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# If xy = – 6, what is the value of xy(x + y)? (1) x – y = 5

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If xy = – 6, what is the value of xy(x + y)? (1) x – y = 5 [#permalink]

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27 Apr 2012, 10:39
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If xy = – 6, what is the value of xy(x + y)?

(1) x – y = 5
(2) xy^2= 18

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-xy-6-what ... 67646.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 05 May 2017, 07:24, edited 2 times in total.
Edited the question and added the OA

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Re: If xy = – 6, what is the value of xy(x + y)? (1) x – y = 5 [#permalink]

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27 Apr 2012, 10:48
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If xy = – 6, what is the value of xy(x + y)?

(1) $$x-y=5$$ --> $$x=y+5$$ --> $$xy=(y+5)y=-6$$ --> $$y^2+5y+6=0$$ --> $$y=-3$$ or $$y=-2$$. If $$y=-3$$ then $$x=y+5=2$$ so $$xy(x+y)=-6(x+y)=6$$. But if $$y=-2$$ then $$x=y+5=3$$ so $$xy(x+y)=-6(x+y)=-6$$. Two different answers, hence not sufficient.

(2) $$x*y^2=18$$ --> $$xy*y=-6*y=18$$ --> $$y=-3$$ --> $$x=2$$ --> $$xy(x+y)=-6(x+y)=6$$. Sufficient.

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Re: If xy = – 6, what is the value of xy(x + y)? (1) x – y = 5 [#permalink]

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27 Apr 2012, 10:52
Bunuel wrote:
If xy = – 6, what is the value of xy(x + y)?

(1) $$x-y=5$$ --> $$x=y+5$$ --> $$xy=(y+5)y=-6$$ --> $$y^2+5y+6=0$$ --> $$y=-3$$ or $$y=-2$$. If $$y=-3$$ then $$x=y+5=2$$ so $$xy(x+y)=-6(x+y)=6$$. But if $$y=-2$$ then $$x=y+5=3$$ so $$xy(x+y)=-6(x+y)=-6$$. Two different answers, hence not sufficient.

(2) $$x*y^2=18$$ --> $$xy*y=-6*y=18$$ --> $$y=-3$$ --> $$x=2$$ --> $$xy(x+y)=-6(x+y)=6$$. Sufficient.

B is OA, thank you!!!

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Re: If xy = – 6, what is the value of xy(x + y)? (1) x – y = 5 [#permalink]

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14 Feb 2013, 09:39
roopika2990 wrote:
If xy = -6 , what is the value of xy(x+y ) ?
(1) x – y = 5
(2)xy^2 = 48

xy=-6 ----> x=-6/y

S1) x-y=5 ---------------> x=y+5 -------------> -6/y=y+5 --------> y^2 + 5y + 6 = 0 -------> y = -3 or -2 two values for y so not sufficiesnt

S2) xy^2 = 48 -------> We know x=-6/y -------> therefore (-6/y)(y^2)=48 -------> -6y = 48 -------> y = -8 and x = 6/8 so now we can calculate the value of xy(x+y) hence statement is sufficient

Regards,

Abhijit
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Re: If xy = – 6, what is the value of xy(x + y)? (1) x – y = 5 [#permalink]

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14 Feb 2013, 11:36
$$(x+y)^2 = (x-y)^2+4xy$$

= 5^2+4(-6) = 25-24 = 1

From F.S 1, we know that (x+y)^2 will have two values 1 and -1. Hence, insufficient.

But, from F.S 2, we know that y will have only one value. Thus sufficient. No need to solve for either of the options.

B.
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Re: If xy = – 6, what is the value of xy(x + y)? (1) x – y = 5 [#permalink]

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Re: If xy = – 6, what is the value of xy(x + y)? (1) x – y = 5 [#permalink]

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09 Feb 2016, 19:05
Looking for advice on this question.

I got this question right but could easily have gotten it wrong. For stmt II I figured out that I could plug in -6 for xy and then calculate the equation of -6y=18, which allowed me to solve for y. However for stmt I I eliminated it because although I recognized that from the stmt I could derive that both x and y are not decimals, one is positive and one is negative, and either x or y could equal -6 and the other 1 - I was suspicious that I could count on those values so I eliminated it.

Looking at the OA I see that the proper approach would have been to use factorization.

My question is - how was I supposed to intuitively know to use factorization with stmt I?

I am decent with algebra when I know the typical rules and common properties... Trouble is I've taught myself algebra from scratch using various gmat guides and so my knowledge is spotty. If anyone has any recommendations for any bare bones algebra books I'd appreciate it. I'm rewriting in 1.5 weeks.

Right now I'm trying to improve my algebra through drilling and examining my right and wrong answers.

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Re: If xy = – 6, what is the value of xy(x + y)? (1) x – y = 5 [#permalink]

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05 May 2017, 07:24
myfish wrote:
If xy = – 6, what is the value of xy(x + y)?

(1) x – y = 5
(2) xy^2= 18

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-xy-6-what ... 67646.html

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-xy-6-what ... 67646.html
_________________

Kudos [?]: 133013 [0], given: 12402

Re: If xy = – 6, what is the value of xy(x + y)? (1) x – y = 5   [#permalink] 05 May 2017, 07:24
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# Comparing Futures vs Spot Prices for WTI Crude Oil

In this post, we will be comparing the Futures vs Spot Prices for WTI Crude Oil, i.e., the WTI Futures series with the WTI Spot series. The futures prices generally show high volatility and they are more volatile than the underlying spot price. For this purpose, it would be interesting to compare future and spot prices on the same asset and see some metrics of volatility. First we will plot both series for the same period and then will implement a function to get some volatility metrics for each series.

WTI Crude Oil Spot and Future Prices

WTI Crude Oil Spot and Future Prices

The pattern for the two series is similar, but strictly it will be good to have volatility measures for both series. We will make an R function that compares series from spot and future prices WTI for the same period and then calculates the Interquartile Range (difference between 75th and 25th percentiles) and Standard Deviation for both  series. With this procedure we can compare volatility measures for the two series.

The future series is the one that we have been using along this tutorial which is the CME_CL_Data_ dataset and we will load oil spot data for WTI from a csv file. The function spotVsFutures will implement this task:

spotVsFutures <- function(file1,cl_future){

oilFuture <- cl_future
# Retrieve the first and last date from oilSpot dataframe
first_date <- oilSpot$date[1] last_date <- tail(oilSpot$date,1)

# Subset the CME_CL_Data_ by the first date and last date of oilSpot data in order
# to have both series with same dates
CL <- subset(CME_CL_Data_,Date >= first_date & Date <= last_date)
# Calculate returns of both series
future_returns <- diff(log(CL$Last)) spot_returns <- diff(log(oil_spot$value))
# Make a dataframe with the returns of both series
df <- data.frame(cbind(spot_returns,future_returns))
# Obtain the standard deviation and the Interquartile Range for both returns series.
IQRSpot <- round(IQR(df$spot_returns),6) IQRFuture <- round(IQR(df$future_returns),6)
SDFuture <- round(sd(df$future_returns),6) SDSpot <- round(sd(df$spot_returns),6)
cat('Interquartile Range for spot and future returns of WTI are', c(IQRSpot,IQRFuture))
cat('\n')
cat('Standard Deviation for spot and future returns of WTI are', c(SDSpot,SDFuture))

}

We can now execute the function with our data.

file1 <- "C:/Users/Nicolas/Documents/Derivatives with R/OilSpot/wti-crude-oil-prices-10-year-daily-chart.csv"

format: plain


text:
## torsdag 11 augusti 2016

### New Quantum Mechanics 16: Relation to Hartree and Hartree-Fock

The standard computational form of the quantum mechanics of an atom with N electrons (Hartree or Hartree-Fock) seeks solutions to the standard multi-dimensional Schrödinger equation as linear combinations of wave functions $\psi (x1,x2,...,xN)$ depending on $N$ 3d space coordinates $x1$,...,$xN$ as a product:
• $\psi (x1,x2,...,xN)=\psi_1(x1)\times\psi_2(x2)\times ....\times\psi_N(x_N)$
where the $\psi_j$ are globally defined electronic wave functions depending on a single space coordinate $xj$.

The new model takes the form of a non-standard free boundary Schrödinger equation in a wave function $\psi (x)$ as a sum:
• $\psi (x)=\psi_1(x)+\psi_2(x)+....+\psi_N(x)$,
where the $\psi_j(x)$ are electronic wave functions with local support on a common partition of 3d space with common space coordinate $x$.

The difference between the new model and Hartree/Hartree-Fock is evident and profound.  A big trouble with electronic wave functions having global support is that they overlap and demand an exclusion principle and new physics of exchange energy.  The wave functions of the new model do not overlap and there is no need of any exclusion principle or exchange energy.

PS Standard quantum mechanics comes with new forms of energy such as exchange energy and correlation energy. Here correlation energy is simply the difference between experimental total energy and total energy computed with Hartree-Fock and thus is not a physical form of energy as suggested by the name, simply a computational /modeling error.


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# Theoretical Yield Calculator

Created by Jack Bowater
Reviewed by Bogna Szyk and Steven Wooding
Last updated: Dec 30, 2022

This theoretical yield calculator will answer all the burning questions you have regarding how to calculate the theoretical yield, such as how to find theoretical yield as well as the theoretical yield definition and the theoretical yield formula.

