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# What does it mean when people say "Physics break down"?

"Physics breaks down" is a bad way of saying what people are trying to say. It's the sort of thing that sounds cool at first, but then it starts misleading people.

What scientists mean is "our best theory produces non-sensical or contradictory results in this situation, so we know the theory doesn't make good predictions there."

They do not mean that there can never be a theory that works, or that somehow there are no laws of physics whatsoever in the situation. It just means we don't know what the law is.

Every physicist fully expects that there are laws of physics that predict what happens at the center of a black hole. Probably something perfectly sensible happens, though it's probably something weird and unlike anything else we know.

Let me give an example of a very, very mild case of 'theory breaks down'.

Boyle's law is stated as follows:

$$P_1V_1 = P_2V_2$$

Expressing that for a given quantity of gas the pressure and volume are inversely proportional to each other.

At low pressures Boyle's law holds good.

The reason that it holds good is that at low pressure the gas molecules take up only a small percentage of the total space.

If you keep increasing the pressure then eventually you reach the point where the molecules are touching all the time, and after that point even a huge increase in pressure leads to only a tiny reduction of the volume.

A dramatic way of saying the same thing is to state that at high pressure 'Boyle's law breaks down'. Of course, in the case of Boyle's law we have a deeper theory that we can move to.

There is another consideration, which is independent of observables.
As stated Boyle's law suggests that as pressure is increased volume can be decreased to an arbitrarily small value. In and of itself Boyle's law doesn't include any restriction. That is susipicious. A theory with a formula that allows infinities, that is an indication that the theory is only an approximation of a deeper theory.

Another comparison:
Coulomb's law of the electrostatic force gives that the magnitude of the force is inversely proportional to the distance.
Let's say that you have a theory that describes particles as entities that are true point particles. Then two charged particles can approach each other infinitly close, which would imply an infinitely large electrostatic force. That is suspicious

My understanding is that the black hole case is in some sense like Boyle's law; it doesn't disallow infinities in such a way that there is a suggestion that there is describable physics to be found, but it is out of scope of the theory.

A dramatic way of saying that is that at the singularity (as implied by the mathematics of GR) the theory breaks down.

"Physics breaks down" sounds good, but it is confusing. A better phrasing would be "known physics breaks down." Physics attempts to model reality using mathematics. In this sense, physics has no "laws." In the famous words of Captain Barbossa, "They're more like guidelines." However, we have many of these guidelines which are so astonishingly reliable on the scale that humans live and breathe at that we find the word "guidelines" is too soft. If you are building a skyscraper, you are more likely to make one that stands up if you think of gravity as a "law" rather than a "guideline." Architects who call gravity a "guideline" tend not to be given the opportunity to design multi-million dollar skyscrapers using other people's money!

As such, we should not be surprised when guidelines break down in extraordinary circumstances.

Personally, I would define two versions of this "physics breaks down" concept. One of them is when these guidelines break down in known ways, changing into new regimes. Supersonic flight is one great example. You can learn lots of guidelines about how fluids (like air) flow, but those guidelines break down at the speed of sound in that fluid. We then have supersonic flow guidelines to help us from there. In this case, this breakdown was something observed, typically on the edge of a phase transition. We figured out that supersonic flow followed different guidelines than sonic flow because we flew something faster than the speed of sound, and it behaved differently.

The other, more complete meaning of "physics breaks down" occurs in things like black holes. These are cases where we have never observed the breakdown, but our guidelines lead us in counterintuitive directions. To the best of all of our study, we have never found an experiment or an astronomical observation which suggests that Einstein's theory of general relativity is anything less than a law. Like everything in physics, its just a guideline, but we have not been able to find any situation where it fails.

However, there's a catch. Mathematically, Einstein's theories make strange predictions about extremely massive dense objects. The equations behind it spiral out of control, and you end up with oddities like a "singularity," with 0 volume and finite mass. This means infinite density, and we generally don't think that kind of infinity happens in the universe. So we say "physics breaks down" here because we honestly believe that if we ever get to actually observe this part of a black hole, we will see one of those phase transitions where physics just stops doing the General Relativity thing and starts doing Something Else. We believe this because we are not comfortable with the possibility that such math involving infinities is "right."

Another place "physics breaks down" is on the Planck scale. Planck time and Planck length are very small units of time and space. They're on the order of $$10^{-43} \text s$$ and $$10^{-35} \text m$$. They are important for another guideline that has shown to be so effective that we call it a law: Quantum Mechanics. We have yet to see anything on the small scale which suggests that Quantum Mechanics is anything less than a law, just like we haven't seen anything on the large scale to suggest that General Relativity is anything less than a law. However, when we get down to these absurdly small scales, the math of quantum mechanics (in particular, quantum field theory) start to break down, failing to yield predictive answers about what happens at that scale because some of the tools simply stop working. In a very handwavy sense, its like cutting things with a knife, smaller and smaller things, until one day you realize that you're trying to cut something thinner than the edge of your knife. Our metaphorical mathematical knife (renormalization) simply stops cutting at this point.

Of course, we have no reason to believe that the universe "ends" at this scale. It's just the point where all of the guidelines which are so successful that they have earned the monicker "law" fall apart. And in the extreme cases, such as the issues at Planck scale or in black holes, its the mathematics of the theory itself which suggests that the theory falls apart.

And so, we spend countless hours devising new experiments with the goal of plumbing these uncertain regions, with the hope of one day writing new physics which doesn't "break down" at these levels.


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# Properties

 Label 2450.2.c.t Level $2450$ Weight $2$ Character orbit 2450.c Analytic conductor $19.563$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.5633484952$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 490) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{2} q^{2} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} - q^{4} + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{6} -\zeta_{8}^{2} q^{8} + ( -3 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + \zeta_{8}^{2} q^{2} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} - q^{4} + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{6} -\zeta_{8}^{2} q^{8} + ( -3 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{9} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{11} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{12} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{13} + q^{16} + ( -\zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{17} + ( 4 \zeta_{8} - 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{18} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{19} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{22} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{23} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{24} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{26} + ( 8 \zeta_{8} - 8 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{27} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{31} + \zeta_{8}^{2} q^{32} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{33} + ( 4 + \zeta_{8} - \zeta_{8}^{3} ) q^{34} + ( 3 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{36} + ( -4 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{37} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{38} + ( 8 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{39} + ( 4 + 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{41} + ( 2 \zeta_{8} - 6 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{43} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{44} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{46} + ( -2 \zeta_{8} - 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{47} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{48} + ( 6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{51} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{52} + ( -6 \zeta_{8} - 2 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{53} + ( 8 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{54} -2 \zeta_{8}^{2} q^{57} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{58} + ( -10 - \zeta_{8} + \zeta_{8}^{3} ) q^{59} + ( -2 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{61} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{62} - q^{64} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{66} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{67} + ( \zeta_{8} + 4 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{68} + 4 q^{69} + ( 4 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{71} + ( -4 \zeta_{8} + 3 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{72} + ( \zeta_{8} - 8 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{73} + ( -2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{74} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{76} + ( -6 \zeta_{8} + 8 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{78} + ( 4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{79} + ( 23 - 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{81} + ( 5 \zeta_{8} + 4 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{82} + ( 3 \zeta_{8} + 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{83} + ( 6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{86} + ( -6 \zeta_{8} + 8 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{87} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{88} + ( 9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{89} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{92} + ( 4 \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{93} + ( 8 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{94} + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{96} + ( 3 \zeta_{8} + 12 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{97} + ( 10 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} - 8q^{6} - 12q^{9} + O(q^{10})$$ $$4q - 4q^{4} - 8q^{6} - 12q^{9} + 8q^{11} + 4q^{16} - 8q^{19} + 8q^{24} + 8q^{26} + 8q^{29} + 16q^{34} + 12q^{36} + 32q^{39} + 16q^{41} - 8q^{44} + 16q^{46} + 24q^{51} + 32q^{54} - 40q^{59} - 8q^{61} - 4q^{64} + 16q^{69} + 16q^{71} - 8q^{74} + 8q^{76} + 16q^{79} + 92q^{81} + 24q^{86} + 32q^{94} - 8q^{96} + 40q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
1.00000i 3.41421i −1.00000 0 −3.41421 0 1.00000i −8.65685 0
99.2 1.00000i 0.585786i −1.00000 0 −0.585786 0 1.00000i 2.65685 0
99.3 1.00000i 0.585786i −1.00000 0 −0.585786 0 1.00000i 2.65685 0
99.4 1.00000i 3.41421i −1.00000 0 −3.41421 0 1.00000i −8.65685 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2450.2.c.t 4
5.b even 2 1 inner 2450.2.c.t 4
5.c odd 4 1 490.2.a.m yes 2
5.c odd 4 1 2450.2.a.bn 2
7.b odd 2 1 2450.2.c.w 4
15.e even 4 1 4410.2.a.bt 2
20.e even 4 1 3920.2.a.bm 2
35.c odd 2 1 2450.2.c.w 4
35.f even 4 1 490.2.a.l 2
35.f even 4 1 2450.2.a.bs 2
35.k even 12 2 490.2.e.j 4
35.l odd 12 2 490.2.e.i 4
105.k odd 4 1 4410.2.a.by 2
140.j odd 4 1 3920.2.a.ca 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.a.l 2 35.f even 4 1
490.2.a.m yes 2 5.c odd 4 1
490.2.e.i 4 35.l odd 12 2
490.2.e.j 4 35.k even 12 2
2450.2.a.bn 2 5.c odd 4 1
2450.2.a.bs 2 35.f even 4 1
2450.2.c.t 4 1.a even 1 1 trivial
2450.2.c.t 4 5.b even 2 1 inner
2450.2.c.w 4 7.b odd 2 1
2450.2.c.w 4 35.c odd 2 1
3920.2.a.bm 2 20.e even 4 1
3920.2.a.ca 2 140.j odd 4 1
4410.2.a.bt 2 15.e even 4 1
4410.2.a.by 2 105.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2450, [\chi])$$:

 $$T_{3}^{4} + 12 T_{3}^{2} + 4$$ $$T_{11}^{2} - 4 T_{11} - 4$$ $$T_{13}^{4} + 24 T_{13}^{2} + 16$$ $$T_{19}^{2} + 4 T_{19} + 2$$ $$T_{31}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$4 + 12 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -4 - 4 T + T^{2} )^{2}$$
$13$ $$16 + 24 T^{2} + T^{4}$$
$17$ $$196 + 36 T^{2} + T^{4}$$
$19$ $$( 2 + 4 T + T^{2} )^{2}$$
$23$ $$64 + 48 T^{2} + T^{4}$$
$29$ $$( -4 - 4 T + T^{2} )^{2}$$
$31$ $$( -8 + T^{2} )^{2}$$
$37$ $$784 + 72 T^{2} + T^{4}$$
$41$ $$( -34 - 8 T + T^{2} )^{2}$$
$43$ $$784 + 88 T^{2} + T^{4}$$
$47$ $$3136 + 144 T^{2} + T^{4}$$
$53$ $$4624 + 152 T^{2} + T^{4}$$
$59$ $$( 98 + 20 T + T^{2} )^{2}$$
$61$ $$( -124 + 4 T + T^{2} )^{2}$$
$67$ $$256 + 96 T^{2} + T^{4}$$
$71$ $$( -56 - 8 T + T^{2} )^{2}$$
$73$ $$3844 + 132 T^{2} + T^{4}$$
$79$ $$( 8 - 8 T + T^{2} )^{2}$$
$83$ $$196 + 44 T^{2} + T^{4}$$
$89$ $$( -162 + T^{2} )^{2}$$
$97$ $$15876 + 324 T^{2} + T^{4}$$

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This section describes how preloads, offsets and scales enter into bushing force computations. You use Preloads, Offsets and Scales to alter the operating point of a bushing. You can offset the bushing displacement in any direction, and scale the input displacement and velocity. You can also offset the bushing force in any direction by adding a preload or scale-output force or moment in any direction.

