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# Magnetic Field from Two Wires

1. Mar 8, 2008

### cse63146

1. The problem statement, all variables and given/known data
In this problem, you will be asked to calculate the magnetic field due to a set of two wires with antiparallel currents as shown in the diagram . Each of the wires carries a current of magnitude . The current in wire 1 is directed out of the page and that in wire 2 is directed into the page. The distance between the wires is 2d. The x axis is perpendicular to the line connecting the wires and is equidistant from the wires.

Which of the vectors best represents the direction of the magnetic field created at point K (see the diagram in the problem introduction) by wire 1 alone?
Enter the number of the vector with the appropriate direction.

2. Relevant equations

Right Hand Rule for a Straight Wire

3. The attempt at a solution

Using the right hand rule for a straight wire, I found out that the direction of the magnetic field is counter - clockwise. But I'm not sure which vector represents that.

Any suggestions?

2. Mar 9, 2008

### Shooting Star

The magnetic field is a vector quantity and has a single direction at a given point. If you draw a magnetic line of force through K, due to the field of wire 1 alone, in which direction does the tangent to the line of force at K point?

3. Mar 9, 2008

### physixguru

The magnetic field surrounding a current-carrying wire is tangent to a circle centered on that wire. Use the right-hand rule to find which way it points. For the current coming out of the page, point your thumb in the direction of current (out) and your fingers curl in the direction of the field (counter clockwise).

4. Mar 9, 2008

### cse63146

Vector #8 best describes the direction of the magnetic field.

There's another question that asks: Which of these vectors best represents the direction of the net magnetic field created at point K by both wires?

Since $$I_2 = I_1$$, the magnetic field due to wire #2 be clock wise and the current due to wire #1 be counter - clockwise, would the vector be #1?

5. Mar 9, 2008

### Shooting Star

Let's settle this one first.

Draw a circle with the centre at wire 1 and passing through K. Now draw the tangent to the circle at K. Do you still think it's vector #8?

6. Mar 9, 2008

### cse63146

If I drew that diagram correctly, the answer would be vector #7.

7. Mar 9, 2008

### Shooting Star

Think about the right hand rule. The current is coming toward you. It should be #3.

Try the other questions now.

8. Mar 9, 2008

### cse63146

Thanks, got the other few questions, but I'm having trouble with this one: Point L is located a distance d$$\sqrt{2}$$ from the midpoint between the two wires. Find the magnitude of the magnetic field created at point L by wire 1.

So I need to use the Biot - Savart Law:

$$\frac{\mu_0 I}{2 \pi d}$$

to find the distance, I need to use the pythogorean thereom and I get distance to be $$\sqrt{3 d^2} = \sqrt{3} d$$ and I would just plug that it for the d in the Biot Savart Law to obtain the equation for the magnetic field of point L by wire 1?

9. Mar 9, 2008

### Shooting Star

That's it.

10. Apr 10, 2011

### sdodson

I need to find the magnitude of the net magnetic field at point L in the figure due to both wires and I am having some trouble. I understand that the y components of the magnetic fields from the wires will cancel and just the x components will be left. Therefore, I found the magnitudes of the B_1 field and the B_2 field (magnetic fields due to wires 1 and 2) and used trig to find only the x-components and added these. However, my answer is off by a some factor and I cannot figure out why.

Using t as the angle K-1-L and K-2-L, I got:
B_1x = B_1(sin(t)) = (($$\mu$$ * I)/(2 $$\pi$$ d $$\sqrt{}3$$) *($$\sqrt{}2$$ d)/$$\sqrt{}3$$d)
B_2x = B_1x

I multiplied B_1x by 2 to get my final answer of:
($$\sqrt{}2$$$$\mu$$I)/(3$$\pi$$d)

* note: mu in equations is supposed to be mu sub not and it strangely appears as if 2 and 3 are raised to the pi, they are multiplied

I believe I must have simply made a math error. Any insights would be greatly appreciated! Thank You!


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# Il x < â‚¬ il x >= (r)f4x - 7,Find vajuc for which the function continucus everywhere_

###### Question:

il x < â‚¬ il x > = (r)f 4x - 7, Find vajuc for which the function continucus everywhere_

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## Mirror Symmetry and Symplectic Geometry Seminar: Mayer-Vietoris sequence for relative symplectic cohomology

Seminar | October 13 | 11 a.m.-12 p.m. | 939 Evans Hall

Umut Varolgunes, MIT

Department of Mathematics

I will first recall the definition of an invariant that assigns to any compact subset $K$ of a closed symplectic manifold $M$ a module $SH_M(K)$ over the Novikov ring. I will go over the case of $M=S^2$ to illustrate various points about the invariant. Finally I will state the Mayer-Vietoris property and explain under what conditions it holds.

auroux@math.berkeley.edu


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# Reply To: OFFICE HOURS – Mon June 22nd– Post Questions by Sun 21st @ 5pm EST

Homepage Forums Sprint To Profit Discussion OFFICE HOURS – Mon June 22nd– Post Questions by Sun 21st @ 5pm EST Reply To: OFFICE HOURS – Mon June 22nd– Post Questions by Sun 21st @ 5pm EST

Honor T
Participant

I tried the link I put in when I attached the link and it worked at the time, I tried again and it doesn’t I am hoping this one will work for my question. Thanks.


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The Cap-Set Problem and Frankl-Rodl Theorem (C)

Update: This is a third of three posts (part I, part II) proposing some extensions of the cap set problem and some connections with the Frankl Rodl theorem. Here is a post presenting the problem on Terry Tao’s blog (March 2007). Here is an open discussion post around Roth theorem (March 2009). Here are two “open discussion” posts on the cap set problem (first, and second) from Gowers’s blog (January 2011). These posts include several interesting Fourier theoretic approaches towards improvement of the Roth-Meshulam bound. The second post describes a startling example by Seva Lev.

Suppose that the maximum size of a cap set in $\Gamma(n)=\{1,2,3\}^n$ is $n/g(n)$.  Here is an obvious fact:

The maximum size of a set $F$ in $G$ with the property that every $x,y\in G$ ($x \ne y$) satisfy $x+y \notin G$ is at most the maximum size of a cap set in $G$

Proof: Indeed the condition for $F$ is stronger than being a cap set. We require for every $x,y \in F$ $x \ne y$ not only that $x+y \notin F$ but even $x+y \notin G$.

Part C.  A more direct relation between Frankl-Rodl’s theorem and the cap set problem and all sorts of fine gradings on subsets of {1,2,…,n}.

In part A we moved from sets avoiding affine lines (cap sets) to sets avoiding “modular lines” and used the Frankl-Rodl theorem to deal with the latter. This may look artificial. Here is a simpler connection between the cap set problem and the Frankl-Rodl theorem.

Around the Cap-Set problem (B)

Part B: Finding special cap sets

This is a second part in a 3-part series about variations on the cap set problem that I studied with Roy Meshulam. (The first post is here.)  I will use here a different notation than in part A to make it compatible with various posts of polymath1 which consider similar objects. We write $\Gamma(n) = \{0,1,2\}^n$ and $\Gamma_{a,b,c}$ denotes all vectors of length $n$ with $a$ 0’s, $b$ 1’s and $c$ 2’s.

An affine line are three vectors $x,y,z$ so that $x_i + y_i + z_i = 0 (mod~3)$ for every $i$. A cap set is a subset of $\Gamma(n)$ without an affine line. The cap set problem asks for the maximum size of a cap set. For information on the cap set problem itself look here.  For questions regarding density of sets avoiding other types of lines look here.

In the first part we considered the case that $n=3m$ and considered “modular lines” , i.e., three vectors  $x,y,z$ so that the number of solutions to $x_i+y_i+z_i=a(mod~3)$  is divisible by $m$ for every $a$. When $n$ is divisible by 9 the Frankl-Rodl theorem implies that a subset of $\Gamma$ with more than $(3-\epsilon)^n$ elements must contain a modular line, and even a modular line with two equal elements. (Question: does the same statement follow from Frankl-Rodl theorem when $n$ is divisible by 3 and not by 9?)

10. Some propositions in the desired direction.

Proposition 6:  Suppose that $n=3m$, and that $m=1 (mod~3)$. If every set of size > $(3-\epsilon)^n$ has a modular line, then every set of size > $(3-\epsilon)^n$ has a modular line which is either an affine  line, or  the number of $i$s such that $x_i + y_i + z_i =a$ is precisely $m$ for $a=0,1,2$.

(So we can get rid of a few “types” of modular lines which are not really affine  lines, but not all of them.)