Before carrying out any kind of lab work, you need to work out what is the theoretical yield so you know how much of your product, be it a molecule or lattice, you can expect from a given amount of starting material. This allows you to work out how efficiently you carried out your reaction (the quantity you can find at the actual yield calculator), which is done by calculating the percent yield. You can also use the theoretical yield equation to ensure that you react with equal moles of your reactants so no molecule is wasted.

IMPORTANT NOTE: Yields can only be found using the limiting reagent. If you are uncertain which of your reagents are limiting, plug in your reagents one at a time, and whichever one gives you the lowest number of moles is the limiting reagent. Remember to hit refresh at the bottom of the calculator to reset it.

## Theoretical yield definition

What is the theoretical yield? It is the amount of a product that would be formed if your reaction was 100% efficient. How to achieve 100% efficiency? Well, it would mean that every molecule reacted correctly (i.e., no side products are formed) at every step and that no molecule was lost on the sides of the glassware.

As a normal reaction deals with quintillions of molecules or atoms, it should be obvious that some of these molecules will be lost. Therefore the percent yield will never be 100%, but it is still useful to know as a metric to base your efficiency of the reaction. For more on this, check out our percent yield calculator (link above). Find out how to calculate theoretical yield with the theoretical yield equation below!

## Theoretical yield formula

Using the theoretical yield equation helps you find the theoretical yield from the moles of the limiting reagent, assuming 100% efficiency. So, to stop you from wondering how to find theoretical yield, here is the theoretical yield formula:

$m_{\text{product}} = m_{\text{mol},\text{product}}\cdot n_{\text{lim}}\cdot c,$

where:

• $m_{\text{product}}$ — Mass of product;
• $m_{\text{mol},\text{product}}$ — Molecular weight of the desired product;
• $n_{\text{lim}}$ — Moles of the limiting reagent; and
• $c$ — The stoichiometry of the desired product.

The number of moles of the limiting reagent in the reaction is equal to:

$n_{\text{lim}} = \frac{m_{\text{lim}}}{m_{\text{mol},\text{lim}}\cdot c_{\text{lim}}},$

where:

• $n_{\text{lim}}$ — Number of moles of the limiting reagent;
• $m_{\text{lim}}$ — Mass of limiting reagent;
• $m_{\text{mol},\text{lim}}$ — Molecular weight of the limiting reagent; and
• $c_{\text{lim}}$ — Stoichiometry of the limiting reagent.

Stoichiometry is defined as the number before the chemical formula in a balanced reaction. If no number is present, then the stoichiometry is 1. The stoichiometry is needed to reflect the ratios of molecules that come together to form a product.

The good thing about this calculator is that it can be used any way you like, that is, to find the mass of reactants needed to produce a certain mass of your product. All this information is hidden in the moles, which can be derived from a solution's molarity or concentration (you can learn how to do so with our molarity calculator and concentration calculator).

## How to calculate theoretical yield?

Now, the theoretical yield formula may seem challenging to understand, so we will show you a quick guide on how to calculate the theoretical yield. The measurements you need are the mass of the reagents, their molecular weights, the stoichiometry of the reaction (found from the balanced equation), and the molecular weight of the desired product. Look no further to know how to find the theoretical yield:

1. First, calculate the moles of your limiting reagent. We do this by using the second equation in the theoretical yield formula section (pro tip: make sure that the units of weight are the same for the correct results: you can use the weight converter if you need help with the factors).

2. Select the reactant with the lowest number of moles when stoichiometry is taken into account. This is your limiting reagent. If both have the same amount of moles, you can use either.

3. Use the first equation to find the mass of your desired product in whatever units your reactants were in.

There you go! If you are still struggling, check the examples below for a more practical approach.

## Examples of yield calculations

Time for some examples. Let's say you are doing a nucleophilic addition reaction, forming hydroxyacetonitrile from sodium cyanide and acetone.

Let's ignore the solvents underneath the arrow (they will both be present in excess and therefore will not be limiting reagents), but also the sodium cation of the sodium cyanide, as it is just a spectator ion. If we react $5\ \text{g}$ of acetone with $2\ \text{g}$ of cyanide, what is the theoretical yield of hydroxyacetonitrile?

1. We need to work out the limiting reagent first. As the stoichiometry of both reagents is 1 (i.e., one molecule of acetone reacts with one molecule of cyanide), we can simply use the mass = molecular weight × mole equation to find this:

• Let's rearrange the equation to find moles. This gives:
$\small\text{mole} = \text{mass}/\text{molecular weight}$

• Acetone has a molecular weight of $58\ \text{g}/\text{mole}$, so:
$\small\text{mole} = 5 / 58 = 0.862\ \text{mol}$

• Cyanide has a molecular weight of $26\ \text{g}/\text{mole}$, so:
$\small\text{mole} = 2 / 26 = 0.0769\ \text{mol}$

• So there are fewer moles of cyanide, meaning this is the limiting reagent.

2. Knowing the limiting reagent and its moles means knowing how many moles the product will form. As the stoichiometry of the product is $1$, $0.0769$ moles will form.

3. We can once again use the $\small\text{mass} = \text{molecular weight} \times \text{mole}$ equation to determine the theoretical mass of the product. The molecular weight of hydroxyacetonitrile is $85\ \text{g}/\text{mol}$:

$\quad\ \ \text{mass} = 85 \cdot 0.0769 = 6.54\ \text{g}$

Now we know that if we carry out the experiment, we would expect $6.54\ \text{g}$ of hydroxyacetonitrile. Not too bad, right?

Let's say you are trying to synthesize acetone to use in the above reaction.

You react $8\ \text{g}$ of calcium carbonate ($100\ \text{g}/\text{mol}$) with $9\ \text{g}$ of acetic acid ($60\ \text{g}/\text{mol}$), how much acetone is formed?

1. Once again, we need to work out which is the limiting reagent first. Let's use the $\small\text{mass} = \text{molecular weight}\cdot \text{mole}$ equation again:

• Let's rearrange the equation to find moles. This gives:
$\small\text{mole} = \text{mass} / \text{molecular weight}$

• Let's find the moles of acetic acid:
$\small\text{mole} = 9 / 60 = 0.15\ \text{mol}$

• And the moles of calcium carbonate:
$\small\text{mole} = 8 / 100 = 0.08\ \text{mol}$

• It looks like calcium carbonate is the limiting reagent. But wait! We haven't considered the stoichiometry. Since we need 2 molecules of acetic acid to form one molecule of acetone, we need to divide the moles of acetic acid by $2$:
$\small\text{mole} = 0.15 / 2 = 0.075\ \text{mol}$

• So it turns out that the acetic acid is the limiting reagent!

2. Now that we know the limiting reagent and its moles, we know how many moles of the product will form. As the stoichiometry of the product is $1$, $0.75$ moles will form.

3. Use the $\small\text{mass} = \text{molecular weight}\cdot\text{mole}$ equation to determine the theoretical mass of the product. The molecular weight of acetone is $58\ \text{g}/\text{mol}$:

$\text{mass} = 58 × 0.075 = 4.35\ \text{g}$

So from this reaction, we should get, theoretically speaking, $4.35\ \text{g}$ of acetone. Nice!

Now go on and conquer the world of theoretical yield calculations, you can do it!

## FAQ

### How do I find theoretical yield?

To find the theoretical yield:

1. Balance the reaction.
2. Identify the limiting reagent, which is the reagent with the fewest moles.
3. Divide the fewest number of reagent moles by the stoichiometry of the product.
4. Multiply the result of Step 3 by the molecular weight of the desired product.

### What is theoretical yield?

The theoretical yield is the amount of product that would be formed from a reaction if it was 100% efficient. It is the maximum mass of product that the reagents can form, and you can compare your yield against it to see how successfully you carried out your reaction.

### Is the limiting reactant the theoretical yield?

No, the limiting reactant is not the theoretical yield. To find the theoretical yield, you must find the number of moles present of the limiting reagent. You can then multiply this number by the stoichiometry of the desired product to find the number of moles formed, then use this to derive the theoretical yield.

### What is the theoretical yield of carbon dioxide?

The theoretical yield of CO2 depends on the reaction taking place and the amount of reagents. To find the theoretical yield, you can follow the steps below:

1. Find the moles of the limiting reagent.
2. Multiply the moles of the limiting reagent by the stoichiometry of carbon dioxide in the reaction to give the moles of CO2 produced.
3. Multiply the moles of CO2 produced by 44, the molecular weight of CO2, to get the theoretical yield of your reaction.
Jack Bowater
Limiting reagent
Mass
g
Molecular weight
g / mol
Moles
mole(s)
Stoichiometry
Desired product
Stoichiometry
Moles
mole(s)
Molecular weight
g / mol
Theoretical yield
g
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## Cryptology ePrint Archive: Report 2013/024

New Cube Root Algorithm Based on Third Order Linear Recurrence Relation in Finite Field

Gook Hwa Cho and Namhun Koo and Eunhye Ha and Soonhak Kwon

Abstract: In this paper, we present a new cube root algorithm in finite field $\mathbb{F}_{q}$ with $q$ a power of prime, which extends the Cipolla-Lehmer type algorithms \cite{Cip,Leh}. Our cube root method is inspired by the work of M\"{u}ller \cite{Muller} on quadratic case. For given cubic residue $c \in \mathbb{F}_{q}$ with $q \equiv 1 \pmod{9}$, we show that there is an irreducible polynomial $f(x)=x^{3}-ax^{2}+bx-1$ with root $\alpha \in \mathbb{F}_{q^{3}}$ such that $Tr(\alpha^{\frac{q^{2}+q-2}{9}})$ is a cube root of $c$. Consequently we find an efficient cube root algorithm based on third order linear recurrence sequence arising from $f(x)$. Complexity estimation shows that our algorithm is better than previously proposed Cipolla-Lehmer type algorithms.