## Displacement Offsets ans Scales

Displacement offsets change the origins of the force-deflection curves. Displacement scales expand or contract the force-deflection curves along the deflection axis.

Let:

${d}_{k}$
be the displacement in the kth displacement direction (x, y, z, ax, ay, az) between the bushing I- and J-Markers reported in the coordinate system of the J marker.
${H}_{k}$
is the displacement (horizontal) scaling factor in the kth direction.
${Q}_{k}$
is the displacement offset in the kth direction.

Then the scaled and offset deflection, ${q}_{k}$ , is given by:

${q}_{k}={H}_{k}·{d}_{k}-{Q}_{k}$

The velocity in the kth direction, is given by:

${\stackrel{˙}{q}}_{k}={H}_{k}·{\stackrel{˙}{d}}_{k}$

Preloads offset the forces or torques at zero displacement, while the force scale applies an amplification factor to the force. If ${G}_{k}$ is the function computing the force in the kth direction, then the force in that direction is given as follows:

Let:

${P}_{k}$
is the preload in the kth direction.
${V}_{k}$
is the force (vertical) scaling in the kth direction.
${G}_{k}$
is the sum of all force methods in the kth direction.
${F}_{k}$
is the preload in the kth direction.

Then:

${F}_{k}={P}_{k}-{V}_{k}·{G}_{k}\left({q}_{k},{\stackrel{˙}{q}}_{k},{x}_{k},t\right)$


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# Click Here!

A bracket sequence is a string containing only characters ” ( ” and ” ) “. A regular bracket sequence is a bracket sequence that can be transformed into a correct arithmetic expression by inserting characters ” 1 ” and ” + ” between the original characters of the sequence. For example, bracket sequences ” () () ” and ” (()) ” are regular (the resulting expressions are: ” (1) + (1) ” and ” ((1 + 1) +1) “), and ” ) ( “, ” ( ” and ” ) ” are not.

You are given an integer nn… Your goal is to construct and print exactlynn different regular bracket sequences of length n2n

### Regular Bracket Sequences solution codeforces

The first line contains one integer tt (≤ ≤ 501t50) – the number of test cases.

Each test case consists of one line containing one integer nn (≤ ≤ 501n50).

### Regular Bracket Sequences solution codeforces

For each test case, print nnlines, each containing a regular bracket sequence of length exactlyn2n… All bracket sequences you output for a testcase should be different (though they may repeat in different test cases). If there are multiple answers, print any of them. It can be shown that it’s always possible.

Example
input

Copy
3
3
1
3

### Regular Bracket Sequences solution codeforces

Copy
() () ()
((()))
(() ())
()
((()))
(()) ()
() (())


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IMP Reference Guide  develop.97bab1d8fd,2021/09/21 The Integrative Modeling Platform
IMP::NonCopyable Class Reference

Base class for all objects that cannot be copied. More...

#include <IMP/NonCopyable.h>

Inheritance diagram for IMP::NonCopyable:

## Detailed Description

Base class for all objects that cannot be copied.

Non-copyable classes cannot be copied, as the name implies. If you need to do delayed initialization, either use an initialize method on them or use a boost::scoped_ptr to allocate them on the heap.

Definition at line 22 of file NonCopyable.h.

The documentation for this class was generated from the following file:


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# Field Guides

(Redirected from Forestry/Tabbing Tips)

In some events (such as Forestry), students are allowed commercially published field guides that can be tabbed or annotated. Note that in recent years most events have begun to move away from allowing field guides. As of the 2018 season, Herpetology allows field guides, but only as a replacement for a three-ring binder; teams may not bring both.

## Using a Field Guide

There are many ways to use a field guide to prepare for an event. They serve as very helpful study tools and can be used to add information to a binder. Even if an event does not allow a field guide in the test, picking one up to prepare with can be a good way to find information. Some teams may get their field guides hole punched and place them in their binder, or remove pages from the guide and place them alongside written notes. The key to using a field guide is learning how it is laid out and where important information is located. Making notes in a field guide is another efficient way to use them. If the guide is missing important information for a specific event, it is easy to either write on the page or add sticky notes with information.

### Choosing a Field Guide

The types of field guides available for an event typically depend on what the event is. Some events such as Rocks and Minerals may only have a handful of field guides available, but events such as Herpetology may have a wide variety of field guides available. Events having to do with animal identification typically have a variety of field guides available on the state and national level. State level competitions tend to focus on specimens that can be found in that area, and the areas that they can be found.

Choosing a field guide to use is something that should be taken especially seriously if a student plans on using the guide at competition. Field guides should have good quality pictures and have information that is easy to parse quickly. Any field guide can be a good field guide if it is worked with enough, but some field guides are easier to start using than others. If the field guide is going to be placed in a binder, pages can also be removed and pages can be added from other field guides. This is useful if a team prefers the pictures from one field guide but the information from another.

### Tabbing

Field guides are typically less organized than written notes are, as students can not control where information is placed. However, placing tabs in a field guide can make it much easier to use and find information. Tabbing is especially useful when a field guide has more information than what is used for the event, or when a field guide is divided into sections. There are a few options that students have for what material they choose to tab with.

#### Post-its/Sticky Notes

Post-its/Sticky Notes

One material is the ordinary Post-it or sticky note. Sticky notes can hold more information than regular flags can, so more information about a section can fit onto the outside of a tab. However, they are easy to tear or remove from a field guide if care is not taken. Sticky notes can also be used for annotating field guides and making notes without messing up the pages.

Pros

• Easily removable
• Thin and shapeable; can cut them to preferred sizes
• Inexpensive (Averages $4.00-$5.00 for one 3x3 cube, 470 sheets)
• Easily found in stores and online

Cons

• Adhesive is not very powerful, so the tab may lose its "stickiness" quickly
• Must be cut down to size, as the sheets are generally too big for most books
• Can curl if the field guide is used frequently

#### Flags

Flags

Flags are similar to sticky notes in that they also have a small strip of adhesive on one side. However, flags are usually smaller and thinner. They may be made of paper or a thin, transparent plastic. Plastic flags are typically very robust and good for quick page turns. They come in a variety of sizes, with thin flags being good for marking smaller sections and large rectangular flags being good for marking important information. Larger plastic flags are typically stiffer than smaller or paper flags.

Pros

• Minimal if any resizing
• Can fit multiple on one page
• Also inexpensive
• Easily located in stores and online
• Transparent flags can be used for highlighting

Cons

• May tear easily, especially paper flags
• Small, so little room to write on them


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From a quality perspective, Six Sigma is defined as 3.4 defects per million opportunities. This figure is called a Six Sigma level of quality. Sigma scores are thrown about so much that you definitely need to be comfortable understanding what they are and how they’re calculated. Basically, a sigma score tells you how many standard deviations can fit between the mean and specification limit of any process or specification.

The sigma score can be applied to the performance of anything that has a specification and a defect rate: the performance of the mail system in delivering letters to the correct address, the ability of an automobile manufacturer to produce a door that fits to the body within a required dimensional tolerance, or a repeated budgeting process that must be completed within its specified schedule window.

How many standard deviations can fit?

The central tendency of the performance distribution is defined by its mean. The amount of variation in the performance, or the width of the distribution, is defined by its standard deviation σ. The question is how many standard deviations can you fit between the process or characteristic’s mean and its specification limit SL?

You can see that four standard deviations can fit between the mean and the specification limit. The exact number can always be calculated by the formula

A low sigma (Z) score means that a significant part of the tail of the distribution extends past the specification limit. So the higher the sigma (Z) score, the fewer the defects. A process or characteristic gets a good sigma (Z) score when the variation distribution is safely away from the edge of the specification cliff. A sigma (Z) score can change in one of three ways:

• The location of the central tendency of the distribution — the mean — moves either closer or farther from the specification limit.

• The width of the distribution, as defined by the standard deviation σ changes.

• The location of the specification limit SL moves either closer or farther from the characteristic or process variation.

Compare short-term to long-term sigma score calculations

From the mean and the standard deviation, you can calculate a sigma (Z) score. A wrinkle here is that you must know what type of standard deviation you’re using to calculate the sigma (Z) score: Is it a short-term standard deviation σST, or is it a long-term standard deviation σLT?

If you’re using a short-term standard deviation, the sigma (Z) score you calculate is a short-term sigma score ZST:

If, however, you have a long-term standard deviation, you can calculate the long-term sigma score ZLT:

Link short-term capability to long-term performance with the 1.5-sigma shift

Short-term variation performance, as quantified by the short-term sigma score ZST, represents the best variation performance that you can expect out of your currently configured process. It’s an idealistic measure of capability. It’s also the easiest type of data to collect; just go and quickly grab a relatively small sample of measurements from the process or characteristic, and you have it.

But in the long term, a process or characteristic doesn’t operate ideally like it does in the short term. Its performance will degrade by shift, drift, and trend influences. At the heart of Six Sigma is a method that combines the best of both worlds. It allows you to leverage the economy of short-term variation data while projecting realistic, long-term performance versus the process’s or characteristic’s specifications.

The characteristic or process stays within specifications during the short term and appears to have no problems. But over the long term, disturbances to the process cause it to expand, and this expansion creates defects beyond the specification limit.

One mathematical way to simulate the effect of these degrading, long-term influences is to artificially move the short-term distribution closer to the specification limit until the amount of defects for the short-term distribution is the same as that for the long-term distribution.

Early practitioners of Six Sigma proposed that mathematically shifting a characteristic’s or process’s short-term distribution closer to its specification limit by a distance of 1.5 times its short-term standard deviation would approximate the number of defects occurring in the long term. This breakthrough idea can be applied directly to the calculation of short-term and long-term sigma (Z) scores.

Because ZST represents the number of short-term standard deviations between the variation center and the specification, the sigma (Z) score of the shifted distribution is

Zshifted = ZST – 1.5

But with the shifted distribution being equivalent, defect-wise, to the long-term distribution, the preceding equation can be rewritten as

ZLT = ZST – 1.5

So what Six Sigma practitioners do is measure the short-term variability of a process or characteristic and calculate its short-term sigma score. Then they immediately translate this score to the expected long-term defect rate performance, using the 1.5 short-term standard deviation shift. This long-term sigma score, ZLT, is communicated in terms of defects per million opportunities, DPMO.


text:
# Kilograms to pounds

This free conversion calculator will convert kilograms to pounds, i.e. kg to lb. Correct conversion between different measurement scales.

Convert

The formula is $M_{\text{lb}} = \frac{100000000}{45359237} M_{\text{kg}}$, where $M_{\text{kg}} = 15$.

Therefore, $M_{\text{lb}} = \frac{1500000000}{45359237}$.

Answer: $15 \text{kg} = \frac{1500000000}{45359237} \text{lb}\approx 33.069339327731637 \text{lb}.$


text:
# Finding shearing force and moment

## Homework Statement

i couldnt understand why the author take the distance of force 129kN for 4<x<6 and 6<x<8 is x-2 ?

## The Attempt at a Solution

IMO , for 4<x<6 , the distance of force 129kN to the point should be x , where x is the distance from the 129kN to the point of moment , correct me if i am wrong [/B]

#### Attachments

• 1.jpg
18.9 KB · Views: 356
• 2.jpg
23.6 KB · Views: 384

is the author wrong ?

