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Tech blog
Multi-channel noise reduction
August 7, 2020

# Noise Reduction for Distant Voice Recognition in Smart Speakers

Saeed Sereshki

Principal Audio Research Engineer, Advanced Technology

Daniele Giacobello

Distinguished Research Engineer, Advanced Technology Team

## Why Noise Reduction?

Multi-channel noise reduction is an integral part of many modern microphone arrays systems with applications ranging from communication systems to human-machine interfaces. The recent advent of smart loudspeakers like the Amazon Echo, Google Home, and Sonos One, has pushed the robustness required in far-field noise reduction, as the user expects the same level of performance in multiple conditions, regardless of their acoustic environments. Differently from single channel approaches, these devices leverage the spatial information captured in different microphone elements to improve the ability to cancel directional and diffused noise. A well designed noise reduction method will enhance the performance of the smart speaker (wake word detection, word error rate, etc) by cleaning the speech signal.

Microphone array is one of the common components in voice-enabled Sonos smart speakers. The added spatial dimension, inherent to the array spatial aperture, results in more degrees of freedom that allows for noise reduction with low or even no speech distortion.

In our case, we are faced with a few key challenges due to the fact that our approach should work for players with different types of design constraints. On the hardware side, we expect to allow for different microphone-array geometries and configurations as the industrial design pushes for different form factors. Our players are also designed with different user experiences in mind, for example, the Sonos Move was designed as a (mostly) outdoor speaker. Furthermore, at Sonos we work with different voice partners (i.e., Google Assistant and Amazon Alexa) for which performance requirements and use cases might vary.

Taking into account all these challenges, the multi-channel noise reduction technique we employ must satisfy the following objectives:

• A robust and scalable far-field multi-channel noise reduction method

• Easy to deploy on different devices, and different microphone-array geometries

• Fast prototyping, testing, and deployment

• Ability to trade-off noise reduction level to optimize device specific performance metrics

• Long life-time → fault tolerant!

• Potential for generalization to distributed applications (e.g., Sonos home sound system)

These requirements point to a noise reduction technique which does not depend on the microphone array geometry.

## What Are the Options?

Given the inherent spatial nature of the multi-channel noise reduction problem, earlier approaches were greatly influenced by the traditional theory of beamforming that was initially developed for sonar and radar applications using antenna arrays (beamforming). Well-known multi-channel noise reduction techniques include the delay and sum beamformer, minimum variance distortionless response (MVDR) beamformer, also known as Capon beamformer, the linearly constrained minimum variance (LCMV) beamformers, and the generalized sidelobe canceler (GSC) (see this page for more details on these beamformers). In all these methods, the general idea is to steer a beam toward the desired speaker while reducing the background noise coming from other directions.

The multi-channel Wiener filter (MWF) is another well-known multi-channel noise reduction technique, providing a minimum mean-squared error (MMSE) estimate of the speech component in one of the microphone signals. The literature offers several extensions to the traditional MWF. In particular, the MVDR, the GSC, and the parametric multi-channel Wiener filter (PMWF) can be formulated into a common frequency-domain framework where a trade-off between noise reduction and speech distortion can be achieved [1]. Furthermore, in contrast to traditional beamforming research, this formulation makes no assumptions on the geometry of the microphone array system. An interesting extension of this approach is that MWF and PMWF can also be formulated as a distributed noise reduction algorithm where the microphone arrays are part of a wireless acoustic sensor network system [2]. This makes the MWF particularly relevant in practical implementation of speech enhancement algorithms in multi-device applications where the relative geometry of multiple sets of microphones is unknown, like multi-device smart loudspeaker systems (e.g., Sonos home sound system).

The MWF and its extensions, require an accurate estimate of the noise power spectral density (PSD). This, in turn, requires a robust estimation of when speech is present. The speech presence probability (SPP), and its multichannel incarnation (MC-SPP), has been known to offer better performance when incorporated in the noise spectrum estimation [3]. The MC-SPP expression is established for a microphone array with arbitrary geometry under the assumption of Gaussian statistical model for speech and noise. The MC-SPP is then used to extend the single-channel noise PSD estimation to the multi-channel case.

## Problem Statement

An example of the signal model at the microphone array (4 microphones in Sonos Beam) is depicted in the following figure. This example is with factory noise where the signal-to-noise ratio (SNR) at the microphone is 5 dB.

The spatial filtering idea is to design a filter to exploit the spatial localization to recover the clean speech from the observed microphone signals. The problem of adaptive beamforming (spatial filtering) is schematically and mathematically described in the following figure with $N$ microphones where each microphone input of the array includes an additive mixture of reverberant speech component (or desired signal), and noise. The noise can represent multiple competing point sources or a spatially incoherent noise. The microphone input is transformed using Short-Time Fourier Transform (STFT) which is a standard approach in speech processing and enhancement. The analysis in STFT enables us to process the signal in smaller time segments and independently in different frequency bands. The time-domain signal is first broken into overlapping frames (typically 10-20ms). Each frame is multiplied by a windowing function and then the FFT of the windowed signal is provided at the output. In the figure below, $\ell$ denotes the frame index and $k$ denotes the index of frequency bin. The number of frequency bins depends on the FFT size. In the following, the derivation will focus on a single frequency bin $k$ and frame $\ell$. However, all the expressions are used for all the frequency bins.

In the above figure, $g_i(t)$ represents the room impulse response (RIR). The objective in adaptive beamforming is to apply a spatial linear filter to the microphone inputs in the STFT domain to extract the enhanced speech signal. The input, and clean speech in the STFT domain is shown in the figure below for the same example shown above.

## Solution

The block diagram of our practical implementation of PMWF, presented in [5, 6, 7], which incorporates an estimate of the multi-channel speech presence probability (MC-SPP) is depicted in the following figure. In the following, we describe the blocks in this figure in more detail and share the ideas that contribute to the development and implementation of this solution.

We first need to explain how the spatial linear filter is derived. The multi-channel Wiener filter (MCWF) is a linear filter that attempts to enhance the output signal-to-noise ratio (SNR) by reducing noise, utilizing the microphone array's input observation. The objective is to reduce the noise and recover the target speech signal in some optimal way (by solving a constraint optimization) where a linear filter is applied to the observation vector. The constraint optimization problem is formed to maximize the local noise reduction factor while limiting the maximum allowable speech distortion below a frequency-dependent threshold. This trade-off between noise reduction and speech distortion naturally satisfies one of our key objectives. In practice, we utilize this capability to tune our method for different players and partners.

The linear filter is derived by solving this optimization problem. In [1], the closed-form solution for the linear spatial filter with respect to microphone $i$ is derived as

$\mathbf{h}_i(\ell,k) = \frac{\mathbf{\Phi}_{vv}^{-1}(\ell,k)\mathbf{\Phi}_{yy}(\ell,k) - \mathbf{I}_N}{\beta(\ell,k) +\xi(\ell,k)}\mathbf{u}_i.$

The expression for the optimal linear filter depends on the noise and noisy input PSD matrices $\mathbf{\Phi}_{vv}(\ell,k)$ and $\mathbf{\Phi}_{yy}(\ell,k)$. In this expression, $\xi(\ell,k)\triangleq\text{tr}\{\mathbf{\Phi}_{vv}^{-1}(\ell,k)\mathbf{\Phi}_{yy}(\ell,k)\}-N$ denotes the multi-channel a priori signal-to-noise ratio (SNR). $\text{tr}\{\cdot\}$ denotes the trace operator (sum of the elements on the main diagonal of a matrix) and $\beta(\ell,k)$ (positive valued) is a time-frequency dependent factor that allows for tuning the signal distortion and noise reduction at the output of the beamformer. PMWF is uniquely based on second-order statistics, and specifically it only depends on the estimation of the noisy input and the noise PSD matrices. Typically, an averaging time window of 2-3 seconds is used to achieve a reliable estimate of the PSD matrices. This suggests that the noise reduction performance of the PMWF depends on the long-term average of the spectra/spatial characteristics of the speech and the noise sources. In practice, this means that the PMWF works well if the long-term spectral and/or spatial characteristics of the speech and the noise are slowly time-varying.

### How to Estimate the PSD Matrices?

As it is evident from the final expression of PMWF, the key to obtain the linear filter is to estimate the PSD matrices. The accuracy of these estimates play a crucial role in the quality of the filter and its final performance. The estimation of the noisy input PSD is relatively straightforward. We use the typical first order smoothing (exponential smoothing) to approximate the mathematical expectation and estimate the PSD matrix. The only parameter of importance is the smoothing coefficient $\alpha_y$ which needs to be tuned properly.

The following expression is used to update the input PSD matrix $\widehat{\mathbf{\Phi}}_{yy}(\ell,k) = \alpha_y\widehat{\mathbf{\Phi}}_{yy}(\ell-1,k) + (1 - \alpha_y)\mathbf{y}(\ell,k)\mathbf{y}^H(\ell,k)$. For noise PSD matrix estimation, we need to take into account the speech presence uncertainty. The key idea is to estimate the speech presence probability and use it to adapt the exponential smoothing technique. Assuming that the speech and noise signals are modelled as complex multivariate Gaussian random variables, the MC-SPP is estimated as follows [4]

$p(\ell,k) \triangleq \left\{ 1 + \frac{q(\ell,k)}{1-q(\ell,k)} [1 + \xi(\ell,k)]\text{exp}\left[ -\frac{\gamma(\ell,k)}{1 + \xi(\ell,k)}\right]\right\}^{-1} ,$

where, $\gamma(\ell,k)\triangleq\mathbf{y}^H(\ell,k)[\mathbf{\Phi}_{vv}^{-1}(\ell,k)\mathbf{\Phi}_{yy}(\ell,k)\mathbf{\Phi}_{vv}^{-1}(\ell,k)-\mathbf{\Phi}_{vv}^{-1}(\ell,k)]\mathbf{y}(\ell,k)$ and $q(\ell, k)$ is the a priori speech absence probability. In our implementation, we have used $q(\ell, k)=0.5$.