Category / Keywords: applications / cube root algorithm, Cipolla-Lehmer algorithm


text:
### TRYING AND ISSUES CONCERNING THE FITOSANITARY TREATMENTS, WITH PERFECTIONATE DISPOSABLE WITH SPRAYING PUBLISHED

Florinel Boja, Nicu Boja, Alin Teuşdea None
In this paper it is studied the possibility of using the watering devices used by man for pest control, with experimental composition in the laboratory for the establishment of some work optimal parameters as the following: the optimization of the liquid norms, the observation of the uniformity level, the number and the diameter of drops, the establishment of the efficient spraying distance for the performance of an optimal handling.
spraying devices; uniformity level; spraying; optimal and handling
Presentation: oral


text:
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Velocity of an electron in Bohr’s stationary orbit is given by: v = $\frac{KZe^{2}}{nhr}$ Total energy of an electron in Bohr’s nth stationary orbit is:. Electron have an energy, and it is the sum of potential and kinetic. Plugging this into the energy equation gives the total energy of the nth level: E = -2p 2 mk 2 Z 2 e 4 /n 2 h 2. 112/.

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nebraska volleyball camp. FUTABA T6K V3S 8-K. 2.4GHz T-Fhss + R3008SB M2/1000098 - EUR 264,70. À VENDRE! Un affranchissement, peu importe le nombre d'articles que vous achetez. En raison 284858430760. FUTABA T6K TRANSMITTER AND RECEIVER $249.99 Add to Cart. Backordered. Add to Wishlist Already In Wishlist. Add to Wishlist. HOLY SMOKES GEN2 SMOKE. . Energy of an electron in the ground state of hydrogen atom . Energy of the electron in the nth orbit of hydrogen atom can be given by the expression$ \displaystyle E_n = -\frac{13.6}{n^2} eV \$ Energy of the electron in the nth orbit of hydrogen like atom with atomic number Z can be given by the expression. Energy of the electron in the nth orbit of hydrogen atom can be given by the expression E n = − 13.6 n 2 e V Energy of the electron in the nth orbit of hydrogen like atom with atomic number Z can be given by the expression. E n = − 13.6 Z 2 n 2 e V Also Read : ∗ Rutherford experiment & Observations ∗ Distance of Closest Approach.

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The photon used to eject the electron had 4.19 x 10^-18 J of energy . Calculate the work function (or threshold energy ) of the unknown metal. You would use the equation : E (photon) - (threshold energy ) = E K (E K is kinetic energy >) Plugging in known values, where E (photon) = 4.19 x 10 -18 J, and E K = 2.35 x 10 -18 J. In physics, the kinetic energy of an object is the energy that it possesses due to its motion. [1] It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes.. Energy of the electron in nth orbit. We know that energy of n th orbit, E n = -K/n 2 (for hydrogen atom), where K is a constant. The energy required to excite the electron from n = 1 to n = 2 is: ΔE (2,1) = E 2 - E 1 = (-K/n 22) - (-K/n 12 ) = K (1/n 12 - 1/n 22 ) = K (1/1 2 - 1/2 2 ) So we have to know the value of 'K'. This can be done by using ionization enthalpy data.

Velocity of an electron in Bohr’s stationary orbit is given by: v = $\frac{KZe^{2}}{nhr}$ Total energy of an electron in Bohr’s nth stationary orbit is:. Electron have an energy, and it is the sum of potential and kinetic. Plugging this into the energy equation gives the total energy of the nth level: E = -2p 2 mk 2 Z 2 e 4 /n 2 h 2. 112/.

maui moana

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13.6 eV is the lowest possible energy of a hydrogen electron E(1). The energy obtained is always a negative number and the ground state n = 1, has the most negative value. The reason being that the energy of an electron in orbit is relative to the energy of an electron that is entirely separated from its nucleus,. The speed in the n = 2 orbit is half the value of the K2 = mož – į m(v,/2)2.


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# How is it decided if $G_1$ and $G_2$ are two “additive” or “multiplicative” cyclic groups?

According to wiki's definition of Bilinear pairing…

Let $G_1$ and $G_2$ be two additive cyclic groups of prime order $q$, and $G_T$ another cyclic group of order $q$ written multiplicatively. A pairing is a map: $e: G_1 \times G_2 \rightarrow G_T$ but, there also have different definition such that $G_1$ and $G_2$ be two multiplicative cyclic groups.

How is the decision made if $G_1$ and $G_2$ are two additive cyclic groups or multiplicative cyclic groups?

Notation is basically a free choice of the author, as they describe functionally the same. And there is no fixed definition for this. However, common practice in mathematical publications is:

• Multiplicative notation for arbitrary groups
• Additive notation for commutative groups

This can be found here: math-SE, wolfram

Wikipedia also states, that additive notation is primarily used for modules and rings. However, that is quite obvious because for those you need both types of notations.

Interestingly, in cryptography we rarely see any non-abelian groups except matrix multiplication, yet multiplicative notation is quite common. The emphasis here might be on the difficulty of the discrete logarithm problem: If a group has the property, that calculating DLOG is difficult, we tend to use multiplicative notation. Additionally, we note generic groups in multiplicative notation mostly.

Additive notation is used mostly just for elliptic curves, or of it is actually a group operation derived from the addition of integers. In that case it might be used to emphasize: In this group the DLOG problem is easy.

• There's probably also a historic note there: DLOG was traditionally (as the name implies) something multiplicative. While we nowadays typically instantiate new protocols using EC, papers are still written multiplicatively as was historically the case. I personally find them more difficult to read too, but that might be on me. – Ruben De Smet Jun 26 '19 at 7:39

For all efficient pairings we are aware of and actually use in cryptography, the groups $G_1$ and $G_2$ are elliptic curve groups (which are traditionally additively written, i.e., additive groups). However, as writing down protocols or schemes using multiplicative notation for groups is more compact and often more convenient to read, many people simply write $G_1$ and $G_2$ multiplicatively.

A group has an operation. We could call this operation by any name, since the name doesn't change what the operation is.

However, there is a convention when naming the operation: we often say that a group is additive (i.e., operation is the "addition") or multiplicative (i.e., its operation is the "multiplication").

The integers number equipped with the usual addition, that is $$(\mathbb Z, +)$$, form a group, which is obviously called an additive group. In the same way, any group that uses operations that we are already used to call "addition" are called additive groups (for instance, the set of $$n \times n$$ matrices equipped with the usual entry-wise addition). And groups derived from additive groups are also said to be additive groups (for instance, $$\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}$$).

Similarly, if a group uses a operation that we are already used to call by "multiplication", then we call it a multiplicative group. For instance, the set of rational numbers without the zero, that is $$\mathbb{Q}^*$$, is a multiplicative group (under the usual multiplication); the set of $$n \times n$$ invertible real matrices is a multiplicative group (under the usual matrix multiplication). And when a group is constructed by borrowing the multiplication of some set, we also call it a multiplicative group (e.g., think of $$\mathbb{Z}_p^*$$, for $$p$$ prime: we operate over its equivalence classes by multiplying the integers that represent such classes...).

Once you have said a group is additive or multiplicative, we will represent a single operation by $$a+b$$ or by $$ab$$, and a sequence of $$n-1$$ operations over a single element $$a$$ as $$na$$ or $$a^n$$, and the inverse element by $$-a$$ or $$a^{-1}$$, respectively.

Finally, there is also a preference to use additive notation when the operation is commutative and multiplicative notation otherwise. Note however that it is just a tendency. In several cases, commutative groups, like $$\mathbb{Z}_p^*$$ or elliptic curves, are represented multiplicatively.

Moreover, note that sometimes it is not clear how to name the group. For instance, for groups whose elements are functions and the operation is the composition of functions. When we compose two functions $$f$$ and $$g$$, obtaining $$f \circ g$$, we are not adding nor multiplying. Even so, a composition of the type $$f \circ f$$ is often denoted as $$f^2$$ instead of $$2f$$.

• Notation of exponents on function symbols has nothing to do with additive or multiplicative notations of groups. – tylo Sep 9 '16 at 11:05
• @tylo Arguably, they do since functions form a monoid under composition. What is missing from this answer, though, is that the additive notation is almost exclusively used for abelian groups. – Aleph Sep 9 '16 at 21:27
• @Aleph this is a good point (: – Hilder Vitor Lima Pereira Sep 11 '16 at 21:45
• @Aleph Well yes. But writing function composition with multiplicative notation is just another convention. Although I've never seen it, writing function composition in additive notation would also work. Unless a ring or field is involved, you can basically always exchange multiplicative notation with additive notation and vice versa. – tylo Sep 12 '16 at 9:17


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# Embedded Young Stellar Clusters

## Summary

Young stellar objects in the Perseus, Orion, and NGC 2264 star forming regions.

