\frac{s}{\pi}).##
Or a bit more detailed: ##p(s)=\gamma(\Phi(s))=\gamma\left(\dfrac{s}{\pi}\right) = \cos(s)\cdot e^{is}##.
I had hoped for the arc length integral and a calculation why ##\gamma## solves the circle equation for our youngsters to follow the argumentation, but o.k.
 

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