The black dot on the left end of the x axis is the origin (x=0), so the moment arm from x (the point about which the moment is being calculated) to the 192kN force is x-2.

The black dot on the left end of the x axis is the origin (x=0), so the moment arm from x (the point about which the moment is being calculated) to the 192kN force is x-2.
one more thing , why at x = 6 , the shearing force is still 49kN as in the calculation ? why shouldn't it be = 0 ?

why at x = 2 , there are 2 values of shearing forces , namely -40kN (0<x<2) and 89kN (2<x<4) ?

SteamKing
Staff Emeritus
Homework Helper
why at x = 2 , there are 2 values of shearing forces , namely -40kN (0<x<2) and 89kN (2<x<4) ?
Because the reaction RA is located at x = 2. As you move along the beam from left to right, the shear force has a sudden jump from V = -40 kN to V = 89 kN, as the reaction RA = 129 kN is added.

SteamKing
Staff Emeritus
Homework Helper
one more thing , why at x = 6 , the shearing force is still 49kN as in the calculation ? why shouldn't it be = 0 ?
Why should the shear be zero at x = 6?

The shear force and the bending moment values are each cumulative starting at the left end of the beam. If there is no additional load applied to the beam, the shear force at a particular point will not be changed.

Because the reaction RA is located at x = 2. As you move along the beam from left to right, the shear force has a sudden jump from V = -40 kN to V = 89 kN, as the reaction RA = 129 kN is added.
sudden jump means there are 2 values of RA at x =2 ? why ?

can you explain why The shear force and the bending moment values are each cumulative starting at the left end of the beam.??

SteamKing
Staff Emeritus
Homework Helper
can you explain why The shear force and the bending moment values are each cumulative starting at the left end of the beam.??
Because they are each integrals. The shear force is the integral of the load curve w.r.t. length, and the bending moment is the integral of the shear curve, also w.r.t. length.

The following slides explain this in more detail:

http://www.ce.memphis.edu/3121/notes/notes_04c.pdf

SteamKing
Staff Emeritus
Homework Helper
sudden jump means there are 2 values of RA at x =2 ? why ?
No, there is only one value of RA, namely RA = 129 kN.

The shear force diagram is a plot of just the shear force values acting on the beam. When a concentrated load is encountered, such as one of the beam reactions or an applied load, the shear force diagram will show a jump discontinuity at that location. The size of the discontinuity = the magnitude on the concentrated force.

foo9008
SteamKing
Staff Emeritus


text:
# GSEB Solutions Class 11 Statistics Chapter 2 Presentation of Data Ex 2.1

Gujarat Board Statistics Class 11 GSEB Solutions Chapter 2 Presentation of Data Ex 2.1 Textbook Exercise Questions and Answers.

## Gujarat Board Textbook Solutions Class 11 Statistics Chapter 2 Presentation of Data Ex 2.1

Question 1.
The data regarding the number of children of 50 families residing in a certain area are given below. Prepare an appropriate frequency distribution.

Here, the number of children in a family is discrete variable. Therefore we will obtain discrete frequency distribution. In the frequency distribution x = No. of children, f = No. of families. In the given data, the minimum value of x is 0 and the maximum value is 3. Hence the discrete frequency distribution of the number of children is obtained as follows :

Discrete frequency distribution showing the number of children in 50 families

Question 2.
The ages (in full years) of 60 employees working in an office are registered as follows.
Prepare a frequency distribution by taking class length as 5 from this information.

In the given data, the minimum age of an employee is 21 years and the maximum age is 58 years. Class length 5 is given. Therefore the lower limit (which is multiple of 5) of the initial class we will take 20 and the upper limit (which is also multiple of 5) of the last class we will take 60. Thus, the initial class and the last class of the frequency distribution will be respectively 20 – 25 and 55-60.

Continuous frequency distribution showing the age of 60 employees of an office

Question 3.
The data regarding the number of mobile phones produced during last 60 days by a mobile phone manufacturing company is given below. Distribute it into 10 classes.

Obtain ‘less than’ and ‘more than’ type cumulative frequency distribution from it.
In the given data, the minimum production of mobile is 188 and the maximum is 1075. Therefore, range of data R = 1075 – 188 = 887 mobiles. The data is to be divided in 10 classes. Therefore, k = 10 range
Now, class length c = $$\frac{\text { range }}{\text { no. } \text { of classes }}=\frac{\mathrm{R}}{k}$$
∴ c = $$\frac{887}{10}$$ = 88.7 ≈ 89
If c = 89, then ck = 89 x 10 = 890, which is greater than R, i.e., ck > R. Hence, we can take c = 89. But, usually the value of c should be taken which is multiple of 10. Hence, c = 100 is taken.

Here, the range of data is large. Hence, it is advisable to prepare inclusive frequency distribution, c = 100. therefore the initial class which includes the lowest observation 188 will be 100-199 and the last class which includes the highest observation 1075 will be 1000-1099. Thus, inclusive continuous frequency distribution is obtained as follows :

Inclusive continuous frequency distribution, showing the production of mobiles during 60 days

Explanation: For ‘less than’ and ‘more than’, cumulative frequency distribution, we have to obtain upper class boundary point and lower class boundary point of the classes respectively.
For initial class 100-199,
Lower class boundary point = $$\frac{99+100}{2}$$ = 99.5;
Upper class boundary point = $$\frac{199+200}{2}$$ = 199.5]
[Alternative method: The upper limit of the class 100-199 is 199 and the lower limit of immediate class 200 – 299 is 200. So their difference is (200 – 199 = ) 1. Dividing this difference by 2, we get($$\frac{1}{2}$$ = ) 0.5. Subtracting
this difference from the lower limit of each class and adding it to the upper limit of each class, we get lower boundary point and upper boundary point respectively for each class.]

Less than’ type cumulative frequency distribution

‘More than’ type cumulative frequency distribution

Question 4.
Rewrite the following frequency distribution by stating class length and Mid Value of each class :

Given frequency distribution is inclusive type. Here, the difference between the upper limit of each class and the lower limit of its immediate adjoining class is 1. Subtracting $$\frac{1}{2}$$ = 0.5 from the lower limit of each class and adding it to the upper limit of each class, we get lower boundary point and upper boundary point respectively for each class, which is shown in the table on page 28.

Question 5.
From the following frequency distribution, obtain ‘less than’ and ‘more than’ type cumulative frequency distributions :

x = No. of errors per page, f = No. of pages

Question 6.
Obtain an inclusive continuous frequency distribution from the following data:

Given data is ‘more than’ type cumulative frequency distribution.
∴ Class length = Difference between the adjoining lower boundary points = 54.5 – 49.5 = 5
Now, upper boundary point of the class = lower boundary point + class length
For initial class lower boundary point = 44.5
∴ initial class is 44.5 – 49.5.

For inclusive continuous frequency distribution lower limit of initial class = 44.5 + 0.5 = 45 and upper limit of initial class = 49.5 – 0.5 = 49.
Thus, for given data the initial class in inclusive form, we get 45-49.
In this manner, we will obtain the class for each lower boundary point.
We will find the frequency of each class from the given ‘more than’ cumulative frequency as follows :

Frequency of a class = (‘more than’ cumulative frequency of a class) – (‘more than cumulative frequency of immediate following class)

Frequency of class for 44.5 = (‘more than’ cumulative frequency of a class for 44.5) – (‘more than cumulative frequency of immediate following class for 49.5)
= (500 – 470) = 30

In this manner we will obtain the frequency for the rest of classes.
For the given data, we get the inclusive continuous frequency distribution as follows :

 Lower Boundary point or more More than’ cumulative frequency Cf Class Frequency f 44.5 500 45-49 500 – 470 = 30 49.5 470 50-54 470 – 390 = 80 54.5 390 55-59           . 390 – 290 = 100 59.5 290 60-64 290 – 240 = 50 64.5 240 65-69 240 – 90 = 150 69.5 90 70-74 90 – 10 = 80 74.5 10 75-79 10 – 0 = 10 79.5 0 80-84 = 0 – – Total n = 500

Question 7.
Obtain an exclusive continuous frequency distribution from the following data:

Given data is ‘less than’ type cumulative frequency distribution.
Class length = Difference between two adjoining upper boundary points
= 35 – 30 = 5
Now, lower boundary point of a class = upper boundary point – class length
∴ for initial class, lower boundary point = 30 – 5 = 25 and we get initial class 25 – 30.

In this manner we will obtain class for each upper boundary point.
We find the frequency of a class as follows :
Frequency of a class = (‘less than ‘ cumulative frequency of a class) – (‘less than’ cumulative frequency of preceeding class)
Class frequency for upper boundary point 35 = 17 – 0 = 17
In this manner, we will obtain the class frequency for each upper boundary point.

For the given data exclusive continuous frequency distribution is obtained as follows:

 Weight ‘less than’ in kg ‘Less than’ cumulative frequency cf Weight(in kg) Frequency f 30 0 25-30 = 0 35 17 30-35 17- 0 = 17 40 25 35-40 25 – 17 = 8 45 40 40-45 40 – 25 = 15 50 48 45-50 48 – 40 = 8 55 54 50-55 54 – 48 = 6 60 57 55-60 57 – 54 = 3 65 59 60-65 59 – 57 = 2 70 60 65-70 60 – 59 = 1 – – Total n = 60

Question 8.

Obtain the original frequency distribution.
In the given data, mid values and class length are given. We will find the lower limit and upper limit of a class as follows :
Lower limit = mid value – $$\frac{\text { class length }}{2}$$
Upper limit = mid value + $$\frac{\text { class length }}{2}$$
For mid value 25, lower limit of the class = 25 – $$\frac{50}{2}$$ = 25 – 25 = 0 and
upper limit of the class = 25 + $$\frac{50}{2}$$ = 25 + 25 = 50
∴ For mid value 25, we get the class 0 – 50.

In this manner, we will obtain the class for each mid value.
Thus, original frequency distribution is obtained as follows:

Question 9.
Information regarding the number of accidents in a city during a year is as under. Find the inclusive continuous frequency distribution from it.

In given data, mid values of class and their frequency are given.
The difference between two adjoining mid values is 10.
∴ class length =10

[Note : For inclusive class : Lower boundary point = Lower limit + 0.5; Upper boundary point = Upper limit – 0.5]

Question 10.
From the following data, obtain class boundary points from the class limits and write the frequency distribution:

Subtracting $$\frac{0.025}{2}$$ = 0.0125 from the lower limit of the class and adding it to the upper limit, we get lower boundary point and upper boundary point respectively of the classes.


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# MS 12: Numerical microlocal analysis

Thu, 30 March, 2017, 16:30–18:30, Room: UC 202DH

## Organizers

Marta Betcke and Jürgen Frikel

## Abstract

One of the defining properties of the Radon transform (and its variants) is the wave front resolution of the singularities of the function under the transformation. This property is underlying many theoretical results in the field, while more recently there have been attempts to design practical computational tools based on the microlocal insights. In particular, in limited data problems, microlocal analysis has proven to provide valuable insights that help facilitate interpretation of images obtained from limited data reconstructions, efficiently design limited data system and even provide ways to improve classical algorithms. In order to exploit microlocal information even further, tools are needed which make this information computationally accessible. Here, we mention the development of directional multiresolution transforms such as directional Wavelets, Contourlets, Curvelets and Shearlets. While those tools have been originally developed in the context of signal and image processing, they allow for exploitation of some more theoretical results and are thus finding more and more applications in inverse problems. In this minisymposium we are going to review some ongoing efforts for development of effective methods for solution of inverse problems informed by microlocal analysis and provide a platform for discussion of new ways how these concepts could be translated into efficient numerical algorithms.

