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## Tuesday, December 14, 2010

### Equivalent description in terms of a Hecke Algebra - Part II

Second in a three part series of me trying to describe the Bost-Connes algebra. Part I: $\mathbb{Q}$-Lattices and the Presentation. Part II: Equivalent description in terms of a Hecke Algebra. Part III: Describing the key Sub-algebra.

In this post I hope to very briefly describe the Hecke Algebra formulation of the Bost-Connes System. I'm embarrassed to say it, but I was initially afraid of the Hecke Algebra description (the wikipedia page didn't include much information I could understand). Now that I've read through it, I've realized that 1) it's not that difficult and 2) it's not really used in the more interesting generalizations to Complex and Real Multiplication (Shimura varieties and ${\mathbb{Q}}$-lattices seem much more important.) Nevertheless, I thought I'd talk about because it was the way the system was formulated in the original 1995 Paper Hecke Algebras, Type III Factors and Phase Transitions with Spontaneous Symmetry Breaking in Number Theory''. So for any ring $R$, we define:

$\displaystyle P_R := \left\{ \begin{pmatrix} 1 & b \\ 0 & a \end{pmatrix} : a, b \in R, a\; \text{invertible} \right\}$

And let $\Gamma_0 = P_\mathbb{Z}^+ = \left\{ \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix} : n \in \mathbb{N} \right\}$ and $\Gamma = P_\mathbb{Q}^+ = \left\{ \begin{pmatrix} 1 & a \\ 0 & k \end{pmatrix} : a, k \in \mathbb{Q}^+ \right\}$. We'll be looking at the coset space $\Gamma/\Gamma_0$ (actually the double coset $\Gamma_0 \char\\ \Gamma/\Gamma_0$, but we aren't so worried about that). You have to be careful to watch the left and right cosets here. First note that the left action of $\Gamma_0$ on $\Gamma/\Gamma_0$ has finite orbits. To see this, let $\gamma = \begin{pmatrix} 1 & a \\ 0 & k \\ \end{pmatrix} \in \Gamma$. Then $\begin{pmatrix} 1 & n \\ 0 & 1 \\ \end{pmatrix} \gamma \begin{pmatrix} 1 & m \\ 0 & 1 \\ \end{pmatrix} = \begin{pmatrix} 1 & n + a + mk \\ 0 & k \end{pmatrix}}$ for $n,m \in \mathbb{N}$. As $m$ varies, $mk$ only takes finitely many values modulo $\mathbb{Z}$, the number depending on the $b$, where $k = \frac{a}{b}$. Thus we do get a finite orbit for $\gamma \Gamma_0$. In fact, the same holds for the right action. (Hecke Algebras, Type III Factors and Phase Transitions with Spontaneous Symmetry Breaking in Number Theory'' pg 17) We define the Hecke algebra $\mathcal{H}_\mathbb{Q} (\Gamma,\Gamma_0)$ as the convolution algebra of finitely supported functions $f: \Gamma_0 \char\\ \Gamma \rightarrow \mathbb{Q}$ such that

$\displaystyle f(\gamma \gamma_0) = f(\gamma), \;\;\; \text{for all}\;\gamma\in\Gamma, \; \gamma_0\in\Gamma_0$

. The convolution is given by:

$\displaystyle f_1 f_2 (\gamma) = \sum_{g \in \Gamma_0 \char\\ \Gamma} f_1(\gamma g^{-1})f_2(g)$

And the involution by:

$\displaystyle f^*(\gamma) = \overline{f(\gamma^{-1})}$

We can complexify this algebra using a tensor product with $\mathbb{C}$ to get:

$\displaystyle \mathcal{H}_\mathbb{C}(\Gamma,\Gamma_0) = \mathcal{H}_\mathbb{Q}(\Gamma,\Gamma_0) \otimes_\mathbb{Q} \mathbb{C}$

Similiar to before, we have a representation on the Hilbert space $\mathcal{H} = \ell^2 (\Gamma_0/\Gamma)$ defined by

$\displaystyle (\pi(f)\psi)(\gamma) = \sum_{g \in \Gamma_0/\Gamma} f(\gamma g^{-1}) \psi(g)$

Where $\psi : \Gamma_0/\Gamma \rightarrow \mathbb{C} \in \ell^2(\Gamma_0/\Gamma)$. This allows the same $C^*$-algebra completition as before, and the time evolution is given by

$\displaystyle \sigma_t(f)(\gamma) = \left(\frac{L(\gamma)}{R(\gamma)} \right)^{-it} f(\gamma)$

. Where $L(\gamma)$ is the cardinality of the left $\Gamma_0$ orbit of $\gamma \in \Gamma/\Gamma_0$ and $R(\gamma) = L(\gamma^{-1})$. The aforementioned paper actually derives the same set of generators and relations that I described in my last post, hence this Hecke algebra is the same as the algebra we built from commensurability on 1dQL's modulo scaling.
Next post will describe the key subalgebra of the BC system.

## Sunday, December 12, 2010

### Presentations of the Bost-Connes Algebra - Part I

First in a three part series of me trying to describe the Bost-Connes algebra. Part I: $\mathbb{Q}$-Lattices and the Presentation. Part II: Equivalent description in terms of a Hecke Algebra. Part III: Describing the key Sub-algebra.

Our goal here is to present the algebra of the Bost-Connes system in terms of generators and relations (Prop 3.23, Noncommutative Geometry, Quantum Fields, and Motives).
Let $\mathcal{G}=\{(r,\rho) : r \in \mathbb{Q}^{*}_{+}, \rho \in \hat{\mathbb{Z}}\; \text{such that} \; r \rho \in \hat{\mathbb{Z}}\}$ be our groupoid of commensurable 1dQL's modulo scaling, and let $C^{\ast}[\mathcal{G}]$ be the $C^{\ast}$-completion of the convolution algebra of continuous complex-valued functions on $\mathcal{G}$ with compact support. It's not too difficult to see that $C^{\ast}[\mathcal{G}]$ contains $C(\hat{\mathbb{Z}})$ (continuous complex-valued function on $\hat{\mathbb{Z}}$). Pontryagin duality (which I do not fully understand, but I'm not worried about that at this point) gives us an isomorphism between $C(\hat{\mathbb{Z}})$ and $C^{\ast}$ group algebra $C^{\ast}[\mathbb{Q}/\mathbb{Z}]$. So let $e_{\gamma}, \gamma \in \mathbb{Q}/\mathbb{Z}$ be the canonical additive bases for $C^{\ast}[\mathbb{Q}/\mathbb{Z}]$. E.g., for an $f \in C^{\ast}[\mathbb{Q}/\mathbb{Z}]$, we can write

$\displaystyle f = \sum_{\gamma \in \mathbb{Q}/\mathbb{Z}} \lambda_{\gamma} e_{\gamma}\;\;\; \lambda_{\gamma} \in \mathbb{C}$

So that $f(\gamma) := \lambda_{\gamma}$. Our $e_{\gamma}$ can define a function $e_{\gamma} : \mathcal{G} \rightarrow \mathbb{C}$ by

$\displaystyle e_{\gamma}(r,\rho) = \begin{cases} \text{exp}(2\pi i \rho(\gamma)) & r=1 \\ 0 & \text{otherwise} \end{cases}$

Hence $e_{\gamma} \in C^{\ast}[\mathcal{G}]$. Moreover, they behave under convolution in $C^{\ast}[\mathcal{G}]$ just as they would as the basis for the C* group algebra, namely, we have:

