text:
Chapter 4 to 6 Review
Chapter
Chapter 6
Section
Chapter 4 to 6 Review
Solutions 44 Videos

Use finite differences to determine whether each relation is linear, quadratic, or neither.

Q1a

Use finite differences to determine whether each relation is linear, quadratic, or neither.

Q1b

Sketch the graph of the quadratic relation. Describe the transformation from the graph of y = x^2.

\displaystyle y =x^2 + 2

Q2a

Sketch the graph of the quadratic relation. Describe the transformation from the graph of y = x^2.

\displaystyle y = (x + 3)^2

Q2b

Sketch the graph of the quadratic relation. Describe the transformation from the graph of y = x^2.

\displaystyle y =- \frac{1}{4}x^2

Q2c

Dianne dove from the 10-m diving board. Her height h, in metres, above the water when she is x metres away from the end of the board is given by h = -(x - 1]^2 + 11.

a) Sketch a graph of her dive.

b) What was her maximum height above the water?

c) What horizontal distance had she travelled when she entered the water? Answer to the nearest tenth of a metre.

Q3

Determine an equation to represent the parabola in the form y = a[x - r)(x - s).

Q4

Evaluate.

2^{-4}

Q5a

Evaluate.

(-3)^{-2}

Q5b

Evaluate.

25^{0}

Q5c

Evaluate.

8^{-1}

Q5d

Evaluate.

(-1)^{12}

Q5e

Evaluate.

(\frac{3}{4})^{-3}

Q5f

Cobalt—60 is a radioactive element that is used to sterilize medical equipment.

Cobalt-60 decays to \frac{1}{2}, or 2^{-1}. of its original amount after every 5.2 years. Determine the remaining mass of 20 g of cobalt-60 after

a) 20.8 years

b) 36.4 years

Q6

Write an algebraic expression to represent the area of the figure. Expand and simplify.

Q7

Expand and simplify.

\displaystyle (n+ 3)(n -3)

Q8a

Expand and simplify.

\displaystyle (h + 5)^2

Q8b

Expand and simplify.

\displaystyle (d-4)(d-2)

Q8c

Expand and simplify.

\displaystyle (m +3)(m + 7)

Q8d

Expand and simplify.

\displaystyle (3t -5)(3t + 5)

Q8e

Expand and simplify.

\displaystyle (x - 7)^2

Q8f

Expand and simplify.

\displaystyle x(3x + 1)(2x - 5)

Q9a

Expand and simplify.

\displaystyle (2k + 3)^2 -k(k+ 2)(k -2)

Q9b

Expand and simplify.

\displaystyle 5(y-4)(3y + 1) + (3y -4)^2

Q9c

Expand and simplify.

\displaystyle 3(2a + 3b)(3a -2b)

Q9d

The area of a rectangle is given by the expression 8x^2 + 4x. Draw diagrams to show the possible rectangles, labelling the length and width of each.

Q10

Factor, if possible.

\displaystyle y^2 + 12y + 27

Q11a

Factor, if possible.

\displaystyle x^2 + 2x -3

Q11b

Factor, if possible.

\displaystyle n^2 + 22n + 21

Q11c

Factor, if possible.

\displaystyle p^2 -8p + 15

Q11d

Factor, if possible.

\displaystyle x^2 + 2x - 15

Q11e

Factor, if possible.

\displaystyle k^2 -5k + 24

Q11f

Factor.

\displaystyle p^2 + 12p + 36

Q12a

Factor.

\displaystyle 9d^2 -6d + 1

Q12b

Factor.

\displaystyle x^2 - 49

Q12c

Factor.

\displaystyle 4a^2 -20a + 25

Q12d

Factor.

\displaystyle 8t^2 -18

Q12e

Factor.

\displaystyle a^2 -4b^2

Q12f

Find the value of k so that each trinomial can be factored over the integers.

\displaystyle m^2 + km + 10

Q13a

Find the value of k so that each trinomial can be factored over the integers.

\displaystyle 9a^2 -a + 4

Q13b

The area of a circle is given by the expression \pi(4x^2 + 36x + 81). What expression represents the diameter of this circle?

Q14

A quadratic relation has roots 0 and -6 and a maximum at (-3, 4). Determine the equation of the relation.

The perimeter of a rectangle is 8 m and its area is 2 m^2. Find the length and width of the rectangle to the nearest tenth of a metre.
A ferry operator takes tourists to an island. The operator carries an average of 500 people per day for a round-trip fare of $20. The operator estimates that for each$1 increase in fare, 20 fewer people will take the trip. What fare will maximize the number of people taking the ferry?


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### Home > A2C > Chapter 13 > Lesson 13.2.1 > Problem13-96

13-96.

An isosceles triangle has sides with lengths 10, 10, and 5 cm. What are the measures of the angles in the triangle? Homework Help ✎

Find BD.

Which trigonometric ratio should you use to find m ∠ B?

$\text{cos}B=\frac{2.5}{10}=0.25$

$m\angle B=\text{cos}^{-1}(0.25)$

m ∠B ≈ 75.52°

Now that you know the measure of angle B, find the measure of angles A and C.

format: plain


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# Function constants and what they mean!

• November 5th 2013, 11:44 AM
aprilrocks92
Function constants and what they mean!
I am having problems interpreting a function. A container is supplied with water at a rate given by the equation in attachment 1 (photo1).

Which denotes a function of the time t, where V and k are positive constants. What does the constant V denote in this function?

Does V denote the initial volume of the water body at time t = 0?
• November 5th 2013, 11:48 AM
SlipEternal
Re: Function constants and what they mean!
$V(0) = \dfrac{1}{\sqrt{2}} V_0 e^{-k(0)}\sqrt{1+e^{-k(0)}} = \dfrac{1}{\sqrt{2}} V_0 \sqrt{1+1} = V_0$

So, yes, $V_0$ is the initial volume.
• November 5th 2013, 03:49 PM
aprilrocks92
Re: Function constants and what they mean!
Thank you. I have been trying to integrate this expression, but unsure of what to do when given several constants. Any advice on how to approach this?


text:
# Why is the twisted cubic given as maximal minors of a matrix?

Today I began reading Hartshorne, and was doing an exercise about the twisted cubic, parametrically given by $t \mapsto (t,t^2,t^3)$. According to Wikipedia, its (non-homogeneous) ideal is $(xz-y^2,y-z^2,x-yz)$. Toying a bit with these equations, I noticed that they are precisely the maximal minors of the matrix

$\begin{pmatrix}x &y &z \\ y& z& 1\end{pmatrix}$

Is there a reason for this? I tend to think of determinants as having to do only with linear equations of some sort, but the twisted cubic does not seem very "linear".

-
There is a whole class of "determinantal varieties", the Segre embedding (used to define the product of two projective varieties) usually being the first one that is explicitly given in this way. –  Dylan Moreland Jan 12 '12 at 20:49
Actually, are these equations in the ideal correct? The second generator, for example, seems off: $t^2 - t^6$ isn't usually zero. –  Dylan Moreland Jan 21 '12 at 8:11
Hm, seems you're right. I just copied the homogeneous equation from Wikipedia (en.wikipedia.org/wiki/Twisted_cubic), and put $w=1$. –  Fredrik Meyer Jan 21 '12 at 19:44


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# m01 Q17

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18 Sep 2008, 03:14
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Is $$M \lt 0$$?

1. $$-M = |-M|$$
2. $$M^2 = 9$$

[Reveal] Spoiler: OA
C

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From statement (1) we have that should be negative or zero. Not enough for the strict condition. From statement (2) can be 3 or -3. Not enough again. If we combine two statements then .

Could somebody please explain how can we ever have modulus of any number as negative ? I beleive modulus of a positive number , negative number or zero is positive.... So isnt statement one completely false ?

From statement 2 we can infer that M can be either negative or positive .... but cannot answer the question...

What should be the answer ?
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Re: m01 - Q17 - Wrong answer [#permalink]

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18 Sep 2008, 06:35
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Hi.
First, modulus of zero is neither positive nor negative.
Second, try plugging a negative number in S1. You're right, the |-M| part is always positive, but the left-hand part isn't. That is why we need both S1 and S2 to answer the question. From S2 we know that M is either -3 or 3. From S1 we know that M is either negative or zero. combining the two Statements we get M=-3. Answer C.

Hope this helps.
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Re: m01 - Q17 - Wrong answer [#permalink]

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25 Sep 2008, 13:10
dzyubam wrote:
Hi.
First, modulus of zero is neither positive nor negative.
Second, try plugging a negative number in S1. You're right, the |-M| part is always positive, but the left-hand part isn't. That is why we need both S1 and S2 to answer the question. From S2 we know that M is either -3 or 3. From S1 we know that M is either negative or zero. combining the two Statements we get M=-3. Answer C.

Hope this helps.

how is S1 possible to be negative. i understand that modulus of zero will always be zero, but the first statement doesn't allow for negative result.
for example lets plug -3 in S1. -3=|-3| , thus -3=3 but this is not true!

so S1 tells me that M=0, what am i missing here?
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25 Sep 2008, 13:18
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elmagnifico wrote:
dzyubam wrote:
Hi.
First, modulus of zero is neither positive nor negative.
Second, try plugging a negative number in S1. You're right, the |-M| part is always positive, but the left-hand part isn't. That is why we need both S1 and S2 to answer the question. From S2 we know that M is either -3 or 3. From S1 we know that M is either negative or zero. combining the two Statements we get M=-3. Answer C.

Hope this helps.

how is S1 possible to be negative. i understand that modulus of zero will always be zero, but the first statement doesn't allow for negative result.
for example lets plug -3 in S1. -3=|-3| , thus -3=3 but this is not true!

so S1 tells me that M=0, what am i missing here?

you took M=3 (+VE) thats not possible
M must be -ve here.. M=-3
-(-3) = |-(-3) --> 3=3
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25 Sep 2008, 13:19
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Wait. Let's see. S1 states
-M = |-M|
Now we plug in -3 as you suggested:
-(-3) = |-(-3)| -->
3 = 3.
Now we plug in a positive number 3:
-(3) = |-(3)| -->
-3 = 3.
This is why we know that S1 tells us the $$M \le 0$$. Do you agree?

Saw the post made but still decided to proceed with mine.
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Re: m01 - Q17 - Wrong answer [#permalink]

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25 Sep 2008, 14:28
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dzyubam wrote:
Wait. Let's see. S1 states
-M = |-M|
Now we plug in -3 as you suggested:
-(-3) = |-(-3)| -->
3 = 3.
Now we plug in a positive number 3:
-(3) = |-(3)| -->
-3 = 3.
This is why we know that S1 tells us the $$M \le 0$$. Do you agree?

Saw the post made but still decided to proceed with mine.

thanks for your post. both, you and x2suresh deserve a tap on the shoulder. (and kudos:)
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Re: m01 - Q17 - Wrong answer [#permalink]

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06 Nov 2008, 14:45
I still didn't get this.

Isn't -3 = 3 invalid ?? we cannot even apply the values as such to arrive at the answer.

For me only answer that fits value of M is -ve values from S1.

-M = |-M| this can be true only if M is -ve right ?? hence sufficient ?

-
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06 Nov 2008, 15:47
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S1 is insufficient by itself. Consider M = -3 and M = 0. We have a strict inequality in the question, so we'll need S2.

mbaobsessed wrote:
I still didn't get this.

Isn't -3 = 3 invalid ?? we cannot even apply the values as such to arrive at the answer.

For me only answer that fits value of M is -ve values from S1.

-M = |-M| this can be true only if M is -ve right ?? hence sufficient ?

-

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Re: m01 - Q17 - Wrong answer [#permalink]

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06 Nov 2008, 22:16
1
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Also look at stmt1 this way.

|-M| is an absolute value and hence >=0.

