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# Image Processing

The first steps in almost any computer vision algorithm usually involve some transformation of the raw image pixels to enhance certain desirable properties, and to suppress undesirable properties. This is referred to as image “filtering. This chapter describes several of the more common filtering operations used in computer vision.

## Linear Filtering

To begin, consider a single pixel in an image. In this chapter and in all that follows, we will use $I(u,v,t)$ to denote the value of an image at location $(u,v)$ at the time $t$. When $t$ is not essential to the discussion, it will be suppressed. The indices $(u,v)$ are to be interpreted as an index into the image array. We will use C language conventions, and so $(0,0)$ will be the upper left corner of the image. Unless otherwise stated, it is assumed that the value of a pixel is scalar (obviously, this is not the case if we are dealing with color images, but that will be the exception).

It is important to realize the the value that is stored in the image array does not necessarily exactly reflect the intensity of light that hit the sensor. Even absent such obvious irregularities such as the blooming described earlier, there are random and systematic variations in the signal. Thus, if we were to model what is going on, we could write

$I(u, v, t) = I*(u,v,t) + r(u,v,t)$

where $I*$ is some ideal signal, and $r$ is a random process. For the purposes of the present discussion, we will assume that $r$ as the following properties:

1. $r(u,v,t)$ and $r(u’,v’,t’)$ are independent random variables so long as at least one of the arguments differ.
2. $r(u,v,t)$ and $r(u’,v’,t’)$ are identically distributed.
3. $r(u,v,t)$ follows a Gaussian distribution with zero mean and variance $\sigma^2$for all values of $u$, $v$, and $t.$

For the moment, let us further assume that $I*$ is stationary (that is, unchanging) with respect to time. It follows from basic statistics that we could reduce the effect of noise (i.e. of the random signal $r$) by averaging. In fact, we if we compute

$\hat{I}(u,v,t) = \sum_{t’= t-n}^{t+n} I(u,v,t’)/(2n+1)$

then it follows that the variance of $\hat{I}$ is $\sigma^2/(2n+1).$ In short, in principle, we could get as close to the ideal signal as we like, simply by averaging values. Note, by the way, that this often true even if the noise is not Gaussian, as we have assumed.

Unfortunately, it is not usually the case that the world is willing to stand still while we gather several observations of the same signal. However, we could instead assume that nearby pixels have about the same ideal value, and so instead we could average over space. That is, we could instead compute

$\hat{I}(u,v,t) = \sum_{u’ = u-N}^{u+N} \sum_{v’ = v-M}^{v+M} I(u’,v’,t)/((2N+1)*(2M+1))$

What is the resulting variance in this case?

Of course, the idea that all of the ideal values for pixels near a given pixel are exactly the same is not really reasonable; indeed if it were exactly the case, then the only images possible would be monotone! However, note that there are spatially non-stationary processes for which this would still be a reasonable strategy. For example, suppose the values of $I$ were varying linearly with $u$ and $v.$ Indeed, one might argue that, in general, for small neighborhoods a linear variation is likely to be a good approximation. So, maybe this is not such a bad idea after all!

Another idea that might make things even better would be to perhaps weight the pixels nearer to the central pixel (i.e, the one of interest) more highly than those further away. That is, we might decide to compute

$\hat{I}(u,v) = \sum_{n = – N}^{N} \sum_{m = -M}^{M} K(n, m) I(u-n,v-m)$

Here, we have made several changes. First, we are no longer interested in time-variation, so we have dropped $t$. Second, we have redone the indexing in terms of variables $n$ and $m$. You might want to verify, though, that what we have written is exactly equivalent to our previous expression, provided we set

$K(n,m) = 1/((2N+1)*(2M+1))$

The operation we have defined above is so fundamental that it has its own name: convolution and its own operator, *. Thus we define now

$(I*K)(u,v) = \sum_{n = – \infty}^{\infty} \sum_{m = -\infty}^{\infty} K(n, m) I(u-n,v-m)$

in the discrete domain and

$(I*K)(u,v) = \int_{ – \infty}^{\infty} \int_{-\infty}^{\infty} K(n, m) I(u-n,v-m) dm dn$

in the continuous domain. We henceforth refer to $K$ as the kernel of the convolution.

Note we have extended the summation (respectively integration) to the entire integer (respectively real) line. We have done so with the understanding that we can just put in zeros if we have a kernel over a finite range, as we had previously.