1. $e_{\gamma_1} e_{\gamma_2} = e_{\gamma_1 + \gamma_2}$
2. $e_{0} = \mu_1$, id est, $e_{0}$ maps to unity in $C^{\ast}[\mathcal{G}]$. (See edit in previous post)
3. For an $f \in C^{\ast}[\mathcal{G}]$, the involution of $f$ is defined as $f^{\ast}(r,\rho) := \overline{f(r^{-1},r \rho)}$. This gives us $e_{\gamma}^{\ast} = e_{-\gamma}$.
The proofs of the above are straightforward, just following from the definitions (keep in mind that the multiplication is convolution in $C^{\ast}[\mathcal{G}]$).
Now, recall that the space of 1dQL's modulo scaling looks like $\hat{\mathbb{Z}}$ and that our groupoid $\mathcal{G}$ describes the commensurability relation of 1dQL's modulo scaling, so that $C(\hat{\mathbb{Z}}) \simeq C^{\ast}[\mathbb{Q}/\mathbb{Z}]$ captures quite a bit of the information in $C^{\ast}[\mathcal{G}]$. The commensurability condition is captured by a semigroup cross product with $\mathbb{N}$. I have not been able to figure out how, nor have I found a description in any of the literature I've read. But I can give a description of the action. For an $f \in C(\hat{\mathbb{Z}})$, let

$\displaystyle \alpha_n(f) (r, \rho) = \begin{cases} f(n^{-1} \rho) & \rho \in n \hat{\mathbb{Z}} \\ 0 & \text{otherwise} \end{cases}$

Recall the $\mu_n$'s from the last post, id est,

$\displaystyle \mu_n(r,\rho) = \begin{cases} 1 & r=n \\ 0 & \text{otherwise} \end{cases}$

By treating our $f \in C(\hat{\mathbb{Z}})$ as $f(r,\rho) := f(\rho)$, we can conjugate $f$ with $\mu_n$ and its involution and get:

$\displaystyle \mu_n f \mu_n^{\ast} = \alpha_n(f)$

Again, the multiplication here is convolution in $C^{\ast}[\mathcal{G}]$, and the proof is straightforward. As discussed in my last post, the $\mu_n$'s also behave nicely under the following relations (proofs are still straightforward):
1. $\mu_n \mu_m = \mu_{nm}$
2. $\mu_n^{\ast} \mu_n = \mu_1$, where $\mu_1$ is the unity/multiplicative identity.
So we see then that out $e_{\gamma}$'s describe $C(\hat{\mathbb{Z}})$, and that our $\mu_n$'s capture the semigroup action to implement commensurability, hence together the two are sufficient to describe $C^{\ast}[\mathcal{G}]$. We're just missing one last relation: while we can conjugate $f \in C(\hat{\mathbb{Z}})$, we haven't shown what happens when we conjugate a basis element $e_{\gamma}$ by $\mu_n$. The proof of this relation is slightly more involved than the previous ones, so I'll describe $\mu_n e_{\gamma} \mu_n^{\ast}$ step by step. Lets do the far right convolution first:

$\displaystyle e_{\gamma}\mu_n^{\ast}(r,\rho) = \sum_{r_1 r_2 = r} e_{\gamma}(r_1,r_2 \rho) \mu_n^{\ast}(r_2,\rho) = \sum_{r_1 r_2 = r} e_{\gamma}(r_1,r_2 \rho) \mu_n(r_2^{-1},r_2 \rho)$

We see that $r_2 = n^{-1}$, $r_1 = 1$ and $r=n^{-1}$ for the sum to be nonzero. So we have:

$\displaystyle e_{\gamma}\mu_n^{\ast}(r,\rho) = \begin{cases} \text{exp}(2\pi i (\frac{1}{n} \rho)(\gamma) ) & r = n^{-1} \\ 0 & \text{otherwise} \end{cases}$

For the full conjugation, we have:

$\displaystyle \mu_n e_{\gamma}\mu_n^{\ast}(r,\rho) = \sum_{r_1 r_2 = r} \mu_n(r_1,r_2 \rho) \; e_{\gamma}\mu_n^{\ast}(r_2,\rho)$

We see again that $r_2 = n^{-1}$, $r_1 = n$ and thus $r=1$ for the sum to be nonzero. So we have

$\displaystyle \mu_n e_{\gamma}\mu_n^{\ast}(r,\rho) = \begin{cases} \text{exp}(2\pi i (\frac{1}{n} \rho)(\gamma) ) & r = 1 \\ 0 & \text{otherwise} \end{cases}$

We can express this purely in terms of our $e_{\gamma}$'s. Note that for $\gamma = \frac{a}{b} + \mathbb{Z}$ (I'll henceforth omit the $+ \mathbb{Z}$), we have $n$ $\delta$'s such that $n\delta = \gamma$, namely $\delta_k = \frac{a + kb}{nb}$ for $k=0, \ldots, n-1$. For such a $\delta$, $e_{\delta}(1,\rho) = \text{exp}(2\pi i \rho (\frac{1}{n} \gamma) ) = \text{exp}(2\pi i (\frac{1}{n} \rho) (\gamma) )$. Hence we have

$\displaystyle \mu_n e_{\gamma}\mu_n^{\ast}(r,\rho) = \frac{1}{n} \sum_{n \delta = \gamma} e_{\delta}(r,\rho) = \begin{cases} \text{exp}(2\pi i (\frac{1}{n} \rho)(\gamma) ) & r = 1 \\ 0 & \text{otherwise} \end{cases}$

That, along with relations 1-5, describe the Bost-Connes algebra. The time evolution in terms of this description is given by:

$\sigma_t(\mu_n) = n^{it}\mu_n, \;\;\; \sigma_t(e_{\gamma}) = e_{\gamma}$

Additionally, we have that $C^{\ast}[\mathbb{G}] \simeq C(\hat{\mathbb{Z}}) \rtimes \mathbb{N}$. It is also stated in Noncommutative Geometry, Quantum Fields, and Motives that $C^{\ast}[\mathbb{G}]$ is Morita equivalent to $C_0(\mathbb{A}_f) \rtimes \mathbb{Q}^{\ast}_{+}$, as shown in Laca's From Endomorphisms to Automorphisms And Back: Dilations and Full Corners'', but I'm not at all worried about that.
I am worried about how the cross product with $\mathbb{N}$ that I've described above implements the commensurability condition on 1dQL's modulo scaling. In fact, I don't even know what it means for the crossed product action to implement commensurability (my initial guess was that $\alpha_n(f)$ would be constant on commensurable 1dQLs, but I've yet to make sense of this condition). The main monograph states the action in terms of a function on 1dQL's:

$\displaystyle \alpha_n(f)(\Lambda,\phi) = f(n\Lambda,\phi)$

For $(\Lambda, \phi)$ divisible by $n$ in a specific sense, $0$ otherwise.. This implementation is mentioned in From Physics to Number theory via Noncommutative Geometry'', $\mathbb{Q}$-Lattices: Quantum Statistical Mechanics and Galois Theory'', and Lectures on Arithmetic Noncommutative Geometry''. I've read a few papers describing the transitions from Hecke algebras to the semigroup crossed product $C(\hat{\mathbb{Z}}) \rtimes \mathbb{N}$, including Laca's Semigroups of $\ast$-Endomorphisms, Dirichlet Series, and Phase Transitions'' and A Semigroup Crossed Product Arising In Number Theory'' but nothing describing the groupoid of the commensurability relation 1dQL's modulo scaling. I would appreciate any help anyone can provide. (As well as any comments on the rest of the maths in this post - I was fumbling about quite a bit.)
Next post will be a brief account of an equivalent formulation via Hecke Algebras.