## List of speakers

 Anuj AbhishekA Support Theorem for Integral Moments of a Symmetric $m$-Tensor Field Matthias EhrhardtStructured guided Total Variation Markus HaltmeierWavelet methods in photoacoustic tomography Kim KnudsenMicrolocal stability and instability in Acousto-Electric tomography Holger KohrTotal Variation Regularization in variable Lebesgue spaces Todd QuintoArtifacts in Arbitrary Limited Data Tomography Problems


text:
40m QIL Cryo_Lab CTN SUS_Lab TCS_Lab OMC_Lab CRIME_Lab FEA ENG_Labs OptContFac Mariner WBEEShop
 40m Log, Page 235 of 335 Not logged in
ID Date Author Type Category Subject
11779   Wed Nov 18 11:22:13 2015 yutaroUpdateASCLoss maps of arm cavities

I got linear relation between these. The results and method are below.

 Quote: In preparation for the measurement of loss maps of arm cavities, I measured the relationship between: the offset just after the demodulation of dithering loop (C1:ASS-YARM_ETM_PIT_L_DEMOD_I_OFFSET and C1:ASS-YARM_ETM_YAW_L_DEMOD_I_OFFSET) and the angle of ETMY measured with oplev (C1:SUS-ETMY_OL_PIT_INMONC1:SUS-ETMY_OL_PIT_INMON and C1:SUS-ETMY_OL_PIT_INMONC1:SUS-ETMY_OL_PIT_INMON) while the dithering script is running. With the angle of ETMY, we can calculate the beam spot on the ETMY assuming that the beam spot on the ITMY is not changed thanks to the dithering. What we have to do is to check the calbration of oplev with another way to measure the angle, to see if the results are reliable or not. I will report the results later.

RESULTS

METHOD

I added offset (C1:ASS-YARM_ETM_PIT_L_DEMOD_I_OFFSET and C1:ASS-YARM_ETM_YAW_L_DEMOD_I_OFFSET) to shift the beam spot on ETMY. For each data point, I measured the difference in angle of ETMY with oplev before and after adding offset. The precedure for each measurement I employed is following:

-- run dither

-- wait until error signals of dithering settle down

-- freeze dither

-- measure angle (10s avg)

-- add offset

-- wait until error signals of dithering settle down

-- freeze dither

-- measure angle (10s avg)

The reason why I measured the angle without offset for each measurement is that the angle which oplev shows changed with time (~several tens of minutes or something).

DISCUSSION

At the maximum values of offsets, the arm transmission power started to degrade, so the range where I can do this measurement is limited by these values as for now. However, we have to shift the beam spot more in order to make loss maps of sufficiently broad area on the mirror (the beam width (w) on ETM; w ~ 5 mm). Then, we have to find out how to shift the beam spot more.

11780   Wed Nov 18 16:51:58 2015 KojiSummaryPSLPMC servo calibration

Summary

The PMC servo error (MIX OUT MON on the panel) and actuation (HV OUT MON) have been calibrated using the swept cavity.

Error signal slope in round-trip displacement: 2.93e9 +/- 0.05e9 [V/m]
HV OUT calibration (round-trip displacement): 5.36e-7 +/- 0.17e-7 [m/V]
PZT calibration (round-trip displacement): 10.8 +/- 0.3 [nm/V]
=> corresponds to ~2.5 fringes for 0~250V full range => not crazy

Measurement condition

The transmission level: 0.743V (on the PMC MEDM screen)
LO level: ~13dBm (after 3dB attenuation)
Phase setting: 5.7
PMC Servo gain: 7dB during the measurement (nominal 3dB)

Method

- Chose PMC actuation "BLANK" to disable servo
- Connect DS345 function generator to EXT DC input on the panel
- Monitor "MIX OUT MON" and "HV OUT MON" with an oscilloscope
- Inject a triangular wave with ~1Vpp@1 or 2Hz with appropriate offset to see the cavity resonance at about the middle of the sweep.
The frequency of the sweep was decided considering the LPF corner freq formed by the output impedance and the capacitance of the PZT. (i.e. 11.3Hz, see next entry)

Result

- 4 sweep was taken (one 2Hz seep, three 1Hz sweep)
- The example of the sweep is shown in the attachment.
- The input triangular wave and the PDH slopes were fitted by linear lines.
- Spacing between the sideband zero crossing corresponds to twice of the modulation frequency (2x35.5MHz = 71MHz)
- The error signal slope was calibrated as V/MHz
- FSR of the PMC is given by google https://www.google.com/search?q=LIGO+pmc.m
=> Cavity round trip length is 0.4095m, FSR is 732.2MHz
- Convert frequency into round-trip displacement

- Convert HV OUT MON signal into displacement in the same way.
- The voltage applied to the PZT element is obtained considering the ratio of 49.6 between the actual HV and the HV OUT MON voltage.

11781   Wed Nov 18 16:53:17 2015 KojiSummaryPSLPMC PZT capacitance

The PMC PZT capacitance was measured.
- Turn off the HV supplies. Disconnect HV OUT cable.
- Make sure the cable is discharged.
- Measure the capacity at the cable end with an impedance meter.
=> The PMC PZT capacitance at the cable end was measured to be 222nF
Combined with the output impedance of 63.3kOhm, the LPF pole is at 11.3Hz

11782   Wed Nov 18 17:09:22 2015 KojiSummaryPSLPMC Servo analysis

Summary

The PMC servo was analysed. OLTF was measured and modeled by ZPK (Attachment 1). The error and actuator signals were calibrated in m/rtHz (Attachment 2)

Measurement methods

OLTF:

- The PMC servo board does not have dedicated summing/monitor points for the OLTF measurement. Moreover the PZT HV output voltage is monitored with 1/49.6 attenuation.
Therefore we need a bit of consideration.

- The noise injection can be done at EXT DC.
- Quantity (A): Transfer function between HV OUT MON and MIX OUT MON with the injection.
We can measure the transfer function between the HV OUT (virtual) and the MIX OUT. (HV OUT->MIX OUT). In reality, HV OUT is attenuated by factor of 49.6.
i.e. A = (HV_OUT->MIX_OUT)*49.6
- Quantity (B): Transfer function between HV OUT MON and MIX OUT MON without the injection.

This is related to the transfer function between the MIX OUT and HV OUT. In reality, HV OUT is attenuated.
i.e. B = 1/((MIX_OUT->HV_OUT)/49.6)

- What we want to know is HV_OUT->MIX_OUT->HV_OUT. i.e. A/B = (HV_OUT->MIX_OUT*49.6)*((MIX_OUT->HV_OUT)/49.6) = HV_OUT->MIX_OUT->HV_OUT

PSD:

- The MIX OUT and HV OUT spectra have been measured. The MIX OUT was calibrated with the calibration factor in the previous entry. This is the inloop stability estimation.
From the calibrated MIX OUT and HV OUT, the free running stability of the cavity was estimated, by mutiplying with |1-OLTF| and |1-1/(1-OLTF)|, respectively, in order to recover
the free running motion.

OLTF Modeling

Here is the model function for the open loop TF. The first line comes from the circuit diagram. The overall factor was determined by eye-fit.
The second and third lines are to reproduce the peak/notch feature at 12kHz. The fourth line is to reproduce 28kHz feature.
The LPF right after the mixer was analyzed by a circuit simulation (Circuit Lab). It can be approximated as 150kHz LPF as the second pole
seems to come at 1.5MHz.

The sixth line comes from the LPF formed by the output resistance and the PZT capacitance.

The seventh line is to reproduce the limit by the GBW product of OP27. As the gain is 101 in one of the stages,
it yields the pole freq of ~80kHz. But it is not enough to explain the phase delay at low frequency. Therefore this
discrepancy was compensated by empirical LPF at 30kHz.

function cmpOLTFc = PMC_OLTF_model(freqOLTFc)

cmpOLTFc = -7e5*pole1(freqOLTFc,0.162).*zero1(freqOLTFc,491)... % from the circuit diagram     .*zero2(freqOLTFc,12.5e3,100)... % eye-fit     .*pole2(freqOLTFc,12.2e3,6)... % eye-fit     .*pole2(freqOLTFc,27.8e3, 12)... % eye-fit     .*pole1(freqOLTFc,150e3)... % Mixer LPF estimated from Circuit Lab Simulation     .*pole1(freqOLTFc,11.3)... % Output Impedance + PZT LPF     .*pole1(freqOLTFc,8e6/101)... % GBW OP27     .*pole1(freqOLTFc,3e4); % Unknown

end

Result

Attachment 1:

The nominal OLTF (Nov 17 data) shows the nominal UGF is ~1.7kHz and the phase margin of ~60deg.

The measured OLTF was compared with the modelled OLTF. In the end they show very sufficient agreement for further calibration.
The servo is about to be instable at 28kHz due to unknown series resonance. Later in the same day, the gain of the PMC loop had to be
reduced from 7dB to 3dB to mitigate servo oscillation. It is likely that this peak caused the oscillation. The notch frequency was measured
next day and it showed no sign of frequncy drift. That's good.

We still have some phase to reduce the high freq peaks by an LPF in order to increase the over all gain.

Attachment 2:

The red curve shows the residual floor displacement of 2~10x10-15 m/rtHz. Below 4Hz there is a big peak. I suspect that I forgot to close
the PSL shutter and the IMC was locked during the measurement. Then does this mean the measured noise corresponds to the residual laser
freq noise or the PMC cavity displacement? This is interesting to see.

The estimated free running motion from the error and actuation signals agrees very well. This ensures the precision of the caibration in the precious entries.

11783   Wed Nov 18 17:32:36 2015 yutaroUpdateOptical LeversBeam centering for the oplev of ITMY

[yutaro, Koji]

We made the beam spot on QPD for the oplev of ITMY centered by changing the orientation of the mirror just before the QPD.

Before doing this, we ran dithering for Y arm and froze the output of ASS for Y arm.

11784   Wed Nov 18 20:49:05 2015 ranaUpdateComputer Scripts / Programsnodus boot getting full

controls@nodus|~ > df -h
Filesystem                   Size  Used Avail Use% Mounted on
/dev/mapper/nodus2--vg-root  355G   69G  269G  21% /
udev                         5.9G  4.0K  5.9G   1% /dev
tmpfs                        1.2G  308K  1.2G   1% /run
none                         5.0M     0  5.0M   0% /run/lock
none                         5.9G     0  5.9G   0% /run/shm
/dev/sda1                    236M  210M   14M  94% /boot
chiara:/home/cds             2.0T  1.5T  459G  77% /cvs/cds
fb:/frames                    13T   11T  1.6T  88% /frames

11785   Wed Nov 18 22:26:33 2015 yutaroUpdateOptical LeversCalibration of oplevs for ITMY/ETMY

Based on elog 1403, I calibrated the oplevs for ITMY/ETMY.

Summary of this method is following:

We lock an arm, and slightly misalign one mirror of the arm. Then, the transmission of the arm starts to decrease quadratically as the misalign angle of the mirror changes. Here, how much the transmission decreases in terms of the misalign angle is determined by geometry of optics, so we can see how much the angle really changes from this quadratic curve.

RESULTS

These are the relationship between misalign angles measured by oplev (the units are based on the calibration for now) and transmission power.

(I updated following figures on Nov 19 2015. You can find the figures I attached once here in the zipped folder attached.)

According to this measurement, ratio of the calibration factor derived with this measurement (NEW) and the calibration factor for now (OLD), i.e. NEW/OLD was:

ETMY_PIT: 5.0265  --->> 5.3922 (without an outlier; the outlier I removed is shown in the figure in zipped folder attached.)

