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most recent 30 from mathoverflow.net 2018-12-10T12:30:06Z

If not functions, then what should be the backbone of the secondary curriculum

Mon, 12/10/2018 - 05:14

In the early 1900s Felix Klein lay out his vision for secondary mathematics curriculum. He wanted schools to teach calculus, so that universities would not be burdened by it. And at the core of the curriculum was to be the notion of function, which was conceptually necessary for the foundations of calculus. (I know I make a very messy story short here, but I think the basic points are more or less correct.)

All of this was over 100 years ago, and to this day we teach according to Klein's vision. My question has to do with whether now, given the current state of the art, we would chose to design the mathematics curriculum in the same way. The choice of function as the central topic is particularly of interest to me-- it seems a bit strange that we teach young students functions, given the difficulty and abstractness of the topic (high school students don't use the full abstractness, yet they are supposed to -- unless they don't read the book-- know about it.

Thank you for indulging me another soft question, if indeed you do. This question relates to earlier discussions here about the work of Felix Klein (e.g. Felix Klein on mean value theorem and infinitesimals), but I am specifically interested in the notion of function, and our modern view about whether it is wise to dedicate nearly the entire secondary curriculum to the calculus.

Prove that a set is relatively compact by a MNC

Mon, 12/10/2018 - 05:08

I am working on an article based with the notion of Measure of noncompactness.

Let $ \mathcal {M}_E $ be the family of all nonempty bounded subsets of $ E $ (a Banach space)

Definition:

The function $ \beta: \mathcal{M}_E \rightarrow \mathbb R^ + $ is a measure of noncompactness if, for all $ C, D\in \mathcal {M}_E $:

1) If $ \beta (C) = 0 $ Then, $ C $ is relatively compact.

2) $ C\subset D \Rightarrow \beta (C) \leq \beta (D)$.

3) $ \beta (D) = \beta (conv (D)) $.

4) $ \beta (D) = \beta (\overline {conv (D)}) $.

5) $ \beta \big(\lambda C + (1- \lambda) D \big) \leq \lambda \beta (C) + (1- \lambda) \beta (D) $, for all $ \lambda \in [0,1] $.

6) If $ D_n \in \mathcal {M}_E $, and $ D_{n+1} \subset D_{n} $ and if $ \lim_{n \to + \infty} \beta (D_n) = 0 $, then: $ D_{\infty} = \bigcap_{n =}^ {+\infty} D_n \neq \emptyset $.

In this paper, Theorem 2.1's proof we have:

A sequence $(u_n)_{n\in\mathbb N}$ and $\mathcal C_n=\overline{conv}\{u_n,u_{n+1},...\} \text{ for } \text{ for } n=0,... \:\:.$ Such that $\mathcal C_n\in \mathcal {M}_E $, and $\beta(\mathcal C_n)\rightarrow 0 \text{ as } n\rightarrow \infty \:\:.$

As $\mathcal C_n\subset \mathcal C_{n-1}$ and by $6)$ of the definition of MNC, we have $\mathcal C_{\infty}=\bigcap_{n=1}^{\infty}\mathcal C_n\neq \emptyset$ and $\mathcal C_{\infty}$ is relatively compact.

Hence, for every $\epsilon>0$ there exists an $n_0$ such that $\beta(\mathcal C_{n})<\epsilon$ for every $n>n_0$.

The authors said that, this concludes that $\mathcal C_{n_0}$ is relatively compact.

I tried to find out why this statement is true but to no avail. Any help is appreciated.

Hadamard matrix research

Mon, 12/10/2018 - 04:22

I need to find out whether something new has happened in the study of Hadamard matrices for the last 10 years. If someone was engaged in matrices, can you recommend literature? K. Horadam is the newest research I found.

Can we derive large cardinals by a principle of cardinals to ordinals isomorphism?

Mon, 12/10/2018 - 04:18

Working in Morse-Kelley set theory

Add to it the following schema:

Cardinals to Ordinals isomprhism: if $\psi(Y)$ is a formula in which only the symbol $``Y"$ appear free [and only free], and the symbol $``x"$ never occurring, and $\psi(x)$ is the formula obtained from $\psi(Y)$ by merely replacing each occurrence of the symbol $``Y"$ by an occurrence of the symbol $``x"$, then:

$\forall x (x \text{ is an ordinal }\to \exists Y(\psi(Y) \wedge x < Y)) \\ \to \{x | x \text{ is a cardinal } \wedge \psi(x) \} \text{ is order isomorphic to } \{x | x \text{ is an ordinal }\} $

is an axiom.