## Sunday, December 5, 2010

### Confusion in the Presentation of the Bost-Connes System

I had planned to write a long glorious post about the generators and relations of the Bost Connes system, the same formulated via Hecke algebras, and the key arithmetic subalgebra. Alas, I got stuck on a rather silly point: verifying the relations. Recall that we're considering the groupoid $\mathcal{G} = \{(r,\rho) : r \in \mathbb{Q}^{*}_{+}, \rho \in \hat{\mathbb{Z}}\; \text{such that} \; r \rho \in \hat{\mathbb{Z}}\}$ and the $C^{\ast}$-completion $C^{\ast}[\mathcal{G}]$ of its convolution algebra:

$\displaystyle \{f:\mathcal{G} \rightarrow \mathbb{C} \;|\; f\; \text{is continuous and with compact support.}\}$

Let me start by stating the generating function I'm having trouble with: for all $n \in \mathbb{N}$, let $\mu_n : \mathcal{G} \rightarrow {\mathbb C}$ be the functions:

$\displaystyle \mu_n(r,\rho) = \begin{cases} 1 & r=n \\ 0 & \text{otherwise} \end{cases}$

These functions, under some relations (and plus another set of functions), are suppose to generate $C^{\ast}[\mathcal{G}]$. Nevermind how that is suppose to happen, I don't even understand the relations! Specifically, it's stated that

$\displaystyle \mu_n^{\ast} \mu_n = 1 \;\text{for all}\; n \in \mathbb{N}$

(There are several other relations, only this one is causing trouble.)

The multiplication here is convolution, not pointwise multiplication; it's clearly 0 under pointwise multiplication. So lets try this out. First recall that for $f \in C^{\ast}[\mathcal{G}]}$, we have $f^{\ast}(g) = \overline{f(g^{-1})}$, and convolution means that $f_1 f_2 (g) = \sum_{g_1 g_2 = g} f_1(g_1) f_2(g_2)$. Next, recall what inversion means in $\mathcal{G}$. A $g \in \mathcal{G}$ means that $g = (r,\rho)$, and if you write down the multiplication $(r_1,\rho_1) \circ (r_2,\rho_2) = (r_1 r_2, \rho_2)$ when $r_2 \rho_2 = \rho_1$, it's not heard to see that $(r,\rho)^{-1} = (r^{-1},r\rho)$. Assuming I didn't make in error in that, we can write down $\mu_n^{\ast} \mu_n$:

$\displaystyle \mu_n^{\ast} \mu_n (r,\rho) = \sum_{(r_1,\rho_1) \circ (r_2,\rho_2) = (r, \rho)} \mu_n^{\ast}(r_1,\rho_1)\mu_n (r_2,\rho_2)$

Now, the condition on the sum, $(r_1,\rho_1) \circ (r_2,\rho_2) = (r, \rho)$ clearly means that $\rho_2 = \rho$, and the condition to perform the multiplication then forces $\rho_1 = r_2 \rho$. So we have $(r_1,r_2 \rho) \circ (r_2,\rho) = (r, \rho)$ and we can restate this as:

$\displaystyle \mu_n^{\ast} \mu_n (r,\rho) = \sum_{r_1 r_2 = r} \mu_n^{\ast}(r_1)\mu_n (r_2) = \sum_{r_1 r_2 = r} \mu_n(r_1^{-1})\mu_n (r_2)$

Where $r_1 r_2 \rho, r_2 \rho \in \hat{\mathbb{Z}}$. I've dropped the $\rho$'s as the $\mu_n$'s don't care about them. The term $\mu_n(r_1^{-1})\mu_n (r_2)$ is 0 except when $n=r_1^{-1}=r_2$, but since $r = r_1 r_2$, we require $r=1$ in order to get a nonzero sum. We end up with:

$\displaystyle \mu_n^{\ast} \mu_n (r,\rho) = \mu_1 (r,\rho) \; \text{instead of} \; \mu_n^{\ast} \mu_n (r,\rho) = 1$

Where am I going wrong?

Edit: I'm not going wrong. The problem is that I don't understand unity in a convolution algebra. Here, $1$ is the function such that for all $f \in C^{\ast}[\mathcal{G}]$, $1f = f1 = f$ under convolution. Writing down the convolution of $\mu_1$ shows that it works. Thanks to Professor Kim for questioning me on that. Right, now back to the rest of the writeup!

Note: definitions for the functions came from Fun with $\mathbb{F}_1$'' page 11 and Noncommutative Geometry, Quantum Fields, and Motives'' page 417-418, 424. I didn't use Hecke algebras here, but that formulation is in Proposition 18 of `Hecke Algebras, Type III Factors and Phase Transitions with Spontaneous Symmetry Breaking in Number Theory'' by Bost and Connes.


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# $\left \langle x,y\right\rangle\subseteq\mathbb{C}[x,y]$ can be shown as $\bigcup_{i\in I}Q_i$ s.t. $Q_i$ is a prime ideal.

Prove that the ideal $$\left \langle x,y\right\rangle\subseteq\mathbb{C}[x,y]$$ can be shown as $$\bigcup_{i\in I}Q_i$$ such that $$Q_i$$ is a prime ideal.

• $\Bbb C[x,y]$ is the polynomial ring with two indeterminates? – Bernard Mar 22 at 16:10
• @Bernard Indeed. – J. Doe Mar 22 at 16:11
• Well, $\langle x,y\rangle$ itself is a maximal ideal. – Bernard Mar 22 at 16:12

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Square Root of Complex Numbers

by littleHilbert
Tags: complex, numbers, root, square
 P: 57 Hi! I've got a question. There is a nice formula for finding square roots of arbitrary complex numbers z=a+bi: $\frac{1}{\sqrt{2}}(\epsilon\sqrt{|z|+a}+i\sqrt{|z|-a})$ where epsilon:=sing(b) if b≠0 or epsilon:=1 if b=0. I've just looked it up and it's nice to use it to find complex roots of quadratic equations with complex coefficients. Where does it come from? I mean, squaring shows that it's true but how can one derive it from other facts? Is there a similar formula for n-th roots (not in polar form but analogous to that above)? Any info, links?
 P: 25 LittileHilbrt; Let y=c+id, where y^2=z. Then; c^2-d^2+2i c*d=a+ib. As a result; a=c^2-d^2 (eq1) and b=2c*d (eq2). Therefore, from eq1 and eq2; c^2 = a + (b/c)^2/4 (eq3); Solving eq3 in c^2, we get c^2={a+sqr(a^2+b^2)}^.5 /2 Therefore, c=(+/-) sqr({a+mod(z)}/2) (eq4 ), {the other solution is rejected as c must be real}. Then; z^1/2 = (+/-) y = (+/-) [c + id] = (+/-) [c+id] = (+/-) [sqr({a+mod(z)})+i sign(b)* sqr({[mod(z)]^2 –a^2} / {a+mod(z)})]/sqr(2) = (+/-) [ sqr( a+mod(z) ) + i sign(b) * sqr({mod(z)–a} ] / sqr(2) = (+/-) [sign(b) * sqr( a+mod(z) ) + i sqr({mod(z)–a} ] / [sqr(2)* sign(b)] ====> (+/-) [sign(b) * sqr( a+mod(z) ) + i sqr({mod(z)–a} ] / [sqr(2)] Solving these equaitons when b=0 will need some modifications. By the way, when b is zero, z is real. Amr Morsi.
 P: 1,520 It can probably be derived with few identities and Demoivre's theorem.
 Sci Advisor HW Helper P: 944 Square Root of Complex Numbers You could always convert the number to e^(x + iy) form, divide x and y by 2, convert back to normal form. I think that should work.
P: 1,160
 Quote by littleHilbert Where does it come from? I mean, squaring shows that it's true but how can one derive it from other facts? Is there a similar formula for n-th roots (not in polar form but analogous to that above)? Any info, links?
littleHilbert,

When you multiply by a complex number, you are rotating the other number
by the angle the complex number makes with the x-axis.