ETMY_YAW: 4.6205

ITMY_PIT: 1.5010

ITMY_YAW: 1.2970

This results show that calibration of oplevs for ITMY is kind of OK, but that for ETMY is so BAD and the calibration factors should be updated.

NOTE

The calibration factors of the oplevs for ETMY/ITMY are   NOT UPDATED YET.  I updated on Dec 11, 2015

If these results are reliable, I will update them tomorrow.

11786   Wed Nov 18 23:18:07 2015 ericqUpdateComputer Scripts / Programsnodus /boot cleared up

The /boot partition was filling up with old kernels. Nodus has automatic security updates turned on, so new kernels roll in and the old ones don't get removed.

I ran apt-get autoremove, which removed several old kernels. (apt is configured by default to keep two previous kernels around when autoremoving, so this isn't so risky)

Now: /dev/sda1                    236M   94M  130M  42% /boot

In principle, one should be able change a setting in /etc/apt/apt.conf.d/50unattended-upgrades that would do this cleanup automatically, but this mechanism has a bug whose fix hasn't propagated out yet (link). So, I've added a line to nodus' root crontab to autoremove once a week, Sunday morning.

11787   Wed Nov 18 23:40:01 2015 ranaUpdateOptical LeversCalibration of oplevs for ITMY/ETMY

OMG. Please try to use larger fonts and PDF so that we can read the plots.

 Quote: Based on elog 1403, I calibrated the oplevs for ITMY/ETMY.

I'm not sure that these calibration measurements are reliable. I would feel better if Steve can confirm them using our low accuracy method of moving the QPD by 1 mm and doing trigonometry.

11788   Thu Nov 19 14:50:34 2015 Max IsiUpdateGeneralNew 2D histogram plot for summary pages

A new type of plot is now available for use in the summary pages, based on EricQ's 2D histogram plots (elog 11210). I have added an example of this to the SandBox tab (https://nodus.ligo.caltech.edu:30889/detcharsummary/day/20151119/sandbox/). The usage is straighforwad: the name to be used in config files is histogram2d; the first channel corresponds to the x-axis and the second one to the y-axis; the options accepted are the same as numpy.histogram2d and pyploy.pcolormesh (besides plot limits, titles, etc.). The default colormap is inferno_r and the shading is flat.

11789   Thu Nov 19 15:16:24 2015 ericqUpdateCDSc1iscey IO chassis now has brackets

[Steve, ericq]

Brackets for the c1iscey IO chassis cards have been installed. Now, I can't unseat the cards by wiggling the ADC or DAC cable.

11790   Thu Nov 19 16:06:54 2015 yutaroUpdateOptical LeversCalibration of oplevs for ITMY/ETMY

I'm sorry. I will be careful about that. And I updated the plots in elog 11785.

 Quote: OMG. Please try to use larger fonts and PDF so that we can read the plots.

Quote:
 Quote: Based on elog 1403, I calibrated the oplevs for ITMY/ETMY.

I'm not sure that these calibration measurements are reliable. I would feel better if Steve can confirm them using our low accuracy method of moving the QPD by 1 mm and doing trigonometry.

In this morning, Steve and I looked at the ETMY table and we found that the measurement you suggested might interfere with other optics or detectors because of space constraint. So, before doing this measurement, I roughly estimated the calibration factors for ETMY oplev by using the rough value of the arm length of the optical lever and the beam width of the light just before the QPD.

How I got the arm length and the beam width:

I measured the length of the optical path between ETMY and the QPD. Then I measured the beam width with an iris to screen the beam. To get the beam width from the decrease of the power of the beam detected by QPD, I used this formula: $P/P_{max}=1-\exp(-2r^2/w^2)$.

Then I got:   (arm length) = 1.8 +/-0.2 m,    w= 0.56 +/- 0.5 mm.

How I estimated the calibration factors from these:

The calibration factors (such as C1:SUS-ETMY_OL_PIT_CALIB; (real angle) / (normalized output of QPDXorY)) can be calculated with: $\sqrt{\pi/32}\times w\,/\,(\mathrm{arm\,length})$. Then, I got

$130\,\pm\,20 \,\mu \mathrm{rad}$,

though the calibration factors, C1:SUS-ETMY_OL_PIT_CALIB C1:SUS-ETMY_OL_YAW_CALIB, right now are 26.0 and 31.0, respectively. (If I express this in the same way as elog 11785, 5.0 and 4.2 for ETMY_PIT and ETMY_YAW, respectively. they are consistent with yesterday's results.)

I believe that the calibration factors I estimated today are not different from the true values by a factor of 2 or something, so this estimation indicates that the oplev calibration measurements I did yesterday are reliable, at least for the oplev for ETMY.

11791   Thu Nov 19 17:06:57 2015 KojiConfigurationCDSDisabled auto-launching RT processes upon FE booting

We want to startup the RT processes one by one on boot of FE machines.

Therefore /diskless/root/etc/rc.local on FB was modified as follows. The last sudo line was commented out.

for sys in $(/etc/rt.sh); do #sudo -u controls sh -c ". /opt/rtapps/rtapps-user-env.sh && /opt/rtcds/cal\ tech/c1/scripts/start${sys}"

    # NOTE: we need epics stuff AND iniChk.pl in PATH     # we use -i here so that the .bashrc is sourced, which should also     # source rtapps and rtcds user env (for epics and scripts paths)

    # commented out Nov 19, 2015, KA     # see ELOG 11791 http://nodus.ligo.caltech.edu:8080/40m/11791     # sudo -u controls -i /opt/rtcds/caltech/c1/scripts/start\${sys} done

11792   Fri Nov 20 09:45:39 2015 steveSummarySUSoplev laser summary

Quote:

 Quote: May  13, 2014             ETMX,  .............laser in place 90 d       May  22, 2012             ETMY,       Oct.  7,  2013             ETMY,  LT  503 d  or  1.4 y............bad beam quality ?      Aug. 8,  2014              ETMY,  .............laser in place   425 days  or  1.2 y

Sept. 5, 2014              new 1103P, sn P893516  installed at SP table for aLIGO oplev use qualification

11793   Fri Nov 20 15:44:12 2015 KojiSummaryPSLAdded 17.5kHz LPF to the PMC servo

As a final tune of the PMC servo, I've added 1nF cap at the error signal amplification stage. The diagram has been updated and uploaded to DCC. https://dcc.ligo.org/D1400221

It should be noted that this modification yielded the error signal to have 17.5kHz roll off.

The openloop TF after the modification has been measured. (Attachment 1)

With the new nominal gain of 9dB, almost the same gain margin for the 28kHz peak has been realized.
=> We have 6dB (factor of 2) more gain at low frequency. Currently, the feature at 8kHz causes the oscillation when the gain is further increased.

Here is the model function for the OLTF.

function cmpOLTFc = PMC_OLTF_model(freqOLTFc)

cmpOLTFc = -9.5e5*pole1(freqOLTFc,0.162).*zero1(freqOLTFc,491)... % from the circuit diagram     .*pole1(freqOLTFc,17.5e3)... % Newly implemented input filter => GBW pole was replaced with this     .*zero2(freqOLTFc,12.5e3,100)... % eye-fit     .*pole2(freqOLTFc,12.2e3,6)... % eye-fit     .*pole2(freqOLTFc,28.8e3, 12)... % eye-fit     .*pole1(freqOLTFc,150e3)... % Mixer LPF estimated from Circuit Lab Simulation     .*pole1(freqOLTFc,11.3)... % Output Impedance + PZT LPF     .*pole1(freqOLTFc,3e4); % Unknown    end

The free-running round-trip displacement (roundtrip) / frequency noise is shown in Attachments. There we compare the spectra with and without IMC locked.

i.e. When the IMC is not locked, we are measuring the laser frequency noise with the sensor (PMC cavity) that is noisy due to the PMC displacement.
When the IMC is locked, the laser frequency is further stabilized while the sensor (PMC) noise is not changed.

- Without IMC locked

Can we see the laser freq noise? It seems that it is visible above 100Hz.
The red curve is the measured noise level. The NPRO (although it is LWE NPRO) noise level from S. Nagano's thesis (see our wiki) is shown there.

- With IMC locked

When the IC is locked, we see the increase of the noise between 1~4Hz. It means that the IMC is not only noisier than the laser, but also noisier than the PMC cavity.
Sounds reasonable. And the PMC is capable to handle this motion.

The reduction of the frequency noise is seen from 100Hz to 30kHz.

The interesting point is that we can see the noise increase above 30kHz when the IMC is locked.
I believe that the phase correction EOM is shared with the PMC modulation. i.e. PMC sees the corrected laser frequency.

We expect that the frequency noise is reduced at this frequency. But in reality not.

In addition, there is a sharp peak at ~35kHz. I wonder If this is caused by the IMC servo. It is worse to investigate.

11794   Sat Nov 21 00:45:30 2015 KojiUpdateIOOIMC fix

Based on the observation of the PMC error signal, I started measuring the IMC OLTF. Immediately, it was found that the overall IMC loop gain was too low.
The UGF was ~40kHz, which was really marginal. It had been >100kHz when I have adjusted it about a year ago. (Next entry for the detail)

The first obvious thing was that the SMA cables around the IMC servo have visible degradation (Attatched photos).
I jiggled the signal cable from the demodulator Q_out to the MC servo. The openloop gain seemed fluctuating (increased) based on the cabling.
I decided to repair these cables by adding solder on the shield.

Even after the repair, the open loop TF didn't show any improvement. I checked the LO level and found that it was -16.7dBm.

I traced the problem down to the frequency generator unit (T1000461). The front panel of the unit indicates the output power for the 29.5MHz output is 13dBm,
while measurement showed it was 6~8dBm (fluctuating). The T1000461 document describes that there is only a wenzel oscillator inside. Does this mean the oscillatorwas degraded??? We need to open the box.

I was not sure what was the LO level. I naively assumed the input is 0dBm. Reducing the attenuation of the dial on the AM Stabilizer unit from 12dB attn to 0dB.
This made the LO level -3.3dBm.

Later at home, I thought this nominal LO level of 0dBm could have been wrong.

The demodulator circuit (D990511) has the amplifier ERA-5 (G=~20dB) at the input. Between the input and the ERA-5 there is a pattern for an attenuator.
Assuming we have no attenuator, the ERA-5 has to spit out 20dBm. That is too much for this chip. I need to pull out the box to see how much is the nominal LO for this box using an active probe.

This decrease/increase of the LO level affects the WFS demod too. According to D980233-B, the input stage has the comparator chip AD96687, which can handle differential voltage of 5.5V.
Therefore the effect is minimal.

11795   Sat Nov 21 00:46:33 2015 KojiUpdateIOOIMC OLTF

Here is the comparison before and after the fix.

Before the work, the UGF was ~40kHz. The phase margin was ~5deg. This caused huge bump of the frequency noise.

After the LO power increase, I had to reduce the MC loop gain (VCO Gain) from 18dB to 6dB. This resulted 4dB (x2.5) increase of the OLTF. This means that my fix increased the optical gain by 16dB (x6.3). The resulting UGF and phase mergin were measured to be 117kHz and 31deg, respectively.

Now I was curious to see if the PMC err shows reasonable improvement when the IMC is locked. Attachment 2 shows the latest comparison of the PMC err with and without the IMC locked. The PMC error has been taken up to 500kHz. The errors were divided by 17.5kHz LPF and 150kHz LPF to compensate the sensing response. The PMC cavity pole was ignored in this calculation. T990025 saids the PMC finesse is 4400 and the cavity pole is 174kHz. If this is true, this also needs to be applied.