Proposition 6 follows from the much more transparent  proposition:

Proposition 7:  There are sets of constant density which do not contain a modular line $\{x,y,z\}$ for which the numbers of solutions of $x_i + y_i + z_i =a(mod~3)$ for $a=0,1,2$ is never a permutation of $(0,m,2m)$.

Proof: Consider all vectors where the sum of the coordinates is zero modulo 3. This property is preserved under sums but it does not hold for the sums we are looking at. (All the quantities: 0+m+4m, 0+0 +4m, 0+2m+2m, 0+2m+0, m, 2m are not divisible by 3.)

11. A strong variant form of the problem

Here are innocent-looking variants of Problem 1: (the existence of modular lines)

Problem 6:  How large can a set $F$ in $\Gamma(7m)$  be without three vectors x, y and z such that $|\{i:x_i+y_i+z_i=a\}|=o(mod~m)$  for $a=0,1,2,3,4,5,6$?

Problem 7:  How large can a set F in $\Gamma (7m)$  be without three vectors x, y and z such that $|\{i: x_1+y_i+z_i=a\}| = m$ for every a=0,1,2,3,4,5,6? Continue reading

An Open Discussion and Polls: Around Roth’s Theorem

Suppose that $R_n$ is a subset of $\{1,2,\dots, n \}$ of maximum cardinality not containing an arithmetic progression of length 3. Let $g(n)=n/|R_n|$.

How does $g(n)$ behave? We do not really know. Will it help talking about it? Can we somehow look beyond the horizon and try to guess what the truth is?

Update 1: for the discussion: Here are a few more specific questions that we can wonder about and discuss.

1) Is the robustness of Behrend’s bound an indication that the truth is in this neighborhood.

2) Why arn’t there analogs for Salem-Spencer and Behrend’s constructions for the cup set problem?

3) What type of growth functions can we expect at all as the answer to such problems?

4) Where is the deadlock in improving the upper bounds for AP-free sets?

5) What new kind of examples one should try in order to improve the lower bounds? Are there some directions that were already extensively explored?

6) Can you offer some analogies? Other problems that might be related?

Polymath1: Success!

polymath” based on internet image search

And here is a link to the current draft of the paper.

Update:  March 26, the name of the post originally entitled “Polymath1: Probable Success!” was now updated to “Polymath1: Success!” It is now becoming clear that there are  three (perhaps four) new emerging proofs of DHJ. (April 2: See this post by Terry Tao. As this update was also based on briefly talking with Terry,  Terry’s new post gives a better description on the state of affairs and relations between the different proofs.)

The proof directly emerged from the project indeed looks conceptually different and simpler than all other proofs, and may indeed lead to the simplest known proof for Szemeredi’s theorem. (But for this we will have to wait for the details.) In addition, there is a new Ergodic proof by Tim Austin, which was partially inspired by and which used (among several other ingredients developed by Austin) some ideas and  results discovered in the polymath project. Both the original  ergodic proof and Austin’s proof were (at least roughly) “combinatorialized”.

In what sense was it a massive open collaboration? It is true that in the crucial phases of polymath, the phases where two concrete strategies for proofs were considered, the number of pivotal participants was not large. But there was an initial phase were probably more than a hundred mathematicians took part as observers and as commentators. The comments in this early phase played some role in the later developments but what is more important is that this stage have led to the (emerging) selection of the team that developed the proof. Among the hundreds, those who felt they have ideas that can be crucial, or methods that could be helpful, and were smart or lucky to be correct,  and had the persistence to follow these ideas and how these ideas can be combined with other ideas, became the pivotal players. The team that played the game was not so large, but the main massive ingredient of the project which accounts for its accessive mathematical power was in the “draft”. The team emerged from a massive number of participants. (So if you believe there were 10 pivotal players out of a hundred, think about the emergence of the team among ${{100} \choose {10}}$  possible (but, of course, not equally plausible) teams, as a point were  polymath had extra power.)

Two related posts: Tim Gowers raised inthis post  interesting questions regarding the possibility of projects were the actual number of provers will be massive. Here on my blog we have a post  with an “open discussion”  on what are the correct bounds for Roth type problems.  The emphasis is on “small-talk discussion” and not on actual “hard-nose researching”.

We took the opportunity to spend three days of “Purim” visiting northern Israel. Coming back I saw two new posts on Tim Gowers’s blog entitled “Problem solved probably” and “polymath1 and open collaborative mathematics.” It appears that “polymath1” has led to a new proof for the density version of Hales-Jewett’s theorem for k=3 which was the original central goal! Also it looks like the open collaboration mode (while not being a massive collaboration) was indeed useful.

Perhaps the most important thing is to make sure that a complete proof for the k=3 case is indeed in place (as these famous problems sometimes “fight back,” as Erdos used to say).  The outline is described here. If everything is OK as Tim and other participants expect, there are already some discussions or even plans about an extension to the general k case. This seems to be the next major step in the project.  There are also other fruits from the various threads of the polymath1 project. Overall, this looks very exciting! The mathematical result is a first-rate achievement, and the mode of cooperation is novel, interesting and appears genuinely useful.

Let me quote what Tim writes about it: “Better still, it looks very much as though the argument here will generalize straightforwardly to give the full density Hales-Jewett theorem. We are actively working on this and I expect it to be done within a week or so. (Work in progress can be found on the polymath1 wiki.) Better even than that, it seems that the resulting proof will be the simplest known proof of Szemerédi’s theorem.” sababa!

Frankl-Rodl’s Theorem and Variations on the Cap Set Problem: A Recent Research Project with Roy Meshulam (A)

Voita Rodl

I would like to tell you about a research project in progress with Roy Meshulam. (We started it in the summer, but then moved to other things;  so far there are interesting insights, and perhaps problems, but not substantial results. Noga Alon contributed an important observation, and Jeff Kahn and Gabor Simonyi helpful comments.)

Roy and I are friends and have been discussing various mathematical problems for decades. In the last few years we have studied together topological Helly-type theorems and have written several papers together. Our most recent success was finding a topological version of a Helly-type theorem by Nina Amenta. This post is on another problem.

As always, and more so in the spirit of Nielsen’s and Gowers’s recent suggestions:  comments, ideas, related problems, lemmas, and discussions  are welcome.  Links or references to earlier works (published and unpublished) are also welcome.  I thought to wait with this story after I discuss the Frankl-Wilson theorem and the Frankl-Rodl theorem in my extremal combinatorics series. But as this project is somewhat related to the density Hales-Jewett project discussed over Gowers’s blog (and even more to this thread at Tao’s blog), I decided to go ahead with it.

Part A: Finding generalized solutions for the cap set problem

1. Modular-lines-free-subsets

Let $n=3m$. A modular line in $T(n)= \{0,1,2\}^n$ is a set of  three elements $x,y,z$ such that the number of coordinates $i$ for which $x_i + y_i + z_i =a$ is zero modulo $m$ for $a=0,1,2$. (The sum $x_i+y_i+z_i$ is taken modulo 3.)

Problem 1:  How large can a subset $F$ of $T(n)$ be without containing a modular line? Prove that $|F| < (3-\epsilon)^n$, for some constant $\epsilon >0$.

2. Why? avoiding lines (3-term arithmetic progressions) in $Z_3^n$..

The following is a well-known problem:

Problem 2: Find the largest size $f(n)$ of a set $A \subset Z_3^n$ without an arithmetic progression of size 3. In this case an arithmetic progression of size three is simply a sequence of three elements $x, y$ and $z$ such that $x+y+z=0$. In other words, an arithmetic progression is simply an affine line.

This problem was discussed by Frankl, Graham, and Rodl in 1987 (and perhaps was posed even earlier) and they showed that $f(n)=o(3^n)$. Roy Meshulam proved that $f(n) \le 3^n/n$. This is a beautiful and clear application of Roth’s argument (for his famous theorem on the density of sets without 3-term arithmetic progressions,) for the case of subsets of $\{1,2,\dots,n\}$. Ten years ago Roy and I tried quite hard to push this Fourier-theoretic argument and to get a better bound, even slightly, even very slightly. We did not succeed. (And so far, nobody else has; a recent impressive result of Tom Sanders for 3-term arithmetic progressions without a repeated element for subsets of  $Z_4^n$ gives reason for optimism.) Let us write $f(n)=3^n/g(n)$. The best lower bound on $g(n)$ is $n$. There is a construction of Edell showing that $f(n) \ge2.2^n$ and thus the best upper bound for $g(n)$ is more than $1.3^n$. The gap is huge! What is the true value? Terry Tao wrote a lovely post on this problem. A set in $Z_3^n$ without a 3-term arithmetic progression (or alternatively not containing an affine line) is called a cap set.