Left side of the equation is -M.

Thus, we are comparing -M to some value that is >=0.
This is possible only in two cases
Case1: when M = 0
Case2, when m<0.

Hence, statement 1 gives M <=0 and is not sufficient to answer the question.
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06 Jul 2010, 06:09
Initially i thought answer to be A but,looking at the explanation by both dzyubam and x2suresh, C seems to be correct answer. Kudos to both of you.
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06 Jul 2010, 13:44
nice explanations thanks. I also thought A at first but you need the information from B
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08 Jul 2010, 02:07
From S1 we can say -M>=0
From S2 we can say M=+3,-3
so Ans:M=-3
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Re: m01 Q17 [#permalink]

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02 Dec 2010, 05:37
The correct answer is C.
(1) is not sufficient because -M = |-M| means M = 0 or M < 0.
For example:
-0 = |-0| : correct
-(-1) = |-(-1)| : correct
M > 0 is impossible because if M > 0, then -M < 0 but |-M| is non-negative.

(2) is not sufficient because M = -3 or M = 3.

(1) & (2) together: sufficient because M = -3, so the answer to the question is YES.
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02 Dec 2010, 06:24
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Expert's post
itiskavikatha wrote:
Is $$M \lt 0$$?

1. $$-M = |-M|$$
2. $$M^2 = 9$$

This question was also posted in DS subforum. Below is my solution from there:

Is $$m<0$$?

(1) $$-m=|-m|$$ --> first of all $$|-m|=|m|$$, (for example: $$|-3|=|3|=3$$), so we have $$-m=|m|$$, as RHS is absolute value which is always non-negative, then LHS, $${-m}$$ must also be non-negative --> $$-m\geq{0}$$ --> $$m\leq{0}$$, so $$m$$ could be either negative or zero. Not sufficient.

(2) $$m^2=9$$ --> $$m=3=positive$$ or $$m=-3=negative$$. Not sufficient.

(1)+(2) Intersection of the values from (1) and (2) is $$m=-3=negative$$, hence answer to the question "is $$m<$$0" is YES. Sufficient.

Hope it's clear.
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08 Jul 2011, 05:59
I initially considered S1 as enough. Apparently not. I didn't consider the case when M = 0. I always make this mistake!
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11 Jul 2011, 09:04
Beware! This is one of the tricks gmat usually pulls on test takers.
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14 Jan 2012, 22:16
Was in trap - forgot to check for zero. Rest Bunnel explains as it
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14 Jul 2012, 08:13
man...silly mistake! chose A as i overlooked M=0 scenario. C is correct.
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12 Jul 2013, 07:04
Post going through all DS questions from OG, I have noticed that most of the DS questions are like this. Mostly, the tricky ones can be classified in the following buckets: (its not an extensive list)

1) Post deduction and simplification ,both equations are same or one of the equation is an information set that was already present in the question itself
2) Missing out the possibility of 0 as both being positive and negative
3) 4-5 variables being removed by use of two equations only (GMAT gives questions having variables more than the number of equations, but the even those equations are linked in such a manner that even lesser number of equations are sufficient)
4) square root of a number (if given by GMAT) only considers positive solution only
Re: m01 Q17   [#permalink] 12 Jul 2013, 07:04
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# Chapter 2 - Solving Equations - 2-2 Solving Two-Step Equations - Practice and Problem-Solving Exercises - Page 92: 55

10.5

#### Work Step by Step

The measures of the two angles are equal so we can put them into an equation. 4x+3=45 (we subtract 3 from both sides of the equation) 4x+3-3=45-3 (simplify) 4x=42 (Divide each side by 4 to isolate the variable) x=42$\div$4=10.5 Thus, x=10.5

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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## [1004.5322] Computing waveforms for spinning compact binaries in quasi-eccentric orbits

Authors: Neil J. Cornish, Joey Shapiro Key

Date: 29 Apr 2010

Abstract: Several scenarios have been proposed in which the orbits of binary black holes enter the band of a gravitational wave detector with significant eccentricity. To avoid missing these signals or biasing the parameter estimation it is important that we consider waveform models that account for eccentricity. The ingredients needed to compute post-Newtonian (PN) waveforms produced by spinning black holes inspiralling on quasi-eccentric orbits have been available for almost two decades at 2 PN order, and this work has recently been extended to 2.5 PN order. However, the computational cost of directly implementing these waveforms is high, requiring many steps per orbit to evolve the system of coupled differential equations. Here we employ a separation of timescales and a generalized Keplarian parameterization of the orbits to produce efficient waveforms describing spinning black hole binaries with arbitrary spin orientations on quasi-eccentric orbits to 1.5 PN order. Our solution includes the spin contributions to the decay of the semi-major axis and eccentricity. We outline a scheme for extending our approach to higher post-Newtonian order.

#### Apr 30, 2010

1004.5322 (/preprints)
2010-04-30, 10:55

## [1004.5057] Revisit on "Ruling out chaos in compact binary systems"

Authors: Xin Wu (Nanchang University), Yi Xie (Nanjing University)

Date: 28 Apr 2010

Abstract: Full general relativity requires that chaos indicators should be invariant in various spacetime coordinate systems for a given relativistic dynamical problem. On the basis of this point, we calculate the invariant Lyapunov exponents (LEs) for one of the spinning compact binaries in the conservative second post-Newtonian (2PN) Lagrangian formulation without the dissipative effects of gravitational radiation, using the two-nearby-orbits method with projection operations and with coordinate time as an independent variable. It is found that the actual source leading to zero LEs in one paper [J. D. Schnittman and F. A. Rasio, Phys. Rev. Lett. 87, 121101 (2001)] but to positive LEs in the other [N. J. Cornish and J. Levin, Phys. Rev. Lett. 89, 179001 (2002)] does not mainly depend on rescaling, but is due to two slightly different treatments of the LEs. It takes much more CPU time to obtain the stabilizing limit values as reliable values of LEs for the former than to get the slopes (equal to LEs) of the fit lines for the latter. Due to coalescence of some of the black holes, the LEs from the former are not an adaptive indicator of chaos for comparable mass compact binaries. In this case, the invariant fast Lyapunov indicator (FLI) of two-nearby orbits, as a very sensitive tool to distinguish chaos from order, is worth recommending. As a result, we do again find chaos in the 2PN approximation through different ratios of FLIs varying with time. Chaos cannot indeed be ruled out in real binaries.

#### Apr 29, 2010

1004.5057 (/preprints)
2010-04-29, 10:02

## [0907.0462] Gravitational-wave detectability of equal-mass black-hole binaries with aligned spins

Authors: Christian Reisswig, Sascha Husa, Luciano Rezzolla, Ernst Nils Dorband, Denis Pollney, Jennifer Seiler

Date: 2 Jul 2009

Abstract: Binary black-hole systems with spins aligned or anti-aligned to the orbital angular momentum provide the natural ground to start detailed studies of the influence of strong-field spin effects on gravitational wave observations of coalescing binaries. Furthermore, such systems may be the preferred end-state of the inspiral of generic supermassive binary black-hole systems. In view of this, we have computed the inspiral and merger of a large set of binary systems of equal-mass black holes with spins parallel to the orbital angular momentum but otherwise arbitrary. Our attention is particularly focused on the gravitational-wave emission so as to quantify how much spin effects contribute to the signal-to-noise ratio, to the horizon distances, and to the relative event rates for the representative ranges in masses and detectors. As expected, the signal-to-noise ratio increases with the projection of the total black hole spin in the direction of the orbital momentum. We find that equal-spin binaries with maximum spin aligned with the orbital angular momentum are more than "three times as loud" as the corresponding binaries with anti-aligned spins, thus corresponding to event rates up to 30 times larger. We also consider the waveform mismatch between the different spinning configurations and find that, within our numerical accuracy, binaries with opposite spins S_1=-S_2 cannot be distinguished whereas binaries with spin S_1=S_2 have clearly distinct gravitational-wave emissions. Finally, we derive a simple expression for the energy radiated in gravitational waves and find that the binaries always have efficiencies E_rad/M > 3.6%, which can become as large as E_rad/M = 10% for maximally spinning binaries with spins aligned with the orbital angular momentum.

#### Apr 29, 2010

0907.0462 (/preprints)
2010-04-29, 10:02

## [1004.4939] On the Gravitational Inverse Problem

Authors: Ulvi Yurtsever, Caren Marzban, Marina Meila

Date: 28 Apr 2010

Abstract: We discuss some mathematical aspects of the problem of inverting gravitational field data to extract the underlying mass distribution. While the forward problem of computing the gravity field from a given mass distribution is mathematically straightforward, the inverse of this forward map has some interesting features that make inversion a difficult problem. In particular, the forward map has an infinite-dimensional kernel which makes the inversion fundamentally non-unique. We characterize completely the kernels of two gravitational forward maps, one mapping mass density to the Newtonian scalar potential, and the other mapping mass density to the gravity gradient tensor, which is the quantity most commonly measured in field observations. In addition, we present some results on unique inversion under constrained conditions, and comment on the roles the kernel of the forward map and non-uniqueness play in discretized approaches to the continuum inverse problem.

#### Apr 29, 2010

1004.4939 (/preprints)
2010-04-29, 10:02

## [1004.4209] Gravitational memory in binary black hole mergers

Authors: Denis Pollney, Christian Reisswig

Date: 23 Apr 2010

Abstract: In addition to the dominant oscillatory gravitational wave signals produced during binary inspirals, a non-oscillatory component arises from the nonlinear "memory" effect, sourced by the emitted gravitational radiation. The memory grows significantly during the late inspiral and merger, modifying the signal by an almost step-function profile, and making it difficult to model by approximate methods. We use numerical evolutions of binary black holes to evaluate the nonlinear memory during late-inspiral, merger and ringdown. We identify two main components of the signal: the monotonically growing portion corresponding to the memory, and an oscillatory part which sets in roughly at the time of merger and is due to the black hole ringdown. Counter-intuitively, the ringdown is most prominent for models with the lowest total spin. Thus, the case of maximally spinning black holes anti-aligned to the orbital angular momentum exhibits the highest signal-to-noise (SNR) for interferometric detectors. The largest memory offset, however, occurs for highly spinning black holes, with an estimated value of hˆtot_20 \approx 0.24 in the maximally spinning case. These results are central to determining the detectability of nonlinear memory through pulsar timing array measurements.

#### Apr 29, 2010

1004.4209 (/preprints)
2010-04-29, 10:01

## [1004.2051] Gravitational waves from first order phase transitions during inflation

Authors: Diego Chialva

Date: 12 Apr 2010

Abstract: We study the production, spectrum and detectability of gravitational waves in models of the early Universe where first order phase transitions occur during inflation. We consider all possible sources: bubble collisions, dynamics of the fluid, thermal fluctuations, turbulence,

The final spectrum of the waves is strongly affected by the constraints deriving from the requirement of efficient inflation (small backreaction of the sources) and by the redshift between the time of emission and the exit from horizon. It results that models with a few phase transitions leave no detectable marks in the CMBR nor a gravitational background sufficient for detection by interferometers such as LIGO, LISA, DECIGO. When the number of phase transitions is instead large, the primordial gravitational waves can be observed both in the CMBR or with LISA (marginally) and especially DECIGO. It is also discussed the nucleosynthesis bound and the constraints it places on the parameters of the models.