Questions:

  1. Is there a clear inconsistency with this principle?

  2. if this is consistent, then what's the consistency strength of the above theory?

Global dimension of a graded algebra

Mon, 12/10/2018 - 03:50

Let $A= \bigoplus\limits_{n=0}^{\infty}{A_n}$ be an $\mathbb{N}$-graded algebra with semisimple $A_0$.

Question: Do we have that the global dimension of $A$ is equal to $\sup \{i \geq 0 | Ext_A^i(A_0,A_0) \neq 0 \}$?

Maybe one should ask this question under some mild further restrictions such as $A$ being noetherian and/or $A_{i+j}=A_i A_j$ for all $i,j \geq 0$.

This question was asked in the special case of quadratic algebras here: Quadratic algebras and Koszul algebras and a positive answer would have a nice applications.

Moduli space of semistable bundles: proof of diagram being commutative

Mon, 12/10/2018 - 03:19

Suppose $C$ is smooth complex projective curve of genus $g$ and $J$ is the Jacobian of $C$. Let $\mathcal{M}= M(n,d) $ be the moduli space of semistable bundles of rank n and degree d on $C$ and let $SM= SM(n, L)$ be the moduli space of those bundles whose determinant is isomorphic to a fixed line bundle $L$ on $C$. Consider the following diagram:

$\begin{array}[c]{ccc} SM \times J &\stackrel{\tau}{\rightarrow}& \mathcal{M}\\ \downarrow\scriptstyle{\alpha}&&\downarrow\scriptstyle{\beta}\\ J&\stackrel{\rho}{\rightarrow}&J \end{array}$

Where $\alpha$ is the projection, $\rho$ is the $n$-th tensor power map, $\beta$ is the composite of $\det : \mathcal{M} \rightarrow J_d$ followed by multiplication by $L^{-1} : J_d \rightarrow J$.

In the paper (Donagi and Tu, page 351) http://www.intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0001/0003/a006/

they are using commutativity of the above diagram. I would like to know the proof of the diagram being commutative.

Computing Bohr Radii

Mon, 12/10/2018 - 02:54

The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as $$R = \sup\limits_{0<r<1} \left\{ r\ |\ \sum\limits_{k=0}^{\infty}|a_k|r^k \leq |f|_\mathbb{D} \text{ for all }f(z)=\sum\limits_{k=0}^{\infty} a_{k}z^{k}\in\mathcal{H}(\mathbb{D}) \right\},$$ where $|f|_{\mathbb{D}}=\sup\limits_{z\in\mathbb{D}} |f(z)|$, $\mathcal{H}(\mathbb{D})$ is the space of functions analytic in $\mathbb{D}$, and $\sum\limits_{k=0}^{\infty}|a_k|r^k$ is known as the majorization of $f$. Harold Bohr and later M. Riesz, I. Shur, and F. Wiener showed that $R=\frac{1}{3}$, and it is known the only extremal functions satisfying $\sum\limits_{k=0}^\infty |a_k|(\frac{1}{3})^k=|f|_{\mathbb{D}}$ are constant. There are several extensions of this result (for instance one can exactly compute the radius in terms of entries in an associated Toeplitz matrix when $\mathcal{H}(\mathbb{D})$ is replaced with $\mathcal{P}_{n}$, polynomials of degree of at most $n$, another interesting example being functions on the bi-disk $f\in\mathcal{H}(\mathbb{D}^2)$, where the radius is unknown).