Let me mention that the set of 0-1 vectors of length $n$ is a very simple example of a cap set of size $2^n$; choosing a set at random works if the size is less than $3^{n/3}$.   Problem 1 is a variation on problem 2.  Continue reading

Mathematics, Science, and Blogs

Michael Nielsen wrote a lovely essay entitled “Doing science online” about  mathematics, science,  and blogs. Michael’s primary example is a post over Terry Tao’s blog about the Navier-Stokes equation and he suggests blogs as a way of scaling up scientific conversation. Michael is writing a book called “The Future of Science.” He is a strong advocate of doing science in the open, and regard these changes as truly revolutionary.  (The term “Science 2.0″ is mentioned in the remarks.)

Michael’s post triggered Tim Gowers to present his thoughts about massive collaboration in mathematics, and this post is also very interesting with interesting follow-up remarks.  Tim Gowers mentioned the n-category cafe as a place where a whole research programme is advanced on a blog. Terry Tao mentioned comments on posts on his open-problems series as having some value. He  mentioned, in particular, the post on Mahler’s conjecture. (Also I think some discussions over Scott Aaronson’s blog had the nature of discussing specific technical math (coming from CS) problems.)

Tim actually proposes an experiment: trying to solve collectively a specific math problem. This would be interesting!!! I suppose we need to give such an effort over a blog a longer than ususal life-span – a few months perhaps. (And maybe not to start with a terribly difficult problem.) (What can be an appropriate control experiment though?)

Ben Webster in “Secret blogging seminar”  mentioned, in this context, earlier interesting related posts about “Working in secret“.

Christian Elsholtz mentioned on Gowers’s blog an intermediate problem (called “Moser’s cube problem”) where you look not for combinatorial lines (where the undetermined coordinates should be 1 in x 2 in y and 3 in z), and not for an affine line (where it should be 1,2,3 in x y and z in any order), but for a line: it can be 1 in x 2 in y and 3 in z or 3 in x 2 in y and 1 in z.

Update: Things are moving fast regarding Gowers’s massive collaboration experiment. He peoposes to study together a new approach to the $k=3$ “density Hales -Jewett theorem”. A background post apears here. Hillel Furstenberg and Izzy Katznelson’s proof of the density Hales-Jewett theorem was a crowning achievement of the ergodic theory method towards Szemeredi’s theorem. Like the case for Furstenberg’s proof of Szemeredi’s theorem itself the case $k=3$ was considerably simpler and had appeared in an earlier paper by Hillel and Izzy. The recent extensions of Szemeredi regularity lemma that led to simpler combinatorial proof of Szemeredi’s theorem did not led so far to simpler proofs for the density Hales-Jewett case. If you look at Tim’s background post let me ask you this: What is the case $k=2$ of the density Hales-Jewett’s theorem? It is something familiar that we talked about!

Here is a particularly silly problem that I suggested at some point along the discussions: How large can a family of subsets of an $n$-element set be without having two sets A and B such that the number of elements in A but not in B is twice the number of elements in B but not in A?

Update: This problem was completely reolved by Imre Leader and Eoin Long, their paper Tilted Sperner families contains also related results and conjectures.

Massive collaboration in art (click the picture for details)

Q: What is the case $k=2$of the density Hales-Jewett’s theorem? A: It is  Sperner’s theorem! (that we discussed in this post.)

I will keep updating about news from Tim’s project. [Last update is from October 21].  More updates: Tim’s project is getting quickly off the ground. A useful wiki was established. More update: It is probably successful


text:
#### Vol. 4, No. 4, 2011

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
The corona theorem for the Drury–Arveson Hardy space and other holomorphic Besov–Sobolev spaces on the unit ball in $\C^n$

### Şerban Costea, Eric T. Sawyer and Brett D. Wick

Vol. 4 (2011), No. 4, 499–550
##### Abstract

We prove that the multiplier algebra of the Drury–Arveson Hardy space ${H}_{n}^{2}$ on the unit ball in ${ℂ}^{n}$ has no corona in its maximal ideal space, thus generalizing the corona theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov–Sobolev space ${B}_{p}^{\sigma }$ has the “baby corona property” for all $\sigma \ge 0$ and $1. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.

##### Keywords
Besov–Sobolev Spaces, corona Theorem, several complex variables, Toeplitz corona theorem
##### Mathematical Subject Classification 2000
Primary: 30H05, 32A37
##### Supplementary material

PDF file containing proofs of formulas and modifications of arguments already in the literature that would otherwise interrupt the main flow of the paper.


text:
# The extension of diffeomorphism

Let ${\Omega _1}$,${\Omega _2}$ be two open sets in $\mathbb R^n$ and $f$ is a diffeomorphism between them. For every $x$ in ${\Omega _1}$, is there an open set $\Omega_{x} \subset \Omega_1$ and a diffeomorphism $g$ of $\mathbb R^n$ such that $g=f$ when restricted to $\Omega_{x}$?

-

Assume $x\in \Omega_1$ such that $Df(x) = A$. Wolg $x=0$ and $f(0) =0$, otherwise the following reasoning needs to be 'translated'. Now let $\phi (x)= \psi(||x||)$ be a smooth radially symmetric cutoff function which is identically $0$ outside of $B_t(0)$ and $=1$ inside, say, $B_{t/2}(0)$, s.t. $\psi^'(t) \le O(1/t)$ . Assume $B_t(0)\subset \Omega_1.$

Now look at

$$F(x):= (1-\phi(x)) Ax + \phi(x) f(x)$$

which is defined for every $x\in\mathbb{R}^n$ (it's easy to see that this is possible).

For $||x|| \ge t$ this is just $Ax$.

For $||x||< t$ denote $r=||x||$ and calculate for $v$ such that $||v||= 1$ $$DF(x) v = -\psi^'(r)<\frac{x}{r},v> Ax +(1-\phi(x))Av +\psi^'(r)<\frac{x}{r}, v> f(x) + \phi(x) Df(x)v\\ = Av + \phi(x)(Df(x)v-Av) +\psi^'(r) <\frac{x}{r}, v>(f(x)-Ax)$$

The first term on the rhs is nonzero for nonzero $v$, since $A$ is invertible. Note that it does not depend on $t$, and since the sphere is compact it's norm is bounded from below by a constant.

The second term is arbitrarily small, since $f$ is continuously differentiable (I do assume continuity of $Df$) and $\phi$ is bounded, if only $t$ is small enough.

Since $f(x)-Ax = o(x)$ the third term also becomes arbitrarily small given the growth rate of $\psi$.

In response to a comment from Leonid Kovalev: from $$F(x)-F(y) =Ax-Ay - \phi(x)(Ax-f(x)) +\phi(y)(Ay - f(y))$$ it is easy to see that $F$ is 1-1, using the $f(x)-Ax=o(||x||)$ property and the fact that $A$ invertible. The fact that $F$ is onto follows, e.g., by elementary degree theory from the fact that $F=A$ on the boundary of $B_t(0)$. It is probably possible to see this with a more elementary reasoning, for which I don't have the time right now.

An alternative reasoning which you may find more satisfying: according to Thm. 1.7 in Hirsch's Differential Topology the set of diffeomorphisms is open in $C^1(\mathbb{R}^n, \mathbb{R}^n)$. The $F$ I have constructed is arbitrarily close to $x\mapsto Ax$ in $C^1$.