#### Apr 29, 2010

1004.2051 (/preprints)
2010-04-29, 10:01

## [1004.3499] Detection, Localization and Characterization of Gravitational Wave Bursts in a Pulsar Timing Array

Authors: Lee Samuel Finn, Andrea N. Lommen

Date: 20 Apr 2010

Abstract: Efforts to detect gravitational waves by timing an array of pulsars have focused traditionally on stationary gravitational waves: e.g., stochastic or periodic signals. Gravitational wave bursts — signals whose duration is much shorter than the observation period — will also arise in the pulsar timing array waveband. Sources that give rise to detectable bursts include the formation or coalescence of supermassive black holes (SMBHs), the periapsis passage of compact objects in highly elliptic or unbound orbits about a SMBH, or cusps on cosmic strings. Here we describe how pulsar timing array data may be analyzed to detect and characterize these bursts. Our analysis addresses, in a mutually consistent manner, a hierarchy of three questions: \emph{i}) What are the odds that a dataset includes the signal from a gravitational wave burst? \emph{ii}) Assuming the presence of a burst, what is the direction to its source? and \emph{iii}) Assuming the burst propagation direction, what is the burst waveform's time dependence in each of its polarization states? Applying our analysis to synthetic data sets we find that we can \emph{detect} gravitational waves even when the radiation is too weak to either localize the source of infer the waveform, and \emph{detect} and \emph{localize} sources even when the radiation amplitude is too weak to permit the waveform to be determined. While the context of our discussion is gravitational wave detection via pulsar timing arrays, the analysis itself is directly applicable to gravitational wave detection using either ground or space-based detector data.

#### Apr 29, 2010

1004.3499 (/preprints)
2010-04-29, 10:01

## [1004.2470] Gravitational Waves III: Detecting Systems

Authors: M.Cattani

Date: 14 Apr 2010

Abstract: In a recent paper we have deduced the basic equations that predict the emission of gravitational waves (GW) according to the Einstein gravitation theory. In a subsequent paper these equations have been used to calculate the luminosities and the amplitudes of the waves generated by binary stars, pulsations of neutron stars, wobbling of deformed neutron stars, oscillating quadrupoles, rotating bars and collapsing and bouncing cores of supernovas. We show here how the GW could be detected in our laboratories. This paper, like the preceding ones, was written to graduate and postgraduate students of Physics.

#### Apr 29, 2010

1004.2470 (/preprints)
2010-04-29, 10:01

## [1004.2276] Gravitational Self-force in a Radiation Gauge

Authors: Tobias S. Keidl, Abhay G. Shah, John L. Friedman, Dong-Hoon Kim, Larry R. Price

Date: 13 Apr 2010

Abstract: In this, the first of two companion papers, we present a method for finding the gravitational self-force in a radiation gauge for a particle moving on a geodesic in a Schwarzschild or Kerr spacetime. The method involves a mode-sum renormalization of a spin-weight $\pm 2$ perturbed Weyl scalar and the subsequent reconstruction from a Hertz potential of the renormalized perturbed metric. We show that the Hertz potential is uniquely specified by the requirement that it have no angular harmonics with $\ell\leq 2$. The resulting perturbed metric is singular only at the position of the particle: It is smooth on the axis of symmetry. An extension of an earlier result by Wald is needed to show that the perturbed metric is determined up to a gauge transformation and an infinitesimal change in the black hole mass and spin. We show that the singular behavior of the metric and self-force has the same power-law behavior in $L=\ell+½$ as in a Lorenz gauge (with different coefficients). We compute the singular Weyl scalar and its mode-sum decomposition to subleading order in $L$ for a particle in circular orbit in a Schwarzschild geometry and obtain the renormalized field. Because the singular field can be defined as this mode sum, the coefficients of each angular harmonic in the sum must agree with the large $L$ limit of the corresponding coefficients of the retarded field. One may compute the singular field by matching the retarded field to a power series in $L$ and subtracting off the leading and subleading terms in this series. We do so, and compare the accuracy of the two methods. Details of the numerical computation of the perturbed metric, the self-force, and the quantity $h_{\alpha\beta}uˆ\alpha uˆ\beta$ (gauge invariant under helically symmetric gauge transformations) are presented for this test case in the companion paper.

#### Apr 29, 2010

1004.2276 (/preprints)
2010-04-29, 10:01

## [1004.4537] Parameter estimation of IMR waves from non-spinning binary black-holes, beyond the Cramer-Rao Lower Bound

Authors: Salvatore Vitale, Michele Zanolin

Date: 26 Apr 2010

Abstract: In this paper we use a new methodology to calculate analytically the error for a maximum likelihood estimate (MLE) of physical parameters from Gravitational Wave (GW) signals, by applying it to IMR waves from non-spinning binary system. While the existing literature focuses on using the Cramer Rao Lower bound. (CRLB) as a mean to approximate the errors for large signal to noise ratios, taking into account only the fist derivative of the signal, we consider also the higher order derivatives, obtaining an improved estimation of parameters' errors. We see how the bias is in general non negligible for high mass systems (200 solar masses and above), due to the nonlinear dipendence of the signal on the parameters, where it can become the most important contributor to the parameters' errors. This new feature will require numerical injections to be proved true.

#### Apr 29, 2010

1004.4537 (/preprints)
2010-04-29, 10:01

## [1004.4007] Gravitational signature of Schwarzschild black holes in dynamical Chern-Simons gravity

Authors: C. Molina, Paolo Pani, Vitor Cardoso, Leonardo Gualtieri

Date: 22 Apr 2010

Abstract: Dynamical Chern-Simons gravity is an extension of General Relativity in which the gravitational field is coupled to a scalar field through a parity-violating Chern-Simons term. In this framework, we study perturbations of spherically symmetric black hole spacetimes, assuming that the background scalar field vanishes. Our results suggest that these spacetimes are stable, and small perturbations die away as a ringdown. However, in contrast to standard General Relativity, the gravitational waveforms are also driven by the scalar field. Thus, the gravitational oscillation modes of black holes carry imprints of the coupling to the scalar field. This is a smoking gun for Chern-Simons theory and could be tested with gravitational-wave detectors, such as LIGO or LISA. For negative values of the coupling constant, ghosts are known to arise, and we explicitly verify their appearance numerically. Our results are validated using both time evolution and frequency domain methods.

#### Apr 29, 2010

1004.4007 (/preprints)
2010-04-29, 10:01

## [1004.4833] Repeating Tidal Disruption of Stars as a Prompt Electromagnetic Signature of Supermassive Black Hole Coalescence

Authors: Nicholas Stone, Abraham Loeb (Harvard)

Date: 27 Apr 2010

Abstract: A precise electromagnetic measurement of the sky coordinates and redshift of a coalescing black hole binary holds the key for using its gravitational wave (GW) signal to constrain cosmological parameters and to test general relativity. Here we show that the merger of ~10ˆ{6-8}M_sun black holes is generically followed over a period of years by multiple electromagnetic flares from tidally disrupted stars. The sudden recoil imparted to the merged black hole by GW emission results promptly in a tidal disruption rate of stars as high as ~0.1-1 per year. The sequential disruption of stars within a single galaxy over a short period provides a unique electromagnetic flag of a recent black hole coalescence event, and can be used on its own to calibrate the expected rate of GW sources for pulsar timing arrays or the proposed Laser Interferometer Space Antenna (LISA).

#### Apr 29, 2010

1004.4833 (/preprints)
2010-04-29, 10:01

## [1004.4697] Measuring orbital eccentricity and periastron advance in quasi-circular black hole simulations

Authors: Abdul H. Mroué, Harald P. Pfeiffer, Lawrence E. Kidder, Saul A. Teukolsky

Date: 27 Apr 2010

Abstract: We compare different methods of computing the orbital eccentricity of quasi-circular binary black hole systems using the orbital variables and gravitational wave phase and frequency. For eccentricities of about a per cent, most methods work satisfactorily. For small eccentricity, however, the gravitational wave phase allows a particularly clean and reliable measurement of the eccentricity. Furthermore, we measure the decay of the orbital eccentricity during the inspiral and find reasonable agreement with post-Newtonian results. Finally, we measure the periastron advance of non-spinning binary black holes, and we compare them to post-Newtonian approximations. With the low uncertainty in the measurement of the periastron advance, we positively detect deviations between fully numerical simulations and post-Newtonian calculations.

#### Apr 29, 2010

1004.4697 (/preprints)
2010-04-29, 10:01

## [0908.3635] Binary compact object coalescence rates: The role of elliptical galaxies

Authors: R. O'Shaughnessy (1), V. Kalogera (2), K. Belczynski (3 and 4) ((1) Center for Gravitational Wave Physics, Penn State University, (2) Northwestern University, (3) Los Alamos National Laboratory, (4) Astronomical Observatory, University of Warsaw)

Date: 25 Aug 2009

Abstract: We estimate binary compact object merger detection rates for LIGO, including the binaries formed in ellipticals long ago. Specifically, we convolve hundreds of model realizations of elliptical- and spiral-galaxy population syntheses with a model for elliptical- and spiral-galaxy star formation history as a function of redshift. Our results favor local merger rate densities of 4\times 10ˆ{-3} {Mpc}ˆ{-3}{Myr}ˆ{-1} for binary black holes (BH), 3\times 10ˆ{-2} {Mpc}ˆ{-3}{Myr}ˆ{-1} for binary neutron stars (NS), and 10ˆ{-2} {Mpc}ˆ{-3}{Myr}ˆ{-1} for BH-NS binaries. Mergers in elliptical galaxies are a significant fraction of our total estimate for BH-BH and BH-NS detection rates; NS-NS detection rates are dominated by the contribution from spiral galaxies. Using only models that reproduce current observations of Galactic NS-NS binaries, we find slightly higher rates for NS-NS and largely similar ranges for BH-NS and BH-BH binaries. Assuming a detection signal-to-noise ratio threshold of 8 for a single detector (as part of a network), corresponding to radii \Cv of the effective volume inside of which a single LIGO detector could observe the inspiral of two 1.4 M_\sun neutron stars of 14 Mpc and 197 Mpc, for initial and advanced LIGO, we find event rates of any merger type of 2.9* 10ˆ{-2} -- 0.46 and 25-400 per year (at 90% confidence level), respectively. We also find that the probability P_{detect} of detecting one or more mergers with this single detector can be approximated by (i) P_{detect}\simeq 0.4+0.5\log (T/0.01{yr}), assuming \Cv=197 {Mpc} and it operates for T years, for T between 2 days and 0.1 {yr}); or by (ii) P_{detect}\simeq 0.5 + 1.5 \log \Cv/32{Mpc}, for one year of operation and for $\Cv$ between 20 and 70 Mpc. [ABRIDGED]

#### Apr 29, 2010

0908.3635 (/preprints)
2010-04-29, 10:01

## [1004.0386] The effect of metallicity on the detection prospects for gravitational waves

Authors: K. Belczynski, M. Dominik, T. Bulik, R. O'Shaughnessy, C.L. Fryer, D.E. Holz

Date: 2 Apr 2010

Abstract: By combining advances in observational astrophysics with recent progress in stellar evolution, we show that there will be a remarkably high number of black holes with compact object (neutron star or black hole) companions in the local Universe. Data from the Sloan Digital Sky Survey (300,000 galaxies) indicates that recent star formation (within the last 1 billion years) is bimodal: half the stars form from gas with high amounts of metals (solar metallicity), and the other half form with small contribution of elements heavier than Helium (20% solar). Theoretical studies of mass loss derive significantly higher stellar-origin black hole masses (30-80 Msun) than previously estimated for sub-solar compositions. We combine these findings to estimate the probability for detection of gravitational waves arising from the inspiral of double compact objects. Our results show that a low metallicity environment significantly boosts the formation of double compact object binaries with at least one black hole. In particular, we find the gravitational-wave detection rate is increased by a factor of 20 if the metallicity is decreased from solar (as in all previous estimates) to a 50-50 mixture of solar and 10% solar metallicity. The current sensitivity of the largest instruments to double neutron-star binaries (VIRGO: 9 Mpc; LIGO: 18) is not high enough to ensure a first detection. However, our results indicate that if a future instrument increased the reach to 50-100Mpc, a detection of gravitational waves would be expected within the first year of observation. It was previously thought that binary neutron stars were the most likely source, but our results indicate that black-hole binaries are 25-times more likely. We are therefore truly on the cusp of seeing gravitational waves, and the first source ever to be seen is likely to be a black hole binary.