## Finding Targets

An object whose APOGEE_TARGET2 value includes one or more of the bitmasks in the following table was targeted for spectroscopy as part of this ancillary target program. See SDSS bitmasks to learn how to use these values to identify objects in this ancillary target program.

APOGEE_TARGET2 bit name Bit Target Description
APOGEE_EMBEDDEDCLUSTER_STAR 13 Young embedded cluster member

## Description

Measuring stellar kinematics in star forming regions enables critical tests of theories that
describe the formation and early evolution of stars, planets, and stellar clusters. The velocities of
young stars can reveal if/how the dynamics of star forming regions influence protostellar mass accretion, the evolution of circumstellar disks, and the structure and ultimate fate of stellar clusters. Similarly, spectroscopically identified binaries provide important tests for pre-main sequence (PMS) evolutionary tracks and help identify how the binary population evolves with age and environment. The INfrared Survey of Young Nebulous Clusters (IN-SYNC) ancillary science project utilizes APOGEE’s unique ability to obtain high-resolution NIR spectroscopy for hundreds of sources in a single field to study the dynamics and stellar populations of young clusters.

## Primary contacts

Kevin Covey
Western Washington University
kevin.covey -at- wwu.edu
Jonathan Tan
University of Florida
jt -at- astro.ufl.edu

## Other contacts

Michiel Cottaar, Michael Meyer, Jonathan Foster, Nicola Da Rio, Kevin Flaherty

## Target Selection Details

### Perseus

Young stellar objects in Perseus were observed in each of two APOGEE fields, placed such that they provide optimal coverage of Perseus’ sub-clusters, IC 348 and NGC 1333, while still spanning the full extent of the Perseus Molecular Cloud. A catalog of potential young stellar objects in Perseus was constructed from:

• mid-infrared excess sources identified by the cores-to-discs (c2d) Spitzer survey team across all of Perseus (Jörgensen et al. 2006; Rebull et al. 2007)
• candidate or confirmed IC 348 members, selected via broad-band photometry and/or X-ray observations, and often confirmed spectroscopically (e.g. Luhman et al. 1998; Luhman 1999; Preibisch & Zinnecker 2001; Preibisch et al. 2003; Luhman et al. 2003, 2005; Muench et al. 2003, 2007, priv. communication)

This catalog of likely Perseus members appears highly complete: Muench et al. 2007 find that their catalog include more than 80% of IC 348 members with H < 16, a finding verified by our independent analysis of 2MASS source counts in the region (see Figure 2, Cottaar et al. 2014). Observations of the Perseus fields were designed to maximize the completeness of the IN-SYNC sample, particularly for sources with 8 < H < 12.5. For these brighter sources, APOGEE's 71.5' fiber collision limit presents the primary obstacle to obtaining a complete sample of cluster members in a single visit, particularly in the densest portions of IC 348 and NGC 1333. Constructing and observing multiple designs for each field mitigated these fiber collisions, however, maximizing the completeness of each cluster sample while also sampling multiple epochs for uncrowded sources, providing some opportunity to identify spectroscopic binaries. The high priority targets in each cluster (i.e., 8 < H < 12.5) were further sorted according to their extinction-corrected H0 magnitudes (with brighter ones at higher priority) to ensure that the survey is as complete as possible for higher mass stars. Once fibers had been allocated as evenly as possible across all visits to all accessible targets with H < 12.5, additional cluster members with H > 12.5 were assigned to all designs for that field, ensuring the maximal S/N possible for these faint targets.

### Orion

The sample of targets for the IN-SYNC Orion dataset is based on the 2MASS catalog. Sources with H<12.5 mag have been restricted within a bent stripe approximately 6x1 degrees in the sky, following the distribution of YSOs in the Orion A region from Megeath et al. (2012). Five different field centers were defined, following the YSO densities along the region, with a total of 15 designs. For each design, targets were selected among all the available sources based on an iterative Monte Carlo procedure that sought to maximize target counts while respecting relative priorities for different targets. The target priorities are calculated from several factors:

1. to encourage coverage in the crowded regions, priorities are increased as a function of the local density of candidate sources;
2. to increase the number of targeted sources faster than the number of epochs for individual sources, a star already targeted in an existing design was de-prioritized in future designs, less so if faint (H>11.5);
3. known cluster members with available stellar parameters from the literature (Da Rio et al. 2010, 2012; Fang et al. 2009, 2013; Hsu et al 2012, 2013) have priorities increased over non-members or sources with unknown memberships; and
4. stars with no available stellar parameters but with IR excess detected by Spitzer (Megeath et al. 2012) have priorities increased over sources with neither stellar parameters nor detected IR excess.

Using this procedure, a total of 2697 individual sources were targeted. Of these, 684 are observed in more than 1 epoch, 1403 have available stellar parameters from the literature, and 1403 have Spitzer IR excess.

### NGC 2264

To test the RV precision that APOGEE could achieve for pre-main sequence stars with significant accretion variability and/or large magnetic spots, 115 candidate NGC 2264 members were selected for APOGEE observations on the basis of:

• optical and/or IR variability identified via CoRoT & Spitzer monitoring of NGC 2264 (Cody et al. 2014)
• IRAC colors consistent with Class I/II YSOs (Sung et al. 2009)
• X-ray and/or Halpha emission consistent with Class III YSOs (Dahm et al. 2007, Sung et al. 2008)

To provide as many observations of each target as possible, and thus provide the richest dataset for examining the RV precision that APOGEE can obtain for pre-main sequence stars, the same set of 115 IN-SYNC targets were observed in each of 6 visits to the APOGEE field containing NGC 2264 (i.e., 203+00).

## REFERENCES

Cody et al. 2014, AJ, 147, 82
Cottaar et al. 2014, ApJ, 794, 125
Dahm et al. 2007, AJ, 134, 999
Da Rio et al. 2010, ApJ, 722, 1092
Da Rio et al. 2012, ApJ, 748, 14
Fang et al. 2013, ApJS, 207, 5
Hsu et al. 2012, ApJ, 752, 59
Hsu et al. 2013, ApJ, 764, 144
Jörgensen et al. 2006, ApJ, 645, 1246
Luhman et al. 1998, ApJ, 508, 347
Luhman 1999, ApJ, 525, 466
Luhman et al. 2003, ApJ, 593, 1093
Luhman et al. 2005, ApJ, 623, 1141
Megeath et al. 2012, AJ, 144, 192
Muench et al. 2003, AJ, 125, 2029
Muench et al. 2007, AJ, 134, 411
Preibisch & Zinnecker 2001, AJ, 122, 866
Preibisch et al. 2003, A&A, 409, 147
Rebull et al. 2007, ApJS, 171, 447
Sung et al. 2008, AJ, 135, 441
Sung et al. 2009, AJ, 138, 1116


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# Statistical Evidence for the Role of Southwestern Indian Ocean Heat Content in the Indian Summer Monsoon Rainfall

## Abstract

This study examines the benefit of using Ocean Mean Temperature (OMT) to aid in the prediction of the sign of Indian Summer Monsoon Rainfall (ISMR) anomalies. This is a statistical examination, rather than a process study. The thermal energy needed for maintaining and intensifying hurricanes and monsoons comes from the upper ocean, not just from the thin layer represented by sea surface temperature (SST) alone. Here, we show that the southwestern Indian OMT down to the depth of the 26 °C isotherm during January–March is a better qualitative predictor of the ISMR than SST. The success rate in predicting above- or below-average ISMR is 80% for OMT compared to 60% for SST. Other January–March mean climate indices (e.g., NINO3.4, Indian Ocean Dipole Mode Index, El Niño Southern Oscillation Modoki Index) have less predictability (52%, 48%, and 56%, respectively) than OMT percentage deviation (PD) (80%). Thus, OMT PD in the southwestern Indian Ocean provides a better qualitative prediction of ISMR by the end of March and indicates whether the ISMR will be above or below the climatological mean value.

## Introduction

Predicting Indian Summer Monsoon Rainfall (ISMR) is a challenging problem. Although attempts are being made to improve the forecasting skill of the monsoon through numerical and statistical modelling, it remains difficult to predict. One reason for this low skill could be the incorrect representation of the ocean thermal energy that may lead to improper assessments of the influence of the oceans in the prediction models for extreme weather events. ISMR has a substantial impact on the Indian agriculture and economic growth. The Indian monsoon is a coupled phenomenon between ocean, land, and atmosphere. In addition to the atmospheric factors, oceans play a critical role in monsoon physics and prediction. The two main processes that affect ISMR are El Niño/Southern Oscillation1 and Indian Ocean Dipole (IOD)2,3. Sea Surface Temperature (SST), representing the thin upper layer of the ocean, has been the main oceanographic parameter historically used to infer the influence of IOD and El Niño/Southern Oscillations, although the thermal energy required for the atmospheric processes comes from the upper ocean heat content (OHC). Many times, the thermal energy available in the upper ocean is not reflected in SST. For example, rapid heating and cooling events can quickly erase the thermal signature of subsurface warm or cold features4. Smith et al.5 observed that El Niño forecast models could be improved by initializing with the observed OHC. In addition, OHC has forecasting skill with more lead time as reported by Ji and Leetma6 and Latif et al.7. They found that forecasting skills improved, even with 6–12 months lead times, by initializing the numerical modelling forecast with the OHC. This predictability increases because interannual anomalies in OHC near the equatorial Pacific lead those in equatorial SST by several months8. This phase relationship is observed both in ocean models and in observations9,10. The atmosphere interacts with the upper ocean rather than with the surface alone, suggesting that ISMR predictions might be improved through the inclusion of upper ocean parameters. In this context, OHC is worthy of consideration for improving ISMR qualitative predictability (meaning the prediction of a weaker or stronger ISMR relative to the climatological mean).