The speech presence probability can be further improved by post-processing. A simple approach is to apply exponential smoothing. In addition, in order to avoid stagnation, we make sure that the probabilities do not suck at 0 or 1. So, eventually we have $\overline{p}(\ell,k) = f(\:p(\ell,k))$. Using the speech presence probability, we follow the same approach we took to estimate $\mathbf{\Phi}_{yy}(\ell,k)$. We use recursive averaging with a smoothing parameter $\alpha_v$ when noise is present by including the new observation vector $\mathbf{y}$ in the averaging, while we do not change the estimate when speech is present. Employing this technique, the noise PSD matrix can be estimated as

$\widehat{\mathbf{\Phi}}_{vv}(\ell,k)=\widetilde{\alpha}_v(\ell,k)\widehat{\mathbf{\Phi}}_{vv}(\ell-1,k)+(1- \widetilde{\alpha}_v(\ell,k))\mathbf{y}(\ell,k)\mathbf{y}^H(\ell,k),$

where the time-frequency dependent smoothing coefficient depends on the speech presence probability as $\widetilde{\alpha}_v(\ell,k) \triangleq \alpha_v + (1-\alpha_v)\overline{p}(\ell,k)$.

The spatial filter also requires the trade-off parameter $\beta(\ell,k)$. In the special case that $\beta(\ell,k)=0$, the proposed filter reduces to the well-known MVDR filter. If $\beta(\ell,k)=1$, the filter is equivalent to MCWF. The advantage of a tunable $\beta(\ell,k)$ is the ability to control the noise reduction and speech distortion based on a high level performance metric of interest. The goal is to provide a flexible solution which can be tuned for specific applications and devices. We propose to use a MC-SPP controlled version of PMWF. The SPP-controlled PMWF outperforms the traditional MWF that uses a fixed trade-off parameter in terms of noise reduction and speech distortion. The idea is to use small trade-off values when MC-SPP is high to reduce speech distortion, and use larger trade-off values when MC-SPP is low to increase the noise reduction. Putting all these techniques together, we can now compute the PMWF spatial filter.

### Extracting the Enhanced Speech Signal

Once the spatial filter is derived, the output is computed as $\mathbf{h}^H_i(\ell,k)\:\mathbf{y}(\ell,k)$. The raw PMWF output is further improved by MMSE estimate of the desired speech signal where the speech presence probability estimation is utilized. The MMSE estimate of the desired speech signal can be obtained according to

$\widehat{X}_i(\ell,k) = \overline{p}(\ell,k)\mathbf{h}^H_i(\ell,k)\mathbf{y}(\ell,k) + (1 - \overline{p}(\ell,k))G_{\text{min}}Y_i(\ell,k),$

where the gain factor $G_{\text{min}}$ determines the maximum amount of noise suppression when speech is not present, $\overline{p}(\ell,k) = 0$. The inclusion of minimum gain in the expression reduces the speech distortion caused due to the estimation error of MC-SPP. The parameter $G_{\text{min}}$ can be tuned to optimize the performance metric of interest (e.g. word error rate in automatic speech recognition systems, wake-word detection rate) in the speech acquisition system.

## Conclusion

We showed how the MC-SPP affects different aspects of a traditional PMWF formulation, i.e., the estimation of the noise PSD matrix, the control of the trade-off between noise reduction and speech distortion, and the estimate of desired speech signal at the output of the PMWF. The proposed method outperforms traditional beamforming techniques in terms of SINR improvement and speech distortion factor (see [5] for the detailed simulation results).

The following two figures show two examples of the method in action with SNR = 0 dB, arguably one of the most challenging scenarios.

#### References

[1] M. Souden, J. Benesty, and S. Affes, “On optimal frequency-domain multichannel linear filtering for noise reduction”. IEEE Transactions on Audio, Speech, and Language Processing, vol. 18, no. 2, pp. 260–276, 2010.

[2] A. Bertrand, and M. Moonen. "Distributed adaptive node-specific signal estimation in fully connected sensor networks—Part I: Sequential node updating". IEEE Transactions on Signal Processing, vol. 58, no. 10, pp. 5277-5291, 2010.

[3] T. Gerkmann and R. C. Hendriks. "Unbiased MMSE-based noise power estimation with low complexity and low tracking delay". IEEE Transactions on Audio, Speech, and Language Processing, vol. 20, no. 4 pp. 1383-1393, 2011.

[4] M. Souden, J. Chen, J. Benesty, and S. Affes, “Gaussian model-based multichannel speech presence probability”. IEEE Transactions on Audio, Speech, and Language Processing , vol. 18, no. 5, pp. 1072–1077, 2010.

[5] S. Bagheri, and D. Giacobello, "Exploiting Multi-Channel Speech Presence Probability in Parametric Multi-Channel Wiener Filter". INTERSPEECH, pp. 101-105, 2019.

[6] S. Bagheri Sereshki, and D. Giacobello. "Linear Filtering for Noise-Suppressed Speech Detection". U.S. Patent Application No. 15/984,073, 2017.

[7] S. Bagheri Sereshki, and D. Giacobello “Linear Filtering for Noise-Suppressed Speech Detection Via Multiple Network Microphone Devices”. US Patent 10,692,518, 2020.

© 2021 by Sonos. Inc.


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# What's the Deal With Yield Inversion?

·10 mins

You may occasionally hear about “yield inversion”, which is usually referring to the difference in yield between the 10 year and the 2 year US government treasury bonds. This number often gets mentioned in the news because when the value goes negative, it’s often interpreted as a sign that we’re either in a recession, or about to be in a recession.

Let’s try to understand what’s going on here. The term yield is used to refer to the current market yield rate for a bond of a specific maturity. The 10 and 2 year yields are some of the most popular ones, but bonds come in all kinds of maturities. US treasury bonds in particular come in all kinds of shapes and sizes, but often cited are the 3 month, 2 year, 10 year, 20 year, and 30 year treasuries. The yields on these bonds are often interpreted as signals about future economic policies.

Wikipedia has a good run-down of all the different kinds of bonds issued by the US federal government. There are also municipal bonds, corporate bonds, and lots of other different bonds.

In general terms, the main components of any bond are as follows:

• The principal, which is the face-value of the band (i.e., $100). • A maturity, which is the date on which the bond expires and the issuer has to repay the principal. • A coupon, which is the interest a bond issuer pays the holder for the bond. • The yield, which is the effective interest rate the bond pays if you buy it today and hold it till maturity, which can fluctuate as bonds are bought and sold with varying maturities at prices at, above, or below their face value. There are some other important details, but those are the most important things. If you buy a bond, you are a bond holder, and the seller (or issuer) is in debted to you. Buying a bond is the opposite of taking out a loan, except when you buy a bond it’s like you’re the bank, and the government (assuming you buy a government bond) is borrowing money from you. Most government bonds are sold at auction, and their value is determined by market participants. The interest rates paid on the bonds are determined at the time of their issuance, which fluctates based on the interest rates set by the central bank. You can also buy bonds directly from the US treasury, but this is generally not how you get the best price on them. For example, you can buy a 2 year note paying 2.75% from TreasuryDirect for$100, but if you instead buy the same bond from public markets you can get a yield of 4.127% as of the time of writing. Because of these spreads, buying and selling bonds on open markets is a huge business.

## How is yield determined #

Okay so let’s talk about that yield word some more: what exactly is the yield? It’s fairly simple: if you buy a 10 year bond from TreasuryDirect, it will cost you $100 and pay a coupon of 2.75% per year. The 2.75% number is not the yield, because if you were to turn around and sell that same bond you will not receive$100 for it on the open market, because the market will discount the face value.

Calculating the yield $$Y$$ of a bond is pretty simple. All you do is take the annual coupon payment, $$C_{payment}$$, and divide that by the current market price, $$P_{market}$$:

$$Y = \dfrac{C_{payment}}{P_{market}}$$

It’s important to note that the yield of a bond is inversely proportional to the market price of that bond. The coupon payment is fixed for the entire duration of the bond, so as the price of a bond goes down, the yield goes up, and vice versa.

In the case of the 10 year treasuries, they pay 2.75% twice a year, which works out to 2 payments of $1.375. Now, as I said, if you buy a 10 year treasury directly from the government today, you won’t be able to sell it for the$100 you paid. We can work out what you’d get for it by taking the current yield (which is 4.127%) and working backwards:

$$P_{market} = \dfrac{\2.75}{4.127\%}$$

Which works out to:

$$P_{market} = \66.63$$

Yikes! So if you pay $100 for a 10 year bond today, and try to turn around and sell it tomorrow, you’ll only get about$66 for it. Of course, people don’t typically do this, unless they plan on holding the bond all the way to maturity. If you did that, then you’d eventually receive the full face value plus all the coupon payments over 10 years, which can be calculated with:

$$V_{present} = V_{face} + C_{payment} \times n_{periods}$$

Where $$V_{present}$$ is the present value, which is also the total value at maturity, $$C_{payment}$$ is the amount of each coupon payment, and $$n_{periods}$$ is the total number of payment periods. Plugging in the values for the 10 year treasury we get:

$$V_{present} = \100 + \1.375 \times 20$$

Which works out to:

$$V_{present} = \127.5$$

Now, if you dig into the details on the TreasuryDirect website, you can find the details for the auctions which happen periodically. The most recent auction for the 10 year note can be found here, which shows the following results:

Interestingly, we can see that the market paid the government $90.45 for bonds yielding 2.75% on$100 maturing in 10 years. That yield works out to about 3.85%, as noted in the auction results.

In other words, you could have bought $100 worth of debt from the US government for$90.45, and for the next 10 years you’d receive $1.375 every six months, and at the end the government would return$100 to you (in addition to the coupon payments), assuming the government doesn’t default on its debts. Neat!

Okay, so, we’re starting to get a grasp on what yield means, but we’ve only discussed bonds that are purchased from their date of issue. How do you figure out the yield of, say, a 10 year bond that someone bought 5 years ago? Well, that math is a bit more tricky. That value is known as yield to maturity, which is generally what people are referring to when they talk about “bond yields”.

Calculating the yield to maturity is a bit more tricky. The reason it’s tricky is that on open markets, the bonds being bought and sold have different maturities, so you have to calculate the yields on a case by case basis, which is pretty easy for computers but a bit tricky for humans. If you want to determine the yield, you would look at the current market price for a bond, and work backwards to determine the yield. That looks like this (assuming we know the current market price, $$V_{present}$$)

$$Y_{maturity} = \dfrac{C_{rate} + \dfrac{V_{face} - V_{present}}{n_{periods}}}{\dfrac{V_{face} + V_{present}}{2}}$$

Where $$C_{rate}$$ is the coupon rate (i.e., the face value yield on the bond). $$n_{periods}$$ can be anywhere between 20 and 1 (in the case of 10 year notes.