#### Apr 16, 2010

1004.0386 (/preprints)
2010-04-16, 11:33

## [1004.1964] Multimessenger astronomy with the Einstein Telescope

Authors: Eric Chassande-Mottin, Martin Hendry, Patrick J. Sutton, Szabolcs Márka

Date: 12 Apr 2010

Abstract: Gravitational waves (GWs) are expected to play a crucial r\ˆole in the development of multimessenger astrophysics. The combination of GW observations with other astrophysical triggers, such as from gamma-ray and X-ray satellites, optical/radio telescopes, and neutrino detectors allows us to decipher science that would otherwise be inaccessible. In this paper, we provide a broad review from the multimessenger perspective of the science reach offered by the third generation interferometric GW detectors and by the Einstein Telescope (ET) in particular. We focus on cosmic transients, and base our estimates on the results obtained by ET's predecessors GEO, LIGO, and Virgo.

#### Apr 14, 2010

1004.1964 (/preprints)
2010-04-14, 11:05

## [1004.1921] LISA extreme-mass-ratio inspiral events as probes of the black hole mass function

Authors: Jonathan R. Gair, Christopher Tang, Marta Volonteri

Date: 12 Apr 2010

Abstract: One of the sources of gravitational waves for the proposed space-based gravitational wave detector, the Laser Interferometer Space Antenna (LISA), are the inspirals of compact objects into supermassive black holes in the centres of galaxies - extreme-mass-ratio inspirals (EMRIs). Using LISA observations, we will be able to measure the parameters of each EMRI system detected to very high precision. However, the statistics of the set of EMRI events observed by LISA will be more important in constraining astrophysical models than extremely precise measurements for individual systems. The black holes to which LISA is most sensitive are in a mass range that is difficult to probe using other techniques, so LISA provides an almost unique window onto these objects. In this paper we explore, using Bayesian techniques, the constraints that LISA EMRI observations can place on the mass function of black holes at low redshift. We describe a general framework for approaching inference of this type — using multiple observations in combination to constrain a parameterised source population. Assuming that the scaling of EMRI rate with black hole mass is known and taking a black hole distribution given by a simple power law, dn/d(ln M) = A (M/M_*)ˆb, we find that LISA could measure the parameters to a precision of D(ln A) ~ 0.08, and D(b) ~ 0.03 for a reference model that predicts ~1000 events. Even with as few as 10 events, LISA should constrain the slope to a precision ~0.3, which is the current level of observational uncertainty in the low-mass slope of the black hole mass function. We also consider a model in which A and b evolve with redshift, but find that EMRI observations alone do not have much power to probe such an evolution.

#### Apr 14, 2010

1004.1921 (/preprints)
2010-04-14, 11:05

## [1004.1600] Misconceptions About General Relativity in Theoretical Black Hole Astrophysics

Authors: David Garofalo, David L. Meier

Date: 9 Apr 2010

Abstract: The fundamental role played by black holes in our study of microquasars, gamma ray bursts, and the outflows from active galactic nuclei requires an appreciation for, and at times some in-depth analysis of, curved spacetime. We highlight misconceptions surrounding the notion of coordinate transformation in general relativity as applied to metrics for rotating black holes that are beginning to increasingly appear in the literature. We emphasize that there is no coordinate transformation that can turn the metric of a rotating spacetime into that for a Schwarzschild spacetime, or more generally, that no coordinate transformation exists that can diagonalize the metric for a rotating spacetime. We caution against the notion of "local" coordinate transformation, which is often incorrectly associated with a global analysis of the spacetime.

#### Apr 14, 2010

1004.1600 (/preprints)
2010-04-14, 11:04

## [1004.0922] Newtonian and relativistic theory of orbits and emission of gravitational waves

Authors: Mariafelicia De Laurentis

Date: 6 Apr 2010

Abstract: This review paper is devoted to the theory of orbits. We start with the discussion of the Newtonian problem of motion then we consider the relativistic problem of motion, in particular the PN approximation and the further gravitomagnetic corrections. Finally by a classification of orbits in accordance with the conditions of motion, we calculate the gravitational waves luminosity for different types of stellar encounters and orbits.

#### Apr 09, 2010

1004.0922 (/preprints)
2010-04-09, 12:09

## [1004.0804] Primordial Gravitational Waves and Cosmic Microwave Background Radiation

Authors: D. Baskaran, L. P. Grishchuk, W. Zhao

Date: 6 Apr 2010

Abstract: This is a summary of presentations delivered at the OC1 parallel session "Primordial Gravitational Waves and the CMB" of the 12th Marcel Grossmann meeting in Paris, July 2009. The reports and discussions demonstrated significant progress that was achieved in theory and observations. It appears that the existing data provide some indications of the presence of gravitational wave contribution to the CMB anisotropies, while ongoing and planned observational efforts are likely to convert these indications into more confident statements about the actual detection.

#### Apr 09, 2010

1004.0804 (/preprints)
2010-04-09, 12:08

## [0905.4089] Upper Limits from Counting Experiments with Multiple Pipelines

Authors: Patrick J. Sutton

Date: 26 May 2009

Abstract: In counting experiments, one can set an upper limit on the rate of a Poisson process based on a count of the number of events observed due to the process. In some experiments, one makes several counts of the number of events, using different instruments, different event detection algorithms, or observations over multiple time intervals. We demonstrate how to generalize the classical frequentist upper limit calculation to the case where multiple counts of events are made over one or more time intervals using several (not necessarily independent) procedures. We show how different choices of the rank ordering of possible outcomes in the space of counts correspond to applying different levels of significance to the various measurements. We propose an ordering that is matched to the sensitivity of the different measurement procedures and show that in typical cases it gives stronger upper limits than other choices. As an example, we show how this method can be applied to searches for gravitational-wave bursts, where multiple burst-detection algorithms analyse the same data set, and demonstrate how a single combined upper limit can be set on the gravitational-wave burst rate.

#### Apr 09, 2010

0905.4089 (/preprints)
2010-04-09, 12:06

## [0908.3824] Search for gravitational-wave bursts associated with gamma-ray bursts using data from LIGO Science Run 5 and Virgo Science Run 1