In addition to interannual climate variations, OHC is useful for predicting tropical cyclone intensification11,12,13. Ali et al.14,15 suggested that OHC could be a better parameter than SST in predicting cyclone intensities in the Indian Ocean. Nagamani et al.16 reported that the OHC from the surface down to the depth of 26 °C isotherm (D26) in the Arabian Sea Mini Warm Pool (The region in the southeastern Arabian Sea from 4°–14°N to 68°−78°E) increased during 1998–2010. They speculated that this increase was responsible for the overall decreasing trend in the ISMR during this period. A widely used metric for the upper ocean heat energy is the tropical cyclone heat potential (used here as OHC), the integrated heat energy from the surface to D26, relative to the energy if that layer were uniformly 26 °C, with a minimum value of zero. However, since OHC cannot be integrated into purely atmospheric models, Ali et al.17 suggested converting OHC to Ocean Mean Temperature (OMT) of the surface to D26. They studied the relationships of ISMR with OMT and SST at each 2.5° box in the North Indian Ocean for different months and concluded that OMT in the southwestern Indian Ocean region (50°E–70°E and 10°S–0°N; delimited by the rectangular box in Fig. 1a) has the highest correlation. They also reported that the correlations and confidence levels are greater for OMT compared to SST. They estimated only one correlation coefficient for each 2.5° box for the entire study period of 1993–2013. They did not study the performance of OMT for year-to-year ISMR variations. There appears to be a very non-linear dependency on ocean-related variables, hence a linear correlation for the entire study period (where year-to-year variations are large) is a poor indication of skill in the prediction. Predicting the sign of anomalies is important because it is linked to more/less rainfall, which is critical for agricultural economy of a country, such as India. Herein, we focus on that skill of predicting the ISMR anomaly for different years by considering the same southwestern Indian Ocean region as in Ali et al.17 to explore the improvements in the predictability of ISMR using OMT of this region in place of SST, in a statistical sense. We further compare the role of the January–March OMT in the southwestern Indian Ocean in predicting ISMR to those using other oceanic indices, which are defined by January–March mean SSTs and area-averaged regions where NINO3.4 (El Niño), El Niño Southern Oscillation Modoki Index (EMI), and Indian Ocean Dipole Mode Index (DMI) are conventionally defined (the EMI as a coupled ocean-atmosphere phenomenon in the tropical Pacific Ocean differs from El Niño Southern Oscillation in its spatial and temporal characteristics18,19,20. A recent study showed that teleconnections associated with EMI also influence the rainfall over India and South Africa21). The present study is useful because although there is typically modest variability in the monsoon season rainfall, small variations are very important for the agriculture and the economy of a country.

Since the Indian Ocean seasonal and intraseasonal variations are coupled22, the OHC and OMT, which depend on D26 in this case, have both spatial and temporal variations in this region. The D26 varies from 10 m to 100 m (Fig. 1a) and the OHC varies from 0 to 100 kJ cm−2(Fig. 1b), depending on the mixed layer processes and the net heat gain at the surface. January–March, a transition period between the winter and the southwest monsoon seasons in which the winds and currents in the tropical Indian Ocean change their patterns, has a better predictive value as shown in the results section of this paper. The surface temperature during this season in the Indian Ocean south of 20°S is less than 26 °C, meaning that D26 is undefined. Hence, OHC estimated with respect to the D26 reference depth is also undefined or arbitrarily set to zero. Both D26 and OHC during January–March are moderate in the southwestern Indian Ocean (the rectangular box in Fig. 1(a)), which is the region of interest in the present study. The January–March climatic mean D26 is 59 m and the corresponding OHC is 50 kJ cm−2 in this region. The Pearson’s correlation coefficient between OHC and D26 in this region is ~0.9, illustrating the strong control D26 exerts on OHC.

In this paper, we use this region and compare the OMT performance in relation to the other oceanic indices mentioned earlier, but focus on the sign of the monsoon rainfall anomaly in terms of the sign of the Percentage Deviation (PD), the difference in rainfall with respect to the mean value, expressed as a percentage, without finding quantitative relation with rainfall anomaly. Nanjundiah et al.23 also assessed the ability of seven atmospheric-ocean models to predict the sign of the ISMR anomaly. Other oceanic parameters such as ODMI, El Niño or EMI also provide a qualitative prediction18,24,25.

## Data and Approach

We analyzed 25 years (1993–2017) of monthly rainfall data provided by the India Meteorological Department (www.imd.gov.in/section/nhac/dynamic/data.htm). This rainfall time series used 6955 rain gauges spread over India, including the hilly regions26,27 (Fig. S1). The delayed time data of OHC and D26 spanning 30°S to 30°N and 40°E to 100°E, with a grid spacing of 0.25° × 0.25°, is taken from the National Oceanic and Atmospheric Administration, Atlantic Oceanographic and Meteorological Laboratory (ftp.aoml.noaa.gov). The OHC field, the available heat energy in the upper layer, has dimensional inequality with SST and cannot be used in place of SST in the numerical models. To overcome this limitation, we converted the OHC fields to OMT fields using a few assumptions17. As the OHC from satellite and in situ derived observations has a good correlation having regression slope of close to one and a y-intercept of almost zero28, we used the following equation to compute OMT from OHC and D26 following Ali et al.17.

$$OHC=\rho {C}_{p}{\int }_{0}^{{D}_{26}}(T-26)dz$$
(1)

where, ρ is the density of the seawater, Cp the specific heat capacity at constant pressure, T, the temperature (°C) of each layer of dz thickness, and D26, the depth of 26 °C isotherm. If we assume a mean temperature of the layer (from the surface to D26) as OMT, the above equation can be simplified as:

$$OHC=\rho {C}_{p}(OMT-{\rm{26}}){D}_{{\rm{26}}}$$
(2)

From the above equation, OMT can be written as:

$$OMT=(\frac{OHC}{\rho {C}_{p}{D}_{{\rm{26}}}})+{\rm{26}}$$
(3)

Since OMT and SST have the same units (unlike SST and OHC), the two parameters have the potential to be easily compared and replaced. OMT values thus obtained have data gaps at a few locations at 0.25 degrees spatial resolution. Hence, the monthly averaged OMT values have been obtained with a grid spacing of 0.5° × 0.5° using Cressman’s29 technique.

The best and most reliable approach involves calculating OHC or OMT from in situ temperature profiles. Since the temporal and spatial coverage of in situ observations is limited, we used satellite estimations providing more coverage in space and time. Nagamani et al.28 compared the satellite-derived OHC with the estimations from in situ hydrographic observations and reported a root-mean-square difference of 20.95 kJ cm−2, with a coefficient of determination, R2, of 0.65 and a bias of 11.27 kJ cm−2. Since we are using OMT in this study, we then further compared the OMT estimated from equation (3) with that estimated from in situ hydrographic observations. The statistical comparison (Fig. S2 and Table S1) between the OMT estimated from in situ measurements (obtained from www.nodc.noaa.gov) and that estimated from the satellite-derived OHC has a scatter index, defined as root mean square error normalized to the in situ data mean, of 0.02 and coefficient of determination (R2) of >0.77 for both the North Indian Ocean, as a whole, and for the southwestern Indian Ocean, during January–March. Other statistical parameters for this comparison are given in Table S1.

We used monthly SST data from the Met Office Hadley Centre’s sea ice and SST data set30 with a grid spacing of 1° × 1°, to compute the PD of January–March mean SST in the southwestern Indian Ocean. To show the importance of OMT compared to other oceanic indices, such as NINO3.4, EMI, and the DMI, these indices are obtained from the respective organizations as discussed below. Hadley monthly SST data30 was used to compute these parameters. We do not compute these indices separately; instead, we use them as they are for our comparison. The monthly NINO3.4 index, which is a central tropical Pacific SST based-index representing different phases of El Niño, is from Royal Netherlands Meteorological Institute climate explorer. The monthly EMI and DMI are from Japan Agency for Marine-Earth Science and Technology. DMI, representing the IOD phases and intensities during its life cycle, is defined as the gradient in SST anomaly between the western equatorial Indian Ocean (50°E–70°E and 10°S–10°N) and the southeastern equatorial Indian Ocean2,3 (90°E–110°E and 10°S–0°N). In addition, we computed another new index called ODMI, which is similar to DMI but using OMT in place of SST (the area average OMT anomaly gradient between the western equatorial Indian Ocean (50°E–70°E and 10°S–10°N) and the southeastern equatorial Indian Ocean2,3 (90°E–110°E and 10°S–0°N). OHC obtained from satellite observations is up to the depth of 26 °C isotherm. To check if a shallower or a deeper layer would have been a better choice, we used Hadley temperature and salinity profiles (version: EN4) data31 to compute OHC of a few more depths. Complete details and description of the data/indices used in the present study can be found in Table 1.