The equation above might look complicated, but it’s actually quite simple: you take the coupon rate, add the difference between the face and present value, and divide that by the mean of the face and present value. It’s the same as the simple form ($$Y = \dfrac{C}{P}$$) except that we account for the difference between the market price and the face value.

The math can get a bit more complicated if you want to, say, become a market maker, but in general it’s not all that complicated. Prices on the open market are set by buyers and sellers, and they can fluctuate quite a bit day to day, especially when political or economic events happen.

## The yield curve: why does it invert? #

The yield curve is what you get when you plot the difference in yield of one bond versus another. As I mentioned, the 10 year minus the 2 year treasury bond is one of the most often cited yield curse, and it’s popular because some people (wrongly) believe this to be a predictor of recessions. In reality, it’s a trailing indicator at best.

When people talk about “yield inversion”, they simply mean that the 2 year has become more valuable in terms of yield than the 10 year. How does this happen? Well, it’s simple

## Is an inverted yield curve a recession predictor? #

In short, no. By the time the yield curve has inverted, the market is past the point of predicting a future recession, and it’s now signalling that the central bank will raise rates in the future.

If the 2 year has a higher yield than the 10 year, that means the market believes that interest rates will go down within the next 2 years or so. In other words, a 2 year bond right now is more valuable than a 10 year bond, because interest rates will be lower than the current yields in the future.

The current federal funds rate is 3-3.25%, but the yield on the 2 year is 4.554% as of today. So what the bond market is telling you is that it expects interest rates to come down from their current point within 2 years.

The reasons for why yield curve inversion is not a predictor of recessions is as follows:

• Yields aren’t determined by recessions, they’re determined by central bank policies which are largely political.
• Markets are forward facing, but they also reflect what we already know: since market prices are determined by participants (i.e., there are no price controls or central pricing authorities), when yields invert it’s merely reflecting all available data.
• People think that yield curve inversion tells us something we don’t already know, but that is misleading. It’s fair to say that inversion is a leading indicator in the sense that it’s a signal from the market, but it’s not some kind of crystal ball.

## Are we in a recession? Is a recession coming? #

The 10 minus 2 yield inversion has historically been brief, and often precedes a recession. Whether or not we’re currently in a recession is still being debated, but we’ve had negative GDB growth for a while now.

Here’s what the yield curve looks like going back to 1976:

The grey vertical areas in the chart above are periods officially denoted as recessions, and as you can see around 1980, 1990, 2003, 2008, right before 2020, and of course now, the yield curve was inverted.

And our real GDP measured as a percentage changed since the previous quarter has been ever so slightly negative the last 2 quarters:

We can also zoom in a bit on the yield curve and see how it lines up with the GDP data, which had negative real growth in Q1 of 2022:

You can reasonably argue that:

1. We’re probably already in a recession.
2. The yield curve is telling us that the federal reserve is going to need to at the very least pause on raising rates, and possibly even lower rates in the future.

The question of whether we’re “in a recession” or not is, as I said before, mostly a political one, but if we go by the definition of “2 consecutive quarters of negative real growth” (which is the accepted definition in most places) then yes, we’re in a recession.

## Okay but what do #

As I mentioned, markets are always forward looking, and right now the bond market believes we’re headed for worse economic times, and the central bank will have no choice but to lower rates (or at least stop raising them). The Federal Reserve, on the other hand, has maintained it’s not too concerned about the recession panic, and is instead focused on getting inflation down. In other words, the bond market is calling the Federal Reserve’s bluff at the moment, and they’ve gotten into a showdown.

I think it’s unlikely that the bond market is foolish, although markets can stay irrational longer than you can remain solvent yadda yadda, so who knows.

In any case, with yields as high as they currently are on bonds, it’s not a bad time to pick some up at a steep discount.

Author
Brenden Matthews

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• Browse all
Measurement of the inclusive isolated photon production cross section in pp collisions at $\sqrt{s}=7$ TeV

The collaboration
Eur.Phys.J.C 79 (2019) 896, 2019.

Abstract (data abstract)
CERN-LHC.The production cross section of inclusive isolated photons has been measured by the ALICE experiment at the CERN LHC in pp collisions at a centre-of-momentum energy of $\sqrt{s}$= 7 TeV. The measurement is performed with the electromagnetic calorimeter EMCal and the central tracking detectors covering a range of |$\eta$| < 0.27 in pseudorapidity and a transverse momentum range of 10 < $p_{T}$ < 60 GeV/c. The result extends the $p_{T}$ coverage of previously published results of the ATLAS and CMS experiments at the same collision energy to smaller $p_{T}$. The measurement is com pared to next-to-leading order perturbative QCD calculations and to the results from the ATLAS and CMS experiments. All measurements and theory predictions are in agreement with each other

• #### Table 1

Figure 8a

10.17182/hepdata.91995.v1/t1

Double $p_{T}$-differential production cross section of isolated photons in pp collisions at $\sqrt{s}$=7 TeV in the rapidity interval -0.27<$\eta$<0.27.

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#### Ask a Question

Your question will be emailed to those involved with the submission. Please mention the relevant table.

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Services & ResourcesWolfram ForumsMathGroup Archive

difficult/unconventional series expansion

• To: mathgroup at smc.vnet.net
• Subject: [mg106259] difficult/unconventional series expansion
• From: Roman <rschmied at gmail.com>
• Date: Wed, 6 Jan 2010 06:01:43 -0500 (EST)

Dear group,

I am trying to do an unconventional series expansion, and find it
difficult with Mathematica. Given the function
S[x_] = Log[E^x-1]/(E^x-1)+(1+1/(E^x-1))Log[1+1/(E^x-1)]
I am looking for the behavior for very large x.
[For the interested: S(x) is the entropy (in units of the Boltzmann
constant) of a harmonic oscillator, with x=\hbar\omega/(kT). So I'm
looking for the low-temperature behavior of the entropy.]

The problem is that simply doing
Series[S[x],{x,Infinity,1}]
does nothing (since this is not a series expansion in the usual
sense). But I know what I would like to find: the lowest terms of the
"series expansion" are
(x+1)E^(-x+(2x+1)/(2(x+1))E^(-x))
Do you know how to find this expression automatically in Mathematica?
I am interested in a general technique, not just the results for this
particular function.

Cheers!
Roman.

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text:
# How do you solve: x^3/2=125?

$x = 5 \sqrt[3]{2} , \setminus 5 \setminus \sqrt[3]{2} {e}^{i \frac{2 \setminus \pi}{3}} , \setminus 5 \setminus \sqrt[3]{2} {e}^{i \frac{4 \setminus \pi}{3}}$

#### Explanation:

${x}^{3} / 2 = 125$

${x}^{3} = 125 \setminus \cdot 2$

${x}^{3} = 250$

${x}^{3} = 250 {e}^{i 0}$

${x}^{3} = 250 {e}^{i 2 k \setminus \pi}$

$x = {\left(250 {e}^{i 2 k \setminus \pi}\right)}^{\frac{1}{3}}$

$x = \setminus \sqrt[3]{250} {e}^{i \frac{2 k \setminus \pi}{3}}$

$x = 5 \setminus \sqrt[3]{2} {e}^{i \frac{2 k \setminus \pi}{3}}$

Where, $k = 0 , 1 , 2$

Now, setting the values of $k$, we get three roots of given cubic equation as follows

$x = 5 \sqrt[3]{2} , \setminus 5 \setminus \sqrt[3]{2} {e}^{i \frac{2 \setminus \pi}{3}} , \setminus 5 \setminus \sqrt[3]{2} {e}^{i \frac{4 \setminus \pi}{3}}$


text:
## What is Transformation?

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Before getting into the concept, let us do an activity. Take a sheet of paper and draw a picture using straight lines. For example, if you are drawing a hut, you will have to use straight lines to make its walls, roof, windows, and doors. Now take a pack of match sticks from the matchbox. Place one stick on each line of the hut. What do you observe from this? Initially, the match sticks were kept horizontally in a box. But you might have kept it vertically over the walls. For the roofs, you might have placed the sticks in a slanting position. This is how we change the position of the stick from one to another. You can shift the sticks to the left or the right as you wish. Similarly, you can move them upwards or downwards.

Similarly, when you draw a line in a graph sheet, you can move the position of the line by adding some units, subtracting some units, multiplying some units, or dividing some units from the original function. Let us see the technical definition given below.

## How to do Transformation on a Graph?

The transformation is done by shifting the functions to the right or left or by stretching or compressing the functions and so on. Some of the ways in which the transformation is done are given below.

## What is Translation of Graph?

The translation is to move an object in such a way that every single point of the object moves in the same direction and for the same distance. The function’s domain remains the same.

There are two types of translation. They are given below.

### Vertical Translations

Suppose that you are holding a ruler in your hand. Stretch your hands fully and hold the ruler horizontally parallel to your body. The ruler will be parallel to your shoulders. Now if someone asks you to shift it a little lower than the actual position in such a way that the ruler is still parallel and horizontal, you may bring the ruler down to your shoulder level. This is called vertical translation because the graph is translating along the vertical axis. That is, the movement is along the y-axis. This is also called a vertical shift.

Let us see how it appears in a graph. Take a graph sheet and draw a line parallel to the x-axis at a distance of k units upwards from the x-axis. The function is defined as f(x) = k for any value of x. You can translate the line by adding or subtracting ‘a’ units to the function to get a new function. This would give f(x) = k + a or f(x) = k – a. This will make the line either shift ‘a’ units upwards or ‘a’ units downwards along the y-axis.

The coordinates of any point on the graph are of the form (x, f(x) + a) or (x, f(x) – a) where ‘a’ is a constant.

### Practice Problem

Consider the function$f\left(x\right)={x}^{2}$. Graph the function and translate it along the vertical axis. (This is a linear function).

Solution

For the given function$f\left(x\right)={x}^{2}$, the vertical translation will be given by the functions $g\left(x\right)=f\left(x\right)+a$ or$h\left(x\right)=f\left(x\right)-a$, where $a$ is any constant.

Consider$a=2$. Then,$g\left(x\right)={x}^{2}+2$ and $h\left(x\right)={x}^{2}-2$.

The red, blue and green curve represents the graph of$f\left(x\right)$, $g\left(x\right)$ and $h\left(x\right)$ respectively.

### Horizontal Translations

Take a graph sheet and draw a line parallel to the y-axis at a distance of k units to the right of the y-axis. You can translate the line by adding or subtracting ‘c’ units. This will make the line either shift ‘c’ units to the right or ‘c’ units to the left along the x-axis.