Usually, we consider the resulting of convolving entire images by some kernel, so we will often write, for example $H = I*K$ to denote the convolution of the image $I$ by the kernel $K.$ Convolution has several useful properties that we state here (and which you may wish to verify for yourself):

• Convolution is commutative: $I*K = K*I$ (Note this also implicitly assumes that the image in question has infinite support).
• Convolution is associative: $I * (K * J) = (I * K) * J.$
• Convolution is a linear operator: $K * (\alpha I + \beta J) = \alpha K*I + \beta K*J.$

Another useful fact is the following: the result of convolving an impulse with a kernel is the kernel itself. That is, if $U(0,0) = 1$ and $U(u,v) = 0$ otherwise, then

$U*K = K$

This is a useful property, since it means that if someone just hands you a convolution operation as a black box, you can discover the form of the convolution just by convolving it with an impulse. Indeed, this is call the impulse response of the filter. Furthermore, note that there was nothing special about placing the impulse at zero. If we, for example, move the impulse one location to the left, all we do is to also move the response one location to the left. However, we haven’t otherwise changed the response. Thus, convolution is also translation invariant.

In fact, suppose now we do the following. Let’s take the image $I$ and create a set of “impulse images” $U_{u,v}$ where $U_{u,v}$ puts a value of 1 at location $(u,v)$ and zero otherwise. Then it is not hard to show (using the linearity of convolution) that

$I*K = \sum_{u,v} I(u,v) (U_{u,v} * K)$

In short, the result of convolution can also be viewed as the summation of the impulse responses at each image location scaled by the value at that location.

With this as a basis, let’s now consider of what we can do with convolution. First, it is important to point out that if we want tocompute’ a convolution, we will generally only care about the non-zero values of the kernel. Second, for convenience, we will usually assume the width and height of the convolution kernel are odd. However, there are still some issues to consider. In particular, our images will always be of finite size. As a result, if we consider a kernel of size $2n+1$ by $2m+1$, there will always be $2n rows$ and $2m$ columns on the border of the image for which the value of the convolution is not well-defined. In general, there are three possibilities:

1. Use only those values which have well-defined values. As a result, images will shrink in size as we apply convolutions to them.
2. Pad the outside of the image with enough zeros (n rows top and bottom and m columns left and right) so that the result of the convolution is the same size as the image we started with. This is often convenient, but we must realize the values on the boundary are suspect.
3. Pad the outside of the image with enough zeros (2n rows top and bottom and 2m columns left and right) so that anyboundary pixel that would have elicits some response to the image is computed.

In Matlab, convolution is implemented using the conv2 operator. It accepts an argument indicating what type of padding to do. These are valid, same, and full, respectively.

### Models for Convolution

There are many ways to view the effect of a convolution operation on an image. We have already introduced one: convolution can be viewed as a way of averaging the image intensity values in a neighborhood.

Here is another idea. Note that we can view the value produced by a particular kernel applied at a given location as the inner product between two vectors: One vector is the kernel itself; the second vector are the intensity values in the neighborhood about the location in question flipped about both axes. Why the latter? Have a close look at the convolution formula and see if you can figure out why? Even better, if you figure it out, then try to see why its important to have the formula the way it is (hint, think about some of the properties of convolution we’ve discussed along the way).

In any case, suppose these two vectors are written as $k$ for the values of the kernel, and $i_{u,v}$ for the intensity values about the location $(u,v).$ Then we can rewrite convolution as

$I*K(u,v) = k \cdot i(u,v)$

Now, let us write $\vec{i}(u,v) = i(u,v)/\|i(u,v)\|$ and $\vec{k} = k/\|k\|.$ That is, anything with an arrow on top is a unit vector. Plugging this in, we have

$I*K(u,v) \propto \|i(u,v) (\vec{k} \cdot \vec{i}(u,v))$

Here, since $k$ is fixed, we have suppressed its magnitude (this will just rescale the result of the convolution).

As a result, it is clear that the response of a convolution depends on two factors:

1. The magnitude image values in the region of interest, and
2. The inner product between the image and kernel considered as unit vectors

The latter will of course achieve a maximum of 1 only if $\vec{k} = \vec{i}(u,v).$ In short, this term is large when the image vector closely resembles the convolution itself. Thus, convolution can be considered as a type of pattern matchingoperation.

In order to understand our final interpretation of convolution, we need to delve a bit deeper into signal processing. At some point, you have probably been exposed to the idea of approximating a function (e.g. a two-dimensional signal like an image) using some type of basis. For example, one can approximate a scalar function using the well-known Taylor series expansion.