Observations:

1. Now we can see improvement of the PMC error in the region between 10kHz to 70kHz.

2. The sharp peak at 8kHz is due to the marginally stable PMC servo. We should implement another notch there. T990025 suggests that the body resonance of the PMC spacer is somewhere around there. We might be able to damp it by placing a lossy material on it.

3. Similarly, the features at 12kHz and 28kHz is coming from the PMC. They are seen in the OLTF of the PMC loop.

4. The large peak at 36kHz does not change with the IMC state. This does mean that it is coming from the laser itself, or anything high-Q of the PMC. This signal is seen in the IMC error too.

5. 72kHz, 108kHz, 144kHz: Harmonics of 36kHz?

6. Broad feature from 40kHz to 200kHz. The IMC loop is adding the noise. This is the frequency range of the PC drive. Is something in the PC drive noisy???

7. The feature at 130kHz. Unknown. Seems not related to IMC. The laser noise or the PMC noise.

Remaining IMC issues:

Done (Nov 23, 2015) - 29.5MHz oscillator output degraded. Possibly unstable and noisy. Do we have any replacement? Can we take a Marconi back from one of the labs?

Done (Nov 23, 2015) - Too high LO?

- Large 36kHz peak in the IMC

- IMC loop shape optimization

- IMC locking issue. The lock streatch is not long.

- IMC PC drive issue. Could be related to the above issue.

Maybe not relevant - PC drive noise?

11796   Sun Nov 22 07:09:01 2015 ranaUpdateIOOIMC fix

On the demod board there is a 10 dB attenuator (AT1), which lowers the level to -10 dBm before the ERA-5. Then it should be 10 dBm before going to the rest of the parts. But I guess the ERA-5 chips which come later on in the circuit could be decaying like the ones in the PMC LO board.

 Quote: Later at home, I thought this nominal LO level of 0dBm could have been wrong.
11797   Sun Nov 22 12:07:09 2015 KojiUpdateIOOIMC fix

TO DO

Done (Nov 23, 2015) - Check if the attenuator is still there in the input chain

Done (Nov 23, 2015) - Check if the actual LO levels at the 17dBm mixers are reasonable.

- Check if the actual LO levels for the LSC demods are OK too

11798   Sun Nov 22 12:12:17 2015 KojiUpdateIOOIMC OLTF

Well. I thought a bit more and now I think it is likely that this is just the servo bump as you can see in the closed-loop TF.

 Quote: 6. Broad feature from 40kHz to 200kHz. The IMC loop is adding the noise. This is the frequency range of the PC drive. Is something in the PC drive noisy???
11799   Mon Nov 23 14:45:39 2015 ericqUpdateComputer Scripts / ProgramsNew software

COMSOL 5.1 has been installed at: /cvs/cds/caltech/apps/linux64/comsol51/bin/comsol

MATLAB 2015b has been installed at: /cvs/cds/caltech/apps/linux64/matlab15b/bin/matlab

This has not replaced the default matlab on the workstations, which remains at 2013a. If some testing reveals that the upgrade is ok, we can rename the folders to switch.

11800   Mon Nov 23 20:32:43 2015 KojiUpdateLSCFrequency source fixed, IMC LO level adjusted

The frequency source was fixed. The IMC LO level was adjusted.

IMC is locked => OLTF measured UGF 144kHz PM 30deg.

11801   Mon Nov 23 21:48:49 2015 KojiUpdateLSCFrequency source fixed, IMC LO level adjusted

The trouble we had: the 29.5 MHz source had an output of 6 dBm instead of 13 dBm.

The cause of the issue: A short cable inside had its shield cut and had no connection of the return.

- The frequency source box was dismantled.
- The power supply voltages of +28 and +18 were provided from bench supplies.
- The 29.5 MHz output of 5~6 dBm was confirmed on the work bench.
- The 11 MHz OCXO out (unused) had an output of 13 dBm.

- Once the lid was opened, it was immediately found that the output cable for the 29.5 MHz source had a sharp cut of the shield (Attachment1).
- OK. This cable was replaced. The output of 13 dBm was recovered.

- But wait. Why is the decoupling capacitor on the 29.5 MHz OCXO bulging? The polarity of the electrolytic capacitor was wrong!
- OK. This capacitor was replaced. It was 100 uF 35 V but now it is 100 uF 50 V.

- I further found some cables which had flaky shields. Some of them were twisted. When the panel cable s connected, the feedthroughs were rotated. This twists internally connected cables. Solder balls were added to the connector to reinforce the cable end.

- When the box was dismantled, it was already noticed that some of the plastic screws to mount the internal copper heat sinks for ZHL-2's were broken.
They seemed to be degraded because of the silicone grease. I didn't try to replace all as it was expected to take too much time, so only the broken screws
were replaced with steel screws with shoulder washers
at the both side of the box.

- After confirming the circuit diagram, the box was returned to the rack. The 29.5 MHz output of 13 dBm there was confirmed.

11802   Mon Nov 23 22:12:10 2015 KojiUpdateIOOLO level check for the IMC demod board

In order to check the proper LO level, the IMC demod board was checked. As a short summary, -8dBm is the proper input for the IMC demod board. This was realized when the variable attenuator of the RF AM Stabilizer was set up be -7dB.

Initially, I tried to do the measurement using the extender board. But every board had the issue of +15V not working. After several extender boards were tried, I noticed that the current draw of the demod board burned the 15V line of the extender board.

Then I moved to the work bench. The signals were checked with the 10:1 probe. It's not properly the 50Ohm system, exactly to say.

I found that the LO signals at the mixers have huge distortion as it reaches the nominal 17dBm, and I wondered if ERA-5s were gone. Just in case I replaced the ERA-5s but didn't see any significant change. Then I thought it is due to the mixer itself. The mixer was removed and replaced with a 50Ohm SMD resister. Then the output of the last ERA-5 became sinusoidal, and the level was adjusted to be ~17dBm (4.52 Vpp) when the input power was measured to be -7.7dBm with the RF power meter. Once the mixer was reinstalled, it was confirmed that the waveform becase rectangular like, with the similar amplitude (4.42Vpp).

Now the module was returned to the rack. The RF level at the LO input was adjusted to be -8dBm by setting the attenuator level to be 7dBm.

Once the IMC is locked with this setting, the open loop transfer function was measured. The optical gain seemed almost unchanged compared with the recent nominal. The UGF and PM were measured to be 144kHz and 30deg.

11803   Mon Nov 23 23:42:56 2015 ericqUpdateLSCALSY recovered

[ericq, gautam]

Gautam couldn't observe a Y green beatnote earlier, so we checked things out, fixed things up, and performance is back to nominal based on past references.

Things done:

• Marconi carrier output switched back on after Koji's excellent RF maintence
• BBPD power supplies switched on
• Removed a steering mirror from the green beatY path to do near/far field alignment.
• Aligned PSL / Y green beams
• Replaced mirror, centered beam on BBPD, moved GTRY camera to get the new spot.
• POY locked, dither aligned, beatnote found, checked ALS out-of-loop noise, found to be in good shape.
11804   Tue Nov 24 01:14:23 2015 KojiUpdateLSCALSY recovered

Sorry, I completely forgot to turn the Marconi on...

11805   Tue Nov 24 11:18:47 2015 yutaroUpdateOptical LeversBeam centering for the oplev of ITMY

I made the beam spot on QPD for the oplev of ITMY centered by changing the orientation of the mirror just before the QPD.

Before doing this, I ran dithering for Y arm and froze the output of ASS for Y arm.

11806   Tue Nov 24 14:58:40 2015 yutaroUpdateLSCITMX misaligned

I misaligned ITMX. The oplev servo for ITMX is now turned off. You can restore ITMX alignment by running "restore".

11807   Wed Nov 25 04:24:21 2015 ranaUpdateIOOLO level check for the IMC demod board

Hmmm. Very non-standard demod. From the photo, looks like someone did some surgery with the attenuators (AT1, AT2, AT3) in the LO path. (might be me from a long time ago).

-8 dBm input to a circuit is a not a low noise situation. It would be best to remove the amplifiers in the I&Q paths and just have a single amplifier in the main path. Ideally we want the LO to never go below -3 dBm and certainly not below 0 dBm while outside of the board.

I doubt that all of the LSC demods were modified in this way - this one ought to get some sharpie or stickers to show its difference.

11808   Wed Nov 25 10:07:10 2015 KojiUpdateIOOLO level check for the IMC demod board

I didn't finish making the DCC entry for this module yet.

But the attenuators are
- AT1: 10dB. There is a sign that it was 3dB before ---  a 3dB chip was also attached on the boardnext to 10dB.
- AT2/3: Removed. They were replaced with 0Ohm resistors.

Currently the input is -8dBm. The input and output of the first ERA-5 are -17dBm and +7dBm, respectively.
Then the input and output of the second ERA-5 are -2dBm and 17dB, respectively.

In order to remove the second amplification stage, the first stage has to produce 26dBm. This is too much for either ERA-5 or any chips that fits on the foot print. If we use low gain but high output amp like GVA-81 (G=10dB, DF782 package), it is doable

Input 0dBm - [ATTN 3] - -3dBm - [ERA-5 G=20dB] - +16~+17dBm - [Circuits -9dB] - +7dBm - [Attn 0dB] - +7dBm - [GVA-81 G=10dB] - +17dBm

I think we should check the conditions of all the LSC demods.

11809   Wed Nov 25 14:46:53 2015 gautamUpdateCDSc1lsc models restarted

I noticed that all the models running on C1LSC had crashed when I came in earlier today. I restarted all of them by ssh-ing into C1LSC and running rtcds restart all. The models seem to be running fine now.

11810   Wed Nov 25 16:40:32 2015 yutaroUpdateLSCround trip loss of Y arm

I measured round trip loss of Y arm. The alignment of relevant mirrors was set ideal with dithering (no offset).

Summary:

round trip loss of Y arm: 166.2 +/- 9.3 ppm

(In the error, only statistic error is included.)

How I measured it:

I compared the power of light reflected by Y arm (measured at AS) when the arm was locked (P_L) and when ETMY was misaligned (P_M). P_L and P_M can be described as

$P_M=P_0(1-T_\mathrm{ITM})$

$P_L=P_0\left[1-(1-\alpha)\frac{4T_\mathrm{ITM}}{T_\mathrm{tot}^2}T_\mathrm{loss}\right]$.

The reason why P_L takes this form is: (1-alpha)*4T_ITM/(T_tot)^2 is intracavity power and then product of intracavity power and loss describes the power of light that is not reflected back. Here, alpha is power ratio of light that does not resonate in the arm (power of mismatched mode and modulated sideband), and T_tot is T_ITM+T_loss. Transmissivity of ETM is included in T_loss. I assumed alpha = 7%(mode mismatch) + 2 % (modulation) (elog 11745)

After some calculation we get

$1-P_L/P_M\simeq \frac{4(1-\alpha) T_\mathrm{loss}}{T_\mathrm{ITM}}-T_\mathrm{ITM}$.

Here, higher order terms of T_ITM and (T_loss/T_ITM) are ignored. Then we get

$(1-\alpha) T_\mathrm{loss}=\frac{T_\mathrm{ITM}}{4}(1-P_L/P_M+T_\mathrm{ITM})$.

Using this formula, I calculated T_loss. P_L and P_M were measured 100 times (each measurement consisted of 1.5 sec ave.) each and I took average of them. T_ETM =13.7 ppm is used.

Discussion:

-- This value is not so different from the value ericq reported in July (elog 10248).