This is called horizontal translation because the graph is translating along the horizontal axis. That is, the movement is along the x-axis. This is also called a horizontal shift.

The coordinates of any point on the graph is of the form (x + c, f(x)) or (x - c, f(x)) where c is a constant.

### Practice Problem

Consider the function$f\left(x\right)={x}^{2}$. Graph the function and translate it along the horizontal axis.

Solution

For the given function$f\left(x\right)$, the horizontal translation will be given by the functions $g\left(x\right)=f\left(x+a\right)$ or$h\left(x\right)=f\left(x-a\right)$, where $a$ is any constant.

Consider$a=2$. Then, $g\left(x\right)={\left(x+2\right)}^{2}$and $h\left(x\right)={\left(x-2\right)}^{2}$.

The red, blue and green curve represents the graph of$f\left(x\right)$, $g\left(x\right)$ and $h\left(x\right)$ respectively.

## Dilation

Dilation means to either stretch or contract a graph in such a way that the direction and shape of the graph are not changed. But the size of the graph is altered. The stretching can be vertical or horizontal. It is done by multiplying or diving the coordinates such that it changes the location and structure of the graph. If the function is defined as f(x) = kg(x) for any value of x and some function g(x), the stretching is vertical.

## Reflection of Graph

Have you seen your reflection in the mirror? Try carrying a board with some words written on it. If you stand in front of the mirror holding the board in your hand, what do you observe? The words in the board appear to be backward and retrace the path from where it travelled.

Reflection means the mirror image of an object. The reflection of a graph is done by multiplying by -1. We can reflect about the y-axis by multiplying x by -1. This gives –x. To reflect about the x-axis, we multiply the function f(x) by -1. This gives –f(x). By adding a minus sign, the positive values become negative values.

When y = f(-x), the graph reflects about y-axis. Whereas for y = -f(x), the graph reflects about x-axis.

## Practice Problem

Find out if the following is a reflection, scaling or if the graph shifts horizontally or vertically:

1. y = f(x + 2)
2. y = f(x) + 2
3. y = 7f(x)
4. y = f(x/4)

Solution

1) For a function$f\left(x\right)$, the horizontal translation will be given by the functions $g\left(x\right)=f\left(x+a\right)$ or$h\left(x\right)=f\left(x-a\right)$, where $a$ is any constant. This implies that the given function shifts 2 units horizontally.

2) For a function$f\left(x\right)=x$, the vertical translation will be given by the functions $g\left(x\right)=f\left(x\right)+a$ or$h\left(x\right)=f\left(x\right)-a$, where $a$ is any constant. This implies that the given function shifts 2 units vertically.

3) If a function$f\left(x\right)$ is multiplied by a constant value$k$, then the function is said to be scaled by a factor of $k$vertically. This implies that the given function represents scaling by a factor of$7$vertically.

4) If the $x$ value in a function$f\left(x\right)$ is multiplied by a constant value$k$, then the function is said to be scaled by a factor of $k$horizontally. This implies that the given function represents scaling by a factor of $\frac{1}{4}$ horizontally.

Try to do some transformations to the constant function (f(x) = c), linear function (f(x) = x), quadratic function, absolute value function, square root function and cubic function. This activity is to enhance your leaning.

## Real-Life Examples of Transformation

• The clock or watch has a minute and hour hand which keeps changing its position.
• Keeping a box on the floor and moving it in a straight line.
• While playing chess, the pieces are translated up and down.
• The rays of the sun reflected by the water.

## Formulas

The formulas for different transformations of a function f(x) are as follows:

• Vertical translation: y=f(x)+c
• Horizontal translation: y=f(x+c)
• Vertical dilation: y=kf(x)
• Horizontal dilation: y=f(kx)
• Vertical reflection: y=-f(x)
• Horizontal reflection: y=f(-x)

Here c is any real number and k is any positive real number.

## Context and Applications

It is used in the K-12 curriculum, undergraduate and post-graduate mathematics, and is asked in entrance examinations.

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

• Bachelors Mathematics
• Masters Mathematics

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Objectives. Using Jackson Heart Study data, we examined whether perceived discrimination was associated with prevalent hypertension in African Americans.

Methods. Everyday discrimination, lifetime discrimination, burden of discrimination, and stress from discrimination were examined among 4939 participants aged 35 to 84 years (women = 3123; men = 1816). We estimated prevalence ratios of hypertension by discrimination, and adjusted for age, gender, socioeconomic status, and risk factors.

Results. The prevalence of hypertension was 64.0% in women and 59.7% in men. After adjustment for age, gender, and socioeconomic status, lifetime discrimination and burden of discrimination were associated with greater hypertension prevalence (prevalence ratios for highest vs lowest quartile were 1.08 [95% confidence interval (CI) = 1.02, 1.15] and 1.09 [95% CI = 1.02,1.16] for lifetime discrimination and burden of discrimination, respectively). Associations were slightly weakened after adjustment for body mass index and behavioral factors. No associations were observed for everyday discrimination.

Conclusions. Further understanding the role of perceived discrimination in the etiology of hypertension may be beneficial in eliminating hypertension disparities.

Previous research reported an association between perceived discrimination and health outcomes.1,2 In particular, hypertension (which is higher among African Americans than Whites) was linked to discrimination,3–6 although findings were not always consistent.4,7–10 Although traditional biobehavioral risk factors might explain some of the African American–White disparity in hypertension, differences in exposure to discrimination by race might also contribute. African Americans’ exposure to discrimination could influence their risk for hypertension through various mechanisms. These include negative coping behaviors, such as unhealthy eating, sedentary lifestyles, and tobacco and alcohol intake. The experience of discrimination could also cause emotional distress,3 which can trigger physiological responses involving the hypothalamic–pituitary–adrenal axis and the sympathetic–parasympathetic systems, which play an important role in the pathophysiology of hypertension.7,11,12

Although exceptions exist, most previous studies of perceived discrimination and hypertension in African Americans employed relatively small samples13,14 and focused on a single measure of discrimination.3,5 The Jackson Heart Study (JHS), the largest prospective study of cardiovascular disease in African Americans, offered a unique opportunity to examine the association between multiple dimensions of discrimination (everyday, lifetime, burden, and stress from discrimination) and hypertension in a large sample of African Americans. The measures of various dimensions of discrimination obtained in JHS included everyday discrimination, which captured daily hassles associated with discrimination,15 lifetime discrimination, which captured acute and observable experiences similar to life events in the stress literature, and perceived burden and stress from discrimination.4 Because discrimination was conceptualized as a multifactorial construct, the inclusion of multiple measures might be necessary to accurately capture the impact of discrimination on health.2,16–18

We hypothesized that perceived discrimination would be positively associated with hypertension, and that health behaviors would partially mediate the association. Because previous work documented associations of discrimination with hypertension among women, but not men,5 and because of differences in hypertension prevalence by gender, we tested for effect modification by gender.

Between 2000 and 2004, 5301 African Americans (1941 men, 3360 women) were recruited from Hinds, Madison, and Rankin counties in the Jackson, Mississippi, metropolitan area. Participants came from 4 recruitment pools: community random sample (17%), volunteer sample (22%), the Atherosclerosis Risk in Communities Study (30%), and family members (31%). Recruitment was limited to noninstitutionalized African Americans aged 35 to 84 years, except in the family sample, in which those aged 21 to 34 years were eligible.19 Details of the design and recruitment were published elsewhere.20–22 The study was approved by the institutional review boards of University of Mississippi Medical Center, Jackson State University, and Tougaloo College. All participants provided informed consent.

The JHS discrimination instrument (JHSDIS) was administered by trained African American interviewers during the visit 1 clinic examination after blood draws and blood pressure measurement; 96% of participants completed the JHSDIS. Questions were asked to identify experiences with and reactions to perceived everyday and lifetime discrimination. Everyday discrimination was based on the scale by Williams et al.23 and had good internal consistency (α = 0.88).18 Participants were asked questions such as, “How often on a day-to-day basis do you have the following experiences? You are treated with less courtesy. You are treated with less respect. You receive poorer service than others at restaurants.” A total of 9 items were asked, and responses ranged from 1 (“never”) to 7 (“several times a day”). The mean of the 9 responses was used as the score for frequency of everyday discrimination.

Lifetime discrimination was adapted from the work of Krieger and Sidney3 and Krieger24 and had an internal consistency (α = 0.78)18 comparable with that observed for major life-event scales.13 Participants were asked about the occurrence of unfair treatment over their lifetime across 9 domains (yes/no): at school, getting a job, at work, getting housing, getting resources or money, getting medical care, on the street or public place, and getting services. The count of the domains (range = 0–9) was the lifetime discrimination score.

Burden of lifetime discrimination was measured by the following: “When you had experiences like these over your lifetime, have they been—very stressful, moderately stressful, or not stressful?” (coded 4, 2.5, and 1, respectively). “Overall, how much has discrimination interfered with you having a full and productive life,” and “Overall, how much harder has your life been because of discrimination? With response options, a lot, some, a little, or not at all?” (coded 4–1). Responses for the 3 items were reverse coded and summed to create a mean score ranging from 1 to 4, where higher scores indicated greater burden. Internal consistency of this scale was 0.63,18 suggesting a fair amount of heterogeneity among the items.13,25 Because previous research suggested that the degree of stress was a key determinant of the adverse health consequences of discrimination,2 the “stress” component of the “burden” index was examined separately (hereafter referred to as stress from discrimination). Burden and stress analyses were restricted to persons who reported at least 1 experience of lifetime discrimination.

Everyday, lifetime, and burden of discrimination were divided into quartiles to investigate whether there were threshold effects. In additional analyses, everyday, lifetime, and burden scores were examined as continuous variables after transforming them into SDs.

Participants were also asked to indicate the main reason for everyday and lifetime discrimination (age, gender, race, height or weight, or other). In secondary analyses, these responses were combined with everyday discrimination scores to create 5 categories:

no discrimination,

low discrimination (below the median) attributed to race,

high discrimination (at or above the median) attributed to race,

low discrimination attributed to nonracial factors, and

high discrimination attributed to nonracial factors.

Identical categories were created for lifetime and burden of discrimination.