-
Something should be said about the global invertibility of $F$. –  user31373 May 23 '12 at 18:42
@LeonidKovalev ok, valid point. I did say something about global invertibility of $F$. –  user20266 May 23 '12 at 19:24
Thanks! I did not doubt that your proof works, just wanted to make this point for the sake of other readers. For the same reason I'll add that the computations would be a bit tidier with the normalization $Df(x)=I$ (achieved by composition with a linear invertible map). –  user31373 May 23 '12 at 19:27
@LeonidKovalev: this time I disagree. If, by accident or transformation, $A=I$, it is (in my opinion, your mileage may vary) less transparent, which term is a derivative and which is evaluation of a function. –  user20266 May 23 '12 at 19:34
Nice proof. BTW, that $F$ is a diffeomorphism is easy if we notice that $F$ is proper. –  Hezudao May 24 '12 at 1:45


text:
option.h 18.4 KB
 Marcin Siodelski committed May 05, 2015 1 // Copyright (C) 2011-2015 Internet Systems Consortium, Inc. ("ISC")  Tomek Mrugalski committed Oct 14, 2011 2 3 4 5 6 7 8 9 10 11 12 13 14 // // Permission to use, copy, modify, and/or distribute this software for any // purpose with or without fee is hereby granted, provided that the above // copyright notice and this permission notice appear in all copies. // // THE SOFTWARE IS PROVIDED "AS IS" AND ISC DISCLAIMS ALL WARRANTIES WITH // REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY // AND FITNESS. IN NO EVENT SHALL ISC BE LIABLE FOR ANY SPECIAL, DIRECT, // INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM // LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE // OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR // PERFORMANCE OF THIS SOFTWARE.  Stephen Morris committed Nov 16, 2012 15 16 17 18 19 #ifndef OPTION_H #define OPTION_H #include  Marcin Siodelski committed Oct 04, 2013 20 #include  Stephen Morris committed Nov 16, 2012 21 #include  Tomek Mrugalski committed Oct 14, 2011 22 23  #include  Stephen Morris committed Nov 16, 2012 24 #include  Tomek Mrugalski committed Oct 26, 2011 25 #include  Tomek Mrugalski committed Oct 14, 2011 26 27 28 29  namespace isc { namespace dhcp {  Tomek Mrugalski committed Feb 17, 2012 30 31 32 /// @brief buffer types used in DHCP code. /// /// Dereferencing OptionBuffer iterator will point out to contiguous memory.  Tomek Mrugalski committed Feb 03, 2012 33 34 35 36 37 38 39 40 41 typedef std::vector OptionBuffer; /// iterator for walking over OptionBuffer typedef OptionBuffer::iterator OptionBufferIter; /// const_iterator for walking over OptionBuffer typedef OptionBuffer::const_iterator OptionBufferConstIter; /// pointer to a DHCP buffer  Tomek Mrugalski committed Feb 17, 2012 42 typedef boost::shared_ptr OptionBufferPtr;  Tomek Mrugalski committed Feb 03, 2012 43 44 45 46 47  /// shared pointer to Option object class Option; typedef boost::shared_ptr


text:
Local observability of state variables and parameters in nonlinear modeling quantified by delay reconstruction

# Local observability of state variables and parameters in nonlinear modeling quantified by delay reconstruction

Ulrich Parlitz Max Planck Institute for Dynamics and Self-Organization, Am Faßberg 17, 37077 Göttingen, Germany Institute for Nonlinear Dynamics, Georg-August-Universität Göttingen, Am Faßberg 17, 37077 Göttingen, Germany    Jan Schumann-Bischoff Max Planck Institute for Dynamics and Self-Organization, Am Faßberg 17, 37077 Göttingen, Germany Institute for Nonlinear Dynamics, Georg-August-Universität Göttingen, Am Faßberg 17, 37077 Göttingen, Germany    Stefan Luther Max Planck Institute for Dynamics and Self-Organization, Am Faßberg 17, 37077 Göttingen, Germany Institute for Nonlinear Dynamics, Georg-August-Universität Göttingen, Am Faßberg 17, 37077 Göttingen, Germany
July 5, 2019
###### Abstract

Features of the Jacobian matrix of the delay coordinates map are exploited for quantifying the robustness and reliability of state and parameter estimations for a given dynamical model using an observed time series. Relevant concepts of this approach are introduced and illustrated for discrete and continuous time systems employing a filtered Hénon map and a Rössler system.

Observability, parameter estimation, nonlinear modelling

For many physical processes dynamical models (differential equations or iterated maps) are available but often not all of their variables and parameters are known or can be (easily) measured. In meteorology, for example, sophisticated large scale models exist, which have to be continuously adapted to the true temporal changes of temperatures, wind speed, humidity, and other relevant physical quantities. To obtain a model that “follows” reality, measured data have to be repeatedly incorporated into the model. In geosciences this procedure is called data assimilation, but the task to track state variables and system parameters by means of estimation methods occurs also in other fields of physics and applications. However, not all observables provide the information required to estimate a particular unknown quantity. In this article, we consider this problem of observability in the context of chaotic dynamics where sensitive dependance on initial conditions complicates any estimation method. A quantitative characterization of local observability employing delay coordinates is used to answer the question where in state and parameter space estimation of a particular state variable or parameter is feasible and where not.

## I Introduction

To describe and forecast dynamical processes in physics and many other fields of science mathematical models are used, like, for example, ordinary differential equations (ODEs), partial differential equations (PDEs), or iterated maps. Some of these models are derived from first principles while others are the result of a general black-box modeling approach (e.g., based on neural networks). These models typically contain two kinds of variables and parameters: those that can be directly measured or are known beforehand (e.g., fundamental physical constants) and others whose values are unknown and very difficult to accessVTK04 (); RBKT10 (). To estimate the latter estimation methods have been devised that aim at extracting the required information from the dynamics, here represented by the model equations and the experimentally observed dynamical evolution of the underlying process. Different approaches for solving this dynamical estimation problem have been devised in the past, including (nonlinear) observer or synchronization schemes PJK96 (); NM97 (); HLN01 (); GB08 (); ACJ08 (); SO09 (); SRL09 (); K05 (); FMG05 (); CK07 (); Amritkar09 (), particle filters L10 (), a path integral formalism A09 (); QA10 (), or optimization based algorithms CGA08 (); B10 (); SBP11 ().

Before applying such an estimation method one may ask whether the available time series (observable) actually contains the required information to estimate a particular unknown value. In control theory this is called observability problem and it can for linear systems of ODEs be analyzed and answered by means of the so-called observability matrix Sontag (); NS90 (). Using derivative coordinates this approach can be generalized for nonlinear continuous systems HK77 (); Sontag (); N82 (). For state estimation of chaotic systems Letellier, Aguirre and Maquet LAM05 (); AL05 (); LAM06 (); LA09 (); FBL12 () considered continuous dynamical systems

 ˙x=f(x) (1)

that generate some observed signal where is the state of the system, is a smooth submanifold of , and denotes a measurement or observation function. Consider now -dimensional derivative coordinates Aeyels (); Takens (); SYC91 (); KS97 (); book_HDIA () of the observed signal

 y=(s,˙s,¨s,…,s(D−1))=F(x)∈RD (2)

where stands for the -th temporal derivative of , is the reconstruction dimension, and is called derivative coordinates map SYC91 (). If this map is (at least locally) invertible, then we can uniquely determine the full state vector from the signal and its higher derivatives LAF09 (). Furthermore, small perturbations in should correspond to small perturbations in and vice versa. Therefore, we want the map to be an immersion, i.e. a smooth map whose derivative map is one-to-one at every point of , or equivalently, whose Jacobian matrix has full rank on the tangent space (here ). This does not imply that the map itself is one-to-one (), since the derivative coordinates (2) may provide states with two (or more) pre-images separated by a finite distance. Therefore, the observability analysis presented in the following is local, only, because it is based on analyzing (the rank) of the Jacobian matrix . The (global) one-to-one property of the map is not checked (what would be necessary, and for compact U also sufficient, to show that is an embedding SYC91 ()).

The Jacobian matrix can be computed by means of the vector field given in Eq. (1). In fact, for linear ODEs the Jacobian matrix conforms with the observability matrix known from (linear) control theory AL05 (). To estimate the rank of Letellier, Aguirre, and Maquet LAM05 (); AL05 () suggest to compute the eigenvalues of the - matrix

 A(x)=DFtr(x)⋅DF(x). (3)

Nonzero eigenvalues indicate full rank of and thus local invertibility of at . To quantify the (local) invertibility of and thus the (local) observability of the full state Aguirre, Letellier, and Maquet LAM05 (); AL05 () introduced the observability index

 δ(x)=μmin(A)μmax(A) (4)

where and denote the smallest and the largest eigenvalue of the matrix , respectively. Time averaging (along the available trajectory for ) yields

 ¯δ=1T∫T0δ(x(t))dt. (5)

Instead of derivative coordinates we consider in the following delay coordinates Aeyels (); Takens (); SYC91 (); KS97 (); book_HDIA (). Furthermore, we extend the observability analysis to parameter estimation and compute a specific measure of uncertainty PSBL () for each state variable or parameter to be estimated. Last not least, we are not only interested in quantifying the average observability (like in Eq. (5)) but also in local variations that can be exploited during the state and parameter estimation process.

## Ii Delay coordinates and observability

To motivate the concepts to be presented in the following we shall first consider a discrete time system (iterated map) where all model parameters are known and only state variables have to be estimated from the observed time series.