Authors: LIGO Scientific Collaboration, Virgo Collaboration: B. P. Abbott, R. Abbott, F. Acernese, R. Adhikari, P. Ajith, B. Allen, G. Allen, M. Alshourbagy, R. S. Amin, S. B. Anderson, W. G. Anderson, F. Antonucci, S. Aoudia, M. A. Arain, M. Araya, H. Armandula, P. Armor, K. G. Arun, Y. Aso, S. Aston, P. Astone, P. Aufmuth, C. Aulbert, S. Babak, P. Baker, G. Ballardin, S. Ballmer, C. Barker, D. Barker, F. Barone, B. Barr, P. Barriga, L. Barsotti, M. Barsuglia, M. A. Barton, I. Bartos, R. Bassiri, M. Bastarrika, Th. S. Bauer, B. Behnke, M. Beker, M. Benacquista, J. Betzwieser, P. T. Beyersdorf, S. Bigotta, I. A. Bilenko, G. Billingsley, S. Birindelli, R. Biswas, M. A. Bizouard, E. Black, J. K. Blackburn, L. Blackburn, D. Blair, B. Bland, C. Boccara, T. P. Bodiya, L. Bogue, F. Bondu, L. Bonelli, R. Bork, V. Boschi, S. Bose, L. Bosi, S. Braccini, C. Bradaschia, P. R. Brady, V. B. Braginsky, J. E. Brau, D. O. Bridges, A. Brillet, M. Brinkmann, V. Brisson, C. Van Den Broeck, A. F. Brooks, D. A. Brown, A. Brummit, G. Brunet, R. Budzyński, T. Bulik, A. Bullington, H. J. Bulten, A. Buonanno, O. Burmeister, D. Buskulic, R. L. Byer, L. Cadonati, G. Cagnoli, E. Calloni, J. B. Camp, E. Campagna, J. Cannizzo, K. C. Cannon, B. Canuel, J. Cao, F. Carbognani, L. Cardenas, S. Caride, G. Castaldi, S. Caudill, M. Cavaglià, F. Cavalier, R. Cavalieri, G. Cella, C. Cepeda, E. Cesarini, T. Chalermsongsak, E. Chalkley, P. Charlton, E. Chassande-Mottin, S. Chatterji, S. Chelkowski, Y. Chen, A. Chincarini, N. Christensen, C. T. Y. Chung, D. Clark, J. Clark, J. H. Clayton, F. Cleva, E. Coccia, T. Cokelaer, C. N. Colacino, J. Colas, A. Colla, M. Colombini, R. Conte, D. Cook, T. R. C. Corbitt, C. Corda, N. Cornish, A. Corsi, J.-P. Coulon, D. Coward, D. C. Coyne, J. D. E. Creighton, T. D. Creighton, A. M. Cruise, R. M. Culter, A. Cumming, L. Cunningham, E. Cuoco, S. L. Danilishin, S. D'Antonio, K. Danzmann, A. Dari, V. Dattilo, B. Daudert, M. Davier, G. Davies, E. J. Daw, R. Day, R. De Rosa, D. DeBra, J. Degallaix, M. del Prete, V. Dergachev, S. Desai, R. DeSalvo, S. Dhurandhar, L. Di Fiore, A. Di Lieto, M. Di Paolo Emilio, A. Di Virgilio, M. Díaz, A. Dietz, F. Donovan, K. L. Dooley, E. E. Doomes, M. Drago, R. W. P. Drever, J. Dueck, I. Duke, J.-C. Dumas, J. G. Dwyer, C. Echols, M. Edgar, M. Edwards, A. Effler, P. Ehrens, E. Espinoza, T. Etzel, M. Evans, T. Evans, V. Fafone, S. Fairhurst, Y. Faltas, Y. Fan, D. Fazi, H. Fehrmann, I. Ferrante, F. Fidecaro, L. S. Finn, I. Fiori, R. Flaminio, K. Flasch, S. Foley, C. Forrest, N. Fotopoulos, J.-D. Fournier, J. Franc, A. Franzen, S. Frasca, F. Frasconi, M. Frede, M. Frei, Z. Frei, A. Freise, R. Frey, T. Fricke, P. Fritschel, V. V. Frolov, M. Fyffe, V. Galdi, L. Gammaitoni, J. A. Garofoli, F. Garufi, G. Gemme, E. Genin, A. Gennai, I. Gholami, J. A. Giaime, S. Giampanis, K. D. Giardina, A. Giazotto, K. Goda, E. Goetz, L. M. Goggin, G. González, M. L. Gorodetsky, S. Goeßzetler, S. Goßler, R. Gouaty, M. Granata, V. Granata, A. Grant, S. Gras, C. Gray, M. Gray, R. J. S. Greenhalgh, A. M. Gretarsson, C. Greverie, F. Grimaldi, R. Grosso, H. Grote, S. Grunewald, M. Guenther, G. Guidi, E. K. Gustafson, R. Gustafson, B. Hage, J. M. Hallam, D. Hammer, G. D. Hammond, C. Hanna, J. Hanson, J. Harms, G. M. Harry, I. W. Harry, E. D. Harstad, K. Haughian, K. Hayama, J. Heefner, H. Heitmann, P. Hello, I. S. Heng, A. Heptonstall, M. Hewitson, S. Hild, E. Hirose, D. Hoak, K. A. Hodge, K. Holt, D. J. Hosken, J. Hough, D. Hoyland, D. Huet, B. Hughey, S. H. Huttner, D. R. Ingram, T. Isogai, M. Ito, A. Ivanov, P. Jaranowski, B. Johnson, W. W. Johnson, D. I. Jones, G. Jones, R. Jones, L. Sancho de la Jordana, L. Ju, P. Kalmus, V. Kalogera, S. Kandhasamy, J. Kanner, D. Kasprzyk, E. Katsavounidis, K. Kawabe, S. Kawamura, F. Kawazoe, W. Kells, D. G. Keppel, A. Khalaidovski, F. Y. Khalili, R. Khan, E. Khazanov, P. King, J. S. Kissel, S. Klimenko, K. Kokeyama, V. Kondrashov, R. Kopparapu, S. Koranda, I. Kowalska, D. Kozak, B. Krishnan, A. Królak, R. Kumar, P. Kwee, P. La Penna, P. K. Lam, M. Landry, B. Lantz, A. Lazzarini, H. Lei, M. Lei, N. Leindecker, I. Leonor, N. Leroy, N. Letendre, C. Li, H. Lin, P. E. Lindquist, T. B. Littenberg, N. A. Lockerbie, D. Lodhia, M. Longo, M. Lorenzini, V. Loriette, M. Lormand, G. Losurdo, P. Lu, M. Lubinski, A. Lucianetti, H. Lück, B. Machenschalk, M. MacInnis, J.-M. Mackowski, M. Mageswaran, K. Mailand, E. Majorana, N. Man, I. Mandel, V. Mandic, M. Mantovani, F. Marchesoni, F. Marion, S. Márka, Z. Márka, A. Markosyan, J. Markowitz, E. Maros, J. Marque, F. Martelli, I. W. Martin, R. M. Martin, J. N. Marx, K. Mason, A. Masserot, F. Matichard, L. Matone, R. A. Matzner, N. Mavalvala, R. McCarthy, D. E. McClelland, S. C. McGuire, M. McHugh, G. McIntyre, D. J. A. McKechan, K. McKenzie, M. Mehmet, A. Melatos, A. C. Melissinos, G. Mendell, D. F. Menéndez, F. Menzinger, R. A. Mercer, S. Meshkov, C. Messenger, M. S. Meyer, C. Michel, L. Milano, J. Miller, J. Minelli, Y. Minenkov, Y. Mino, V. P. Mitrofanov, G. Mitselmakher, R. Mittleman, O. Miyakawa, B. Moe, M. Mohan, S. D. Mohanty, S. R. P. Mohapatra, J. Moreau, G. Moreno, N. Morgado, A. Morgia, T. Morioka, K. Mors, S. Mosca, V. Moscatelli, K. Mossavi, B. Mours, C. MowLowry, G. Mueller, D. Muhammad, H. zur Mühlen, S. Mukherjee, H. Mukhopadhyay, A. Mullavey, H. Müller-Ebhardt, J. Munch, P. G. Murray, E. Myers, J. Myers, T. Nash, J. Nelson, I. Neri, G. Newton, A. Nishizawa, F. Nocera, K. Numata, E. Ochsner, J. O'Dell, G. H. Ogin, B. O'Reilly, R. O'Shaughnessy, D. J. Ottaway, R. S. Ottens, H. Overmier, B. J. Owen, G. Pagliaroli, C. Palomba, Y. Pan, C. Pankow, F. Paoletti, M. A. Papa, V. Parameshwaraiah, S. Pardi, A. Pasqualetti, R. Passaquieti, D. Passuello, P. Patel, M. Pedraza, S. Penn, A. Perreca, G. Persichetti, M. Pichot, F. Piergiovanni, V. Pierro, M. Pietka, L. Pinard, I. M. Pinto, M. Pitkin, H. J. Pletsch, M. V. Plissi, R. Poggiani, F. Postiglione, M. Prato, M. Principe, R. Prix, G.A. Prodi, L. Prokhorov, O. Punken, M. Punturo, P. Puppo, V. Quetschke, F. J. Raab, O. Rabaste, D. S. Rabeling, H. Radkins, P. Raffai, Z. Raics, N. Rainer, M. Rakhmanov, P. Rapagnani, V. Raymond, V. Re, C. M. Reed, T. Reed, T. Regimbau, H. Rehbein, S. Reid, D. H. Reitze, F. Ricci, R. Riesen, K. Riles, B. Rivera, P. Roberts, N. A. Robertson, F. Robinet, C. Robinson, E. L. Robinson, A. Rocchi, S. Roddy, L. Rolland, J. Rollins, J. D. Romano, R. Romano, J. H. Romie, D. Rosińska, C. Röver, S. Rowan, A. Rüdiger, P. Ruggi, P. Russell, K. Ryan, S. Sakata, F. Salemi, V. Sandberg, V. Sannibale, L. Santamaría, S. Saraf, P. Sarin, B. Sassolas, B. S. Sathyaprakash, S. Sato, M. Satterthwaite, P. R. Saulson, R. Savage, P. Savov, M. Scanlan, R. Schilling, R. Schnabel, R. Schofield, B. Schulz, B. F. Schutz, P. Schwinberg, J. Scott, S. M. Scott, A. C. Searle, B. Sears, F. Seifert, D. Sellers, A. S. Sengupta, D. Sentenac, A. Sergeev, B. Shapiro, P. Shawhan, D. H. Shoemaker, A. Sibley, X. Siemens, D. Sigg, S. Sinha, A. M. Sintes, B. J. J. Slagmolen, J. Slutsky, M. V. van der Sluys, J. R. Smith, M. R. Smith, N. D. Smith, K. Somiya, B. Sorazu, A. Stein, L. C. Stein, S. Steplewski, A. Stochino, R. Stone, K. A. Strain, S. Strigin, A. Stroeer, R. Sturani, A. L. Stuver, T. Z. Summerscales, K. -X. Sun, M. Sung, Patrick J. Sutton, B. Swinkels, G. P. Szokoly, D. Talukder, L. Tang, D. B. Tanner, S. P. Tarabrin, J. R. Taylor, R. Taylor, R. Terenzi, J. Thacker, K. A. Thorne, K. S. Thorne, A. Thüring, K. V. Tokmakov, A. Toncelli, M. Tonelli, C. Torres, C. Torrie, E. Tournefier, F. Travasso, G. Traylor, M. Trias, J. Trummer, D. Ugolini, J. Ulmen, K. Urbanek, H. Vahlbruch, G. Vajente, M. Vallisneri, J.F.J. van den Brand, S. van der Putten, S. Vass, R. Vaulin, M. Vavoulidis, A. Vecchio, G. Vedovato, A. A. van Veggel, J. Veitch, P. Veitch, C. Veltkamp, D. Verkindt, F. Vetrano, A. Viceré, A. Villar, J.-Y. Vinet, H. Vocca, C. Vorvick, S. P. Vyachanin, S. J. Waldman, L. Wallace, R. L. Ward, M. Was, A. Weidner, M. Weinert, A. J. Weinstein, R. Weiss, L. Wen, S. Wen, K. Wette, J. T. Whelan, S. E. Whitcomb, B. F. Whiting, C. Wilkinson, P. A. Willems, H. R. Williams, L. Williams, B. Willke, I. Wilmut, L. Winkelmann, W. Winkler, C. C. Wipf, A. G. Wiseman, G. Woan, R. Wooley, J. Worden, W. Wu, I. Yakushin, H. Yamamoto, Z. Yan, S. Yoshida, M. Yvert, M. Zanolin, J. Zhang, L. Zhang, C. Zhao, N. Zotov, M. E. Zucker, J. Zweizig

Date: 26 Aug 2009

Abstract: We present the results of a search for gravitational-wave bursts associated with 137 gamma-ray bursts (GRBs) that were detected by satellite-based gamma-ray experiments during the fifth LIGO science run and first Virgo science run. The data used in this analysis were collected from 2005 November 4 to 2007 October 1, and most of the GRB triggers were from the Swift satellite. The search uses a coherent network analysis method that takes into account the different locations and orientations of the interferometers at the three LIGO-Virgo sites. We find no evidence for gravitational-wave burst signals associated with this sample of GRBs. Using simulated short-duration (<1 s) waveforms, we set upper limits on the amplitude of gravitational waves associated with each GRB. We also place lower bounds on the distance to each GRB under the assumption of a fixed energy emission in gravitational waves, with typical limits of D ~ 15 Mpc (E_GWˆiso / 0.01 M_o cˆ2)ˆ½ for emission at frequencies around 150 Hz, where the LIGO-Virgo detector network has best sensitivity. We present astrophysical interpretations and implications of these results, and prospects for corresponding searches during future LIGO-Virgo runs.

#### Apr 09, 2010

0908.3824 (/preprints)
2010-04-09, 12:06

## [1004.1016] Mathematical general relativity: a sampler

Authors: Piotr T. Chruściel, Gregory J. Galloway, Daniel Pollack

Date: 7 Apr 2010

Abstract: We provide an introduction to selected significant advances in the mathematical understanding of Einstein's theory of gravitation which have taken place in recent years.

#### Apr 09, 2010

1004.1016 (/preprints)
2010-04-09, 12:06

## [1004.0961] Observing mergers of non-spinning black-hole binaries

Authors: Sean T. McWilliams, Bernard J. Kelly, John G. Baker

Date: 6 Apr 2010

Abstract: Advances in the field of numerical relativity now make it possible to calculate the final, most powerful merger phase of binary black-hole coalescence for generic binaries. The state of the art has advanced well beyond the equal-mass case into the unequal-mass and spinning regions of parameter space. We present a study of the nonspinning portion of parameter space, primarily using an analytic waveform model tuned to available numerical data, with an emphasis on observational implications. We investigate the impact of varied mass ratio on merger signal-to-noise ratios (SNRs) for several detectors, and compare our results with expectations from the test-mass limit. We note a striking similarity of the waveform phasing of the merger waveform across the available mass ratios. Motivated by this, we calculate the match between our 1:1 (equal mass) and 4:1 mass-ratio waveforms during the merger as a function of location on the source sky, using a new formalism for the match that accounts for higher harmonics. This is an indicator of the amount of degeneracy in mass ratio for mergers of moderate-mass-ratio systems.

#### Apr 09, 2010

1004.0961 (/preprints)
2010-04-09, 12:05

## [1004.0998] Methods for Reducing False Alarms in Searches for Compact Binary Coalescences in LIGO Data

Authors: J. Slutsky, L. Blackburn, D. A. Brown, L. Cadonati, J. Cain, M. Cavaglià, S. Chatterji, N. Christensen, M. Coughlin, S. Desai, G. González, T. Isogai, E. Katsavounidis, B. Rankins, T. Reed, K. Riles, P. Shawhan, J. R. Smith, N. Zotov, J. Zweizig

Date: 7 Apr 2010

Abstract: The LIGO detectors are sensitive to a variety of noise transients of non-astrophysical origin. Instrumental glitches and environmental disturbances increase the false alarm rate in the searches for gravitational waves. Using times already identified when the interferometers produced data of questionable quality, or when the channels that monitor the interferometer indicated non-stationarity, we have developed techniques to safely and effectively veto false triggers from the compact binary coalescences (CBCs) search pipeline.