For an easy comparison between OMT, SST, and ISMR, the PDs of these three parameters are calculated. For example, the ISMR PD is calculated as the ratio of the deviation of a given year seasonal mean value from its June–September climatological mean value, divided by the climatological mean, and then multiplied by 100 as shown below:

$${\rm{ISMR}}\,{\rm{PD}}=({\rm{R}}-{\rm{R}}\_{\rm{clim}})/{\rm{R}}\_{\rm{clim}}\,\ast \,100$$
(4)

Here, R is the June–September ISMR in a given year and R_clim is June–September climatological mean of ISMR (887.5 mm in this case). Similarly, we computed OMT and SST PDs for different months from January to May and for different 3-month combinations starting from January–March to March–May. Then, the monthly DMI, ODMI, NINO3.4, and EMI indices are used to compute the January–March average values.

## Results

### OMT and SST variabilities

Spatial distributions of January–March climatic mean SST and OMT in the Indian Ocean during 1993–2017 are shown in Fig. 2(a,b), respectively. Since OMT is an averaged temperature of a layer, it is, therefore, less than SST in the entire region. Hence, for a better comparison, we computed the coefficient of variation, defined as the relative magnitude of the standard deviation to the average value, for these two parameters. The coefficient of variability of OMT (Fig. S3 (a)) is less than that of SST (Fig. S3 (b)) almost in the entire region. This is evident because SST is affected by atmospheric parameters, such as winds and clouds/radiation, and is, therefore, more variable than OMT. The average coefficient of variation for monthly SST (0.02) for January–March during 1993–2017 (75 values at each grid point) is double that for OMT (0.01) both for the North Indian Ocean, as a whole, and for the southwestern Indian Ocean. On the contrary, these values remain same for both SST and OMT if January–March seasonal averages (25 values at each grid point) are used in computing the coefficient of variation, as the shorter timescale variations average out. Thus, SST variations that have shorter timescales compared to those of OMT introduce more noise and cause lower correlations with ISMR. Hence, OMT variations, which are more stable and consistent with a lower spread, have a better predictability compared to SST, having more noise.

### Comparison of ISMR with OMT and SST PDs

We obtained the correlation coefficient for ISMR PD versus SST and OMT PDs separately for different months from January to May and for different 3-month combinations starting from January–March to March-May (Table S2). OMT PD correlates better with ISMR PD compared to SST PD for the three seasons and five months. Similarly, we compared the number of mismatching years if ISMR is predicted using SST and OMT PDs for the above months and 3-month seasons (Table S3). The number of mismatching years is least (5 out of 24) if January–March OMT PD is used. From these two analyses, we conclude that January–March OMT PD would be a better indicator of ISMR. As interannual anomalies in OHC near the equatorial Pacific lead those in equatorial SST by several months8, similar phase difference may also be present in the Indian Ocean. We further analyzed SST and OMT PDs in this study region to understand the predictability of this parameter for ISMR. For the purpose of easy visual comparison with ISMR PD, both SST and OMT PDs are multiplied by 10. A comparison of SST PD (red) and OMT PD (yellow) in the southwestern Indian Ocean with ISMR PD (green), shown in Fig. 3, indicates that the OMT PD has better predictability of the sign of the ISMR anomaly than SST PD. The OMT PD failed to predict the positive/negative ISMR (more/less than 887.5 mm) in only five years (1995, 1997, 2002, 2011, and 2016) out of the 25 years, marked as blue rectangles in Fig. 3. Except in 1995, the SST PD criterion also failed whenever OMT PD criterion failed, besides failures in other years. There are altogether 10 failures using SST PD (1994, 1996, 1997, 2002, 2004, 2005, 2009, 2011, 2015, and 2016); these are marked as red rectangles in Fig. 3. Thus, the success rate to predict whether ISMR is above- or below-average for OMT PD is 80% (20/25), whereas the success rate for SST PD is 60% (15/25). However, the magnitude of ISMR is not well predicted by either SST or OMT in a few years. Since the aim of this paper is to forecast the sign of ISMR, OMT will be shown to be better than SST for this goal. The ISMR PD is the greatest (12.50%) in 1994, followed by 2007 and 2013, each with 5.70%. The fact that only nine of 25 years have ISMR that is greater than the mean value (i.e., 887.5 mm) indicates that the overall performance (strength) of the monsoon has been declining. Nagamani et al.16 also reported an overall decreasing trend in ISMR, which is attributed to the increasing OHC of the Arabian Sea Mini Warm Pool region. Furthermore, the lowest negative deviations in ISMR are −21.80% and −19.20% during 2009 and 2002, respectively, whose magnitudes are much larger than the highest positive deviations of 12.50% in 1994 and 5.7% in both 2007 and 2014. This indicates that the rainfall departures from the climatological mean during the years with below-average values of rainfall are greater overall than rainfall departures during years with above-average values. This is critically important, particularly for water resources development in the region, because the deficit rainfall in below-average years is not compensated by excess rainfall in above-average years even without consideration of the flood water going to the oceans. During the moderate below-average years (e.g., 2004, 2015 and 2014), the ISMR PDs are −13.80%, −13.52% and −11.87%, respectively. These values are still more prominent than those during the above-average rainfall years. The maximum difference in ISMR PD between above- and below-average years is 34.30%, implying that the ISMR has high inter-annual variability, although normally the absolute value of this PD is much smaller than this extreme.

There are 16 (9) below- (above-) average rainfall years (green rainfall bars having negative (positive) values in Fig. 3) during the 25-year study period. The January–March OMT PD successfully predicts the ISMR for 13 out of 16 below-average rainfall years (1993, 1999, 2000, 2001, 2004, 2005, 2006, 2008, 2009, 2012, 2014, 2015, and 2017) but fails for three years (1995, 2002 and 2016). Out of these three below-average rainfall years, the ISMRs during 1995 and 2016 are influenced by IOD and El Niño Southern Oscillation; during 2002, the lower atmosphere over western India had an anomalous circulation32. Similarly, the OMT PD is successful in predicting the ISMR for 7 out of the 9 above-average rainfall years (Fig. 3 and Table 2) but fails for 2 years (1997 and 2011). By contrast, the SST PD successfully predicts 10 out of the 16 below-average rainfall years and 5 out of 9 above-average rainfall years (Fig. 3 and Table 2). Therefore, OMT PD is a better parameter than SST PD in predicting ISMR during our analysis period (1993–2017). These results are summarized in Table 2. However, modelling efforts considering other phenomena such as IOD and El Niño Southern Oscillation are required to improve the predictability of the ISMR.

### Statistical analyses

We carried out the following statistical analyses to assess whether OMT PD is a better ISMR predictor than SST PD:

#### Significance test

A 1000-sample permutations test revealed that the success rate for OMT PD is significant at the 1% level and that for SST PD it is significant at the 10% level. The difference between the success rates is significant at the 10% level.

#### The equitable threat score analysis

The equitable threat score accounts for the likelihood of random successes, with values > 0 indicating some skill, and a score of 1 indicating a perfect skill. We used separate equitable threat score to examine the effectiveness of determining the positive and negative signs. For SST, the equitable threat score is 0.020 for positive anomalies, and 0.094 for negative anomalies. Thus, SST has a little skill, but not much. In contrast, OMT has an equitable threat score of 0.295 and 0.405 respectively for positive and negative anomalies respectively. These differences are in part due to the probability of a false detection for an OMT-based estimate being about half that of an SST-based estimate.

#### Confounder analysis

To test whether OMT PD is a confounder (A confounding variable is one that may affect the dependent variable, other than the independent variable already in consideration) or not, first the coefficient of SST PD with ISMR is obtained in a linear regression (ISMR = a0 + b1* SST PD). Then OMT PD is added to the regression (ISMR = a0 + b1* SST PD + b2* OMT PD). The coefficient of SST (b1) has significantly changed by adding OMT PD, indicating that OMT PD is a confounder.

#### Comparison of standardized regression coefficients

A standardized beta coefficient compares the strength of the effect of each individual independent variable to the dependent variable. The higher the absolute value of the beta coefficient, the stronger the effect. Standardized beta coefficients are the coefficients we get if the variables in the regression were all converted to z-scores before running the analysis. The multiple regression analysis of ISMR with OMT and SST PDs has the standardized regression coefficients of 0.92 (significant at 3% level for OMT PD and −0.71 (significant at 8% level) for SST PD. The student T-test T values for these two parameters are 2.43 and −1.86 respectively. These two results show that OMT is better correlated to ISMR than SST.

### Comparison with other indices

In addition, the DMI and ODMI were compared with ISMR PD in Fig. 4. If we consider the annual average, 1996 and 2005 have negative DMI values, whereas the January–March average has negative values for 2005, 2006, and 2015. It is shown that the signs of DMI and ISMR do not match for 13 years, with a success rate of 48% (12/25). The success rate for ODMI is 60% (15/25) with 10 mismatching years. Similarly, we compared January-March mean EMI (red) and NINO3.4 index (yellow) with ISMR PD (green) in Fig. 5 and found that EMI and NINO3.4 have 11 mismatching years (i.e., a success rate of 14/25, or 56%) and 12 mismatching years (i.e., a success rate of 13/25, or 52%), respectively. Thus, of the six indices compared, OMT PD is better at predicting the sign of ISMR, with a success rate of 80%. The success rate of other indices varies between 48% and 60% (Table 3). The number of mismatches with OMT PD (5) is almost 50% less than other parameters (10–13). We obtain the January–March average values of DMI, ODMI, NINO3.4, and EMI monthly indices for the purpose of comparing January–March OMT PD, which has better predictability compared to SST PD (Tables S2 and S3). However, we also compared these monthly indices with SST and OMT PDs (Table S4). Except for ODMI (again an index computed with OMT) in January, OMT PD from January to March outperformed other indices even on monthly basis. This highlights that the OMT in the southwestern Indian Ocean during January–March is a key parameter for improved prediction of ISMR.