### ii.1 Estimating state variables of a filtered Hénon map

For an dimensional discrete system

 x(n+1)=g(x(n)) (6)

which generates the times series with where we can construct dimensional delay coordinates Aeyels (); Takens (); SYC91 (); KS97 (); book_HDIA () with reconstructed states

 y(n) = (s(n),s(n+1),....,s(n+D−1)) = G+(x(n))∈RD.

Again we assume that all states of interest lie within a smooth manifold . Here we consider delay coordinates forward in time. The function is therefore called forward delay coordinates map . It is also possible to use delay coordinates backward in time, or mixed forward and backward, and we shall address this issue in Sec. II.2.

As already discussed with derivative coordinates in the previous section a state is locally observable from the time series if is an immersion, i.e. if the Jacobian matrix has maximal (full) rank at . The corresponding Jacobian matrix can be computed using the iterated map Eq. (6). If the Jacobian matrix has maximal rank (assuming ) then is locally invertible (on ). “Local” means that still a delay vector could possess different pre images (separated by a finite distance).

To motivate and illustrate this analysis we consider the Hénon map

 x1(n+1) = 1−ax21(n)+bx2(n) (8) x2(n+1) = x1(n) (9)

with parameters and . In the following we shall assume that the dynamics of this system is observed via a filtered signal provided by an FIR-filter

 s(n) = x1(n)+cx1(n−1) (10) = x1(n)+cx2(n)=h(x(n))

with filter parameter .

For two dimensional delay coordinates the delay coordinates map reads

 G(x(n)) = (s(n),s(n+1)) = (x1(n)+cx2(n), 1−ax21(n)+bx2(n)+cx1(n))

or

 G(x)=(x1+cx2,1−ax21+bx2+cx1). (11)

The Jacobian matrix of the map is given by:

 DG(x)=(1c−2ax1+cb) (12)

and its determinant

 det(DG(x))=2acx1+b−c2 (13)

vanishes for all states on the singular line

 xs1=c2−b2ac. (14)

For the FIR-filter is (asymptotically) deactivated and the critical line disappears (). For , however, the critical line crosses the chaotic attractor as shown for in Fig. 1a.

### ii.2 Forward, backward, and mixed delay coordinates

Instead of using state space reconstruction based on forward delay coordinates (II.1) one could also use backward delay coordinates

 y(n) = (s(n),s(n−1),…,s(n−D+1)) = G−(x(n))∈RD

or more general, a combination of forward and backward components

 y(n) = (s(n−D−),…s(n−1),s(n),s(n+1), …,s(n+D+)) = G±(x(n);D−,D+)∈RD

called mixed delay coordinates in the following, with reconstruction dimension . To obtain the backward components the inverse map and its Jacobian matrix are required (here we assume that the dynamics is time invertible). For discrete time systems (like the Hénon example) the underlying map (6) has to be inverted and for continuous time systems the inverse of the flow can in principle be computed by integrating the system ODEs (1) backward in time. In both cases, however, problems may occur in practice, because an explicit form of the inverse map may not exist and backward integration of dissipative systems results in diverging solutions and numerical instabilities (for longer integration times). Despite these difficulties inclusion of backward components turns out to be beneficial for the estimation task as will be demonstrated in the following for the Hénon examples and the Rössler system.

### ii.3 Noisy observations and uncertainty

At states with the delay coordinates map is in principle invertible, but the inverse can be very susceptible to perturbations in like measurement noise. To quantify the robustness and the sensitivity of the inverse with respect to noise we consider the singular value decomposition of the Jacobian matrix of the delay coordinates map

 DG=U⋅S⋅Vtr (17)

where is an diagonal matrix containing the singular values and and are orthogonal matrices, represented by the column vectors and , respectively. is the transposed of coinciding with the inverse . Analogously, and the inverse Jacobian matrix reads

 DG−1=V⋅S−1⋅Utr (18)

where . Multiplying by from the right we obtain or

 DG−1u(m)=1σmv(m)     (m=1,…,M). (19)

This transformation is illustrated in Fig. 2 and it describes how small perturbations of in delay reconstruction space result in deviations from in the original state space. Most relevant for the observability of the (original) state is the length of the longest principal axis of the ellipsoid given by the inverse of the smallest singular value (see Fig. 2). Small singular values correspond to directions in state space where it is difficult (or even impossible) to locate the true state given a finite precision of the reconstructed state . For the filtered Hénon map we find that the closer the state is to the critical line given by (14) the more severe is this uncertainty. This is illustrated in Fig. 1a,b where at some points the ellipses spanned by the column vectors of the matrix are plotted. Figure 3 shows (color coded) the logarithm of the ratio smallest singular value (here: ) divided by the largest singular value vs. state variables and in a range of coordinates containing the chaotic Hénon attractor. In Fig. 3a dimensional forward delay (II.1) is considered where at the smallest singular value vanishes indicating the singularity (14) illustrated in Fig. 1a,b. If the reconstruction dimension is increased from to the singularity disappears as can be seen in Fig. 3b showing the ratio (color coded) for dimensional forward delay coordinates. For comparison, Figs. 3c,d show results obtained with mixed delay coordinates (II.2) and backward delay coordinates (II.2), respectively. The white areas in Fig. 3d correspond to ratios indicating poor observability (due to fast divergence of backward iterates of the Hénon map). Further increase of the reconstruction dimension ( or ) results in even larger values of (not shown here).

To assess the observability on the Hénon attractor we computed histograms of ratios at points. Figure 4 shows these histograms for the same coordinates used to generate the corresponding diagrams in Fig. 3. The best results (large ratios) provide mixed delay coordinates (Fig. 4c). We speculate that this is due to the fact that forward and backward components cover different directions in state space (similar to Lyapunov vectors corresponding to positive and negative Lyapunov exponents).

If the perturbations of the reconstructed state are due to normally distributed measurement noise they can be described by a symmetric Gaussian distribution centered at

 Q(~y)=exp[−12(~y−y)trΣ−1y(~y−y)]√(2π)Ddet(Σy) (20)

where denotes the covariance matrix ( stands for the -dimensional unit matrix) and the standard deviation quantifies the noise amplitude. For (infinitesimally) small perturbations this distribution is mapped by the (pseudo) inverse of the delay coordinates map to the (non-symmetrical) distribution

 P(~x)=exp[−12(~x−x)trΣ−1x(~x−x)]√(2π)Mdet(Σx) (21)

centered at with the inverse covariance matrix

 Σ−1x = DGtr⋅Σ−1y⋅DG=1ρ2DGtr⋅DG (22) = 1ρ2V⋅S2⋅Vtr. (23)

The marginal distribution of a given state variable centered at is given by

 Pj(~xj)=1ρj√2πexp[−(~xj−xj)22ρ2j] (24)

where the standard deviation is given by the square root of the diagonal elements of the covariance matrix

 ρj=√Σx,jj. (25)

Using Eq. (22) the standard deviation of the marginal distribution can be written

 ρj=ρ√[DGtr⋅DG]−1jj=ρ√[V⋅S−2⋅Vtr]jj (26)

and we consider in the following the factor

 νj=√[DGtr⋅DG]−1jj=√[V⋅S−2⋅Vtr]jj (27)

as our measure of uncertainty when estimating , because it quantitatively describes how the initial standard deviation is amplified when estimating variable PSBL ().

Figure 5 shows the uncertainties and vs. for different dimensional delay coordinates. In Fig. 4a,b results obtained with dimensional forward delay coordinates (II.1) are given. Large uncertainties occur mainly in a vertical stripe located near the singularity at (Eq. (14)) occurring for . Figures 5c,d,e,f show uncertainties of and obtained with mixed delay coordinates ((c),(d)) and backward delay coordinates ((e), (f)). For mixed delay coordinates (Fig. 5c,d) areas with very high uncertainties occur near the origin, but along the attractor and take only relatively low values. This is also confirmed by the -histograms on the attractor given in Fig. 6 for the same delay coordinates as used in Fig. 4. Again, the mixed delay coordinates turns out to be superior to the purely forward or backward coordinates. Furthermore, the dependance of the range of uncertainty values on the type of coordinates is different for different variables. While the uncertainty of increases when changing from forward to backward delay coordinates (Figs. 6a,e), the uncertainty of exhibits the opposite trend (Figs. 6b,f).