One question that is interesting is how one could effectively numerically estimate the Bohr radius for a space of functions (polynomials of degree $m$ say of one or more variables). For example, how might one numerically compute a sequence of polynomials $p_{k}(z_1,\dots,z_n)$ each of degree at most $m$, and the corresponding values of $r_{k}$ so that the majorization of $p_{k}(z_1,\dots,z_n)$, $p^{*}_{k}$ attains the maximum value of $p^{*}_{k}(r,r,...,r)$ on $\mathbb{D}^n$ for value $r_k$, and $r_k\rightarrow R^{n}_{m}$, the corresponding Bohr radius of this space, as $k\rightarrow \infty$ where $$R^{n}_{m}=\sup {r}\ \hspace{3 mm} s.t.\hspace{3 mm} \ |\sum\limits_{|\alpha|\leq m} c_{\alpha}z^{\alpha}|<1\ \text{for }(z_1,\dots,z_{n})\ \text{ s.t. } \|z\|_{\infty}=\max\limits_{1\leq j\leq n} |z_{j}|<1 \implies \sum\limits_{|\alpha|\leq m} |c_{\alpha}z^{\alpha}|<1\ \text{for } \|z\|_{\infty}<r$$

Second order necessary and sufficient conditions for convex nonsmooth optimization

Mon, 12/10/2018 - 02:13

For convex smooth optimization, first and second order necessary and sufficient conditions are well known. Does such standard second order necessary and sufficient conditions exist for convex nonsmooth optimization. For first order, we have the well known condition that zero must belong to the subgradient set of the function. In that sense what is a second order analogue of Hessian being positive semidefinite in convex nonsmooth settings. What will differ in constrained and unconstrained settings?

Note: I had asked the same question on Mathematics Stack Exchange but did not get any answer. I felt like MathOverflow is a better place to ask this question as it probably involves advanced research.

Show the spectral radius of a matrix is smaller than 1

Mon, 12/10/2018 - 01:38

Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (empty places are zero):

The infinite norm of $\hat{\bf H}$ is obviously 1, and I know spectral radius is no larger than any natural norm. My question is how do I prove the spectral radius of this matrix is smaller than 1?

I did a simple numerical experiment, and found the claim should hold. If $p \to \infty$ and $\hat{N} \to \infty$, then the spectral radius should approach to 1.

If we fix $p = 5$ and let $\hat{N}$ go from 5 to 100, we have

If we fix $\hat{N} = 10$ and let $p$ go from 5 to 100, we have

k-ary necklaces with conserved/fixed indexes

Mon, 12/10/2018 - 01:27

I asked a question in a previous post about enumerating all possible k-ary bracelets with certain positions fixed to a specific bead/character. I asked a few mathematicians, and it seems its quite difficult to do so in a reasonable way.

However, it turns out that I am only looking for necklaces, not bracelets. Eg I only care about rotations, not reflections.

Hopefully its ok to ask the basically the same question again, Im thinking it might be more feasible to enumerate if you only have to factor in rotations.

To restate the question, say I have k possible integers to pick from, and my program randomly picks from the set of k when generating necklaces of length n, I know I can enumerate the maximum possible unique necklaces that could be generated for n/k using the equation listed here. This is great because now I have an upper bound on my programs duration and know when to stop running if I have already found all possible unique necklaces for a given n/k

However, my program also has the option to conserve, or fix, any particular value from the set of k, at any position in the necklace. It allows this to be done for as many positions of length n that you want.

For example, given n = 4, and k = [0, 1, 2, 3, 4, 5, 6], and considering the following necklaces equivalent:

[0,0,0,1] and [1,0,0,0]

I can use the formula as mentioned here to calculate easily how many unique necklaces I can possibly generate.

But if the user of my program specifies that they want the generated necklaces to always have the value 1 at position 1 and the value 2 at position 4, eg the random necklace generator will only create sequences such as:

[1,0,0,2] or [1,6,4,2] or [1,2,3,2]

The question is: given necklaces of length n containing any selection of characters from set k with f positions in n being fixed to c different values from k, is it possible to enumerate the total unique necklaces that could be made

Non reduced projective schemes

Mon, 12/10/2018 - 01:20

Let $X$ be separated noetherian scheme. Suppose that $X_{red}$ (the associated reduced scheme) is projective. Does it follow that $X$ is also projective ? Otherwise what are the minimal conditions on $X$ ?

Is every finite graph isomorphic to the proximity graph of some $S\subseteq \mathbb{R}^n$?

Mon, 12/10/2018 - 00:44

This is the question that I should have asked before asking this older question.