## Discussions

Among the six parameters studied, OMT PD averaged over January–March, performed better with two months lead time for a prediction. This statement is strongly supported by the differences in equitable threat scores, a 1000-sample permutation test, confounder analysis and comparison of standard regression coefficients indicating much more skill in the OMT-based forecast. Hence, OMT could be a better parameter as compared to SST, DMI, ODMI or EMI. This may be due to the fact that OMT better represents the upper ocean thermal energy conditions whose variations are mainly responsible for ISMR activity, whereas SST accounts only for the temperature of a very thin layer influenced by meteorological factors such as strong winds, evaporation, or thick clouds. In addition, compared to SST, OMT variations are more stable and consistent with less spread. Rajeevan and McPhaden33 reported that the relationship between warm water volume in the tropical Pacific and El Niño Southern Oscillation has a much stronger lead time than the El Niño Southern Oscillation-SST relationship in predicting ISMR. Lopez et al.34 related the variability of the heat content/transport in the South Atlantic ocean and variability of monsoon at global scales. Zhu et al.35 studied the seasonality between the thermocline depth and SST relationship in the eastern equatorial Pacific. They observed that the thermocline-SST relationship is clearly weaker in spring (with deeper thermocline) than that in other seasons. The cause, according to them, is even though OHC (or the thermocline depth) has the strongest persistence during spring, its variations may not be communicated to the surface (represented by SST). Thus, the link between OHC and SST is depth dependent. This, probably, would not have been the case if OMT were considered.

An important point worth discussing is the use of the mean temperature of the ocean layer down to the depth of 26 °C isotherm (the depth used for OHC computations). Although the use of OMT of this layer makes sense for TC studies as TC-induced mixing and cooling are weaker for higher values of OMT or OHC, such a reduced cooling during the pre-monsoon season cannot be ruled out. It could be related to vertical mixing. When OMT is high, mixing during the winter and spring does not cool SST as much, leading to warmer southwestern Indian Ocean waters, more moisture in the atmosphere, more moisture advected to the continent, and higher monsoon rainfall. Furthermore, there could be a remote oceanic influence associated with changes in OMT that SST does not capture. Westward-propagating Rossby waves could appear in OMT but not in SST, and then affect SST at the western boundary. The ocean thermal energy stored in the upper ocean represented by OMT in the southwestern Indian Ocean in January–March is generally a key parameter for the genesis and development of the following southwest monsoons. The variations of the stored ocean energy in different January–March may control the strength of the following ISMR by providing energy to the overlying atmosphere. This induces anomalous atmospheric circulation and moisture availability in the western Indian Ocean, including the Arabian Sea, a region where wind strength, direction, and moisture availability can significantly control the strength of ISMR. Thus, the use of SST is likely to provide poorer predictability of the ISMR relative to the use of OMT. These issues need to be addressed in a separate study, specifically making use of numerical models.

In this study, we compared only the positive/negative ISMR years with respect to the climatological mean of 887.5 mm. We did not categorize rainfall as the deficit, below normal, normal, above normal, and excess years as the India Meteorological Department does. We also did not attempt to quantitatively predict or compare the ISMR PD with OMT PD. The aim of the present paper is to show that OMT could be a better parameter than SST and other climate indices for monsoon prediction. One possible way to make quantitative predictions could be to replace SST with OMT in numerical weather prediction models, which, however, is beyond the scope of the present study.

To address if a much shallower or deeper layer than the depth of 26 °C would have been a better choice, we compute OMT PDs in southwestern Indian Ocean region for different layers using Hadley interpolated monthly mean temperature profiles with 1° spatial resolution. These temperature profiles are at 42 depth levels from 5 m to 5250 m. Out of the 25 years studied, the range of the annual mean D26 is 35–85 m with 21 (84%) observations varying between 40 m and 70 m. We computed OMT PDs at seven other depths of the surface (5 m) to 15.1 m, 25.2 m, 35.3 m, 45.4 m, 55.7 m, 100 m and 150 m. The number of years mismatching with ISMR PD varies from 11 (correlation: 0.11) with OMT PD of the surface to 100 m to 18 matching or seven years mismatching (correlation: 0.23) with OMT PD of the surface to 15.1 m. Thus, we conclude that using OHC down to D26 to compute OMT PD is a better choice, possibly because this depth varies depending upon the atmospheric factors.

Here, we demonstrated statistical evidence that OMT is a better predictive parameter for the sign of ISMR compared to SST, DMI, ODMI, EMI and Nino3.4. The success rate for OMT in predicting the sign of the ISMR anomaly is 80% and significant at greater than 1%, far better than the other parameters considered.

## Conclusion

Since the OHC, representing the mean temperature of the surface to D26 isotherm layer, has been recognized as an important parameter in cyclone forecasting, we extended the scope of this parameter to predict whether ISMR is above- or below-average value. We converted OHC to OMT so that the two parameters (SST and OMT) have the potential to be easily compared and replaced. Comparison of the correlation coefficient for ISMR PD versus SST and OMT PDs separately for different months from January to March and for 3-month averages from January-March to March-May revealed that OMT PD correlates better with ISMR PD compared to SST PD. Further, the number of mismatching years is least if OMT PD of January-March is considered. OMT PD is showing better performance than SST PD because of the following reasons: (1) The rapid heating or cooling events can quickly eradicate the thermal signatures of subsurface warm or cold features with which the atmosphere interacts rather than with the top thin layer of the ocean represented by SST alone and (2) SST variations that have smaller time scales introduce more noise and cause lower correlations with ISMR compared to OMT variations, which are more stable and consistent with lower spread. In addition, we also compared the OMT PD performance with ODMI another new index, which is similar to DMI but using OMT in place of SST, and other conventionally used climate indices such as NINO3.4, EMI, and DMI.

Equitable threat scores, confounder analysis, standardized regression analysis and a 1000-sample permutation test suggest that among the six parameters/indices used, the January–March mean OMT PD in the southwestern Indian Ocean has the best relationship with ISMR PD. Since OMT PD of January–March has a better predictive value, by the end of March, we expect to predict qualitatively whether the monsoon would be above or below its climatological mean (i.e., 887.5 mm). Similar to other indices or parameters such as DMI, or El Niño/NINO3.4 or EMI, OMT PD helps in a qualitative prediction of ISMR — whether the monsoon is above or below the long period average. Comparison of this parameter estimated for other depths revealed that OMT PD of D26 is a better choice, which could be because of the influence of the atmospheric factors on D26. The physical evidence for OMT in the southwestern Indian Ocean having a higher predictive value, and its ability to provide an improved quantitative prediction, need to be explored using numerical/dynamical models; here we present only the statistical evidence.

Although we have already provided some reasons why OMT failed to predict ISMR in some years, the failure of predicting ISMR using OMT in some years implies the complexity of monsoon system determining the vagaries of ISMR, which are not always controlled by the ocean energy alone represented by OMT in the southwestern Indian Ocean. For example, the influence from other ocean basins associated with ENSO, atmospheric processes related to Madden-Julian Oscillation, Tropical Intraseasonal Oscillation in the time scales of 10–20 days, and the impact from the extra-tropic regions, which are at least partly independent of OMT in Indian Ocean, may also be important contributors to the variability of ISMR in both time and space.

The OMT PD during January–March 2018 is −14.32, which may lead to below average rainfall during June–September 2018 with 80% probability.

## Availability of Data

OMT is estimated using the freely available data. While the other data are available in public domain, OMT data estimated by us can be shared via Dropbox.

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## Acknowledgements

Authors acknowledge the Physical Oceanography Division, Atlantic Oceanographic and Meteorological Laboratory, National Oceanic and Atmospheric Administration for providing the OHC and D26data. The DMI and EMI indices are from Japan Agency for Marine-Earth Science and Technology. The NINO3.4, Hadley datasets are from Royal Netherlands Meteorological Institute and UK Met office, respectively. Authors also wish to acknowledge the ISMR data provided by India Meteorological Department. We thank the two anonymous referees for their critical and constructive comments due which the quality of the paper has significantly improved. MMA acknowledges the support provided at the Indian Institute of Tropical Meteorology at the later stage of this work. The suggestions provided by Francoise Guichard and Soren Merser are gratefully acknowledged. MAB and YZ acknowledge the financial support from NASA/JPL under the Ocean Vector Winds Science Team project (Contact # 1419699). They also acknowledge the support by the Ocean Observing and Monitoring Division, Climate Program Office (Fund Ref number 100007298), National Oceanic and Atmospheric Administration, U.S. Department of Commerce and administered by the Northern Gulf Institute and partial support from NASA PO and NASA NEWS. The authors wish to acknowledge use of the Ferret program for analysis and graphics (Figs 1 and 2) in this paper. Other figures are produced using Libre Office Version: 5.1.6.2.