For example, multiplication by "i" makes a rotation of 90 degrees. Therefore
a multiplication by the square of "i" makes a rotation of 180 degrees or -1.
Hence "i" is the square root of -1.

If you want the "n-th root" of a number; express it in polar form, and
divide the angle by "n". You will get the polar expression for the n-th
root. Use trigonometric identities to transform back into the non-polar
form in terms of the components of the original complex number.
Normalize appropriately if the modulus of the number is not unity.

That's how the formula you cite above is derived.

Dr. Gregory Greenman
Physicist
 P: 134 the question is..take the identity: $$(\sqrt (-2)+1)(\sqrt (-2)-1)=-2$$ and expand it by a continous fraction..what would we get..

 Related Discussions Precalculus Mathematics Homework 11 Calculus & Beyond Homework 0 Precalculus Mathematics Homework 14 Calculus & Beyond Homework 1 Precalculus Mathematics Homework 3


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Corpus ID: 237940616

Revisiting minimum description length complexity in overparameterized models

@inproceedings{Dwivedi2020RevisitingMD,
title={Revisiting minimum description length complexity in overparameterized models},
author={Raaz Dwivedi and Chandan Singh and Bin Yu and Martin J. Wainwright},
year={2020}
}
Complexity is a fundamental concept underlying statistical learning theory that aims to inform generalization performance. Parameter count, while successful in low-dimensional settings, is not well-justified for overparameterized settings when the number of parameters is more than the number of training samples. We revisit complexity measures based on Rissanen’s principle of minimum description length (MDL) and define a novel MDL-based complexity (MDL-COMP) that remains valid for… Expand

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This article reviews the principle of minimum description length (MDL) for problems of model selection. By viewing statistical modeling as a means of generating descriptions of observed data, the MDLExpand
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A novel regularization selection method, in which a tight upper bound of LNML (uLNML) is minimized with local convergence guarantee, and the experimental results show that MDL-RS improves the generalization performance of regularized estimates specifically when the model has redundant parameters. Expand
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A characterization of linear regression problems for which the minimum norm interpolating prediction rule has near-optimal prediction accuracy shows that overparameterization is essential for benign overfitting in this setting: the number of directions in parameter space that are unimportant for prediction must significantly exceed the sample size. Expand
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The normalized maximized likelihood, mixture, and predictive codings are each shown to achieve the stochastic complexity to within asymptotically vanishing terms. Expand


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# Ninh Văn Thu

##### Ninh Văn Thu, Doctor
VNU mail:
thunv@vnu.edu.vn
Website:
thunv.wordpress.com
Research Fields:
Several Complex Variables, Nevanlinna Theory
Education :
• 2005-2010: PhD. in Mathematics at Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam.
• 2002-2004: M.Sc. in Mathematics, VNU University of Science, Vietnam, Hanoi, Vietnam.
• 1998-2002: B.S. in Mathematics, VNU University of Science, Vietnam, Hanoi, Vietnam.
Teaching:
• Analysis (Giải tích)
• Complex Analysis (Giải tích phức)
Science Activities:
• 10/2006-11/2006: Short Visiting Scholar, Labo Emile Picard, University of Toulouse, Toulouse, France
• 09/2010-12/2010: Short Visiting Scholar, Math department, University of Washington, Seattle, USA
• 03/2012-02/2013: Postdoctoral Fellow at Center for Geometry and its Applications (GAIA), POSTECH, Pohang, Korea
• 03/2013-02/2015: Research Assistant Professor at Center for Geometry and its Applications (GAIA), POSTECH, Pohang, Korea
• 04/2015-09/2015: Senior Researcher at VIASM, Hanoi, Vietnam
Awards:
• 2000: VNU-UFJ Foundation Scholarship, Vietnam National University at Hanoi, Vietnam.

## Publications

1. Iterates of holomorphic self-maps on pseudoconvex domains of finite and infinite type in {$\Bbb{C}^n$}. Proc. Amer. Math. Soc. 2016;144:5197–5206. doi:10.1090/proc/13138.
2. Hayashimoto A, Ninh VThu. Infinitesimal CR automorphisms and stability groups of infinite-type models in $\mathbb C^2$. Kyoto Journal of Mathematics. 2016;Volume 56, Number 2 (2016), 441-464.
3. On limit {B}rody curves in {$\Bbb C^n$} and {$(\Bbb C^*)^2$}. Kyushu J. Math. 2015;69:111–123. doi:10.2206/kyushujm.69.111.
4. Ninh VThu. On the CR automorphism group of a certain hypersurface of infinite type in $\mathbb C^2$. Complex Var. Elliptic Equ. 60 (2015), pp. 977-991. 2015.
5. On the tangential holomorphic vector fields vanishing at an infinite type point. Trans. Amer. Math. Soc. 2015;367:867–885. doi:10.1090/S0002-9947-2014-05917-5.
6. On the existence of parabolic actions in convex domains of {$\Bbb{C}^{k+1}$}. Czechoslovak Math. J. 2015;65(140):579–585. doi:10.1007/s10587-015-0197-y.
7. Kim H, Ninh VThu, Yamamori A. The automorphism group of a certain unbounded non-hyperbolic domain. J. Math. Anal. Appl. 409 (2014), pp. 637-642. 2014.
8. Ninh VThu, Chử VTiệp. On the nonexistence of parabolic boundary points of certain domains in $\mathbb C^2$. J. Math. Anal. Appl. 389 (2012), pp. 908–914. 2012.
9. Đỗ Đức T, Ninh VThu. The second main theorem for hypersurfaces. Kyushu J. Math. 65 (2011), no. 2, pp. 219–236. 2011.
10. A remark on the {K}im's theorem. Acta Math. Vietnam. 2009;34:285–297.


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LS法信道估计

Directory: Communication develop
Plat: matlab
Size: 2KB
Description:   The OFDM algorithm is used to estimate the channel of OFDM. The algorithm estimates that the calculation amount of the road is small, but is susceptible to noise.

File list:
LS法信道估计\channel_LS_estimation.m, 3152 , 2019-06-27
LS法信道估计\interpolate.m, 743 , 2019-05-27
LS法信道估计\LS_CE.m, 426 , 2019-07-20
LS法信道估计, 0 , 2019-07-20


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Solution - A uniform steel rod of 5 mm^2 cross section is heated from 0°C to 25°C. Calculate the force which must be exerted to prevent it from expanding. Also calculate strain. - Definition of Stress and Strain

Question

A uniform steel rod of 5 mm2 cross section is heated from 0°C to 25°C. Calculate the force which must be exerted to prevent it from expanding. Also calculate strain.

(α for steel = 12 x 10-6/°C and γ for steel = 20 x 101ON/m2)

Solution

Given that α steel = 12 x 10-6 /°C, γ steel = 20 x 1010 N/m2

Area of cross-section of the rod, A = 5 mm2 = 5 × 10−6 m

Change in temperature (ΔT), (T2 − T1) = 25°C

Force exerted by the rod due to heating = Thermal stress × Area

Thermal stress = γ x Strain

= γsteel x αsteel x ΔT

Therefore, the force exerted by the rod due to heating is

= γsteel x αsteel x ΔT x A

= 20 x 1010 x 12 x 10-6 x 25 x 5 x 10-6

= 300 N

"Strain" = "Change in length"/ "Original length"= alphaDelta"T"

Strain = αsteelΔT

= 12 x 10-6 x 25

= 3 x 10-4

Is there an error in this question or solution?