#### Apr 09, 2010

1004.0998 (/preprints)
2010-04-09, 12:05

## [1004.0324] Search for gravitational waves from known pulsars using $\mathcal{F}$ and $\mathcal{G}$ statistics

Authors: Piotr Jaranowski, Andrzej Królak

Date: 2 Apr 2010

Abstract: In searches for gravitational waves emitted by known isolated pulsars in data collected by a detector one can assume that the frequency of the wave, its spindown parameters, and the position of the source in the sky are known, so the almost monochromatic gravitational-wave signal we are looking for depends on at most four parameters: overall amplitude, initial phase, polarization angle, and inclination angle of the pulsar's rotation axis with respect to the line of sight. We derive two statistics by means of which one can test whether data contains such gravitational-wave signal: the $\mathcal{G}$-statistic for signals which depend on only two unknown parameters (overall amplitude and initial phase), and the $\mathcal{F}$-statistic for signals depending on all four parameters. We study, by means of the Fisher matrix, the theoretical accuracy of the maximum-likelihood estimators of the signal's parameters and we present the results of the Monte Carlo simulations we performed to test the accuracy of these estimators.

#### Apr 05, 2010

1004.0324 (/preprints)
2010-04-05, 11:14

## [1004.0284] Constraining the mass of the graviton using coalescing black-hole binaries

Authors: Drew Keppel, P. Ajith

Date: 2 Apr 2010

Abstract: We study how well the mass of the graviton can be constrained from gravitational-wave (GW) observations of coalescing binary black holes. Whereas the previous investigations employed post-Newtonian (PN) templates describing only the inspiral part of the signal, the recent progress in analytical and numerical relativity has provided analytical waveform templates coherently describing the inspiral-merger-ringdown (IMR) signals. We show that a search for binary black holes employing IMR templates will be able to constrain the mass of the graviton much more accurately (about an order of magnitude) than a search employing PN templates. The best expected bound from GW observatories (lambda_g > 7.8 x 10ˆ13 km from Adv. LIGO, lambda_g > 7.1 x 10ˆ14 km from Einstein Telescope, and lambda_g > 5.9 x 10ˆ17 km from LISA) are several orders-of-magnitude better than the best available model-independent bound (lambda_g > 2.8 x 10ˆ12 km, from Solar system tests). Most importantly, GW observations will provide the first constraints from the highly dynamical, strong-field regime of gravity.

#### Apr 05, 2010

1004.0284 (/preprints)
2010-04-05, 11:14


text:
# Kalman filter example visualised with R

January 6, 2015
By

(This article was first published on mages' blog, and kindly contributed to R-bloggers)

At the last Cologne R user meeting Holger Zien gave a great introduction to dynamic linear models (dlm). One special case of a dlm is the Kalman filter, which I will discuss in this post in more detail. I kind of used it earlier when I measured the temperature in my room.

Over the last week I came across the wonderful quantitative economic modelling site quant-econ.net, designed and written by Thomas J. Sargent and John Stachurski. The site not only provides access to their lecture notes, including the Kalman filer, but also code in Python and Julia.

I will take their example of the Kalman filter and go through it with R. I particularly liked their visuals of the various steps of the Kalman filter.

### Motivation

Suppose I have a little robot that moves autonomously over my desk. How would the robot know where it is? Well, suppose the robot can calculate from its wheel spin how far it went and an additional sensor (e.g. GPS-like) provides also information about its location. Of course both pieces of information will be imprecise. How can the various signals be combined?

This is a classic scenario for the Kalman filter. Its key assumptions are that the errors/noise are Gaussian and that the state space evolution xt from one time step to the next is linear, so is the mapping to the sensor signals yt. For the example I will use below it reads:

xt+1ytx1=Axt+w,wN(0,Q)=Gxt+ν,νN(0,R)N(x0,Σ0)

With A,Q,G,R,Σ matrices of appropriate dimensions. The Kalman filter provides recursive estimators for xt:

Ktx^t+1Σt+1=AΣtG(GΣtG+R)1=Axt^+Kt(yGx^)=AΣtAKtGΣtA+Q

Let’s say at time t0 the robot has the expected position x^=(0.2,0.2). That means, the robot doesn’t know its location on the table with certainty. Further I shall assume that the probability density function of the position follows a Normal distribution with covariance matrix Σ=[0.40.30.30.45]
The prior distribution of the robot’s position can be visualised in R with a contour plot.

library(mnormt)
xhat <- c(0.2, -0.2)
Sigma <- matrix(c(0.4, 0.3,
0.3, 0.45), ncol=2)
x1 <- seq(-2, 4,length=151)
x2 <- seq(-4, 2,length=151)
f <- function(x1,x2, mean=xhat, varcov=Sigma)
dmnorm(cbind(x1,x2), mean,varcov)
z <- outer(x1,x2, f)
mycols <- topo.colors(100,0.5)
image(x1,x2,z, col=mycols, main="Prior density",
xlab=expression('x'[1]), ylab=expression('x'[2]))
points(0.2, -0.2, pch=19)
text(0.1, -0.2, labels = expression(hat(x)), adj = 1)

### Sensor information

The ‘GPS’ sensor provides also data about the location of the robot, again the sensor signal can be regarded as the expected position with some noise. Suppose the reading of the sensor is y=(2.3,1.9). I will assume that the sensor signal has an error following a Normal distribution, with a covariance matrix R=0.5Σ.

Conceptually I think about it like this, given that the robot is at position x, what would be the likelihood that the sensor reading is y|x? The Kalman filter assumes a linear mapping from x to y

y=Gx+ν with νN(0,R)

In my case G will be the identity matrix. Let’s add the sensor information to the plot.

R <- 0.5 * Sigma
z2 <- outer(x1,x2, f, mean=c(2.3, -1.9), varcov=R)
image(x1, x2, z2, col=mycols, main="Sensor density")
points(2.3, -1.9, pch=19)
text(2.2, -1.9, labels = "y", adj = 1)
points(0.2, -0.2, pch=19)
text(0.1, -0.2, labels = expression(hat(x)), adj = 1)

### Filtering

I have two pieces of information about the location of the robot, both with a Normal distribution, my prior is xN(x^,Σ) and for the distribution of my sensor I have y|xN(Gx,R)

What I would like to know is the location of the robot given the sensor reading:

p(x|y)=p(y|x)p(x)p(y)

Now, because x,y are Normal and G is the identity matrix, I know that the posterior distribution of x|y is Normal again, with parameters (see conjugate prior):

x^f=(Σ1+R1)1(Σ1x^+R1y)Σf=(Σ1+R1)1

Using the matrix inversion identity

(A1+B1)1=AA(A+B)1A=A(A+B)1B

I can write the above as:

x^fΣf=(ΣΣ(Σ+R)1Σ)(Σ1x^+R1y)=x^Σ(Σ+R)1x^+ΣR1yΣ(Σ+R)1ΣR1y=x^+Σ(Σ+R)1(yx^)=(1.667,1.333)=ΣΣ(Σ+R)1Σ=[0.1330.1000.100.15]

In the more general case when G is not the identity matrix I have

x^fΣf=x^+ΣG(GΣG+R)1(yx^)=ΣΣG(GΣG+R)1ΣG

The updated posterior probability density function p(x|y)=N(x^f,Σf) is visualised in the chart below.

I can see how the prior and likelihood distributions have influenced the posterior distribution of the robot’s location. The sensor information is regarded more credible than the prior information and hence the posterior is closer to the sensor data.

G = diag(2)
y <- c(2.4, -1.9)
xhatf <- xhat + Sigma %*% t(G) %*% solve(G %*% Sigma %*% t(G) + R) %*% (y - G %*% xhat)
Sigmaf <- Sigma - Sigma %*% t(G) %*% solve(G %*% Sigma %*% t(G) + R) %*% G %*% Sigma
z3 <- outer(x1, x2, f, mean=c(xhatf), varcov=Sigmaf)
image(x1, x2, z3, col=mycols,
xlab=expression('x'[1]), ylab=expression('x'[2]),
main="Filtered density")
points(xhatf[1], xhatf[2], pch=19)
text(xhatf[1]-0.1, xhatf[2],
labels = expression(hat(x)[f]), adj = 1)
points(0.2, -0.2, pch=19, col=lb)
text(0.1, -0.2, labels = expression(hat(x)), adj = 1, col=lb)
points(2.3, -1.9, pch=19, col=lb)
text(2.2, -1.9,labels = "y", adj = 1, col=lb)

### Forecasting

Ok, I have a better understanding of the robot’s position at time t0, but the robot is moving and what I’d like to know is where the it will be after one time step.

Suppose I have a linear model that explains how the state evolves over time, e.g. based on wheel spin. Again, I will assume a Gaussian process. In this toy example I follow the assumptions of Sargent and Stachurski, although this doesn’t make much sense for the robot:

xt+1=Axt+wt+1, where wtN(0,Q)

with

A=(1.20.00.00.2),Q=0.3Σ

As all the processes are linear and Normal the calculations are straightforward:

E[Axf+w]=AExf+Ew=Ax^f=Ax^+AΣG(GΣG+R)1(yGx^)
Var[Axf+w]=AVar[xf]A+Q=AΣfA+Q=AΣAAΣG(GΣG+R)1GΣA+Q

The matrix AΣG(GΣG+R)1 is often written as KΣ and called the Kalman gain. Using this notation, I can summarise the results as follows:

x^newΣnew:=Ax^+KΣ(yGx^):=AΣAKΣGΣA+Q

Adding the prediction to my chart gives me:

A <- matrix(c(1.2, 0,
0, -0.2), ncol=2)
Q <- 0.3 * Sigma
K <- A %*% Sigma %*% t(G) %*% solve(G%*% Sigma %*% t(G) + R)
xhatnew <- A %*% xhat + K %*% (y - G %*% xhat)
Sigmanew <- A %*% Sigma %*% t(A) - K %*% G %*% Sigma %*% t(A) + Q
z4 <- outer(x1,x2, f, mean=c(xhatnew), varcov=Sigmanew)
image(x1, x2, z4, col=mycols,
xlab=expression('x'[1]), ylab=expression('x'[2]),
main="Predictive density")
points(xhatnew[1], xhatnew[2], pch=19)
text(xhatnew[1]-0.1, xhatnew[2],
labels = expression(hat(x)[new]), adj = 1)
points(xhatf[1], xhatf[2], pch=19, col=lb)
text(xhatf[1]-0.1, xhatf[2], col=lb,
labels = expression(hat(x)[f]), adj = 1)
points(0.2, -0.2, pch=19, col=lb)
text(0.1, -0.2, labels = expression(hat(x)), adj = 1, col=lb)
points(2.3, -1.9, pch=19, col=lb)
text(2.2, -1.9,labels = "y", adj = 1, col=lb)

That’s it, having gone through the updating process for one time step gives me the recursive Kalman filter iteration mentioned above.

Another way to visualise the four steps can be achieved with the lattice package:

library(lattice)
grid <- expand.grid(x=x1,y=x2)
grid$Prior <- as.vector(z) grid$Likelihood <- as.vector(z2)
grid$Posterior <- as.vector(z3) grid$Predictive <- as.vector(z4)
contourplot(Prior + Likelihood + Posterior + Predictive ~ x*y,
data=grid, col.regions=mycols, region=TRUE,
as.table=TRUE,
xlab=expression(x[1]),
ylab=expression(x[2]),
main="Kalman Filter",
panel=function(x,y,...){
panel.grid(h=-1, v=-1)
panel.contourplot(x,y,...)
})

You can find the code of this post on Github.

### Session Info

R version 3.1.2 (2014-10-31)
Platform: x86_64-apple-darwin13.4.0 (64-bit)

locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8

attached base packages:
[1] stats graphics grDevices utils datasets methods base

other attached packages:
[1] lattice_0.20-29 mnormt_1.5-1

loaded via a namespace (and not attached):
[1] grid_3.1.2  tools_3.1.2

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text:
Each of the questions given below consists of three statements, numbered I, II and II respectively. Please read the question and the statements carefully and decide which of the statement(s) is/are necessary to answer the question.

1)

What is the speed of the train ?