## Author information

Authors

### Contributions

T.V. Data processing and analysis, made all figures and tables. Prepared first draft. M.M.A. Conceived the concept modified the first draft, improved the manuscript at different stages, and coordinated. All other authors significantly contributed to giving suggestions for more analysis and in improving the manuscript at various stages.

### Corresponding author

Correspondence to M. M. Ali.

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

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Venugopal, T., Ali, M.M., Bourassa, M.A. et al. Statistical Evidence for the Role of Southwestern Indian Ocean Heat Content in the Indian Summer Monsoon Rainfall. Sci Rep 8, 12092 (2018). https://doi.org/10.1038/s41598-018-30552-0

• Accepted:

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#### Citations contained in zbMATH Open

20 Publications have been cited 64 times in 38 Documents Cited by Year
The Björling problem for non-minimal constant mean curvature surfaces. Zbl 1211.53009
Brander, David; Dorfmeister, Josef F.
2010
Holomorphic representation of constant mean curvature surfaces in Minkowski space: consequences of non-compactness in loop group methods. Zbl 1208.53065
Brander, David; Rossman, Wayne; Schmitt, Nicholas
2010
The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps. Zbl 1275.53050
Brander, David; Svensson, Martin
2013
Loop group decompositions in almost split real forms and applications to soliton theory and geometry. Zbl 1166.22013
Brander, David
2008
Generalized DPW method and an application to isometric immersions of space forms. Zbl 1167.37031
Brander, David; Dorfmeister, Josef
2009
Constant Gaussian curvature surfaces in the 3-sphere via loop groups. Zbl 1309.53004
Brander, David; Inoguchi, Jun-Ichi; Kobayashi, Shimpei
2014
Spherical surfaces. Zbl 1346.53007
Brander, David
2016
Singularities of spacelike constant mean curvature surfaces in Lorentz-Minkowski space. Zbl 1218.53064
Brander, David
2011
Results related to generalizations of Hilbert’s non-immersibility theorem for the hyperbolic plane. Zbl 1143.53334
Brander, David
2008
Constant mean curvature surfaces in Euclidean and Minkowski three-spaces. Zbl 1159.53336
Brander, David; Rossman, Wayne; Schmitt, Nicholas
2008
Grassmann geometries in infinite dimensional homogeneous spaces and an application to reflective submanifolds. Zbl 1167.53047
Brander, David
2007
Curved flats, pluriharmonic maps and constant curvature immersions into pseudo-Riemannian space forms. Zbl 1127.37048
Brander, David
2007
Approximation by planar elastic curves. Zbl 1367.41009
Brander, David; Gravesen, Jens; Nørbjerg, Toke Bjerge
2017
Timelike constant mean curvature surfaces with singularities. Zbl 1303.53078
Brander, David; Svensson, Martin
2014
$$k$$-symmetric AKS systems and flat immersions into spheres. Zbl 1153.53039
Brander, David
2008
Deformations of constant mean curvature surfaces preserving symmetries and the Hopf differential. Zbl 1333.53013
Brander, David; Dorfmeister, Josef F.
2015
Pseudospherical surfaces with singularities. Zbl 1371.53003
Brander, David
2017
Remarks on the boundary curve of a constant mean curvature topological disc. Zbl 1376.53011
Brander, David; Lopéz, Rafael
2017
On the Björling problem for Willmore surfaces. Zbl 1387.53016
Brander, David; Wang, Peng
2018
Families of spherical surfaces and harmonic maps. Zbl 1418.53009
Brander, David; Tari, Farid
2019
Families of spherical surfaces and harmonic maps. Zbl 1418.53009
Brander, David; Tari, Farid
2019
On the Björling problem for Willmore surfaces. Zbl 1387.53016
Brander, David; Wang, Peng
2018
Approximation by planar elastic curves. Zbl 1367.41009
Brander, David; Gravesen, Jens; Nørbjerg, Toke Bjerge
2017
Pseudospherical surfaces with singularities. Zbl 1371.53003
Brander, David
2017
Remarks on the boundary curve of a constant mean curvature topological disc. Zbl 1376.53011
Brander, David; Lopéz, Rafael
2017
Spherical surfaces. Zbl 1346.53007
Brander, David
2016
Deformations of constant mean curvature surfaces preserving symmetries and the Hopf differential. Zbl 1333.53013
Brander, David; Dorfmeister, Josef F.
2015
Constant Gaussian curvature surfaces in the 3-sphere via loop groups. Zbl 1309.53004
Brander, David; Inoguchi, Jun-Ichi; Kobayashi, Shimpei
2014
Timelike constant mean curvature surfaces with singularities. Zbl 1303.53078
Brander, David; Svensson, Martin
2014
The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps. Zbl 1275.53050
Brander, David; Svensson, Martin
2013
Singularities of spacelike constant mean curvature surfaces in Lorentz-Minkowski space. Zbl 1218.53064
Brander, David
2011
The Björling problem for non-minimal constant mean curvature surfaces. Zbl 1211.53009
Brander, David; Dorfmeister, Josef F.
2010
Holomorphic representation of constant mean curvature surfaces in Minkowski space: consequences of non-compactness in loop group methods. Zbl 1208.53065
Brander, David; Rossman, Wayne; Schmitt, Nicholas
2010
Generalized DPW method and an application to isometric immersions of space forms. Zbl 1167.37031
Brander, David; Dorfmeister, Josef
2009
Loop group decompositions in almost split real forms and applications to soliton theory and geometry. Zbl 1166.22013
Brander, David
2008
Results related to generalizations of Hilbert’s non-immersibility theorem for the hyperbolic plane. Zbl 1143.53334
Brander, David
2008
Constant mean curvature surfaces in Euclidean and Minkowski three-spaces. Zbl 1159.53336
Brander, David; Rossman, Wayne; Schmitt, Nicholas
2008
$$k$$-symmetric AKS systems and flat immersions into spheres. Zbl 1153.53039
Brander, David
2008
Grassmann geometries in infinite dimensional homogeneous spaces and an application to reflective submanifolds. Zbl 1167.53047
Brander, David
2007
Curved flats, pluriharmonic maps and constant curvature immersions into pseudo-Riemannian space forms. Zbl 1127.37048
Brander, David
2007
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#### Cited by 51 Authors

 9 Brander, David 3 Dorfmeister, Josef F. 3 Honda, Atsufumi 3 Naokawa, Kosuke 3 Saji, Kentaro 2 Gálvez, Jose Antonio 2 Kobayashi, Shimpei 2 Mira, Pablo 2 Pámpano, Álvaro 2 Umehara, Masaaki 2 Yamada, Kotaro 1 Altın, Ayşe 1 Bærentzen, Jakob Andreas 1 Boumuki, Nobutaka 1 Bücking, Ulrike 1 Bueno, Antonio 1 Bulca, Betül 1 Burstall, Francis Everett 1 Cartier, Sébastien 1 Cintra, Adriana Araujo 1 Dajczer, Marcos 1 Fisker, Ann-Sofie 1 Folha, Abigail 1 Gravesen, Jens 1 Gromov, Mikhael Leonidovich 1 Gruber, Anthony 1 Hauswirth, Laurent 1 Ji, Qingchun 1 Jiménez, Asun 1 Khemar, Idrisse 1 Liang, Mingheng 1 Markina, Irina 1 Márquez, M. Carmen 1 Matthes, Daniel 1 Mercuri, Francesco 1 Milán, Francisco 1 Milousheva, Velichka 1 Musso, Emilio 1 Nicolodi, Lorenzo 1 Onnis, Irene Ignazia 1 Onti, Christos-Raent 1 Palmer, Bennett 1 Peñafiel, Carlos 1 Raffaelli, Matteo 1 Svensson, Martin 1 Tari, Farid 1 Teramoto, Keisuke 1 Toda, Magdalena 1 Traizet, Martin 1 Ugail, Hassan 1 Vlachos, Theodoros
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#### Cited in 26 Serials

 4 Annali di Matematica Pura ed Applicata. Serie Quarta 3 Journal of Geometry and Physics 2 Mathematische Annalen 2 Annals of Global Analysis and Geometry 2 Journal of Knot Theory and its Ramifications 2 Mediterranean Journal of Mathematics 1 Computers & Mathematics with Applications 1 Mathematical Proceedings of the Cambridge Philosophical Society 1 Nonlinearity 1 Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1 Advances in Mathematics 1 Archiv der Mathematik 1 Geometriae Dedicata 1 Manuscripta Mathematica 1 Mathematische Zeitschrift 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Results in Mathematics 1 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 1 Computer Aided Geometric Design 1 The Journal of Geometric Analysis 1 Bulletin of the American Mathematical Society. New Series 1 Journal of Nonlinear Science 1 Experimental Mathematics 1 Comptes Rendus. Mathématique. Académie des Sciences, Paris 1 Journal of Singularities 1 Cogent Mathematics
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#### Cited in 11 Fields

 35 Differential geometry (53-XX) 6 Manifolds and cell complexes (57-XX) 5 Dynamical systems and ergodic theory (37-XX) 4 Partial differential equations (35-XX) 4 Global analysis, analysis on manifolds (58-XX) 3 Convex and discrete geometry (52-XX) 2 Topological groups, Lie groups (22-XX) 2 Numerical analysis (65-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 General topology (54-XX)