APPEARS IN

2014-2015 (March)
Question 3.4 | 3 marks

Video TutorialsVIEW ALL [3]

Solution for question: A uniform steel rod of 5 mm^2 cross section is heated from 0°C to 25°C. Calculate the force which must be exerted to prevent it from expanding. Also calculate strain. concept: Definition of Stress and Strain. For the courses HSC Science (Computer Science), HSC Science (Electronics), HSC Science (General)
S


text:
# ItemPick¶

## Introduction¶

The ItemPick component is an optional on-board component of the rc_visard.

Note

The ItemPick component is optional and requires a separate license to be purchased.

It provides an out-of-the-box and model-free perception solution for robotic pick-and-place applications with suction grippers. The component extracts flat surfaces by clustering points in the scene and computes 3D grasp poses for the suction device.

In addition, ItemPick features:

• a dedicated page on the rc_visard Web GUI for easy setup, configuration, testing, and application tuning
• the definition of regions of interest to select relevant volumes in the scene
• a load carrier detection functionality for bin-picking applications, to provide grasps only for items inside a bin
• the definition of compartments inside a load carrier to provide grasps only for specific volumes of the bin
• support for static and robot-mounted rc_visard devices and optional integration with the on-board Hand-eye calibration component, to provide grasps in the user-configured external reference frame
• a quality value associated to each suggested grasp and related to the available grasping surface
• sorting of grasps according to gravity so that items on top of a pile are grasped first.

Note

In the ItemPick component, cluster and surface are used as synonyms and identify a set of points (or pixels) with defined geometrical properties.

Note

In the ItemPick component, load carrier and bin are used as synonyms and identify a container with four walls, a floor and a rectangular rim.

## Data types¶

### Region of Interest¶

A region of interest defines a volume in space which is of interest for a specific user-application. The ItemPick component currently supports regions of interest of the following types:

• BOX, with dimensions box.x, box.y, box.z.
• SPHERE, with radius sphere.radius.

The user can specify the region of interest pose in the camera or the external coordinate system (see Hand-eye calibration).

The ItemPick component can persistently store up to 10 different regions of interest, each one identified by a different id. The configuration of regions of interest is normally performed offline (e.g., on the ItemPick page of the rc_visard Web GUI), during the set up the desired application.

Note

As opposed to the component parameters, the configured regions of interest are persistent even over firmware updates and rollbacks.

The region of interest can narrow the volume that is searched for a load carrier model, or select a volume which only contains items to be grasped.

Note

If the region of interest filter is not applied, the ItemPick component processes the whole scene visible to the camera. In case of grasp computation, this means that the whole scene includes items available for grasping.

### Load Carrier¶

A load carrier (bin) is a container with four walls, a floor and a rectangular rim. It is defined by its outer_dimensions and inner_dimensions.

The load carrier detection algorithm is based on the detection of the load carrier rectangular rim. By default, the rectangular rim_thickness is computed from the outer and inner dimensions. As an alternative, its value can also be explicitly specified by the user.

Note

Typically, outer and inner dimensions of a load carrier are available in the specifications of the load carrier manufacturer.

The load carrier reference frame is defined such that its origin is at the center of the load carrier outer box, its z axis is perpendicular to the load carrier floor and its x axis is directed along the direction of maximum elongation of the load carrier.

Fig. 50 Load carrier models and reference frame.

The user can optionally specify a prior for the load carrier pose. The detected load carrier pose is guaranteed to have the minimum rotation with respect to the load carrier prior pose. If no prior is specified, the algorithm searches for a load carrier whose floor is perpendicular to the estimated gravity vector.

The ItemPick component can persistently store up to 10 different load carrier models, each one identified by a different id. The configuration of a load carrier model is normally performed offline (e.g., on the ItemPick page of the rc_visard Web GUI), during the set up the desired application.

Note

As opposed to the component parameters, the configured load carrier models are persistent even over firmware updates and rollbacks.

The ItemPick module enables the computation of grasps for a specific volume of the load carrier (load_carrier_compartment). The compartment is a box whose pose is defined with respect to the load carrier reference frame.

Fig. 51 Compartment inside a load carrier.

### Suction Grasp¶

A grasp provided by the ItemPick represents the recommended pose of the TCP (Tool Center Point) of the suction gripper. The grasp orientation is a right-handed coordinate system and is defined such that its z axis is normal to the surface pointing inside the object at the grasp position and its x axis is directed along the maximum elongation of the surface.

The computed grasp pose is the center of the biggest ellipse that can be inscribed in each surface.

Fig. 52 Illustration of suction grasp with coordinate system and ellipse representing the maximum suction surface.

Each grasp includes the dimensions of the maximum suction surface available, modelled as an ellipse of axes max_suction_surface_length and max_suction_surface_width. The user is enabled to filter grasps by specifying the minimum suction surface required by the suction device in use.

The grasp definition is complemented by a uuid (Universally Unique Identifier) and the timestamp of the oldest image that was used to compute the grasp.

### Item model¶

The ItemPick component allows to specify a model for the items to be picked. Each item model includes the dimensions of the maximum and minimum bounding box totally enclosing the item.

The ItemPick component currently supports item models of the following types:

• UNKNOWN, for flexible and/or deformable items.
• BOX, for box-shaped rigid items.

## Interaction with other components¶

Internally, the ItemPick component depends on, and interacts with other on-board components as listed below.

Note

All changes and configuration updates to these components will affect the ItemPick component’s performance.

### Stereo camera and Stereo matching¶

The ItemPick component makes use internally of the following data:

• Rectified images from the Stereo camera component (rc_stereocamera);
• Disparity, error, and confidence images from the Stereo matching component (rc_stereomatching).

### Sensor dynamics¶

For each load carrier detection and grasp computation, the ItemPick component estimates the gravity vector by subscribing to the IMU data stream from the Sensor dynamics component (rc_dynamics).

Note

The gravity vector is estimated from linear acceleration readings from the on-board IMU. For this reason, the ItemPick components requires the rc_visard to remain still while the gravity vector is being estimated.

### IO and Projector Control¶

In case the rc_visard is used in conjunction with an external random dot projector and the IO and Projector Control component (rc_iocontrol), the output mode for the GPIO output in use should be se to ExposureAlternateActive, as explained in the Description of run-time parameters of the IO and Projector Control component.

No additional changes are required to use the ItemPick component in combination with a random dot projector.

### Hand-eye calibration¶

In case you have the rc_visard calibrated to a robot, you can choose two different pose_frame values:

1. Camera frame (camera). All poses provided to and from the component are in the camera frame, and no prior knowledge about the pose of the rc_visard in the environment is required. This means that the configured regions of interest and load carriers move with the camera. It is the user’s responsibility to update the configured poses if the camera frame moves (e.g., with a robot-mounted sensor).
2. External frame (external). All poses provided to and from the component are in the external frame, configured by the user during the hand-eye calibration process. The component relies on the on-board Hand-eye calibration component to retrieve the sensor mounting (static or robot mounted) and the hand-eye transformation. If the sensor mounting is static, no further information is needed. If the sensor is robot-mounted, the robot_pose is required to transform poses to and from the external frame.