-- This method of measuring arm loss is NOT sensitive to T_ITM.  In contrast, the method in which loss is obtained from finesse (for example, elog 11740) is sensitive to T_ITM.

In the method I'm now reporting,

$\Delta T_\mathrm{loss}/T_\mathrm{loss}\simeq\Delta T_\mathrm{ITM}/T_\mathrm{ITM}$,

but in the method with finesse,

$\Delta T_\mathrm{loss}\simeq\Delta T_\mathrm{ITM}$.

In the latter case, if relative error of T_ITM is 10%, error of T_loss would be 1000 ppm.

So it would be better to use power of reflected light when you want to measure arm loss.

11811   Wed Nov 25 16:46:10 2015 KojiUpdateIOOLO level check for the IMC demod board

Awwww. I found that the demod board has the power splitter (PSCJ-2-1) with one output unterminated.
This power splitter should be removed.

https://dcc.ligo.org/D1500443

11812   Wed Nov 25 20:07:35 2015 ranaUpdateIOOLO level check for the IMC demod board

Perhaps we can replace T1 with a mini-circuits hybrid 0-90 deg splitter and then remove the trim caps. (JSPQ-80, JYPQ-30, SCPQ-50)

11813   Wed Nov 25 22:37:12 2015 yutaroUpdateLSCremoval of Gautam's cable in 1Y2 and restoration of POYDC

[yutaro, Koji]

We disconnected the cable that was connected to CH5 of the whitening filter in 1Y2, then connected POYDC cable to there (CH5). This channel is where POYDC used to connect.

Then we turned on the whitening filter for POYDC (C1:LSC-POYDC FM1) and changed the gain of analog whitening filter for POYDC from 0 dB to 39 dB (C1:LSC-POYDC_WhiteGain).

11814   Wed Nov 25 22:59:42 2015 yutaroUpdateLSCAS table optics realignment

I slightly changed the orientation of a few mirrors on AS table that are used to make the AS light get into PDs, in order to confirm that the strange behavior of ASDC (I will report later) is not caused by clipping related to these mirrors or miscentering on PDs.

Then output level of ASDC, AS55, and AS165 could have changed.

So take care of this possible change when you do something related to them. But the relative change of them would be at most several %, I think.

11815   Wed Nov 25 23:17:34 2015 yutaroUpdateLSCstrange behavior of ASDC

[yutaro, Koji]

I noticed that ASDC level changes depending on the angle of ITMY when trying to take some data for loss map of YARM. We finally found that ASDC level behaves strangely when the angle of ITMY in yaw direction is varied, as you can see in Attachment 1. Now, AS port recieved only the reflection of ITMY.

NOTE: This behavior indicates that angular motion could couple to length signal in AS port.

Koji suggested that this behavior might be caused by interference at SR2 or SR3 between main path light and the light reflected by the AR surface. By rough estimation, we confirmed that this scenario would be possible. So it would be better to measure AR reflection of the same mirror to ones used for SR2 and SR3 in term of incident angle.

Ed by KA: This senario could be true if the AR reflection of teh G&H mirrors have several % due to large angle of incidence. But then we still need think about the overlap between the ghost beam and the main beam. It's not so trivial.

11816   Wed Nov 25 23:34:52 2015 yutaroUpdateLSCround trip loss of Y arm

[yutaro, Koji]

Due to the strange behavior (elog 11815) of ASDC level, we checked if it is possible to use POYDC instead of ASDC to measure the power of reflected light of YARM. Attached below is the spectrum of them when the arm is locked. This spectrum shows that it is not bad to use POYDC, in terms of noise. The spectrum of them when ETMY is misaligned looked similar.

So I am going to use POYDC instead of ASDC to measure arm loss of YARM.

Ed by KA:
The spectra of POYDC and ASDC were measured. We foudn that they have coherence at around 1Hz (good).
It told us that POYDC is about 1/50 smaller than ASDC. Therefore in the attached plot, POYDC x50 is shown.
That's the meaning of the vertical axis unit "ASDC".

11817   Thu Nov 26 19:39:27 2015 KojiUpdateLSCCurrent state of the frequency source, and possible improvement

Uploaded on T1000461 too.

11818   Fri Nov 27 03:38:23 2015 yutaroSummaryLSCround trip loss of Y arm

Tonight I measured "loss map" of ETMY. The method to calculate round trip loss is same as written in elog 11810, except that I used POYDC instead of ASDC this time.

How I changed beam spot on ETMY is: elog 11779.

I measured round trip loss for 5 x 5 points. The result is below.

(unit: ppm)

494.9 +/- 7.6       356.8 +/- 6.0       253.9 +/- 7.9       250.3 +/- 8.2       290.6 +/- 5.1
215.7 +/- 4.8       225.6 +/- 5.7       235.1 +/- 7.0       284.4 +/- 5.4       294.7 +/- 4.5
205.2 +/- 6.0       227.9 +/- 5.8       229.4 +/- 7.2       280.5 +/- 6.3       320.9 +/- 4.3
227.9 +/- 5.7       230.5 +/- 5.5       262.1 +/- 5.9       315.3 +/- 4.7       346.8 +/- 4.2
239.7 +/- 4.5       260.7 +/- 5.3       281.2 +/- 5.8       333.7 +/- 5.0       373.8 +/- 4.9

The correspondence between the loss shown above and the beam spot on ETMY is shown in the following figure. In the figure, "downward" and "left" indicate direction of shift of the beam spot when you watch it via the camera (ex. 494.9 ppm corresponds to the lowest and rightest point).

Edited below on 28th Nov.

To shift the beam spot on ETMY, I added offset in YARM dither loop. The offset was [-30,-15,0,15,30]x[-10,-5,0,5,10] for pitch and yaw, respectively.  How I calibrated the beam spot is basically based on elog 11779, but I multiplied 5.3922 for vertical direction and 4.6205 for horizontal direction which I had obtained by caliblation of oplev (elog 11785).

Edited above on 28 th Nov.

I will report the detail later.

11819   Fri Nov 27 22:20:24 2015 yutaroUpdateLSCround trip loss of Y arm

Here, I upload data I took last night, including the power of reflected power (locked/misaligned) and transmitted power for each point (attachement 1).

And I would like to write about possible reason why the loss I measured with POYDC and the loss I measured with ASDC are different by about 60 - 70 ppm (elog 11810 and 11818). The conclusion I have reached is:

It could be due to the strange bahavior of ASDC level.

This difference corresponds to the error of ~2% in the value of P_L/P_M. As reported in elog 11815, ASDC level changes when angle of the light reflected by ITMY changes, and 2% change of ASDC level corresponds to 10 urad change of the angle of the light according to my rough estimation with the figure shown in elog 11815 and attachment 2. This means that 2% error in P_L/P_M could occur if the angle of the light incident to YARM and that of resonant light in YARM differ by 10 urad. Since the waist width $w_0$ of the beam is ~3 mm, with the 10 urad difference, the ratio of the power of TEM10 mode is $(10\,\mu \mathrm{rad}/ \theta_0)^2\sim0.01$, where $\theta_0=\lambda/\pi w_0$. This value is reasonable; in elog 11743 Gautam reported that the ratio of the power of TEM10 was ~ 0.03, from the result of cavity scan. Therefore it is possible that the angle of the light incident to YARM and that of resonant light in YARM differ by 10 urad and this difference causes the error of ~2% in P_L/P_M, which could exlain the 60 - 70 ppm difference.

11820   Sat Nov 28 11:46:40 2015 yutaroUpdateLSCpossible error source of loss map measurement

I found that TRY level degraded and the beam shape seen with CCD camera at AS port was splitted when the beam spot on ETMY was not close to the center. This was because dither started not working well. I suspect so because in such a case TRY level went up when I did iteration with TT1 and TT2 after freezing dither. Splitted beam shape indicates that incident light did not match well with the cavity mode.

TRY level for each point was this:

TRYDC
[[ 0.6573      0.8301      0.8983      0.8684      0.6773    ]
[ 0.7555      0.8904      0.9394      0.8521      0.6779    ]
[ 0.6844      0.8438      0.9318      0.8834      0.6593    ]
[ 0.7429      0.8688      0.9254      0.8427      0.6474    ]
[ 0.7034      0.8447      0.8834      0.8147      0.6966    ]]

In the worst case, TRY level was 70 % of the maximum level. Assuming that this degrade was totally due to the mode mismatch, this corresponds to ~50 urad difference between the angle of incident light and resonant lighe in the arm (see elog 11819).

11821   Sun Nov 29 05:23:57 2015 ranaUpdateLSCCurrent state of the frequency source, and possible improvement

I need some more hints to understand the improvement, although its generally good to re-build it considering the sad state of the assembly/installation that you found.

I see that the current design brings the 11 MHz signal to -2 dBm before intering the first ZHL-2+, but since that has a NF of 9 dB, that seems to only degrade the phase noise to -2 - (-174 +9) = -163 dBc. That seems OK since we only need -160 dBc from this system. Probably the AM noise is worse than this already (we should remember to hook up a simple AM stabilizer in 2016, as well as the ISS).

What else are the main features of this improvement? I can reward a good summary with some Wagonga.

11822   Sun Nov 29 12:32:26 2015 KojiUpdateLSCCurrent state of the frequency source, and possible improvement

I'm not claiming we need to modify the frequency source immediately as we are not limited by the oscillator amplitude or phase noise.
I just wanted to note something in mind before it goes away quickly.

Alberto's T1000461 tells us that the oscillator and phase noise are degraded by factor of ~3 and ~5 due to the RF chanin.
My diagram is possible removal of up-down situation of the chain.

Maybe more direct improvement would be:

- Removal of two amplifiers out of four. The heat condition of the box is touch thought it is not critical.

- The modification will allow us to have a spare 11MHz channel at 1X2 rack that would be useful for 3f modulation.

11823   Mon Nov 30 10:41:45 2015 yutaroUpdateLSCDoes a baffle in front of ETMY have effect on loss map measurement?

It might have, so I think I need to estimate shift of beam spot more preciely.

According to Steve's drawing, radius of the hole of the baffle is 19.8 mm.

Intensity distribution of fundamental mode in x axis direction is this (y is integrated out):

$I(x)=\frac{1}{\sqrt{2\pi(w/2)^2}}e^{-\frac{1}{2}\left(\frac{x}{w/2}\right)^2}$

With the radius of curvature of ETMY of 60 m and the arm length of 37.78 m, the beam width $w$ on ETMY is estimated to be 5.14 mm. From this expression of the intensity, $\int^\infty_{x=9.56\mathrm{mm}}\mathrm{d}xI(x)=100\,\mathrm{ppm},\,\int^\infty_{x=10.00\mathrm{mm}}\mathrm{d}xI(x)=50\,\mathrm{ppm}$, for example. If round trip loss is considered, these values are doubled.

Although maximum shift of beam spot from the ideal spot on ETMY is estimated to be sqrt(6.0^2+(1.7+1.7)^2)=6.9 mm, this value could have error of several tens of % because I am not sure to what exten the calibration is precise, which means that the maximum shift could be ~10 mm and seperation between the baffle and the beam could be ~10 mm.

Therefore, I need to check how much the beam spot shifts with another way, maybe with captured image of the CCD camera.

11824   Mon Nov 30 12:19:38 2015 yutaroUpdateComputer Scripts / Programsimage capture

On VIDEO.adl, Image Capture and Video Capture did not seem to work and gave me some errors, so I fixed following two things:

1. just put one side of a USB cable to Pianosa the other side of which was connected to Sensoray; I don't know why but this was unconnected.