### ii.4 State and parameter estimation

Until now only the state variables and are considered as unknowns to be estimated and the parameters and of the Hénon map and of the FIR filter are assumed to be known. We shall now consider the general case including unknown parameters of the dynamical system and unknown parameters of the measurement function . Let the dynamical model be a -dimensional discrete

 x(n+1)=g(x(n),p) (28)

or a continuous

 ˙x=f(x,p) (29)

dynamical system generating a flow

 ϕt:RM→RM (30)

with discrete or continuous time. Furthermore, let’s assume that a time series of length is given observed via the measurement function

 s(t)=h(ϕt(x,p),q) (31)

from a trajectory starting at .

This provides the -dimensional delay coordinates

 y = (s(−D−τ−),…,s(−τ−),s(0),s(τ+),…,s(D+τ+)) (32) = G(x,p,q;D−,D+,τ−,τ+)∈RD

with . Here the delay coordinates map is considered as a function of: (i) the state and the parameters of the underlying system, (ii) the parameters of the measurement function, (iii) the dimension parameters and , and (iv) the delay times and in backward and forward direction, respectively. The option to use different delay times, and for the backward and forward iterations is motivated by the fact that for dissipative systems backward solutions quickly diverge and therefore a choice may be more appropriate. For the same reason has typically to be smaller than . Since the reconstruction dimensions and the delay times are chosen a priori and are not part of the estimation problem they shall not be listed as arguments of to avoid clumsy notation. The Jacobian matrix of has the structure

 DG(x,p,q)=(A,B,C) (33)

where:

 A=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝∂h(ϕ−D−τ−(x,p),q)∂x1…∂h(ϕ−D−τ−(x,p),q)∂xM⋮⋮⋮∂h(ϕ−τ−(x,p),q)∂x1…∂h(ϕ−τ−(x,p),q)∂xM∂h(x,q)∂x1…∂h(x,q)∂xM∂h(ϕτ+(x,p),q)∂x1…∂h(ϕτ+(x,p),q)∂xM⋮⋮⋮∂h(ϕD+τ+(x,p),q)∂x1…∂h(ϕD+τ+(x,p),q)∂xM⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠
 B=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝∂h(ϕ−D−τ−(x,p),q)∂p1…∂h(ϕ−D−τ−(x,p),q)∂pP⋮⋮⋮∂h(ϕ−τ−(x,p),q)∂p1…∂h(ϕ−τ−(x,p),q)∂pP0…0∂h(ϕτ+(x,p),q)∂p1…∂h(ϕτ+(x,p),q)∂pP⋮⋮⋮∂h(ϕD+τ+(x,p),q)∂p1…∂h(ϕD+τ+(x,p),q)∂pP⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠
 C=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝∂h(ϕ−D−τ−(x,p),q)∂q1…∂h(ϕ−D−τ−(x,p),q)∂qL⋮⋮⋮∂h(ϕ−τ−(x,p),q)∂q1…∂h(ϕ−τ−(x,p),q)∂qL∂h(x,q)∂q1…∂h(x,q)∂qL∂h(ϕτ+(x,p),q)∂q1…∂h(ϕτ+(x,p),q)∂qL⋮⋮⋮∂h(ϕD+τ+(x,p),q)∂q1…∂h(ϕD+τ+(x,p),q)∂qL⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

and it can also be written as

 DG= (34)

where

 ∇xh(x,q) = (∂h∂x1,…,∂h∂xM)(x,q) (35) ∇qh(x,q) = (∂h∂q1,…,∂h∂qQ)(x,q) (36)

and and denote the Jacobian matrices of the flow whose elements are derivatives with respect to the state variables and the parameters , respectively. For discrete dynamical systems (28) the Jacobians and can be computed using the chain rule and the recursion schemes

 Dxϕt+1(x,p) = Dxg(ϕt(x,p),p)⋅Dxϕt(x,p) (37) Dpϕt+1(x,p) = Dxg(ϕt(x,p),p)⋅Dpg(ϕt(x,p),p) (38) +Dpϕt(x,p)

with ( unit matrix and . If backward iterations are required ( and ) similar recursion schemes exist based on the inverse map (providing ) and its Jacobian matrice and . Instead of recursion schemes one may also use symbolic or automatic differentiation AD (). For continuous systems (29) the required Jacobians can be obtained by simultaneously solving linearized systems equations as will be discussed in Sec. II.6. Inverse maps ( and ) may be computed via backward integration of the ODEs (at least for short periods of time before solutions diverge significantly). An extension for multivariate time series is straightforward.

### ii.5 Parameter estimation for the Hénon map

We shall now extend the discussion to include not only state estimation but also parameter estimation. For better readability only forward delay coordinates are considered in the following, but all steps can also be done with mixed or backward delay coordinates, of course. We first consider the case where and are assumed to be known and only has to be estimated. In this case, unknown variables and unknown system parameter exists (while ). Therefore, delay coordinates with dimension or higher will be used. Figure 7 shows the ratio of singular values vs. in a plane in given by (and fixed parameters and ). For reconstruction dimension the ration is very small for extended subsets of the plane (white stripes in Fig. 7a). If the delay reconstruction dimension is increased to (Fig. 7b). these regions shrink or disappear. If the dimension is increased furthermore, the delay coordinates map is locally invertible in the full range of and values shown in Fig. 7 (results not shown here).

Now we include in the list of quantities to be estimated. Figure 8 shows the ratio of singular values for reconstruction dimensions and . For curves with very low singular value ratios exist crossing the Hénon attractor which disappear for .

Figure 9 shows the uncertainties (Eq. (27)) for . As can be seen, the values of uncertainties vary strongly in the - plane and still some islands with rather large uncertainties exist.

Similar results are obtained if we include the remaining parameters in the estimation problem. Scanning the two-dimensional - subspace (plane) of the dimensional estimation problem for with fixed , , and indicates (almost) vanishing smallest singular values as long as . With dimensional delay coordinates the Jacobian matrix has clearly full rank almost everywhere within the chosen range as can be seen in Fig.. 10a. Figures 10b,c,d,e,f illustrate the uncertainties (Eq. (27)) of , respectively.

### ii.6 Continuous dynamical systems

To compute the Jacobian matrix (34) of the delay coordinates map we have to compute the gradients (35) and (36) of the observation function and the Jacobian matrices and containing derivatives of the flow generated by the dynamical system (29) with respect to variables and parameters , respectively. The -matrix can be computed by solving the linearized dynamical equations in terms of a matrix ODE

 ddtY=Dxf(ϕt(x,p),p)⋅Y (39)

where is a solution of Eq. (29) with initial value and is an matrix that is initialized as , where denotes the identity matrix. Similarly, the -matrix is obtained as a solution of the matrix ODE Kawakami ()

 ddtZ=Dxf(x(t),p)⋅Z+Dpf(x(t),p) (40)

with . Solving (39) and (40) simultaneously with the system ODEs (29) we can compute , , etc. and use these matrices to obtain the Jacobian matrix of the delay coordinates map (34). For mixed or backward delay coordinates the required components can be computed by integrating the system ODE and the linearized ODEs backward in time.

#### ii.6.1 The Rössler system

To demonstrate the observability analysis for continuous systems we follow Aguirre and Letellier AL05 () and consider the Rössler system

 ˙x1 = −x2−x3 ˙x2 = x1+ax2 (41) ˙x3 = b+x3(x1−c)

with , , and .

Time series of different observables , or , or are considered, all of them consisting of values sampled with . Figure 11 shows the Rössler attractor where color indicates the uncertainty of estimating the variable (first column), or (second column), or (third column) using forward delay coordinates. The results in the first row are obtained when observing , while the diagrams in the second and third row show results for or time series. The reconstruction dimension equals and the delay time is . The bright yellow bullet indicates the state with the lowest uncertainty. This state and the following states plotted as thick red bullets underly the time series values that are used for the delay reconstruction. They span a window in time of length which is about one half of the mean period of the chaotic oscillations . The lowest uncertainties are obtained for states whose reconstruction involves trajectory segments following the vertical -excursion on the attractor. In contrast, trajectory segments starting from states with poor observability (large uncertainty) are located in the flat part of the Rössler attractor. Figures 11a and b show that using time series low values of occur on parts for the attractor where is high (and vice versa). Interestingly, this is not the case for delay reconstructions based on time series as can be seen in Figs. 11d and e, where low uncertainties of and occur in similar regions on the attractor.