Note

If no hand-eye calibration is available, all pose_frame values should be set to camera.

All pose_frame values that are not camera or external are rejected by the component.

## Parameters and Status Values¶

The ItemPick component is called rc_itempick in the REST-API. The user can explore and configure the rc_itempick component’s run-time parameters, e.g., for development and testing, using the rc_visard Web GUI or Swagger UI.

This component offers the following run-time parameters.

Table 32 The rc_itempick component’s application parameters
Name Type Min Max Default Description
max_grasps int32 1 20 5 Maximum number of provided grasps
Table 33 The rc_itempick component’s load carrier detection parameters
Name Type Min Max Default Description
load_carrier_crop_distance float64 0.0 0.02 0.005 Safety margin in meters by which the load carrier inner dimensions are reduced to define the region of interest for grasp computation
load_carrier_model_tolerance float64 0.003 0.025 0.008 Indicates how much the estimated load carrier dimensions are allowed to differ from the load carrier model dimensions in meters
Table 34 The rc_itempick component’s surface clustering parameters
Name Type Min Max Default Description
cluster_max_dimension float64 0.05 0.8 0.3 Diameter of the largest sphere enclosing each cluster in meters. Clusters larger than this value are filtered out before grasp computation.
cluster_max_curvature float64 0.005 0.5 0.11 Maximum curvature allowed within one cluster. The smaller this value, the more clusters will be split apart.
clustering_patch_size int32 3 10 4 Size in pixels of the square patches the depth map is subdivided into during the first clustering step
clustering_max_surface_rmse float64 0.0005 0.01 0.004 Maximum root-mean-square error (RMSE) in meters of points belonging to a surface
clustering_discontinuity_factor float64 0.5 5.0 1.0 Factor used to discriminate depth discontinuities within a patch. The smaller this value, the more clusters will be split apart.
item_model_tolerance float64 0.003 0.05 0.005 Indicates how much the estimated item dimensions are allowed to differ from the item model dimensions in meters

This component reports the following status values.

Table 35 The rc_itempick component’s status values
Name Description
state The current state of the rc_itempick node
last_timestamp_processed The timestamp of the last processed dataset
data_acquisition_time Time in seconds required by the last active service to acquire images. Standard values are between 0.5 s and 0.6 s with High depth image quality.
load_carrier_detection_time Processing time of the last load carrier detection in seconds
grasp_computation_time Processing time of the last grasp computation in seconds

The reported state can take one of the following values.

Table 36 Possible states of the ItemPick components
State name Description
IDLE The component is idle.
RUNNING The component is running and ready for load carrier detection and grasp computation.
FATAL A fatal error has occurred.

## Services¶

The user can explore and call the rc_itempick component’s services, e.g., for development and testing, using Swagger UI or the rc_visard Web GUI.

Each service component provides a return_code, which consists of a value plus an optional message.

A successful service returns with a return_code value of 0. Negative return_code values indicate that the service failed. Positive return_code values indicate that the service succeeded with additional information. The smaller value is selected in case a service has multiple return_code values, but all messages are appended in the return_code message.

The following table contains a list of common codes:

Table 37 Return codes of the ItemPick services
Code Description
0 Success
-1 An invalid argument was provided
-4 Data acquisition took longer than the maximum allowed time of 1.5 seconds
-301 More than one item model provided to the compute_grasps service, but only one is supported
-302 More than one load carrier provided to the detect_load_carriers service, but only one is supported
100 The requested load carriers were not detected in the scene
101 No valid surfaces or grasps were found in the scene
102 The detected load carrier is empty
200 The component is in IDLE state
300 A valid robot_pose was provided as argument but it is not required
400 No item_models were provided to the compute_grasps service request
500 The region of interest visualization images could not be generated during the call to set_region_of_interest
600 An existent persistent model was overwritten by the call to set_load_carrier or set_region_of_interest

The ItemPick component offers the following services.

start

Starts the component. If the command is accepted, the component moves to state RUNNING. The current_state value in the service response may differ from RUNNING if the state transition is still in process when the service returns.

This service requires no arguments.

This service returns the following response:

{
"accepted": "bool",
"current_state": "string"
}

stop

Stops the component. If the command is accepted, the component moves to state IDLE. The current_state value in the service response may differ from IDLE if the state transition is still in process when the service returns.

This service requires no arguments.

This service returns the following response:

{
"accepted": "bool",
"current_state": "string"
}

set_region_of_interest

Persistently stores a region of interest on the rc_visard. All configured regions of interest are persistent over firmware updates and rollbacks.

See Region of Interest the definition of the region of interest type.

This service requires the following arguments:

{
"region_of_interest": {
"box": {
"x": "float64",
"y": "float64",
"z": "float64"
},
"id": "string",
"pose": {
"orientation": {
"w": "float64",
"x": "float64",
"y": "float64",
"z": "float64"
},
"position": {
"x": "float64",
"y": "float64",
"z": "float64"
}
},
"pose_frame": "string",
"sphere": {
"radius": "float64"
},
"type": "string"
},
"robot_pose": {
"orientation": {
"w": "float64",
"x": "float64",
"y": "float64",
"z": "float64"
},
"position": {
"x": "float64",
"y": "float64",
"z": "float64"
}
}
}

This service returns the following response:

{
"return_code": {
"message": "string",
"value": "int16"
}
}

get_regions_of_interest

Returns the configured regions of interest with the requested region_of_interest_ids. If no region_of_interest_ids are provided, all configured regions of interest are returned.

This service requires the following arguments:

{
"region_of_interest_ids": [
"string"
]
}

This service returns the following response:

{
"regions_of_interest": [
{
"box": {
"x": "float64",
"y": "float64",
"z": "float64"
},
"id": "string",
"pose": {
"orientation": {
"w": "float64",
"x": "float64",
"y": "float64",
"z": "float64"
},
"position": {
"x": "float64",
"y": "float64",
"z": "float64"
}
},
"pose_frame": "string",
"sphere": {
"radius": "float64"
},
"type": "string"
}
],
"return_code": {
"message": "string",
"value": "int16"
}
}

delete_regions_of_interest

Deletes the configured regions of interest with the requested region_of_interest_ids. All regions of interest to be deleted must be explicitly stated in region_of_interest_ids.

This service requires the following arguments:

{
"region_of_interest_ids": [
"string"
]
}

This service returns the following response:

{
"return_code": {
"message": "string",
"value": "int16"
}
}

set_load_carrier

Persistently stores a load carrier on the rc_visard. All configured load carriers are persistent over firmware updates and rollbacks.

See Load Carrier the definition of the load carrier type.

This service requires the following arguments:

{
"load_carrier": {
"id": "string",
"inner_dimensions": {
"x": "float64",
"y": "float64",
"z": "float64"
},
"outer_dimensions": {
"x": "float64",
"y": "float64",
"z": "float64"
},
"pose": {
"orientation": {
"w": "float64",
"x": "float64",
"y": "float64",
"z": "float64"
},
"position": {
"x": "float64",
"y": "float64",
"z": "float64"
}
},
"pose_frame": "string",
"rim_thickness": {
"x": "float64",
"y": "float64"
}
}
}

This service returns the following response:

{
"return_code": {
"message": "string",
"value": "int16"
}
}

get_load_carriers

Returns the configured load carriers with the requested load_carrier_ids. If no load_carrier_ids are provided, all configured load carriers are returned.