2. slightly fixed /users/sensoray/sdk_2253_1.2.2_linux/imsub/display-image.py as fpllows

L52:       pix[j, i] = R, G, B     ->     pix[j, i] = int(R), int(G), int(B)

It seems to work, at least for some cameras including ETMYF and ITMYF.

11825   Mon Nov 30 14:12:14 2015 ericqUpdateLSCLO level check for the LSC RF distribution box

I checked the RF levels at the LSC LO distribution box, with the agilent scope and a handful of couplers. This was all done with the Marconi at +13dBm.

I only checked the channels that are currently in use, since the analyzer only measures 3 channels at a time, and rewiring involves walking back and forth to the IOO rack to make sure unpowered amps aren't driven, and I was getting hungry.

For the most part, the LO levels coming into the LSC demod boards are all around +1.5dBm (i.e. I measured around -18.0dBm out of the ZFDC-20-5 coupler, which has a nominal 19.5dB coupling factor)

The inputs piped over from the IOO rack, labeled as "+6dBm" were found to be 4.7dBm and 2.9dBm for 11Mhz and 55MHz, respectively.

The 2F signals were generally about 40dB lower, with two exceptions:

• REFL165's ~332MHz signal was around -18dBc
• POP22 had many more visible harmonics than any other LO signal
• 11MHz: -32 dBc
• 33MHz: -32 dBc
• 44Mhz: -15dBc

Here are the raw numbers I measured out of the couplers, all in dBm:

• 11MHz in: -14.8
• 55MHz in: -16.6
• POX11:    -18.7
• POY11:    -18.0
• REFL11:   -18.0
• REFL33:   -18.3
• POP110:   -17.9
• AS110:    -18.1
• POP22:    -19.9
• REFL165:  -18.5
• AS55:     -18.6
• POP55:    -18.8 (this port is used as the REFL55 LO)
11826   Mon Nov 30 15:17:57 2015 KojiUpdateLSCLO level check for the LSC RF distribution box

T1000461 tells us that the nominal LO input is 2dBm although we don't know what's the LO level is at the mixers in the demod boards.

11827   Mon Nov 30 16:33:06 2015 ericqUpdateLSCstrange behavior of ASDC

One possible explanation of this behavior is simply poor centering of the AS beam on AS55 (whose DC level provides ASDC, if memory serves me correctly).

I misaligned ETMY, and moved ITMY through its current nominal alignment while looking at the POYDC and ASDC levels.

In both pitch and yaw, the nominal alignment is fairly close to the "plateau" in which the AS beam is fully within the PD active surface. I.e. it doesn't take much angular motion to start to lose part of the beam, and thus introduce a first order coupling of angle to power. (Look at the plateaus at around -2min and -0.5min, and where the rapidly changing oplev trace crosses zero)

Furthermore, POYDC seems to be in some weird condition where it is actually possible to increase the reported powerwhen misaligning in pitch, but somehow there is more angular coupling in this state.

In any case, I would advise that the POY11 and AS55 RFPDs have their spots recentered with optics in their nominal aligned states. In fact, given how we found REFL11 alingment to be less-than-ideal not so long ago, all of the RFPDs could probably use a checkup.

11828   Mon Nov 30 17:17:30 2015 yutaroUpdateLSCDoes a baffle in front of ETMY have effect on loss map measurement?

With captured images of ETMYF, I measured the shift of the beam spot on ETMY.

The conclusions are:

the baffle would have almost no effect on loss map measurement and

the calibration of beam spot shift is confirmed to be not so bad.

What I did:

I captured ETMYF images in the cases that (i)beam spot is centered on ETMY, beam spot is at the rightest and lowest point of my loss map measurement (corresponding to [0,0] component of the matrix shown in elog 11818), and beam spot is at the leftest and highest point of my loss map measurement ([4,4] component). Each captured image is attached.

Then using ImageJ, I measured the shift of the beam spot. I calibrated lengh in horizontal direction and vertical direction with the diameter of the mirror.

Results:

The amount of the beam shift was 7.2 mm and 8.0 mm for each case.

These values indicate that clapping loss due to the baffle is less than 10 ppm in a round trip.

Today's results support the previous calibration with oplev, which says the amount of the beam shift is 7.0 mm. Two values derived by different calibrations coincide within ~10 % though they are totally different methods. This also support the calibration of the oplev for ETMY (elog 11785) indirectly.

ELOG V3.1.3-


text:
A Keanu Reeves

Description:
After playing Neo in the legendary "Matrix" trilogy, Keanu Reeves started doubting himself: maybe we really live in virtual reality? To find if this is true, he needs to solve the following problem.

Let's call a string consisting of only zeroes and ones good if it contains different numbers of zeroes and ones. For example, 1, 101, 0000 are good, while 01, 1001, and 111000 are not good.

We are given a string s of length n consisting of only zeroes and ones. We need to cut s into minimal possible number of substrings s1,s2,…,sk such that all of them are good. More formally, we have to find minimal by number of strings sequence of good strings s1,s2,…,sk such that their concatenation (joining) equals s, i.e. s1+s2+⋯+sk=s.

For example, cuttings 110010 into 110 and 010 or into 11 and 0010 are valid, as 110, 010, 11, 0010 are all good, and we can't cut 110010 to the smaller number of substrings as 110010 isn't good itself. At the same time, cutting of 110010 into 1100 and 10 isn't valid as both strings aren't good. Also, cutting of 110010 into 1, 1, 0010 isn't valid, as it isn't minimal, even though all 3 strings are good.

Can you help Keanu? We can show that the solution always exists. If there are multiple optimal answers, print any.

Input
The first line of the input contains a single integer n (1≤n≤100) — the length of the string s.

The second line contains the string s of length n consisting only from zeros and ones.

Output
In the first line, output a single integer k (1≤k) — a minimal number of strings you have cut s into.

In the second line, output k strings s1,s2,…,sk separated with spaces. The length of each string has to be positive. Their concatenation has to be equal to s and all of them have to be good.

If there are multiple answers, print any.

Examples
input

1
1

output

1
1

input

2
10

output

2
1 0

input

6
100011

output

2
100 011

Note

In the first example, the string 1 wasn't cut at all. As it is good, the condition is satisfied.
In the second example, 1 and 0 both are good. As 10 isn't good, the answer is indeed minimal.
In the third example, 100 and 011 both are good. As 100011 isn't good, the answer is indeed minimal.

Code:

#include <bits/stdc++.h>
using namespace std;
int o[105], z[105];
int main()
{
int    n, a, b;
a = b = 0;
string s;
cin >> n >> s;
for (int i = 0; i < s.size(); i++)
{
if (s[i] == '1')
a++;
if (s[i] == '0')
b++;
o[i] = a;
z[i] = b;
}
int ans = 0;
if (a != b)
{
cout << "1\n" << s << endl;
}
else
{
cout << "2\n";
for (int i = 0; i < s.size()-1; i++)
cout << s[i];
cout << " " << s[s.size() - 1] << endl;
}
}

B Number Circle

Description:
You are given n numbers a1,a2,…,an. Is it possible to arrange them in a circle in such a way that every number is strictly less than the sum of its neighbors?

For example, for the array [1,4,5,6,7,8], the arrangement on the left is valid, while arrangement on the right is not, as 5≥4+1 and 8>1+6.

Input
The first line contains a single integer n (3≤n≤105) — the number of numbers.

The second line contains n integers a1,a2,…,an (1≤ai≤109) — the numbers. The given numbers are not necessarily distinct (i.e. duplicates are allowed).

Output
If there is no solution, output "NO" in the first line.

If there is a solution, output "YES" in the first line. In the second line output n numbers — elements of the array in the order they will stay in the circle. The first and the last element you output are considered neighbors in the circle. If there are multiple solutions, output any of them. You can print the circle starting with any element.

Examples
input

3
2 4 3

output

YES
4 2 3

input

5
1 2 3 4 4

output

YES
4 4 2 1 3

input

3
13 8 5

output

NO

input

4
1 10 100 1000

output

NO

Note

One of the possible arrangements is shown in the first example:
4<2+3;
2<4+3;
3<4+2.
One of the possible arrangements is shown in the second example.
No matter how we arrange 13,8,5 in a circle in the third example, 13 will have 8 and 5 as neighbors, but 13≥8+5.
There is no solution in the fourth example.

Code:

#include <bits/stdc++.h>
using namespace std;
const int maxn = 1e5 + 10;
typedef long long ll;
ll        a[maxn];
int main()
{
queue<ll> f;
stack<ll> e;
ll        n;
cin >> n;
for (ll i = 0; i < n; i++)
{
cin >> a[i];
}
sort(a, a + n);
if (a[n - 1] >= (a[n - 3] + a[n - 2]))
{
puts("NO");
return 0;
}
puts("YES");
int flag = 1;
for (int i = n - 1; i >= 0; i--)
{
if (flag)
{
f.push(a[i]);
if (a[i - 1] != a[i])
flag = 0;
}
else
{
e.push(a[i]);
if (a[i - 1] != a[i])
flag = 1;
}
}
while (!f.empty())
{
cout << f.front() << " ";
f.pop();
}
while (!e.empty())
{
cout << e.top() << " ";
e.pop();
}
}


text:
# To Derive: The algorithm for the square root of the given equation by Newton’s method.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 4.7, Problem 23E

(a)

To determine

## To Derive: The algorithm for the square root of the given equation by Newton’s method.

Expert Solution

The derivation is xn+1=12(xn+axn)

### Explanation of Solution

Given:

The equation is x2a=0.

Formula used:

The Newton’s method is xn+1=xnf(xn)f(xn).

Calculation:

Let f(x)=x2a and obtain the derivative of f(x).

Let the derivative of f(x)=df(x)dx.

f(x)=d(x2a)dx=d(x2)dxd(a)dx=2x0=2x

To derive the xn+1 use the above mentioned formula.

xn+1=xnx2na2xn=2x2nx2n+a2xn=x2n+a2xn=12(xn+axn)

Hence, the derivation.

(b)

To determine

### To Compute: The root of 1000 correct to six decimal places by using part (a).

Expert Solution

The root correct to six decimal places is 31.6227762.

### Explanation of Solution

Given:

The equation is x2a=0.

Choose the initial point x1=30.

Calculation:

Calculate f(x1) at the point x1=30 and a=1000.

f(30)=(30)21000=9001000=100

Calculate f(x1) at the point x1=30.

f(30)=2×30=60

Substitute n=1 in the above mentioned result in part (a) and obtain x2

x2=12(x1+ax1)

Substitute x1=30,f(x1)=100 and f(x1)=60.

x2=30(100)60=30+10060=30+1.66666631.666666

Calculate f(x2) at the point x2=31.666666 and a=1000.

f(31.666666)=(31.666666)21000=1002.7777351000=2.777735

Calculate f(x2) at the point x2=31.666666.

f(31.666666)=2×31.666666=63.333332

Substitute n=2 in the above mentioned result in part (a) and obtain x3.

x3=12(x2+ax2)

Substitute x2=31.666666,f(x2)=2.777735 and f(x2)=63.333332.

x3=31.6666662.77773563.333332=31.6666660.04385931.622807

Calculate f(x3) at the point x3=31.622807 and a=1000.

f(31.622807)=(31.666666)21000=1000.0019241000=0.001924

Calculate f(x3) at the point x3=31.622807.

f(31.622807)=2×31.622807=62.666664

Substitute n=3 in the above mentioned result in part (a) and obtain x4.

x4=12(x3+ax3)

Substitute x3=31.622807,f(x3)=0.001924 and f(x3)=62.666664.

x4=31.6228070.00192462.666664=31.6228070.00003031.6227762

Thus, the root corrected to six decimal places is 31.6227762_.

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