If $(X,d)$ is a metric space, we associate with it a simple, undirected graph, called its proximity graph $G(X,d)$ given by $V(G(X,d)) = X$ and $$E(G(X,d)) = \big\{\{x,y\}:x\neq y\in X \text{ and } d(x,y)\leq 1\big\}.$$

As MO user @YCor pointed out in a comment to a recent deleted question, given any (not necessarily finite) simple, undirected graph $G=(V,E)$, the map $d:V\times V\to \mathbb{R}$ given by $d(v,v) = 0$ for $v\in V$, $d(v,w)=1$ iff $\{v,w\}\in E$ and $d(v,w) = 2$ otherwise gives a metric on $V$ such that $G\cong G(V,d)$.

Question. If $G=(V,E)$ is a finite graph, is there a positive integer $n\in\mathbb{N}$ and a finite subset $S\subseteq \mathbb{R}^n$ such that $G \cong G(S,||\cdot||),$ where $||\cdot||$ denotes the Euclidean metric that $S$ inherits from $\mathbb{R}^n$?

Bifurcations due to a nonlinearity parameter

Sun, 12/09/2018 - 23:21

Suppose we want to analyze the behavior of the system $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},t;\varepsilon),\quad \mathbf{x}\in\mathbb{R}^n,\quad t\in\mathbb{R}^+,\quad\varepsilon\in\mathbb{R}^+, $$ in the case where $\mathbf{f}(\mathbf{x},t;\varepsilon)$ is smooth and depends in some way on the variable $x_i^{1+\varepsilon}$ for some $1\leq i \leq n.$ Have bifurcations due to changes in $\varepsilon$ in systems such as these (perhaps also in the discrete analogue) been studied extensively, and if so, where can I find literature on this?

Test the null hypothesis [on hold]

Sun, 12/09/2018 - 22:25

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A new treatment was advertised to reduce the incidence of a common illness. It was possible to get the illness more than once. Consider 500 individuals who participated in an experiment and were randomly assigned to treatment and placebo groups. Test the null hypothesis that the treatment and placebo populations are equivalent.

Finite subgroups of $PSU(3)$

Sun, 12/09/2018 - 20:55

I'm looking for a reference to a classification or description of finite subgroups of $SU(3)$ that contain the center, or equivalently $PSU(3)$. Can anyone point me in the right direction?

A conjectural formula for the "minimal degree function", $k\rightarrow d\min(k)$, attached to recursion, $f\rightarrow A(f)$, in char $3$

Sun, 12/09/2018 - 18:50

THE RECURSION: $f\rightarrow A(f)$

$A: t(\mathbb{Z}/3)[t^3]\rightarrow t(\mathbb{Z}/3)[t^3]$ is the $\mathbb{Z}/3$-linear map with $A(t)=0, A(t^4)=t, A(t^7)=t^4$, and $A((t^9)f)=(2t^9)A(f)+(t^3)A((t^3)f)$.

An induction shows that $A(t^k)$ is either $t^{k-3} + O(t^{k-6})$ or $2t^{k-3} + O(t^{k-6})$ when $k$ is $4$ or $7 \bmod 9$, and is $O(t^{k-6})$ when $k$ is $1 \bmod 9$. (Here $O(t^m)$ is shorthand for an element of degree at most $m$ in $t$)

$\mathbf{d\min (k)}$

Let $N'$ consist of all positive integers that are $1\bmod 3$. For $k$ in $N'$, $d\min(k)$ is the smallest degree of $A(f)$, $f$ running over the elements of $t(\mathbb{Z}/3)[t^3]$ of exact degree $k$. Note that $d \min$ is a function from $N'$ to the union of $N'$ with $-\infty$. By the paragraph above $d\min (k)$ is $k-3$ when $k$ is $4$ or $7\bmod 9$. Suppose however that $k$ is $1 \bmod 9$; the last sentence shows that $d\min(k)$ cannot be $1$ or $4\bmod 9$ and so must be $7\bmod 9$. Also by the last paragraph $d\min(k)$ is at most $k-6$; we conclude that $d\min(k)$ is at most $k-12$.