This service requires the following arguments:

{
"load_carrier_ids": [
"string"
]
}

This service returns the following response:

{
"load_carriers": [
{
"id": "string",
"inner_dimensions": {
"x": "float64",
"y": "float64",
"z": "float64"
},
"outer_dimensions": {
"x": "float64",
"y": "float64",
"z": "float64"
},
"pose": {
"orientation": {
"w": "float64",
"x": "float64",
"y": "float64",
"z": "float64"
},
"position": {
"x": "float64",
"y": "float64",
"z": "float64"
}
},
"pose_frame": "string",
"rim_thickness": {
"x": "float64",
"y": "float64"
}
}
],
"return_code": {
"message": "string",
"value": "int16"
}
}

delete_load_carriers

Deletes the configured load carriers with the requested load_carrier_ids. All load carriers to be deleted must be explicitly stated in load_carrier_ids.

This service requires the following arguments:

{
"load_carrier_ids": [
"string"
]
}

This service returns the following response:

{
"return_code": {
"message": "string",
"value": "int16"
}
}

detect_load_carriers

Triggers a load carrier detection. All images used by the node are guaranteed to be newer than the service trigger time.

This service requires the following arguments:

{
"load_carrier_ids": [
"string"
],
"pose_frame": "string",
"region_of_interest_id": "string",
"robot_pose": {
"orientation": {
"w": "float64",
"x": "float64",
"y": "float64",
"z": "float64"
},
"position": {
"x": "float64",
"y": "float64",
"z": "float64"
}
}
}

This service returns the following response:

{
"load_carriers": [
{
"id": "string",
"inner_dimensions": {
"x": "float64",
"y": "float64",
"z": "float64"
},
"outer_dimensions": {
"x": "float64",
"y": "float64",
"z": "float64"
},
"pose": {
"orientation": {
"w": "float64",
"x": "float64",
"y": "float64",
"z": "float64"
},
"position": {
"x": "float64",
"y": "float64",
"z": "float64"
}
},
"pose_frame": "string",
"rim_thickness": {
"x": "float64",
"y": "float64"
}
}
],
"return_code": {
"message": "string",
"value": "int16"
},
"timestamp": {
"nsec": "int32",
"sec": "int32"
}
}

Required arguments:

pose_frame: only if a pose frame is not already specified by the selected region of interest or load carrier pose prior. If a pose_frame is provided but not required, its value must be consistent with the selected region of interest pose frame and load carrier prior pose frame, otherwise the service will return with an invalid argument error.

load_carrier_ids

robot_pose: only if working in pose_frame="external" and the rc_visard is robot-mounted.

Optional arguments:

region_of_interest_id: delimits the volume of space where to search for the load carrier. The processing time for load carrier detection increases with the size of the selected region of interest.
compute_grasps

Triggers the computation of grasping poses for a suction device. All images used by the node are guaranteed to be newer than the service trigger time. If successful, the service returns a sorted list of grasps and (optionally) the detected load carriers.

This service requires the following arguments:

{
"item_models": [
{
"box": {
"max_dimensions": {
"x": "float64",
"y": "float64",
"z": "float64"
},
"min_dimensions": {
"x": "float64",
"y": "float64",
"z": "float64"
}
},
"type": "string",
"unknown": {
"max_dimensions": {
"x": "float64",
"y": "float64",
"z": "float64"
},
"min_dimensions": {
"x": "float64",
"y": "float64",
"z": "float64"
}
}
}
],
"load_carrier_compartment": {
"box": {
"x": "float64",
"y": "float64",
"z": "float64"
},
"pose": {
"orientation": {
"w": "float64",
"x": "float64",
"y": "float64",
"z": "float64"
},
"position": {
"x": "float64",
"y": "float64",
"z": "float64"
}
}
},
"load_carrier_id": "string",
"pose_frame": "string",
"region_of_interest_id": "string",
"robot_pose": {
"orientation": {
"w": "float64",
"x": "float64",
"y": "float64",
"z": "float64"
},
"position": {
"x": "float64",
"y": "float64",
"z": "float64"
}
},
"suction_surface_length": "float64",
"suction_surface_width": "float64"
}

This service returns the following response:

{
"grasps": [
{
"max_suction_surface_length": "float64",
"max_suction_surface_width": "float64",
"pose": {
"orientation": {
"w": "float64",
"x": "float64",
"y": "float64",
"z": "float64"
},
"position": {
"x": "float64",
"y": "float64",
"z": "float64"
}
},
"pose_frame": "string",
"quality": "float64",
"timestamp": {
"nsec": "int32",
"sec": "int32"
},
"type": "string",
"uuid": "string"
}
],
"load_carriers": [
{
"id": "string",
"inner_dimensions": {
"x": "float64",
"y": "float64",
"z": "float64"
},
"outer_dimensions": {
"x": "float64",
"y": "float64",
"z": "float64"
},
"pose": {
"orientation": {
"w": "float64",
"x": "float64",
"y": "float64",
"z": "float64"
},
"position": {
"x": "float64",
"y": "float64",
"z": "float64"
}
},
"pose_frame": "string",
"rim_thickness": {
"x": "float64",
"y": "float64"
}
}
],
"return_code": {
"message": "string",
"value": "int16"
},
"timestamp": {
"nsec": "int32",
"sec": "int32"
}
}

Required arguments:

pose_frame: only if a pose frame is not already specified by the selected region of interest or load carrier pose prior. If a pose_frame is provided but not required, its value must be consistent with the selected region of interest pose frame and load carrier prior pose frame, otherwise the service will return with an invalid argument error.

suction_surface_length: length of the suction device grasping surface.

suction_surface_width: width of the suction device grasping surface.

robot_pose: only if working in pose_frame="external" and the rc_visard is robot-mounted.

Optional arguments:

region_of_interest_id: delimits the volume of space where to search for the load carrier or selects a volume which contains items to be grasped if no load_carrier_id is set. The processing time for load carrier detection and grasp computation increases with the size of the selected region of interest.

load_carrier_id: limits grasp computation to the content of the detected load carrier.

load_carrier_compartment: selects a compartment within the detected load carrier.

item_models: defines a list of items of interest. Currently only one item model is supported for each call.

save_parameters
This service saves the currently set parameters persistently. Thereby, the same parameters will still apply after a reboot of the sensor. The node parameters are not persistent over firmware updates and rollbacks.
reset_to_defaults
This service resets all parameters of the component to its default values, as listed in above table. The reset does not apply to regions of interest and load carriers.


text:
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# 9.5: Angle Classification

Difficulty Level: At Grade Created by: CK-12
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### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

Acute Angle

An acute angle is an angle with a measure of less than 90 degrees.

Obtuse angle

An obtuse angle is an angle greater than 90 degrees but less than 180 degrees.

Perpendicular

Perpendicular lines are lines that intersect at a $90^{\circ}$ angle. The product of the slopes of two perpendicular lines is -1.

Right Angle

A right angle is an angle equal to 90 degrees.

Straight angle

A straight angle is a straight line equal to $180^{\circ}$.

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# Tag Info

I believe the simplest change to your code is to replace the Return[\$Failed] expressions with Sets that restore the default values of the options: ClearAll[func] Options[func] = {Method -> Automatic, WorkingPrecision -> MachinePrecision, Order -> 2}; func::badval = "1 is not a valid value of option 2"; func[arg1_, arg2_, opts : ...


text:
1. Singularities and integrating

Hi, I have some past papers with no answers and not much idea on how to find them, any help appreciated!!
For part a) is it something about$\displaystyle z^4$ lies on the circle with radius $\displaystyle R^4$and so its minimal distance to -sqrt(2)a i$\displaystyle s R^2-4a^4$?
part b) use the M-L lemma to show int(f)$\displaystyle <$$\displaystyle \Pi R z e^iz$/$\displaystyle R^4-4a^4$ which tends to zero as $\displaystyle R$ tends to infinity?
c) singularities would be $\displaystyle z^4=-4a^4$though not sure what these are . Think I may have to brake $\displaystyle z^4+4a^4$down into composite parts e.g.$\displaystyle (z^2-2ia^2)(z^2+2ia^2)$
part d) I have no idea!