Trained staff obtained seated blood pressure measurements with a Hawksley random-0 sphygmomanometer (Hawksley & Sons Ltd., Lansing, Sussex, UK) in the right arm of seated participants after a 5-minute rest. The average of 2 measures taken 1 minute apart was used in these analyses. Hypertension was defined as systolic blood pressure greater than or equal to 140 millimeters mercury, diastolic blood pressure greater than or equal to 90 millimeters mercury, or being on antihypertensive medications.10 Medications taken in the past 2 weeks were brought to the examination and transcribed verbatim, with subsequent medication coding by a pharmacist using the Medispan Dictionary.26

Education, income, and occupation were used to characterize socioeconomic status (SES). Education was measured as years of schooling completed, and included 4 categories: less than high school; high school graduate or general equivalency diploma; some college (1–3 years), vocational school, or associate degree; and college graduate or higher (≥ 4 years). Persons were assigned to 1 of 4 income categories (poor, lower-middle, upper-middle, and affluent) based on family size, US Census poverty levels, and year of baseline clinic visit (2000–2004). Income ranges for the categories were: poor: less than poverty level; lower-middle: 1 to 1.5 times the poverty level; upper-middle: more than 1.5 but less than 3.5 times the poverty level; and affluent: 3.5 or more times the poverty level. Occupation was coded according to the 2000 US standard occupational codes27 and classified as managerial or professional; service, sales and office; farming, fishing, and forestry; construction, extraction, and maintenance; and production, transportation, and material moving.

Additional covariates included age, gender, and possible behavioral or anthropometric mediators, including body mass index (BMI, defined as weight in kilograms divided by the square of height in meters), physical activity, cigarette smoking, alcohol consumption and diet. Physical activity was measured as the sum of the 4 index scores (active living, work or occupational, home life, and sport) from the JHS physical activity instrument.28 Cigarette smoking was classified as current, former, and never smokers. Alcohol consumption was classified as persons who abstained, had 1 to 7 drinks per week, 8 to 14 drinks per week, or more than 14 drinks per week. Diet was assessed using the Delta Nutrition Intervention Research Initiative, a short-form food frequency questionnaire.29 Selected dietary measures related to components of the Dietary Approaches to Stop Hypertension trial diet30 (percent calories from fat, sodium [mg/1000 kcal], potassium [mg/1000 kcal], calcium [mg/kcal], and fiber [g/1000 kcal]) were examined.

Of the 5301 participants who completed the examination, 362 were excluded because of missing data on hypertension (n = 59), all discrimination measures (n = 283), or education (n = 20). Unknown income and occupation data were coded as a separate category so participants could be retained in analyses, leaving 4939 participants. Regression models for each measure of discrimination were restricted to 4924, 4919, 4302, and 4337 participants for everyday, lifetime, burden, and stress from discrimination, respectively.

We examined the distribution of key variables by gender and discrimination and tested the differences using χ2 tests or t tests. We also examined age-adjusted differences in discrimination and risk factors between hypertensives and normotensives by gender. We estimated associations of discrimination with hypertension before and after adjustment for age, gender, SES, and risk factors. We also performed gender-stratified analyses because hypertension prevalence varies by gender,10 and because the impact of discrimination on hypertension was reported to vary by gender.5 However, because there were no interactions between gender and discrimination (all P > .05), only pooled results are given. Because the prevalence of hypertension was high in our cohort, Poisson regression31 was used to estimate prevalence ratios (PRs) of hypertension by discrimination before and after adjustment for covariates. Trend tests were conducted by including categories of discrimination as ordinal variables. Similar analyses were conducted using discrimination measures as continuous variables. We also tested for interactions between discrimination and age and SES. All tests were 2-tailed, and a P value < 0.05 was considered statistically significant. All statistical analyses were performed using SAS (version 9.1; SAS Institute Inc, Cary, NC).

Table 1 presents baseline characteristics of the sample. Women made up 63.4% of the sample. Hypertension prevalence was higher among women than men (64% vs 60%, respectively). Although men and women had similar levels of education, more men had affluent income, and more women were in management occupations. Men reported slightly higher levels of everyday, lifetime, and burden of discrimination, but differences were small. Women were more likely to report that lifetime discrimination made their lives very stressful. Among those reporting at least some discrimination, 43% of women and 55% of men attributed everyday discrimination primarily to race, and 57% of women and 70% of men attributed lifetime discrimination primarily to race.

TABLE 1— Baseline Characteristics by Gender in Perceived Discrimination and Hypertension: The Jackson Heart Study, Jackson, MS, 2000-2004

TABLE 1— Baseline Characteristics by Gender in Perceived Discrimination and Hypertension: The Jackson Heart Study, Jackson, MS, 2000-2004

 Women (n = 3123), % or Mean (SD) Men (n = 1816), % or Mean (SD) Pa Age, y 55.1 (12.7) 53.9 (12.8) .001 Education .649 < high school 17.4 18.7 High school grad 20.2 19.3 Some college 29.3 29.4 ≥ bachelor's degree 33.1 32.7 Incomeb Poor 18.3 10.4 < .001 Lower-middle 26.5 21.7 Upper-middle 30.1 30.4 Affluent 25.1 37.5 Occupation Transportation/other 1.1 0.8 < .001 Production 9.3 23.5 Construction/extract 0.7 15.7 Farming 0.1 0.2 Sales/office 20.4 12.9 Service 29.8 15.9 Management/professional 38.6 31.1 Everyday (n = 4924)c 2.1 (1.0) 2.2 (1.0) < .001 Lifetime (n = 4919)d 2.9 (2.1) 3.3 (3.1) < .001 Burden (n = 4328)ef 2.3 (0.8) 2.4 (0.8) .066 Stress from discriminationf Not stressful 20.1 24.5 < .001 Moderately stressful 52.7 55.9 Very stressful 27.2 19.6 Attribution of everyday discrimination < .001 Nonracial/high 21.8 19.1 Nonracial/low 25.9 19.2 Race/high 18.4 27.1 Race/low 18.0 19.3 No discrimination 15.9 15.4 Attribution of lifetime discrimination < .001 Nonracial /high 21.1 15.0 Nonracial/low 16.2 10.4 Race/high 33.7 45.8 Race/low 16.5 17.3 No discrimination 12.5 11.4 Attribution of burden < .001 Nonracial/high 19.5 13.0 Nonracial/low 23.2 15.8 Race/high 29.8 40.3 Race/low 27.5 31.0 Hypertension 64.0 59.7 .002

Note. The sample size was (n = 4939).

aP values based on χ2 and t tests.

bIncome categories were based on family size, US Census poverty levels, and year of baseline clinic visit (2000–2004): poor < poverty level; lower-middle = 1–1.5 times the poverty level; upper-middle > 1.5 but < 3.5 times the poverty level; and affluent ≥ 3.5 times the poverty level.

cMedian (25th, and 75th percentiles): women: 1.8 (1.3, 2.6); men: 2.0 (2.0, 2.8).

dMedian (25th, and 75th percentiles): women: 3.0 (1.0, 4.0); men: 3.0 (2.0, 5.0).

eMedian (25th, and 75th percentiles): women: 2.2 (1.7, 2.8); men: 2.3 (1.8, 2.8).

fBurden and stress from discrimination are restricted to persons who reported at least 1 instance of lifetime discrimination.

Table 2 shows selected covariates by categories of discrimination. The average age decreased from the lowest to highest quartile for everyday and lifetime discrimination, which was consistent with previous research.7 Higher levels of lifetime discrimination and higher levels of SES were positively associated, but patterns were not as clear for the other 2 measures. Current smoking was positively associated with reports of everyday discrimination and burden from discrimination. Greater everyday discrimination was associated with higher BMI, and everyday and lifetime discrimination were associated with more physical activity. Dietary fiber and sodium intake were associated with greater everyday discrimination and burden from discrimination.

TABLE 2— Selected Covariates by Categories of Discrimination in Perceived Discrimination and Hypertension: The Jackson Heart Study, Jackson, MS, 2000–2004

TABLE 2— Selected Covariates by Categories of Discrimination in Perceived Discrimination and Hypertension: The Jackson Heart Study, Jackson, MS, 2000–2004