In Fig. 12 distributions of uncertainty values of the Rössler system are shown that were obtained along an orbit of states sampled with . The distributions are shown as color coded histograms, estimated from the relative frequency of occurrence of the corresponding (in %). All diagrams show the dependance of the histograms on the delay time chosen for forward delay coordinates (horizontal axis). The reconstruction dimension is for all cases and all three parameters are assumed to be known (and are not part of the estimation task, i.e. ). In the first row (Figs. 12a,d,g) estimations are based on a time series from the Rössler system, and in rows two and three, and time series are used, respectively. The uncertainties of the given observable (Figs. 12a,e,i) mostly equal one () or are smaller (due to the additional information provided by the delay coordinates). In general, lowest uncertainties for all variables are obtained when using time series (Figs. 12a,d,g) while data provide highest uncertainties (Figs. 12c,f,i).

Figure 13 shows the same diagrams but now computed using four dimensional mixed delay coordinates with and . Similar to the results obtained with the Hénon map the uncertainties computed for mixed delay coordinates are typically smaller than those obtained with forward coordinates. Furthermore, the histograms shown in Fig. 13 suggest that for mixed delay coordinates using as measured variable provides the best results, followed by the time series. This is in contrast to forward coordinates (Fig. 12) where data yield the smallest uncertainties for the other variables ( and ). Similar results have been obtained with three dimensional forward or mixed coordinates.

The fact that mixed delay coordinates provide the lowest uncertainties when using time series is consistent with results for derivative coordinates obtained by Letellier et al. LAM05 () who found a ranking (for a different set of model parameters). For better comparison with their results we computed the (attractor) average

 ¯γ=1T∫T0γ(x(t))dt (42)

of the ratio

 γ(x)=σ2min(DG(x))σ2max(DG(x)) (43)

that provides the delay reconstruction analog of the observability index (4). Figure 14 shows vs. the delay time for different delay coordinates (rows) and different measured time series (columns). While for forward delay coordinates the largest values of the observability index occur if is measured (Fig. 14c), time series provide best observability if mixed delay coordinates are used (Fig. 14d). Note that in most cases high observability occurs for which is very close to the first zero of the autocorrelation function (that is often used as preferred value for delay reconstruction).

Fig. 15 shows similar histograms but now for the full estimation problem ( variables and parameters). Forward delay coordinates are used and the reconstruction dimension is increased to and an time series of length is used (with sampling time ). For delay times that are an integer multiple of half of the mean period relatively high uncertainties of occur, in particular for , , , and . This is due to the well known fact that for these delay times the attractor reconstruction results in points scattered near a straight (diagonal) line (an effect that also occurs when considering delay coordinates of a sinusoidal signal).

## Iii Conclusion

Starting from the question “Does some particular (measured) time series provide sufficient information for estimating a state variable or a model parameter of interest” we revisited the observability problem for nonlinear (chaotic) dynamical systems. In particular we considered delay coordinates and the ability to recover not measured state variables and parameters from delay vectors. This requires to “invert” the delay coordinates construction process which is at least locally possible, if the Jacobian matrix of the delay coordinates map has maximum (full) rank. Furthermore, we investigated how states near the delay vector are mapped back to the state and parameter space of the systems. In this way it is possible to quantify the amplification of small perturbations in delay reconstruction space in different directions of the state and parameter space. This reasoning gave rise to the concept of uncertainties of estimated variables and parameters. Both, observability and uncertainties may vary considerably in state space and on a given (chaotic) attractor. This feature was demonstrated with a discrete time system (filtered Hénon map) and a continuous system (Rössler system). Local observability and uncertainties also depend on the available measured variable (time series) and the type of delay coordinates. Best results were obtained with mixed delay coordinates, containing at least a one step backward in time.

The obtained information about (local) uncertainties in state and parameter estimation can be used in several ways for subsequent analysis. First of all, it may help to decide whether the planned estimation task is feasible at all or whether another observable has to be measured instead. For continuous time systems relevant time scales (delay times) can be identified where uncertainties are minimal.

The strong variations of local uncertainty values in state space (along a trajectory) occurring with the examples shown here are typical and should be taken into account by any estimation method. If the system is in a state where, for example, the uncertainty of the first variable is high then it might be better not to try to estimate this variable in this state or close to it, because the estimate might be poor and may spoil the overall results. Instead it makes more sense to wait until the trajectory enters a region of state space where can be estimated more reliably from the given time series.

The concrete implementation of such an adaptive approach depends on details of the estimation algorithm. For Newton-like algorithms, for example, it may consist of a simple strategy decreasing correction step sizes. Another potential application of uncertainty analysis is the identification of redundant parameters, i.e., parameter combinations that provide the same dynamical output.

###### Acknowledgements.
The research leading to these results has received funding from the European CommunityÕs Seventh Framework Program FP7/2007-2013 under grant agreement no HEALTH-F2-2009-241526, EUTrigTreat. We acknowledge financial support by the German Federal Ministry of Education and Research (BMBF) Grant No. 031A147, by the Deutsche Forschungsgemeinschaft (SFB 1002: Modulatory Units in Heart Failure), and by the German Center for Cardiovascular Research (DZHK e.V.).

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text:
# Math-accent symbol over parentheses enclosing accented symbol (amsmath)

I'm trying to write an expression in which an accented expression enclosed by parentheses has an accent over the parentheses. For example,

\vec{(\dot{x})}

This typesets fine when amsmath is not loaded, but with amsmath present, the x is typeset to the right of the dot, as if the inner expression is being treated as

\dot{}x

This behavior only happens when a closing parentheses (or bracket or brace) is part of the argument to the outer accent.

The best workaround I've been able to find is to use overset for the outer accent instead of the actual accent command (or accentset).

Is this behavior expected/documented anywhere?

I believe that this question was previously asked at amsmath possible bug: \dot over bracketed \dot{x}, but was incorrectly marked as a duplicate of \dot over \vec over : amsmath bug/feature?. The latter question was about the differences in typesetting between dotting a subscripted character and subscripting a dotted character.

• I guess this is rather related to tex.stackexchange.com/q/30327/4427. On the other hand, (\vec{\dot{x}) seems better syntax. – egreg Mar 30 at 16:14
• Thanks. On the syntax question, putting the parentheses outside the \vec command is semantically different from applying the \vec to the whole contents of the parentheses. My \vec{(\dot{x})} minimal example probably would not be used in practice, but applying accents to parenthetically-delimited expressions or to matrices has multiple use-cases. – RLH Mar 30 at 16:27
• How's the reader supposed to know the difference between \vec{(\dot{x})} and (\vec{\dot{x}})? And, if you have \vec{(blahblah)} the problem is even worse. – egreg Mar 30 at 16:29
• As I said, I don't think that my minimal example will be used in practice. An expression like \dot{\begin{bmatrix} x \\ \dot{x} \end{bmatrix}} has a use-case (in explaining basic controls systems principles), and gets mangled by this bug. As a second example, in \vec{(\exp \dot{x})} vs (\vec{\exp \dot{x}}), the \vec is set higher on the first expression, clearly indicating that it applies to the contents of the parentheses. – RLH Mar 30 at 16:37
• Funny thing, I wrote a package to shift the accents of \mathaccentV (the macro amsmath uses internally for \dot, etc.), which doesn't suffer from this issue. I'll try to provide a patch from that. – Skillmon Mar 30 at 18:34

This has to do with a long-standing amsmath issue, see Why do arguments to nested \tilde or \breve commands reappear when amsmath is used?

I can offer a workaround.