2. Hint For R sufficiently large we have

$\displaystyle \left |{\dfrac{1}{z^4+4a^4}}\right |\leq \dfrac{1}{R^4-4a^4}$

By a well known property

$\displaystyle \left |{\displaystyle\int_{\Gamma_R}f(z)dz}\right |\leq \dfrac{\pi R}{R^4-4a^4}$

Now, take limits in both sides as $\displaystyle R\to +\infty$ .


text:
# problem when inserting figure

I have been trying to insert an image to my LaTeX test book so I can further experiment with it, but the only thing which I see is the file path's name. I have tried some proposed solutions on both stackoverflow and other websites, but none worked -- what am I missing?

What I have tried:

1. Including the image as follows:
\begin{figure}[h]
\centering
\includegraphics[width=0.25\textwidth]{Plot}
\caption{$J(\theta_0, \theta_1)$ as regards to $\theta_0$ and $\theta_1$}
\label{fig:Fig1}
\end{figure}

1. Instead of Plot in the previouds code, use Plot.png

2. Removing the \graphicspath{ {Images/} } line

Notes:

1. The image (Plot.png) is in the folder Image.

2. I use overleaf, however, the overleaf's sample from the official website worked as a charm.

Full code:

\documentclass[12pt, letterpaper]{book}
\usepackage[utf8]{inputenc}
\usepackage{graphicx}
\usepackage{grffile}
\usepackage[utf8]{amsmath, amssymb}
\graphicspath{ {Images/} }

\title{LINEAR REGRESSION IN DETAIL}
\author{Alexandros Voliotis \thanks{University of Thessaly}}
\date{May 2020}

\begin{document}

\maketitle
Test title of my first \textbf{ \LaTeX} document!

Italics: \textit{test}

Underline: \underline{test}

\chapter{Chapter 1}

\section{Introduction}

In the gradient descent algorithm we are trying to minimize the value of $J(\theta_0, \theta_1)$ for some $\theta_0, \theta_1 \in \mathbb{R}$.

This is quite tough to achieve in a few steps, however it is not entirely impossible. In order to make $J(\theta_0, \theta_1)$ equal to $0$, we must first set the step variable accordingly, and measure every time the output. This process should, and will, create a table with two columns as shown below:
\begin{itemize}
\item 1\textsuperscript{st} column -- The values for the step variable.
\item 2\textsuperscript{nd} column -- The values for the total tries variable.

\end{itemize}

A better visualization of the function $J(\theta_0, \theta_1)$ is as follows:

\begin{figure}[h]
\centering
\includegraphics[width=0.25\textwidth]{Plot}
\caption{$J(\theta_0, \theta_1)$ as regards to $\theta_0$ and $\theta_1$}
\label{fig:Fig1}
\end{figure}
\end{document}

Fast answer: include \usepackage{float} and use [H] instead of [h]

Explained answer: There's the errors or recommendations that I have founded:

1. Use lowercase letters for labels, path of folders, and others that usually refers object. For example use \label{tbl:its an example} instead of \label{tbl:Its An Example}. Why? because usually we forgot uppercase letters.
2. Include \usepackage{float} to use [H] or [h] in \begin{figure}[H] on your documents.
3. Use \usepackage{amsmath} \usepackage{amssymb} instead of your code if you have no errors with this.

I fixed some common errors in the code below:

\documentclass[12pt, letterpaper]{book}
\usepackage[utf8]{inputenc}
\usepackage{graphicx}
\usepackage{grffile}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{float} % For placement of figures, tables, and other objects
\graphicspath{{imagesfolder/}} % use lowercase to avoid errors for typing

\title{LINEAR REGRESSION IN DETAIL}
\author{Alexandros Voliotis \thanks{University of Thessaly}}
\date{May 2020}

\begin{document}

\maketitle
Test title of my first \textbf{ \LaTeX} document!

Italics: \textit{test}

Underline: \underline{test}

\chapter{Mathematics}

\section{Introduction}

In the gradient descent algorithm we are trying to minimize the value of $J(\theta_0, \theta_1)$ for some $\theta_0, \theta_1 \in \mathbb{R}$.

This is quite tough to achieve in a few steps, however it is not entirely impossible. In order to make $J(\theta_0, \theta_1)$ equal to $0$, we must first set the step variable accordingly, and measure every time the output. This process should, and will, create a table with two columns as shown below:
\begin{itemize}
\item 1\textsuperscript{st} column -- The values for the step variable.
\item 2\textsuperscript{nd} column -- The values for the total tries variable.

\end{itemize}

A better visualization of the function $J(\theta_0, \theta_1)$ is as follows (\ref{fig:figure 1}):

\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{image.png}
\caption{$J(\theta_0, \theta_1)$ as regards to $\ theta_0$ and $\theta_1$}
\label{fig:figure 1}
\end{figure}

\end{document}

• Thanks for your input! I managed to solve this problem by actually switching the compile mode from "fast" to "normal" -- even your code wouldn't run without this change. However, it would be great if you could expand a bit more on why I would need to use the float package, and if I could utilise \floatplacement for more efficiency or better graphing (I guess?). – Alexander Voliotis May 10 '20 at 0:00
• Ohh I see, it's necessary because LaTeX recognizes "tables" and "figures" as floats objects Full documentation here – Pablo Díaz May 10 '20 at 2:04


text:
# Ring spectra whose homotopy is the ring of ordinary differential operators

Consider $$R=\mathbb{C}[[x]][\frac{d}{dx}]$$ as a graded ring (assign $$x$$ zero weight, and $$\frac{d}{dx}$$ some non-zero weight). Are there some $$A_n$$-ring spectra naturally arising in homotopy theory such that $$\pi_*X\approx R$$?

P.S.: the answer should depend on the weight assigned to $$\frac{d}{dx}$$. I am not sure what the weight should be (most probably either 1 or 2). I also do not mind adjoining the inverse of $$\frac{d}{dx}$$ if that is necessary.

• Do you really want $\mathbb C$? This ring is a bit large from an algebraic perspective. The ring spectrum $[MU,MU]$ is likely close to what you want. – Phil Tosteson Feb 21 at 18:21
• @PhilTosteson honestly I only recently began studying this so I am not completely clear about what I want. If switching the field helps, I do not mind (I would prefer to stay in char 0, but if that is not possible, char p is ok) – rori Feb 21 at 18:29
• The ring of differential operators is not a graded ring when x is placed in degree 0 and d/dx in some non-zero degree. The commutation relation [d/dx, x] = 1 is not homogenous. It is a filtered ring though, and one can consider the corresponding Rees algebra where you adjoin an element h so that [d/dx, x] = h. This is naturally a graded ring. – Sam Gunningham Feb 21 at 18:42
• You might find the paper "Differential operators and the Steenrod algebra" by R.M.W Wood to be relevant. It seems related to Phil Tosteson's comment as well. – Sam Gunningham Feb 21 at 23:20
• If you want to work in characteristic zero then the homotopy category of rational spectra is just equivalent to the category of graded rational vector spaces, so there isn't really anything interesting to say. – Neil Strickland Feb 22 at 7:21