EXAMPLES

$A(t^{10} - t^{7})=0$, and $d\min(10)=-\infty$

$A(t^{19} + t^{16} + t^{13})=2t^{7}$, and $d\min(19)=7$.

SOME RESULTS FOR $d \min(k)$

We've seen that when $k$ is $4$ or $7\bmod 9$, $d\min(k)=k-3$. More labor shows:

(a) When $k$ is $19\bmod 27$, $d\min(k)=k-12$

(b) When $k$ is $37$ or $64\bmod 81$, $d\min(k)=k-21$

(c) When $k$ is $55\bmod 81$, $d\min(k)=k-30$.

The question remains--what is $d\min(k)$ for $k$ in the three remaining congruence classes $\bmod 81$--the classes of $1,10,$ and $28$. (These are the only classes containing $k$ with $d\min(k)=-\infty$). We now present a conjectural answer.

THE QUESTION

Do the following hold?

  1. $d\min(81n+1)=9d\min(9n+1) -11$

  2. $d\min(81n+10)=9d\min(9n+1) +7$

  3. $d\min(81n+28)=9d\min(9n+1) +16$

Tim Hickey has kindly provided me with a computer program that establishes that 1,2, and 3 hold for $n$ up to $100$.

MOTIVATION

An elementary proof of 1,2 and 3 should lead an alternative proof of the Bellaiche-Khare-Medvedovsky result about the structure of the Hecke algebra attached to the space of $\bmod 3$ elliptic modular forms of level 1. And there are similar empirically verified conjectures for (much!) more complicated recursions whose proof would lead to an understanding of the structure of related Hecke algebras in levels $\Gamma_0(5)$ and $\Gamma_0(13)$.

Definition of $\in_c$ for power objects

Sun, 12/09/2018 - 16:50

On the nLab page for power objects, the object $\in_c$ is defined as the domain of a monomorphism $\in_c\hookrightarrow c\times\Omega^c$, and it is mentioned at the end of the article that in any topos we have that the power object of an arbitrary object is (isomorphic to) its exponential with the subobject classifier.

It is also mentioned that in the case $c\cong{\bf 1}$ the power object becomes a subobject classifier. This is easy to see considering the diagram

in the case $c\cong{\bf 1}$ since ${\bf1}\times\Omega^{\bf1}\cong\Omega$ and ${\bf 1}\times d\cong d$, which means that $\in_{\bf 1}\cong{\bf 1}$ with the mono in question being 'true' $\top:{\bf 1}\to \Omega$, $\chi_m:d\to\Omega$ the characteristic function of $r$, and the top of the square being $!:r\to{\bf 1}$.

I'm trying to understand the object $\in_c$ when $c\ncong{\bf 1}$, both in ${\bf Sets}$ and more generally in any topos. I've tried 'fusing' the universal properties of an exponential and a subobject classifier to produce $\in_c$, but it is unclear how the domain of $\top$ changes to yield something besides ${\bf 1}$ -- it seems like the monomorphism $\in_c\hookrightarrow c\times\Omega^c$ is perhaps a currying of some sort?

For a more precise version of the question:

Let $\mathcal{C}$ be a closed category with finite limits and a subobject classifier. How can we define $\in_c$ for an arbitrary $c\in{\bf Ob}_\mathcal{C}$ using just the above structure, and what set is $\in_c$ in the case $\mathcal{C}={\bf Sets}$?

What to do if you notice a substantial improvement to a result in a paper whilst refereeing it?

Sun, 12/09/2018 - 08:55

What would you do/have you done in such a situation?

1) Hand out the improvement for free in your report

2) Wait until the result is published and then submit elsewhere

3) Inform the editor about the situation and ask for advice

The paper is not posted publicly so contacting the authors directly informing them and asking what they want to do is out of the question.

Quadratic algebras and Koszul algebras

Sun, 12/09/2018 - 07:35

Let $A$ be a quadratic algebra and $B$ the Ext-algebra of $A$. In case $A$ is a Koszul algebra, we should have that the global dimension of $A$ plus one is equal to the Loewy length of $B$ (is there a reference for this?).