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\newcommand{\dblacc}[2]{\mathpalette\dblacc@{{#1}{#2}}}
\newcommand{\dblacc@}[2]{\dblacc@@#1#2}
\newcommand{\dblacc@@}[3]{%
\begingroup
\sbox\z@{$\m@th#1#3$}%
#2{\box\z@}%
\endgroup
}
\makeatother

\begin{document}

\begin{gather*}
\dblacc\vec{(\dot{x})}
\\
\dblacc\dot{\begin{bmatrix} x \\ \dot{x} \end{bmatrix}}
\\
\dblacc\vec{(\exp \dot{x})}
\end{gather*}

\end{document}

A version that works also with accents:

\documentclass{article}
\usepackage{amsmath}
\usepackage{accents}

\makeatletter
\newsavebox{\dblacc@boxa}
\newsavebox{\dblacc@boxb}
\DeclareRobustCommand{\dblacc}[2]{\mathpalette\dblacc@{{#1}{#2}}}
\newcommand{\dblacc@}[2]{\dblacc@@#1#2}
\newcommand{\dblacc@@}[3]{%
\begingroup
\sbox\dblacc@boxa{$\m@th#1#3$}%
\sbox\dblacc@boxb{$\m@th#1#2{\copy\dblacc@boxa}$}%
\box\dblacc@boxb
\endgroup
}
\makeatother

\begin{document}

\begin{gather*}
\\
\dblacc\dot{\begin{bmatrix} x \\ \dot{x} \end{bmatrix}}
\\
\dblacc\vec{(\exp \dot{x})}
\end{gather*}

\end{document}

• Thanks very much. If it's not too much trouble, is there a way to make this compatible with the accents package? I'm using the \accentset command in my document for a custom accent, and with the accents package turned on, the arguments to the outer accent all disappear. – RLH Mar 30 at 17:13
• @RLH \newcommand{\myaccent}[1]{\accentset{<accent>}{#1}} and \dblacc\myaccent{whatever} should work – egreg Mar 30 at 17:51
• Sorry for the miscommunication: The problem I'm running into is that if I add \usepackage{accents} to the preamble of the example document you posted, only the outer accent character appears -- the rest of the expression is not typeset. – RLH Mar 30 at 18:07
• @RLH Problem solved. :-D – egreg Mar 30 at 21:33

The following redefines an internal of amsmath. After that the output of your example looks good. I don't know whether this has any possibility of breaking other stuff. I've added every test case egreg used in his answer and the output looks good. It doesn't work with the accents package, though.

\documentclass[]{article}

\usepackage[]{amsmath}

\makeatletter
\protected\def\mathaccentV#1#2#3#4#5%
{%
\ifmmode
\mathaccentV@do{#2}{#3}{#4}{#5}%
\else
\@xp\nonmatherr@\csname #1\endcsname
\fi
}
\def\mathaccentV@do#1#2#3#4%
{%
\global\let\macc@nucleus\@empty
\mathaccent"\accentclass@#1#2#3{#4}\macc@nucleus
}
\makeatother

\begin{document}
\begin{gather*}
\vec{(\dot{x})}
\\
\dot{\begin{bmatrix} x \\ \dot{x} \end{bmatrix}}
\\
\vec{(\exp \dot{x})}
\end{gather*}
\end{document}

• Thanks very much. Is there something fundamental about the accents package that breaks solutions of the kind that you and egreg suggested? – RLH Mar 30 at 19:01
• @RLH accents redefines all those accents macros to no longer use \mathaccentV, so my fix is bound to not work there. I didn't dig too deep into why the accents package fails to do the alignment right, so I can't really state anything else about the issue. – Skillmon Mar 30 at 19:05
• Thanks again for your help! – RLH Mar 30 at 19:16
• @RLH note that this patch changes the way \vec{\dot{x}} would be typeset (the arrow is moved slightly to the right which yields imperfect placement, but not severely misplacement). In the original use case of my package where something like this is done, that is not that bad, because the default placement of the accents would be very bad there. – Skillmon Mar 30 at 19:27
• @RLH if you only use accents for \accentset, you can use the package option single. That way my fix above will still be applied. – Skillmon Mar 30 at 19:37


text:
MathSciNet bibliographic data MR885114 22E40 (30F40 32G15 57M05 57N10) Ratcliffe, John G. Euler characteristics of $3$$3$-manifold groups and discrete subgroups of ${\rm SL}(2,{\bf C})$${\rm SL}(2,{\bf C})$. Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985). J. Pure Appl. Algebra 44 (1987), no. 1-3, 303–314. Article

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
:::

Activities

:::
Beginning 2009-11-09 00:00:00 2009-11-09 00:00:00 Linear orthogonality preservers of Hilbert C*-modules Prof. Ngai-Ching Wong,National Sun Yat-sen University) 3rd General Building R734 Click here to read Let \$A\$ be a C*-algebra. A Banach space \$X\$ is called a Hilbert \$A\$-module if \$X\$ is a left \$A\$-module, and there is an \$A\$-valued inner product defined on \$X\$ such that \$\|x\|^2 = \|\|\$. When \$A\$ is the complex numbers, Hilbert \$\mathbb{C}\$-modules are simply Hilbert spaces. It is know that the norm, the inner product and the orthogonality structures of a Hilbert space determine each other. Recently, we (Leung, Ng and Wong) successfully extended these equivalences to general Hilbert C*-modules. This in particular includes the case of Hilbert bundles, in which \$A\$ is an abelian C*-algebra. More precisely, two Hilbert bundles are unitarily isomorphic if and only if there is a bijective linear map between them preserving the orthogonality of continuous vector sections. We also get a complete solution for the non-abelian case, although it might be too technical to be covered in this hourly talk. 2009-11-05 17:15:20


text:
FICO Xpress Optimization Examples Repository
 FICO Optimization Community FICO Xpress Optimization Home

Description

Add a new row and an indicator constraint

Further explanation of this example: Xpress R Reference Manual

Source Files
By clicking on a file name, a preview is opened at the bottom of this page.

Data Files

indicator.R

#####################################
# This file is part of the          #
# Xpress-R interface examples       #
#                                   #
#   (c) 2021 Fair Isaac Corporation #
#####################################
#' ---
#' ---
#'

#'
#' An indicator constraint is made of a condition and a linear constraint. The
#' condition is of the type bin = value, where bin is a binary variable and
#' value is either 0 or 1. The linear constraint is any linear row. During global
#' search, a row configured as an indicator constraint is enforced only when
#' condition holds, that is only if the indicator variable bin has the specified
#' value.
#'
#' We work with the facility location running
#' example. Please familiarize yourself with this example first.
#'
#' We load the facility location example and optimize it first.
## ----Solving the Unmodified Facility Location Problem-------------------------
library(magrittr)
suppressMessages(library(xpress))
# quickly optimize the Facility location problem from a pipe
p <- createprob() %>% readprob("flp.lp") %>% mipoptimize()

print("MIP solution for the 5 facilities")
xprs_getsolution(p)[1:5]

#'
#' # Adding a New Row
#'
#' We would like to express that if facility $F_5$ is open, then both $F_2$ and
#' $F_4$ must be open, as well. Mathematically, we want to impose the implication
#'
#' #' \begin{align} #' x_5 = 1 \Rightarrow x_2 + x_4 = 2 #' \end{align} #'
#'
#' Since $x_5$ is currently open in the basic example problem, this additional
#' restriction will be very expensive. Can we do without opening $x_5$? First, we
#' now add the new row to the existing constraint matrix that expresses the
#' cardinality restriction on variables $x_3$ and $x_5$
#'
#' We use addrows to add the new row to the existing instance.
#' The addrows function can be used to add only a single
#' or even multiple new rows.
#' For each row, we have to indicate its type and right hand side.
#' Since we only add a single row, the rowtype and rhs can be passed
#' as scalars.
#'
#' In addition, we pass the nonzero row coefficients in sparse row major format.
#' When we pass column indices of these entries, we have to remember
#' to use 0-based indexing, because these values are directly passed
#' to Xpress.
#' The 0-based column indices that correspond to variables $x_2$ and $x_4$
#' are 1 and 3. Here we use the syntax c(2,4) - 1 which subtracts
#' 1 from each 1-based index to obtain a 0-based index.
#'
## ----Adding a New Row to the Problem------------------------------------------
rowtype = "E",
rhs = 2,
start=c(0,2), # 0-based indices
colind = c(2,4) - 1,# 0-based indices
rowcoef = c(1,1)
)
)

#'
#' We verify that the new linear constraint changes the
#' solution to the problem (the linear constraint is now active)
#'
summary(mipoptimize(p))
print(xprs_getsolution(p)[1:5])

#' # Declaration as Indicator Row
#'
#' Now we are ready to add the indicator implication between variable $x_5$ and the
#' newly created row.
#' Recall 0-based indexing:
#' This newly created row has index xpress:::ROWS - 1.
#' Similarly variable x_5 has column index 5 in R, but 4 in C
#'
#' The print statement assures us that we now have 1 indicator
#' constraint in our formulation.
#'
## ----Marking the New Row As Indicator-----------------------------------------
print(setindicators(p,
# 0-based. use the last added row
rowind=getintattrib(p, xpress:::ROWS) - 1L,
colind = 5L - 1L,
complement = 1L # row is active when x_5 = 1
))

#'
#' After resolving the indicator formulation, we immediately notice that $F_5$ is
#' not opened, and therefore, the restriction on opening both $F_2$ and $F_4$ is
#' lifted.
#'
## ----Resolve with the Indicator Constraint------------------------------------
summary(mipoptimize(p))
print("MIP solution for the 5 facilities after adding the indicator constraint")
print(xprs_getsolution(p)[1:5])

#'