I. The train crosses $300$ metres long platform in $21$ seconds.

II. The train crosses another stationary train of equal length in $19\frac{1}{2}$ seconds.

III. The train crosses a signal pole in $9\frac{3}{4}$ seconds.

A) I and II only

B) I and either II or III only

C) II and either I or II only

D) III and either I or II only

E) None of these

Option B

### Explanation:

Let the speed of the train be $x$ m/sec.

Time taken to cross a platform = (Length of train + Length of platform) / Speed of the train

Time taken by the train to cross a stationary train = (Sum of the lengths of the train) / Speed of moving train

Time taken to cross a signal pole = Length of train / Speed of train

I gives, $21=\frac{(l+300)}{x}$; II gives, $\frac{39}{2}$ $=\frac{2l}{x}$; III gives, $\frac{39}{4}$ $=\frac{l}{x}$

Thus, (I and II) or (I and III) give $x$.

$\therefore$ Correct answer is (B).


text:
Impact Factor: 3.158

Dec.  2021

Article Contents

# Why Does Extreme Rainfall Occur in Central China during the Summer of 2020 after a Weak El Niño?

Fund Project:

This study was jointly supported by grants from the Strategic Priority Research Program of the Chinese Academy of Sciences (CAS) (Grant No. XDB40000000), the CAS (Grant No. QYZDJ–SSW–DQC021), the National Natural Science Foundation of China (Grant No. 41630531), and the State Key Laboratory of Loess and Quaternary Geology. We thank the supercomputer center of the Pilot Qingdao National Laboratory for Marine Science and Technology and Beijing Super Cloud Computing Center, who offered computing services. We also thank Dr. X.Z. LI, H. LIU, and L. LIU from the Institute of Earth Environment, CAS, who offered suggestions for our numerical experiments

• In summer 2020, extreme rainfall occurred throughout the Yangtze River basin, Huaihe River basin, and southern Yellow River basin, which are defined here as the central China (CC) region. However, only a weak central Pacific (CP) El Niño happened during winter 2019/20, so the correlations between the El Niño–Southern Oscillation (ENSO) indices and ENSO-induced circulation anomalies were insufficient to explain this extreme precipitation event. In this study, reanalysis data and numerical experiments are employed to identify and verify the primary ENSO-related factors that cause this extreme rainfall event. During summer 2020, unusually strong anomalous southwesterlies on the northwest side of an extremely strong Northwest Pacific anticyclone anomaly (NWPAC) contributed excess moisture and convective instability to the CC region, and thus, triggered extreme precipitation in this area. The tropical Indian Ocean (TIO) has warmed in recent decades, and consequently, intensified TIO basinwide warming appears after a weak El Niño, which excites an extremely strong NWPAC via the pathway of the Indo-western Pacific Ocean capacitor (IPOC) effect. Additionally, the ENSO event of 2019/20 should be treated as a fast-decaying CP El Niño rather than a general CP El Niño, so that the circulation and precipitation anomalies in summer 2020 can be better understood. Last, the increasing trend of tropospheric temperature and moisture content in the CC region after 2000 is also conducive to producing heavy precipitation.
摘要: 2020夏季在整个长江流域、淮河流域以及黄河流域南部[本文定义为中国中部（CC）地区]均发生了极端强降水事件。但2019/20年冬赤道太平洋仅发生了一次弱的中太平洋型（CP）El Niño，因而根据厄尔尼诺–南方涛动（ENSO）指数与ENSO激发的环流异常间的相关关系难以解释此次极端降水事件。本研究利用再分析资料和数值模拟试验找出并验证了造成此次极端降水事件与ENSO相关的主要因素。水汽收支和湿静能分析表明：2020年夏季异常强烈的西北太平洋反气旋异常（NWPAC）西北侧的西南气流异常向CC地区输送大量水汽并提供对流不稳定条件，从而引发了该地区的极端降水。近几十年来，热带印度洋（TIO）变暖，使得弱El Niño后也能出现强的TIO海盆增暖，进而通过印度–西太平洋电容器效应激发出极端强烈的NWPAC。此外，2019/20年的ENSO事件应被视为一次快速衰减的CP El Niño，而不是一次一般的CP El Niño，以便更好地理解2020年夏季的环流和降水异常。最后，2000年后CC区域对流层大气温度和水汽含量的上升趋势也有利于更多降水产生。
• Figure 1.  (a) Precipitation anomaly percentages (shaded contours, units: %) in summer (JJA) 2020 based on observed data by meteorological stations. The gray (black) dots indicate the stations where the values exceed the 95th percentile or are below the 5th percentile (are the highest or lowest) during 1979–2020. The green box indicates the CC region; the purple lines indicate the Yellow River, Huaihe River, and Yangtse River respectively from the north to the south (the same in the following). (b) Pacific SST anomalies (units: °C) in winter (D(0)JF(1)) 2019/20, where the hatched areas indicate the anomalies exceed the 95th percentile or are below the 5th percentile during 1979–2020. The red and yellow boxes indicate the regions for defining the Niño-3 index and El Niño Modoki index (EMI), respectively. The SST data are from ERSSTv5. (c) The time series of the area-weighted average JJA precipitation anomaly percentages in the central China (CC) region (bars, units: %), D(0)JF(1) Niño-3 index (red line), and D(0)JF(1) EMI (yellow line) during 1979–2020.

Figure 2.  Monthly precipitation anomaly percentages (shaded contours, units: %) during JJA (a–c) 1980, (d–f) 1998, (g–i) 2016, and (j–l) 2020, respectively, where (a, d, g, j), (b, e, h, k), (c, f, i, l) are for June, July, and August, respectively. The gray (black) dots indicate the stations where the values exceed the 95th percentile or are below the 5th percentile (are the highest or lowest) during 1979–2020. The green boxes indicate the CC region. The data are derived from station observation.

Figure 3.  JJA-mean wind (vectors, units: m s−1) and geopotential height (shaded contours, units: gpm) anomalies in 2020 at (a) 850 hPa and (b) 500 hPa, respectively; (c) JJA-mean SST (shaded contours, units: °C) anomalies in 2020; (d) JJA-mean wind at 200 hPa (vectors, units: m s−1) and TT (shaded contours, units: gpm) anomalies in 2020; (e) the vectors are composites of JJA-mean meridional wind (units: m s−1) and vertical velocity (units: 10−2 Pa s−1) anomalies in 2020 at the meridional-vertical section averaged within 101°–123°E, and the shaded contours are only for vertical velocity anomalies; (f) is similar to (e) but for composites of zonal wind and vertical velocity anomalies at the zonal-vertical section averaged within 27°–36°N. The vectors in black (gray) denote that the anomalies either (neither) exceed the 95th percentile or (nor) are below the 5th percentile during 1979–2020; the hatched areas indicate where the anomalies exceed the 95th percentile or are below the 5th percentile during 1979–2020 for the shaded contours. The black box in (a), (c), and (d) indicate the NWP, TIO, and MC regions, respectively. The green boxes indicate the CC region. The SST data are from ERSSTv5, and the other variables are from ERA5.

Figure 4.  (a) JJA-mean vertical integrated water vapor flux (WVF) (vectors, units: kg m−1 s−1) and moisture content (that is, specific humidity) (shaded contours, units: kg m−2) anomalies in 2020 and climatological JJA-mean summer moisture content (blue contours, units: kg m−2) during 1979–2020; (b) JJA-mean divergence anomalies of integrated WVF (shaded contours, units: 10−5 kg m−2 s−1) in 2020. The meaning of the black and gray vectors and hatched areas are the same as in Fig. 3. The green boxes indicate the CC region. The wind and specific humidity data are from ERA5.

Figure 5.  Scatterplots between the FD EMI and the JJA-mean values of each term in the moisture equation, Eq. (1), for each year during 1979–2020, where (a–g) correspond to each term from left to right in Eq. (1) in sequence. All the values are averaged within the CC region. The thick gray lines are linear fittings of each scatter; the correlation coefficients (r) and corresponding significance levels (p) are marked at the top-left corner of each panel. The vertical and horizontal dashed lines indicate the critical values of FD CP El Niño/La Niña definitions and ±1 standard deviation values of each term, respectively. The scatters of 1979–99 (2000–19) are marked in blue (pink); the years 1980, 1998, 2016, and 2020 are marked with squares, triangles, diamonds, and pentagrams, respectively. The data to calculate the terms in Eq. (1) are from ERA5 except for the precipitation data, which is obtained from station observation.

Figure 6.  Vertical profiles of vertical velocity anomalies (ω', blue line, units: 10−2 Pa s−1) in summer 2020 and climatological JJA-mean MSE ($\overline{{\phi }_{m}}$, pink line, units: J kg−1) during 1979–2020 averaged within the CC region.

Figure 7.  Similar to Fig. 5 but for the FD EMI with each term in the MSE equation, Eq. (2).

Figure 8.  Scatterplots for the (a) D(0)JF(1) Niño-3 index, (b) D(0)JF(1) EMI, and (c) FD EMI with JJA-mean TIO SST anomalies, respectively. Scatterplots between JJA-mean (d) TIO SST and MC TT, (e) MC TT and NWPACI, (f) NWPACI and CC precipitation anomalies, respectively. The dashed lines indicate the critical values defining El Niño/La Niña for the Niño indices and ±1 standard deviation for other factors, respectively.

Figure 9.  (a) Time series of the JJA-mean TIO SST anomalies (blue line, units: °C), MC TT anomalies (pink line, units: gpm), and NWPACI (yellow line, units: 10−6 s−1) during 1979–2020. The thick lines are linear fittings, respectively. The green points and red crosses indicate the years that positive NWPACI matches (1983, 1987, 1988, 1991, 1998, 2003, 2007, 2010, 2020) and mismatches (1980, 1993, 1995, 1996, 2013, 2015, 2016) positive TIO SST and MC TT anomalies, respectively. (b) Time series of JJA-mean vertically-integrated moisture content (blue line, units: kg m−2) and TT (pink line, units: gpm) anomalies in the CC region; the thick lines are quadratic fittings.

Figure 10.  (a) Bars of the boundary condition JJA-mean TIO SST (blue bars, units: °C) and simulated JJA-mean MC TT (pink bars, units: gpm) and NWPACI (yellow bars, units: 10−6 s−1) for each sensitivity experiment minus CLIM; one special case is for ELCP_FD_ALL_TIOW minus CLIM_TIOW (the last group of bars). (b) JJA-mean wind (vectors, units: m s−1) and precipitation (shaded contours, units: mm d−1) at 850 hPa, (c) wind (vectors, units: m s−1) at 200 hPa and TT (shaded contours, units: gpm) for the results of 20_ALL minus CLIM. (d, e) and (f, g) are similar to (b, c) but for ELCP_FD_ALL minus ELCP_ALL and ELCP_FD_ALL_TIOW minus CLIM_TIOW, respectively. The vectors in black (gray) denote wind differences above (below) the 95% confidence level based on Student’s t-test; the hatched areas indicate the differences above the 95% confidence level based on Student’s t-test for the shaded contours. The black boxes in (b, d, f) and (c, e, g) indicate the NWP and MC regions, respectively. The green boxes indicate the CC region.