Namely we should have $gldim(A)= \sup \{ i \geq 0 | Ext_A^i(A_0,A_0) \neq 0 \} = LL(B)-1$, where LL stands for Loewy length and $A_0$ is the degree zero part of the graded algebra $A$. Im not sure in general about the first equality here (it should at least hold for $A$ finite dimensional), but the second equality should be correct since $B$ is generated in degree 0 and 1.

Thus $gldim(A)+1=LL(B)$.

Question 1: Is $gldim(A)= \inf \{ i \geq 0 | Ext_A^i(A_0,A_0)=0 \}$ true in general or under some restrictions? Is there a reference?

Question 2: Is a quadratic algebra Koszul iff $gldim(A)+1=LL(B)$ holds?

Maybe on needs to assume further restrictions for question 1, but I think in some form it will be true. One should be able to apply this in two nice examples:

a) $A=K[x_1,...,x_n]$ the polynomial ring in $n$ variables. Here $B$ is the Grassmann algebra in $n$ variables which has Loewy length $n+1$ and this shows that $A$ has global dimension $n$.

b)$A=kQ$ the quiver algebra of an arbitrary quiver with finitely many points and at least one arrow (that may be infinite dimensional). Then $B$ is the algebra with the same quiver and radical square zero and thus Loewy length 2. Thus the formula would give here that $A$ has global dimension one and I think the proof of this is actually quite complicated without those tools.

Are inclusions "canonical" injections?

Sun, 12/09/2018 - 04:56

[Background: I asked a vague question the other day, but as a result of the answers, particularly Andrej Bauer's, I now have a precise question]

Summary of question: the inclusions are a particularly "good" class of morphisms in the category of sets. I've written down a bunch of properties they have, and reserve the right to write down more. Are there other classes of morphisms which share these properties?

Technical point. in ZFC if you define a function to be a collection of ordered pairs $(x,f(x))$ then two functions with different codomains can be equal as sets (and hence as functions). In this question, when I talk about a morphism $f:X\to Y$ in the category of sets, I mean the data of $f$ and $X$ and $Y$, as is normal in category theory. A function has a well-defined domain and codomain.

Set-up

I am looking for the following piece of data. For each set $X$ I want a set $P(X)$ of injections $f_i:A_i\to X$ in the category of sets, called the "good morphisms to $X$", with the following properties.

1) [representing injections]. For every injective map $g:Y\to X$ in the category of sets, there is a unique good $f:A\to X$ such that $g$ is isomorphic to $f$ in the sense that there's an isomorphism $Y\to A$ which makes the obvious triangle commute.

For conditions (2) and (3) we have injections $f:A\to B$ and $g:B\to C$ with composition $h:=g\circ f:A\to C$.

2) [closure] If $f:A\to B$ is good and $g:B\to C$ is good then $h := g\circ f:A\to C$ is good.

3) [g,h good implies f good] If $h:A\to C$ and $g:B\to C$ are both good, then $f$ is good too.

An example of a good class of maps is the set of all inclusions $i:A\to B$ where $A$ is a subset of $B$.

The question

The question (for which the answer is surely "yes of course") is: are there any other ways to choose a good class of maps with these properties?

I hesitate to put any more conditions, for example conditions about products of maps, because an object like $X\times Y$ is only defined up to unique isomorphism in the category of sets. This question is not at all "natural" in the category-theory sense, because if I replace my category by an equivalent category then I can't easily move my data from one to the other (as far as I can see). On the other hand, there are "canonical" (whatever that weasel word means!) constructions of products and limits in the category of sets, so I reserve the right to add more conditions about the behaviour of "nice" maps under limits if this question gets spiked too easily. In fact the more I think about it the more I wonder whether adding more criteria using these "canonical" constructions of limits (as an actual subset of a product, with the crucial observation being that subsets are being used) can actually turn this question into one with a positive answer, i.e. classifying the inclusions as those morphisms satisfying a bunch of properties, not all of them as "canonical" as one might like...

NB the word "canonical" does not have a definition in my mind, and mathematicians sometimes use it in a way where it can actually be replaced by a formal definition, but sometimes they use it to mean something which just looks like a good idea. I am trying to work out if inclusions are "canonical" monomorphisms, and this is a great example of a poor usage of the word in the sense that once you start digging you realise that you cannot supply a definition. I am attempting to supply a definition and still strongly suspect that I have failed.

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