 Everydaya Lifetimeb Burdenc Q1(n = 1163) Q2(n = 1192) Q3(n = 1305) Q4(n = 1264) P for Trendd Q1(n = 1288) Q2(n = 832) Q3(n = 1576) Q4(n = 1223) P for Trendd Q1(n = 1074) Q2(n = 1064) Q3(n = 1222) Q4(n = 968) P for Trendd Demographic and SES factors Age, y, mean 59.7 55.3 53.4 50.7 < .001 57.2 54.7 54.1 52.8 < .001 53.5 52.6 54.4 56.3 < .001 Female, % 64.0 68.5 61.1 60.0 .021 66.1 66.2 63.8 57.6 < .001 64.5 65.8 60.5 61.7 .045 ≥ bachelor's degree, % 26.3 37.2 37.2 30.1 .153 22.1 28.4 35.3 44.0 < .001 26.9 35.6 37.9 33.6 .047 Income  > $50 000, % 20.0 28.2 29.2 22.8 .184 18.0 22.3 27.8 31.3 < .001 20.4 28.8 28.0 25.6 .078 Professional, % 29.9 40.1 40.3 32.3 .419 24.6 32.9 38.4 46.0 < .001 29.4 41.2 39.6 35.8 .227 Behavioral factors Current smoker, % 11.6 11.1 12.9 15.8 .006 13.0 12.1 13.5 12.6 .802 12.5 10.5 14.0 14.8 .02 > 14 drinks/wk, % 2.2 1.3 2.5 2.5 .205 2.3 1.9 2.5 1.7 .446 2.5 1.8 1.9 2.1 .517 BMI, mean 31.3 31.9 31.7 32.2 .021 31.7 31.7 31.8 32.0 .335 32.0 32.0 31.5 32.4 .533 Physical activity, mean 8.1 8.4 8.6 8.4 < .001 8.0 8.2 8.5 8.6 < .001 8.4 8.4 8.5 8.4 .467 Total dietary fiber, g, mean 15.9 14.9 16.3 16.8 < .001 16.1 15.4 15.9 16.5 .215 15.6 15.8 16.3 16.4 .009 Sodium, mg, mean 4019 3865 4167 4469 < .001 4135 4019 4147 4212 .277 4081 4097 4161 4305 .024 Note. BMI = body mass index (defined as weight in kg divided by the square of height in m); Q = Quartile; SES = socioeconomic status. All variables except age are adjusted to the age distribution of the full sample. The sample size was (n = 4939). aEveryday discrimination is the mean of 9 items scored 1–7. Ranges for Everyday Score Q1: 1.0–1.2; Q2: 1.3–1.7; Q3: 1.8–2.5; Q4: 2.6–7.0. bLifetime discrimination is the count of 9 items. Ranges for Lifetime Score Q1: 0–1.0; Q2: 2.0–2.0; Q3: 3.0–4.0; Q4:5.0–9.0. cBurden is the mean of 3 items scored 1–4. Ranges for Burden Score Q1: 1.0–1.6; Q2: 1.8–2.1; Q3: 2.3–2.8; Q4: 3.0–4.0. dP values for trend across discrimination categories. Table 3 displays covariates by hypertension status. Being hypertensive was associated with lower SES in both women and men, although associations were weaker and not always statistically significant in men. For women and men, each measure of discrimination was higher for hypertensives than normotensives, although the differences were small and not statistically significant. Hypertensives had a higher BMI and lower level of physical activity in both genders. TABLE 3— Selected Characteristics of Study Participants by Hypertension Status and Gender in Perceived Discrimination and Hypertension: The Jackson Heart Study, Jackson, MS, 2000–2004 TABLE 3— Selected Characteristics of Study Participants by Hypertension Status and Gender in Perceived Discrimination and Hypertension: The Jackson Heart Study, Jackson, MS, 2000–2004  Women Men Hypertension (n = 2000) No Hypertension (n = 1123) Pa Hypertension (n = 1084) No Hypertension (n = 732) Pa Age, y, mean 59.3 47.8 < .001 57.7 48.3 < .001 ≥ bachelor's degree, % 30.4 38.0 < .001 31.6 33.4 .47 Income >$50 000, % 19.0 25.4 .002 30.6 33.7 .2 Professional, % 35.5 44.0 < .001 30.0 32.5 .26 Discrimination measures Everyday 2.1 2.0 .46 2.2 2.2 .67 Lifetime 3.0 2.9 .13 3.3 3.2 .26 Burdenb 2.4 2.3 .25 2.4 2.4 .17 Stress from discriminationb 1.1 1.0 .12 0.98 0.91 .06 Behavioral factors Current smoker, % 10.3 9.5 .52 18.4 17.2 .55 > 14 drinks/wk, % 0.38 0.72 .23 5.4 4.3 .28 BMI, mean 34.1 30.7 < .001 31.0 28.4 < .001 Physical activity, mean 8.1 8.4 .005 8.5 8.9 .009 Total dietary fiber, g, mean 14.7 15.1 .18 17.9 18.2 .43 Calcium, mg, mean 839 859 .34 916 864 .05 Sodium, mg, mean 3771 3882 .2 4619 4800 .16 Potassium, mg, mean 2588 2576 .83 3113 3096 .84 Calories from fat, % 34.8 35.1 .41 34.9 35.1 .51

Note. BMI = body mass index (defined as weight in kg divided by the square of height in m). All variables except age are adjusted to the age distribution of the full sample. The sample size was (n = 4939).

aP values via χ2 tests or t tests.

bBurden and stress from discrimination are restricted to persons who reported at least 1 instance of lifetime discrimination.

Table 4 displays the prevalence ratios of hypertension by dimensions of discrimination. Everyday discrimination was not associated with hypertension, and no gradients in hypertension across categories were observed. Greater lifetime discrimination was associated with a slightly higher prevalence of hypertension in a graded fashion after adjustment for age, gender, and SES (P for trend = .006; PR for highest vs lowest quartile = 1.08; 95% confidence interval [CI] = 1.02, 1.15). The prevalence of hypertension increased by 4% for each 1 SD increase in lifetime discrimination (PR = 1.04; 95% CI = 1.01, 1.06). This pattern was slightly weakened, but persisted after adjustment for behavioral risk factors. Gender-stratified analyses (data not shown) suggested that age- and SES-adjusted associations were stronger in women (PR for highest vs lowest quartile = 1.10; 95% CI = 1.00, 1.20) than in men (PR for highest vs lowest quartile = 1.04; 95% CI = 0.94, 1.10), but tests for heterogeneity by gender were not statistically significant.

TABLE 4— Prevalence Ratios of Hypertension by Categories of Discrimination in Perceived Discrimination and Hypertension: The Jackson Heart Study, Jackson, MS, 2000–2004

TABLE 4— Prevalence Ratios of Hypertension by Categories of Discrimination in Perceived Discrimination and Hypertension: The Jackson Heart Study, Jackson, MS, 2000–2004

 Age, Gender Adjusted, Ratio (95% CI) Age, Gender, SES Adjusted,a Ratio (95% CI) Age, Gender, SES, Risk Factors Adjusted,bc Ratio (95% CI) Everyday Q1 (Ref) 1.00 1.00 1.00 Q2 0.99 (0.93, 1.04) 0.99 (0.94, 1.05) 0.97 (0.92, 1.03) Q3 0.99 (0.94, 1.04) 1.00 (0.94, 1.05) 0.98 (0.93, 1.04) Q4 1.00 (0.95, 1.06) 1.00 (0.94, 1.06) 0.98 (0.92, 1.04) P trend .947 .970 .519 SD units 1.01 (0.99, 1.03) 1.01 (0.99, 1.03) 1.01 (0.98, 1.03) Lifetime Q1 (Ref) 1.00 1.00 1.00 Q2 1.00 (0.94, 1.06) 1.00 (0.94, 1.07) 1.00 (0.94, 1.06) Q3 1.03 (0.98, 1.08) 1.04 (0.99, 1.10) 1.02 (0.97, 1.08) Q4 1.06 (1.00, 1.12) 1.08 (1.02, 1.15) 1.05 (0.99, 1.12) P trend .043 .006 .088 SD units 1.03* (1.01, 1.05) 1.04** (1.01, 1.06) 1.03* (1.00, 1.05) Burdend Q1 (Ref) 1.00 1.00 1.00 Q2 1.04 (0.98, 1.11) 1.06 (0.99, 1.13) 1.04 (0.98, 1.11) Q3 1.04 (0.98, 1.11) 1.06 (0.99, 1.12) 1.04 (0.97, 1.10) Q4 1.08 (1.01, 1.15) 1.09 (1.02, 1.16) 1.06 (1.00, 1.13) P trend .023 .013 .084 SD units 1.02* (1.00, 1.04) 1.02* (1.00, 1.05) 1.02 (0.99, 1.04) Stressd Not stressful (Ref) 1.00 1.00 1.00 Moderately stressful 0.99 (0.93, 1.04) 0.99 (0.94, 1.05) 0.99 (0.94, 1.05) Very stressful 1.08 (1.02, 1.15) 1.08 (1.02, 1.15) 1.06 (1.00, 1.13) P trend .009 .008 .05 Attribution of lifetime No discrimination (Ref) 1.00 1.00 1.00 Race/low 1.02 (0.95, 1.09) 1.02 (0.95, 1.10) 1.01 (0.94, 1.09) Race/high 1.05 (0.99, 1.12) 1.07 (1.01, 1.14) 1.04 (0.97, 1.11) Nonracial/low 1.02 (0.95, 1.10) 1.02 (0.95, 1.10) 1.00 (0.92, 1.08) Nonracial/high 1.07 (0.99, 1.15) 1.08 (1.00, 1.16) 1.04 (0.96, 1.12) Attribution of burden Race/low (Ref) 1.00 1.00 1.00 Race/high 1.07 (1.01, 1.13) 1.08 (1.02, 1.14) 1.06 (1.00, 1.12) Nonracial/low 1.06 (0.99, 1.13) 1.04 (0.98, 1.12) 1.03 (0.97, 1.11) Nonracial/high 1.04 (0.97, 1.12) 1.03 (0.96, 1.10) 1.01 (0.94, 1.08)

Note. CI = confidence interval; Q = Quartile; SES = socioeconomic status. The sample size was (n = 4939).

aSES: education—< high school, high school graduate, some college, and ≥ bachelor's degree; income categories were based on family size, US Census poverty levels, and year of baseline clinic visit (2000-2004): poor < poverty level; lower-middle = 1-1.5 times the poverty level; upper-middle > 1.5 but < 3.5 times the poverty level; and affluent ≥ 3.5 times the poverty level; occupation—transportation/other, production, construction/extract, farming, sale/office, service.

bRisk factors: body mass index, physical activity, cigarette smoking, alcohol consumption, and diet.

cAnalysis was restricted to participants who had no missing risk factors (n = 4523, 4516, 3976, and 3985 for everyday, lifetime, burden, and stress, respectively).

dBurden and stress from discrimination are restricted to persons who reported at least 1 instance of lifetime discrimination.

*P ≤ .05; **P ≤ .01

Greater burden from discrimination (among those who reported at least 1 instance of lifetime discrimination) was also associated with greater prevalence of hypertension after adjustment for age, gender, and SES (P for trend = 0.01; PR for highest vs lowest quartile = 1.09; 95% CI = 1.02, 1.16). Each 1 SD increase in burden from discrimination was associated with a 2% greater prevalence of hypertension (PR = 1.02; 95% CI = 1.00, 1.05). This pattern was slightly weakened and was no longer statistically significant after adjustment for behavioral risk factors. There was no evidence of heterogeneity of effects by gender, although stratified analyses (data not shown) suggested slightly weaker associations in women than in men (PR for highest vs lowest quartile = 1.05; 95% CI = 0.97, 1.10 in women, and 1.13;95% CI = 1.00, 1.30 in men). Similar results were observed when stress from discrimination was examined in place of the overall burden score.

Attribution analyses showed that higher levels of lifetime discrimination were similarly associated with hypertension regardless of whether discrimination was attributed to racial or nonracial factors (age-, gender-, and SES-adjusted PR 1.07; 95% CI = 1.01, 1.14 for racial attribution and 1.08; 95% CI = 1.00, 1.16 for nonracial attribution). In the case of burden of discrimination, however, only high burden attributed to racial factors was associated with a higher prevalence of hypertension (age-, gender-, and SES-adjusted PR = 1.08; 95% CI = 1.02, 1.14 for high burden attributed to race, and 1.03; 95% CI = 0.96, 1.10 for high burden attributed to nonracial factors).