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## Manuscript History

Manuscript revised: 14 March 2021
Manuscript accepted: 15 April 2021
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

## Why Does Extreme Rainfall Occur in Central China during the Summer of 2020 after a Weak El Niño?

###### Corresponding author: Yu LIU, yuliu2018@ieecas.cn;
• 1. The State Key Laboratory of Loess and Quaternary Geology, Institute of Earth Environment, Chinese Academy of Sciences, Xi’an, 710061, China
• 2. CAS Center for Excellence in Quaternary Science and Global Change, Chinese Academy of Sciences, Xi’an, 710061, China
• 3. Laboratory for Ocean Dynamics and Climate, Pilot Qingdao National Laboratory for Marine Science and Technology, Qingdao, 266000, China

Abstract: In summer 2020, extreme rainfall occurred throughout the Yangtze River basin, Huaihe River basin, and southern Yellow River basin, which are defined here as the central China (CC) region. However, only a weak central Pacific (CP) El Niño happened during winter 2019/20, so the correlations between the El Niño–Southern Oscillation (ENSO) indices and ENSO-induced circulation anomalies were insufficient to explain this extreme precipitation event. In this study, reanalysis data and numerical experiments are employed to identify and verify the primary ENSO-related factors that cause this extreme rainfall event. During summer 2020, unusually strong anomalous southwesterlies on the northwest side of an extremely strong Northwest Pacific anticyclone anomaly (NWPAC) contributed excess moisture and convective instability to the CC region, and thus, triggered extreme precipitation in this area. The tropical Indian Ocean (TIO) has warmed in recent decades, and consequently, intensified TIO basinwide warming appears after a weak El Niño, which excites an extremely strong NWPAC via the pathway of the Indo-western Pacific Ocean capacitor (IPOC) effect. Additionally, the ENSO event of 2019/20 should be treated as a fast-decaying CP El Niño rather than a general CP El Niño, so that the circulation and precipitation anomalies in summer 2020 can be better understood. Last, the increasing trend of tropospheric temperature and moisture content in the CC region after 2000 is also conducive to producing heavy precipitation.

Reference

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## Projective envelopes October 2, 2009

Posted by Akhil Mathew in algebra, category theory.
Tags: , ,

Ok, today we are interested in finding a projective cover of a given ${R}$-module, which can be done under certain circumstances. (Injective hulls, by contrast, always exist.) The setting in which we are primarily interested is the case of ${R=k[G]}$ for ${k}$ a field. If the characteristic ${k}$ doesn’t divide ${|G|}$, then ${R}$ is semisimple and every module is projective, so this is trivial. But in modular representation theory one does not make that hypothesis.  Then taking projective envelopes of simple objects gives the indecomposable projective objects.

Projective Covers

So, fix an abelian category ${\mathcal{A}}$ that has enough projectives (i.e. for ${A \in \mathcal{A}}$ there is a projective object ${P}$ and an epimorphism ${P \rightarrow A}$) where each object has finite length.  Example: the category of finitely generated modules over an artinian ring.

An epimorphism ${f:A \rightarrow B}$ is called essential if for each proper subobject ${A' \subset A}$, ${f(A') \neq B}$. A projective cover of ${A}$ is a projective ${P}$ with an essential map ${P \rightarrow A}$

Theorem 1 Each object in ${\mathcal{A}}$ has a projective cover.

Pick ${A \in \mathcal{A}}$ and write ${A}$ as a quotient of some projective ${L}$, via a map ${L \rightarrow A}$. Consider the collection of subobjects ${R \subset L}$ such that ${im(R) = A}$; since ${L}$ has finite length, we can choose a minimal element ${F}$.

We thus have an epimorphism ${L \rightarrow L/F \rightarrow A}$. I claim that ${L/F}$ is the projective cover. To see this, choose a subobject ${G \subset L}$ such that the map ${G \rightarrow L/F}$ is epi, and ${G}$ is the minimal such subobject. There is a commutative diagram

Since ${L}$ is projective, we can find a lifting ${L \rightarrow G}$ making the following diagram commutative:

So if we take the kernel ${K}$ of ${L \rightarrow G}$, we have a commutative diagram (note that ${L \rightarrow G}$ is epi by minimality of ${G}$):

By commutativity, we have ${F \supset K}$. Since ${G \simeq L/K}$ and ${G \rightarrow L/F}$ is essential, so is ${L/K \rightarrow L/F}$ and consequently ${L/K \rightarrow L/F \rightarrow A}$ essential. This implies ${K=F}$ by the above assumptions, so ${G \simeq L/F}$ in the above commutative diagram. It now follows that since we have the monomorphism ${G \rightarrow L}$ of which ${L \rightarrow L/F \rightarrow G}$ is a retraction, that ${G}$ is a direct summand of ${L}$ and consequently projective. We’re consequently done. There is also uniqueness:

Proposition 2 The projective cover is unique up to isomorphism.

Given two projective covers ${P, P'}$ of ${A}$ with essential maps ${P \rightarrow A}$, ${P' \rightarrow A}$, we can do a lifting both ways to get a commutative diagram.

By the essentiality, the map ${P \rightarrow P'}$ is epi and consequently by projectivity, split. But the extra direct factor in ${P}$ aside from ${P'}$ means that ${P \rightarrow A}$ isn’t essential after all, contradiction.

Uniqueness works in any abelian category by the above proof.  Source: Jean-Pierre Serre, Linear Representations of Finite Groups, Part III.

By the way, xymatrix is definitely a more versatile package for commutative diagrams than amscd. See, e.g., James Milne’s manual.


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# OperatorSchmidtDecomposition

 Other toolboxes required OperatorSchmidtDecomposition Computes the operator Schmidt decomposition of a bipartite operator none FilterNormalFormIsProductOperatorOperatorSchmidtRankSchmidtDecompositionTensorSum Entanglement and separability

OperatorSchmidtDecomposition is a function that computes the operator Schmidt decomposition of a bipartite operator. The user may specify how many terms in the operator Schmidt decomposition they wish to be returned.

## Syntax

• S = OperatorSchmidtDecomposition(X)
• S = OperatorSchmidtDecomposition(X,DIM)
• S = OperatorSchmidtDecomposition(X,DIM,K)
• [S,U,V] = OperatorSchmidtDecomposition(X,DIM,K)

## Argument descriptions

### Input arguments

• X: A bipartite operator to have its operator Schmidt decomposition computed.
• DIM (optional, by default has both subsystems of equal dimension): A specification of the dimensions of the subsystems that X lives on. DIM can be provided in one of three ways:
• If DIM is a scalar, it is assumed that the first subsystem has dimension DIM and the second subsystem has dimension length(X)/DIM.
• If $X \in M_{n_1} \otimes M_{n_2}$ then DIM should be a row vector containing the dimensions (i.e., DIM = [n_1, n_2]).
• If the subsystems aren't square (i.e., $X \in M_{m_1, n_1} \otimes M_{m_2, n_2}$) then DIM should be a matrix with two rows. The first row of DIM should contain the row dimensions of the subsystems (i.e., the mi's) and its second row should contain the column dimensions (i.e., the ni's). In other words, you should set DIM = [m_1, m_2; n_1, n_2].
• K (optional, default 0): A flag that determines how many terms in the operator Schmidt decomposition should be computed. If K = 0 then all terms with non-zero operator Schmidt coefficients are computed. If K = -1 then all terms (including zero terms) are computed. If K > 0 then the K terms with largest operator Schmidt coefficients are computed.

### Output arguments

• S: A vector containing the operator Schmidt coefficients of X.
• U (optional): A cell containing the left Schmidt operators of X.
• V (optional): A cell containing the right Schmidt operators of X.

## Examples

### A random example

The following code computes the operator Schmidt decomposition of a random density matrix in $M_2 \otimes M_4$ and verifies that it is indeed a valid tensor decomposition of that density matrix:

>> rho = RandomDensityMatrix(8);
>> [s,U,V] = OperatorSchmidtDecomposition(rho,[2,4])

s =

0.3986
0.2310
0.1637
0.1118

U =

[2x2 double]    [2x2 double]    [2x2 double]    [2x2 double]

V =

[4x4 double]    [4x4 double]    [4x4 double]    [4x4 double]

>> norm(s(1)*kron(U{1},V{1}) + s(2)*kron(U{2},V{2}) + s(3)*kron(U{3},V{3}) + s(4)*kron(U{4},V{4}) - rho)

ans =

1.8449e-016

As an alternative (and easier) way to verify that we have a valid tensor decomposition of rho, we can use the TensorSum function:

>> norm(TensorSum(s,U,V) - rho)

ans =

1.8449e-016

## Source code

Click on "expand" to the right to view the MATLAB source code for this function.

1. %%  OPERATORSCHMIDTDECOMPOSITION   Computes the operator Schmidt decomposition of a bipartite operator
2. %   This function has one required argument:
3. %     X: a bipartite operator
4. %
5. %   S = OperatorSchmidtDecomposition(X) is a vector containing the non-zero
6. %   operator Schmidt coefficients of the bipartite operator X, where the
7. %   two subsystems are each of size sqrt(length(X)) and X is assumed to be
8. %   square.
9. %
10. %   This function has two optional input arguments:
11. %     DIM (default has both subsystems of equal size)
12. %     K (default 0)
13. %
14. %   [S,U,V] = OperatorSchmidtDecomposition(X,DIM,K) gives the operator
15. %   Schmidt coefficients S of the operator X and the corresponding left and
16. %   right tensor operators in the cells U and V. DIM is a 1x2 vector containing
17. %   the dimensions of the subsystems that X lives on. If X is non-square,
18. %   different row and column dimensions can be specified by putting the row
19. %   dimensions in the first row of DIM and the column dimensions in the
20. %   second row of DIM. K is a flag that determines how many terms in the
21. %   operator Schmidt decomposition should be computed. If K = 0 then all
22. %   terms with non-zero operator Schmidt coefficients are computed. If
23. %   K = -1 then all terms (including zero terms) are computed. If K > 0
24. %   then the K terms with largest operator Schmidt coefficients are
25. %   computed.
26. %
27. %   URL: http://www.qetlab.com/OperatorSchmidtDecomposition
28.
29. %   requires: opt_args.m, PermuteSystems.m, SchmidtDecomposition.m, Swap.m
30. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)
31. %   package: QETLAB
32. %   last updated: November 12, 2014
33.
34. function [s,U,V] = OperatorSchmidtDecomposition(X,varargin)
35.
36. dX = size(X);
37. sdX = round(sqrt(dX));
38.
39. % set optional argument defaults: dim=sqrt(length(X)), k=0
40. [dim,k] = opt_args({ [sdX(1) sdX(1);sdX(2) sdX(2)], 0 },varargin{:});
41.
42. % allow the user to enter a single number for dim
43. if(length(dim) == 1)
44.     dim = [dim,dX(1)/dim];
45.     if abs(dim(2) - round(dim(2))) >= 2*dX(1)*eps
46.         error('OperatorSchmidtDecomposition:InvalidDim','If DIM is a scalar, X must be square and DIM must evenly divide length(X); please provide the DIM array containing the dimensions of the subsystems.');
47.     end
48.     dim(2) = round(dim(2));
49. end
50.
51. % allow the user to enter a vector for dim if X is square
52. if(min(size(dim)) == 1)
53.     dim = dim(:)'; % force dim to be a row vector
54.     dim = [dim;dim];
55. end
56.
57. % The operator Schmidt decomposition is just the Schmidt decomposition of a
58. % related vector obtained by moving matrix elements around.
59. [s,u,v] = SchmidtDecomposition(Swap(reshape(X,prod(prod(dim)),1),[2,3],[dim(2,:),dim(1,:)]),prod(dim),k);
60.
61. % Now reshape things into the proper output format.
62. sr = length(s);
63. U = mat2cell(reshape(u,dim(1,1),dim(2,1)*sr),dim(1,1),dim(2,1)*ones(1,sr));
64. V = mat2cell(reshape(v,dim(1,2),dim(2,2)*sr),dim(1,2),dim(2,2)*ones(1,sr));


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# The Basics

Rox has a curly-bracket syntax, with blocks defined using the curly brackets { and } like C, Java and Rust. Setting signals also uses the C-like assignment operator =.

This chapter describes the basics of programming in Rox.