Previous studies of discrimination and hypertension have often been limited to a single dimension of discrimination.5,32 This study examined the association of hypertension with multiple dimensions of discrimination (everyday, lifetime, burden, and stress) in a large sample of African Americans. Both greater lifetime discrimination and greater burden of discrimination were associated with greater hypertension prevalence. Being in the highest quartile of lifetime discrimination or burden was associated with an 8% to 9% higher hypertension prevalence in age-, gender-, and SES-adjusted models. These associations were slightly reduced after adjustment for BMI and behavioral risk factors, suggesting that these factors could play a mediating role. In the case of lifetime discrimination, the association was similar, regardless of whether discrimination was attributed to racial or nonracial factors. However, in the case of burden, the association between high burden of discrimination and hypertension appeared to be largely restricted to those who attributed the discrimination to race.

Previous studies that investigated associations of lifetime discrimination with hypertension have not always had consistent findings. One study of discrimination and hypertension reported complex interactions between lifetime discrimination, social class, and responses to discrimination. Among professional African Americans, systolic blood pressure was 9 to 10 millimeters mercury lower among those reporting that they challenged unfair treatment and had not experienced lifetime racial discrimination.3 In contrast, among working class African Americans, systolic blood pressure was higher among those who reported that they typically accepted unfair treatment, but did not report experiencing discrimination compared with those who typically challenged unfair treatment but did report experiences of discrimination.3 Two other studies with African American samples reported no associations between lifetime discrimination and blood pressure,4,33 but sample sizes in these studies were small (< 400). A large study of African American women found no associations between lifetime discrimination (as assessed by a subset of the items we used) and incident hypertension.34 We added to the existing work by documenting associations between lifetime discrimination and hypertension prevalence in a large sample of African Americans living in the US South.

The examination of burden and stress from discrimination was a novel feature of our study. We found that higher burden from discrimination was associated with greater prevalence of hypertension. Because research suggested that the stressfulness of life events is important for predicting health outcomes,2,11,35,36 we examined the stress-related item of the burden scale separately and found similar associations. At least 1 other study also found that stress from discrimination was associated with hypertension in African Americans, even when the number of domains for lifetime discrimination was not reported.4 Our results suggested that not only the experience of lifetime discrimination, but also the burden and stress derived from the experience, might be relevant to the health consequences of discrimination.

In contrast to results for lifetime discrimination and burden, we found no associations of everyday discrimination with hypertension. Previous studies of everyday discrimination and hypertension yielded mixed results.5,37–39 Everyday discrimination was found to be associated with diastolic blood pressure reactivity, but was unrelated to baseline blood pressure in a small sample of Black women.37 A larger study found that greater everyday discrimination was associated with higher diastolic blood pressure in older African Americans (n = 2800) but not in older Whites.38 A study of 1110 African Americans in Pitt County, NC, found no association between everyday discrimination and hypertension in men or women, although everyday discrimination attributed to nonracial factors was associated with greater odds of hypertension in women.5 Analyses based on the Black Women's Health Study found no evidence that everyday discrimination was related to incident hypertension.34 Brown et al.39 also found no associations of everyday discrimination with systolic or diastolic blood pressure in a large sample from the Study of Women's Health Across the Nation. It was plausible that everyday experiences of unfair treatment, which are designed to capture long-term current levels of minor instances of discrimination, would be more related to short-term changes in blood pressure than to high blood pressure, which is likely to develop over the life course. By contrast, lifetime measures attempted to capture the extent of lifetime exposure to discriminatory experiences in multiple domains of life.

Another novel aspect of our study was the examination of whether the factors to which participants attributed their unfair treatment modified the association between discrimination and hypertension. For lifetime discrimination, associations were similar regardless of attribution. However, burden analyses suggested that high burden from discrimination might be especially unhealthy when it is attributed to racial factors. Few studies investigated differential effects depending on the attribution. One small study found that the associations of everyday discrimination with diastolic blood pressure reactivity appeared stronger when the mistreatment was attributed to racial or ethnic discrimination.37 Roberts et al.5 found that everyday discrimination attributed to nonracial factors was associated with hypertension prevalence in African American women, whereas racial discrimination was not. Our findings of possible greater effect of burden from discrimination attributed to racial factors suggested that more work is needed to understand the psychological and physiological consequences of discrimination, especially when it is attributable to racial factors.

The JHS was restricted to a single site, which limits generalizability beyond the southeastern region. Given the sampling strategy, JHS was not a representative sample of the African American population. The location of the sample in the US South might have affected the validity of the self-reported measures as well as their implications for health. Our research design was cross-sectional, which limited our ability to draw causal inferences, although it was unlikely that hypertension caused discrimination. As in all observational studies, residual confounding was a possibility for the weak associations we observed. There was some evidence that coping strategies might modify the health consequences of discrimination.1,3 This topic deserves further examination. If exposure to discrimination was ubiquitous in this study, our ability to detect associations might have been limited. As suggested by a recent review, clinic measures of hypertension might be insufficient to detect associations with discrimination,6 and ambulatory blood pressure measurements might be necessary, especially to detect the impact of recent exposures linked to everyday experiences.40,41

The JHS was the largest study of cardiovascular disease in African Americans. The JHSDIS instrument enabled us to analyze the association of hypertension with multiple dimensions of discrimination and to contrast associations when discrimination was attributed to racial or nonracial factors. The study of discrimination and health was challenging. Perceptions of discrimination might be strongly affected by contextual beliefs about fairness, resulting in differential validity of measures as proxies for true health-related discrimination exposures. The aspects of discrimination that affect health might not be fully understood, resulting in crude and imprecise measurement. Finally, there was likely to be a complex causal chain between exposure to discrimination and hypertension, with intervening factors making the detection of effects difficult in observational studies with limited measurement. Institutional discrimination, which can shape exposures to a range of socioenvironmental factors related to blood pressure, could have a greater effect than interpersonal discrimination.6

Identifying the health consequences of discrimination remains an important public health concern, as it could explain health disparities within and between race/ethnic groups. Cross-sectional studies like ours were a reasonable first approximation. However, different study designs, including longitudinal studies and ones with improved measurement of discrimination that capture the timing of exposure and the proximal and short-term behavioral and physiological consequences of experiences of discrimination as well as the moderating effects of coping strategies, might be needed to better determine if and how interpersonal discrimination affects blood pressure.

## Acknowledgments

The Jackson Heart Study (JHS) was supported by the National Institutes of Health (contracts N01-HC-95170, N01-HC-95171, and N01-HC-95172, provided by the National Heart, Lung, and Blood Institute and the National Center for Minority Health and Health Disparities). This research was also supported by the National Heart, Lung, and Blood Institute (award number K01HL08468-04 to M. Sims and award number 1 K01 HL88735-03 to M. A. Bruce) and the Michigan Center for Integrative Approaches to Health Disparities (award number P60MD002249, funded by the National Center on Minority Health and Health Disparities, to A. V. Diez-Roux).

We thank the participants and staff at the JHS.

## Human Participation Protection

All study procedures and protocols were approved by the institutional review boards of University of Mississippi Medical Center, Jackson State University, and Tougaloo College. All participants provided informed consent.

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No related items


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# CLASSIFICATION OF CROPS

## CLASSIFICATION OF CROPS

Based upon the use and nature, the crops can be classified as given in Table 5.1.

## Similar Posts

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# Create sets with a Goldbach-like property

Goldbach's conjecture states that every even number > 4 is the sum of two prime numbers. Although this conjecture is not yet proven, similar results have been proven for other series of numbers. For example, every even number can be written as the sum of two practical numbers.

# The Challenge

Given a number N, return a set of positive integers such that every even number in the interval [4,N] can be written as the sum of two numbers in the set. Furthermore, the set generated must contain as few elements as possible, otherwise this challenge would be trivial.

# Detailed Example

For N=24, one possible output is {2,4,8,10,12}. This is because it contains the fewest number of elements possible (5). Also, every even number {4,6,8,... 24} can be written as the sum of exactly two of the members of the set.

4=2+2
6=2+4
...
12=8+4
14=10+4
...
24=12+12

# Examples

Here are some examples. There may be many valid outputs for each input, but only a few are listed.

4:{2}
6:{2,4} or {1,3}
9:{2,4}  #pay attention to how the odd number is handled
10:{2,4,6} or {2,3,5}
12:{2,4,6}
24:{2,4,8,10,12}
26:{1,3,7,11,13}

~2/),3>[2][1 3]]({2*:x;{:S{x<},{x\-S+.&}/}%}/{,}$0=n* A major speed optimisation results from a bit of duplicate-trimming, at a cost of 3 chars: ~2/),3>[2][1 3]]({2*:x;{:S{x<},{x\-S+.&$}/}%.&}/{,}$0=n* NB Let f(2n) be the size of the optimal set for [4..2n]. Then f(2n) <= f(2n+2) <= f(2n) + 1. The first inequality is obvious from the definition; the second is easily seen by taking any optimal set S_{2n} for [4..2n] and taking the union with {n+1}, or indeed with {2n+2-x} for any x in S_{2n}. This suggests a greedy algorithm, which the first revision of this answer implements. However, that greedy algorithm turns out to be incorrect. The unique optimal set for [4..20] is {2,4,8,10}, which is not a subset of either of the two optimal sets for [4..26]: {1,3,7,11,13} and {2,4,8,12,14}. My current solution assumes that every optimal set will contain only odd numbers or only even numbers. I haven't proven this, but given that odd + even = odd a set which mixes parities is wasting possible pairs. • {2,3,5} is optimal set for 10, but contains both odd and even numbers – Egor Skriptunoff Sep 1 '13 at 7:56 • @EgorSkriptunoff, true, but there are also two all-odd and two all-even sets for the same score. {even, odd, odd} can produce 4 numbers, so can't get beyond 10, whereas {2,4,8} and {2,4,6} are solutions for 12. If you look at four-element sets then a 2-2 split in parity limits to 6 pairs (so up to 14); a 3-1 split limits to 7 pairs (up to 16); a 4-0 split limits to 10 pairs, and {2,4,8,10} achieves 9 (up to 20). The further you go, the more disadvantage there is to mixing parities. – Peter Taylor Sep 1 '13 at 12:09 # J block, 106 characters f=.3 :0 n=.0 while.1 do.for_z.>,{n#<>:i.y do.if.0=$(+:2+i.<:<.-:y)-.+/~z
do.z
return.end.end.n=.>:n
end.
)

As usual for J programs, very memory hungry. The algorithm is as simple as can be: for each length n, for each combination z of n values, if that combination covers the whole set, return that combination.

# Ruby, 116 112 characters

f=->x{0.upto(x){|n|(1..x).to_a.combination(n){|z|return z if
[]==4.step(x,2).to_a-z.flat_map{|i|z.map{|j|i+j}}}}}

(whitespace insignificant)

Ruby version of the algorithm, built-in deduplication, much more memory-efficient due to the use of enumerators in place of arrays (though J seems to optimise this somewhat), surprisingly not much larger than the J version.

results:

4-5 :[2]
6-9 :[2, 4]
10-13:[2, 4, 6]
14-21:[2, 4, 8, 10]
22-29:[2, 4, 8, 12, 14]
30-39:[2, 4, 8, 12, 16, 18]
40-45:[2, 4, 6, 12, 18, 20, 22]
• upto(1/0.0)? It's trivial to prove that upto(x) is pessimistic. – Peter Taylor Aug 26 '13 at 10:12


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## If a is an integer greater than 1, then aᵈ-1 divides aⁿ-1 if and only if d divides n

Fix an integer $a > 1$. Prove that for all $d,n \in \mathbb{N}$, $d$ divides $n$ if and only if $a^d-1$ divides $a^n-1$. Conclude that $\mathbb{F}_{p^d} \subseteq \mathbb{F}_{p^n}$ if and only if $d|n$.

If $d|n$, then by this previous exercise, $x^d-1$ divides $x^n-1$ in $\mathbb{Z}[x]$, and so $a^d-1$ divides $a^n-1$ in $\mathbb{Z}$.

Conversely, suppose $a^d-1$ divides $a^n-1$. Using the Division Algorithm, say $n = qd+r$. Now $a^n-1 = (a^d)^qa^r - a^r + a^r - 1$ $= a^r((a^d)^q-1) + (a^r-1)$. If $q = 1$, then we have $a^r-1 = 0$, so that $r = 0$ and $d|n$. If $q > 1$, then $a^n-1 = a^r(a^d-1)(\sum_{i=0}^{q-1} (a^d)^i) + (a^r-1)$, and again we have $r = 0$, so that $d|n$ as desired.

Recall that $\mathbb{F}_{p^k}$ is precisely the roots (in some splitting field) of $x^{p^k}-x$. Now $d|n$ if and only if $p^d-1$ divides $p^n-1$ by the above argument, if and only if $x^{p^d-1}-1$ divides $x^{p^n-1}-1$ by a previous exercise, if and only if $x^{p^d}-x$ divides $x^{p^n}-x$, if and only if $\mathbb{F}_{p^d} \subseteq \mathbb{F}_{p^n}$.


text:
# Analytic conformal bootstrap and Virasoro primary fields in the Ashkin-Teller model

### Submission summary

 As Contributors: Sylvain Ribault Arxiv Link: https://arxiv.org/abs/2106.15132v1 (pdf) Date submitted: 2021-07-22 14:05 Submitted by: Ribault, Sylvain Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory

### Abstract

We revisit the critical two-dimensional Ashkin-Teller model, i.e. the $\mathbb{Z}_2$ orbifold of the compactified free boson CFT at $c=1$. We solve the model on the plane by computing its three-point structure constants and proving crossing symmetry of four-point correlation functions. We do this not only for affine primary fields, but also for Virasoro primary fields, i.e. higher twist fields and degenerate fields. This leads us to clarify the analytic properties of Virasoro conformal blocks and fusion kernels at $c=1$. We show that blocks with a degenerate channel field should be computed by taking limits in the central charge, rather than in the conformal dimension. In particular, Al. Zamolodchikov's simple explicit expression for the blocks that appear in four-twist correlation functions is only valid in the non-degenerate case: degenerate blocks, starting with the identity block, are more complicated generalized theta functions.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 2106.15132v2 on 2 September 2021

Submission 2106.15132v1 on 22 July 2021

## Reports on this Submission

### Strengths

1. Contains novel conjectures regarding the computation of degenerate conformal blocks that may find other applications.
2. Very thorough in its presentation of technical results.

### Weaknesses

1. Rather technical subject matter.

### Report

In this paper the authors study correlation functions of local operators in the Ashkin-Teller model, the $\mathbb{Z}_2$ orbifold of the $c=1$ free boson CFT.

The authors begin with a review of correlation functions in the unorbifolded free boson theory. This includes an extended discussion of correlation functions of degenerate Virasoro primary operators that are not affine primaries. Although the conformal data of the latter are completely determined by those of the current algebra primaries via the affine symmetry, extracting this conformal data in this way in practice is highly tedious. The authors circumvent this by imposing crossing symmetry of the correlators of these Virasoro primaries directly, although this still turns out to be a rather technical exercise due to analytic subtleties of the conformal blocks when the dimensions correspond to those of degenerate representations.

The authors then proceed to study correlation functions in the orbifolded theory. This includes a, to the best of my knowledge, novel determination of structure constants involved in mixed four-point functions of affine primaries in the untwisted sector and twist fields. Together with the known four-point functions of affine primaries in the untwisted sector and of twist fields, this establishes consistency of the theory on the sphere. The authors also study correlation functions of degenerate Virasoro primaries and current algebra descendants of the twist fields, encountering similar subtleties as in the degenerate correlators of the free boson.

In studying the correlators of degenerate Virasoro primaries, the authors must compute the corresponding degenerate conformal blocks (or fusion kernels). In practice, the computation of degenerate blocks is subtle. The reason for this is that methods for the computation of generic conformal blocks view the blocks as meromorphic functions of (say) the central charge, with poles appearing when internal states correspond to degenerate representations of the Virasoro algebra (signalling the presence of null states). The authors propose a concrete method to determine degenerate conformal blocks as limits of generic conformal blocks in the form of two conjectures. These proposals might be of broader interest than the results on crossing symmetry in the Ashkin-Teller model.

The paper concerns rather technical subject matter but is well-written and very thorough in the presentation of its results. I recommend the publication of this paper in SciPost.

### Requested changes

1. The notation $k_{123}$ and $k^1_{23}$ are undefined when used in (3.49). Although they are defined after (5.42), these definitions should be moved up to their first use in the main text.

2. The authors should discuss the extent to which they have checked their conjectures regarding degenerate conformal blocks as limits of generic conformal blocks.

• validity: high
• significance: ok
• originality: good
• clarity: high
• formatting: perfect
• grammar: perfect

### Strengths

1. Very thorough and detailed.
2. Contains a conjecture which might be of interest also to readers who are not necessarily interested in the specific model.

### Weaknesses

1. Very very technical.
2. Presentation could be improved.

### Report

In this papers the authors study the Ashkin-Teller model. This has been solved from the point of view of the affine algebra, meaning that the conformal data in terms of the affine primaries is known. Conformal data for the Virasoro primaries can then be obtained, in principle. However, this becomes rather technical; the authors study directly correlation functions of Virasoro primaries. By imposing crossing symmetry of four point functions, the three point function coefficients are found.
The paper also contains a pair of conjectures on how to use Zamolodchikov’s recursion relation for the Virasoro blocks in case of degenerate operators. Naively, this is ill defined. The authors instead propose a precise way of deforming the theory so that the correct Virasoro blocks can be obtained. This might be of interest also to readers who are not necessarily interested in the Ashkin-Teller model.
I find the paper very thorough and rather technical. Most of the edits I suggest concern notation and presentation of the paper. Exceptions are the more important points 5 and 6, which deal with the precise recipe of Conjecture 2.

### Requested changes

1. I have some remarks about the notation of section 2: starting from (2.11), the summation variable $s$ should be renamed since $s$ appears also as the letter indicating that we are expanding in the $s$-channel. (2.19) could be made leaner by replacing $\Delta_j \to j$, as is done in most of the section. Should (2.20) include $\Delta_t,\bar \Delta_t \in \text{spectrum}$ or something of the kind? Does (2.26) need a $\pm$ in front?

2. I find the minus sign difference between (3.39) and (3.46) confusing. Since the sign is not determined, can we choose it to be the same in these two expressions?

3. A clearer explaination of the notation of (3.41) would help (I mean which operator is at which coordinate in $F_{0,\epsilon} \big[\ldots\big]$)

4. I imagine that (4.19) could also be zero for some values of the $(n_i,m_j)$, given equation (4.10). Is that right? If so, it should be indicated in (4.19).

5. The recipe of Conjecture 2 doesn't specify which degenerate representation we should consider. There are several $\mathcal{R}^{(c)}$ who have the same number of states as $\mathcal{R}$ at generic central charge. The question of which $\mathcal{R}^{(c)}$ one must choose should be addressed, but it isn't. For example, above eq (5.16), the authors choose one of the two possible structures, but it's not explained why.

6. Probably related to the previous point, in which cases did the authors check the validity of the conjectures? Which conformal blocks can be computed explicitly and then compared to the conjecture? Is the check up to some level? It would be good to include this.

7.It should be explained what the pentagon diagram of (5.32) symbolizes so that a reader not familiar with it could understanding without looking at reference [9].

• validity: good
• significance: good
• originality: ok
• clarity: good
• formatting: perfect
• grammar: excellent


text:
# Find the conditional distribution of $X$ given that $Y=y$

Let $$X$$ and $$Y$$ be two random variable with density $$f_{X,Y}(x,y)=\begin{cases} \frac{1}{y}, & \text{for } 0.

To find the conditional distribution of $$X$$ given that $$Y=y$$ first I calculated: $$f_Y(y)=\int_0^y \frac{1}{y}dx = 1$$ for $$0. Then $$f_{X|Y}(x|y)=\begin{cases} \frac{\frac{1}{y}}{1}, & \text{for } 0. Is that correct? I am very unsure about that.

Yes, just like for 2 events $$A,B$$ you have $$\mathbb{P}[A|B] = \frac{\mathbb{P}[A \cap B]}{\mathbb{P}[B]}$$ so too more generally in terms of random variables, $$f_{X|Y}(x,y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}$$
• Is $f_{X|Y}(x|y)$ already the conditional distribution? Or is it only the conditional density and I have to integrate $f_{X|Y}(x|y)$? – tommy_m Apr 23 at 18:25
• @tommy_m Not sure what you mean by conditional distribution. The usual notation denotes the pdf as $f(\cdot)$ and the cdf as $F(\cdot)$, both can characterize the distribution, and so can the mgf $M_X(t)$... To be specific, in this case $f_{X|Y}(x,y)$ denotes the conditional pdf of $X$ given $Y$, but if you need the conditional cdf, you would indeed need to integrate it. – gt6989b Apr 23 at 18:41