In signal processing, it is useful to think about decomposing a signal in terms of its frequency structure. We will represent this by using sine functions as approximation basis — that is, we will try to express an image as a sum of just the right combinations of sine waves. For a scalar function (let’s keep it simple for now), we could write

$f(x) = \sum_{i=1}^{\infty} \alpha_i sin(\beta_i x + \gamma_i)$

for the right choices of $\alpha_i,$ $\beta_i$ and $\gamma_i.$ Indeed, one can show that for almost all reasonable functions, there is a choice which, in the limit, converges exactly to $f.$

The method of finding these coefficients is the Fourier transform (continuous case) and the Discrete Fourier transform (discrete case). Both make use of the following trick. We can write sines and cosines using [http://en.wikipedia.org/wiki/Euler%27s_formula Euler’s formula] which states that

$e^{i x} = \cos x + i \sin x$

The continuous Fourier transform in one dimension is then defined as

$F(\eta) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \eta} d x$

It closely resembles the inverse transform which is given by

$f(x) = \int_{-\infty}^{\infty} F(\eta) e^{2\pi i x \eta} d \eta$

If we expand the Fourier transform using Euler’s formula, we have

$F(\eta) = \int_{-\infty}^{\infty} f(x) \cos(-2\pi x \eta) + i f(x) \sin(-2\pi x \eta) d x$

Note that this looks much like our convolution operator above. If we rely on the intuition we developed there, we can see the following general properties:

• The Fourier transform of a real function is a complex function.
• The value of $\eta$ governs the frequency of the underlying sine and cosine functions.
• Based on our previous discussion of convolution, we see that the response of each of the terms will be proportional to the similarity of the sin or cosine function at a given frequency.
• Because sin and cosine are $\pi/2$ radians out of phase, the difference in the components of the response will tell us something about the phase ($previously, our \gamma_i)$ at which the signal responds.

Given the Fourier representation of signal, there are two common quantities of interest. The first is the power spectrum, defined as

$| F(\eta) | = \sqrt{Re(F(\eta))^2 + Im(F(\eta))^2}$

Intuitively, the power at a given value frequency $\eta$ is the relative strength of that frequency in the signal,independent of phase. The second quantity is the phase spectrum given by

$\angle F(\eta) = \pm \arctan (Im(F(\eta))/Re(F(\eta))$

An intuitive interpretation of phase is much harder to come by. Suffice to say that we can interpret this quantity as indicatingaverage phase at a given frequency. It is interesting to note that, although the power spectrum is much easier to understand intuitively, there is strong evidence that the phase spectrum of an image actually carries much more of the content we interpret as form.

To extend these ideas to images, we make two steps at once. First, we move from the continuous domain to the discrete. Here, it is useful to note that there is a maximum frequency that any discretely sampled signal can represent (the so-called Nyquist frequency), and so the values of $\eta$ will be both bounded and also themselves discrete. Second, we extend the DFT to two dimensions. The result is:

$DF(\eta, \gamma) = \sum_{u=0}^{N-1} \sum_{v=0}{N-1} I(u,v) e^{-2\pi i (u \eta + v \gamma)} $

and the inverse is $I(u, v) = \frac{1}{N^2} \sum_{\eta=0}^{N-1} \sum_{\gamma=0}{N-1} DF(\eta,\gamma) e^{-2\pi i (u \eta + v \gamma)} $

The easiest way to visual this is to think of $(\eta, \gamma)$ as a vector in the plane, the direction of which defines the direction of the periodic function, and the magnitude of which defines the frequency (phase is, as usual, encoded in the resulting complex value). Thus, by sweeping out the entire plane, we cover frequencies from 0 (D.C.) to the maximum possible and also all possible directions.

In earlier days, the DFT was often used as a way of characterizing texture. Indeed, it is quite interesting to see what the DFT of many common scenes is. We do so in the accompanying notes.

So, why the huge detour — what does this have to do with convolution? Well, there is a particularly elegant statement we can now make. Namely, given two signals $f$ and $g$ (scalar sound recordings, two-dimensional images and kernals, or whatever), their convolution can be expressed as follows:

$\mathcal{F}\{f * g\} = \mathcal{F}\{f\}\cdot \mathcal{F}\{g\}$

In short, the Fourier transform of the convolution of two signals is equal to the product of their Fourier transform. If we consider the power spectrum, we see that convolving a signal $f$ with $g$ accentuates or suppresses the signal frequencies in $f$ corresponding to high or low values of frequencies in the filter $g$.

Details at Wikipedia and Math World.

### Kernels for Smoothing

Given what we have established above, we can now think about what types of kernels would be most appropriate for smoothing. In our introduction, we have seen that one possible kernel is one with constant values, which corresponds to computing a moving average of the image. This is also referred to as the box filter (from the fact that when you plot the function in 1-D it looks like a box.

Averaging with a box filter takes out small perturbations by averaging across a pixel with all the neighborhood around it. However, if we consider the FT of the box function, it is possible to show it is the sync function

$sync(x) = sin(x)/x$

This implies that smoothing with a box filter has the undesirable effect of accentuating a series of higher order frequencies (this is referred to as ringing.). In short, some higher order frequencies of our noise will also creep into the signal.

Gaussian smoothing is a type of averaging which uses a 2D Gaussian as its kernel. The 2D Gaussian is given by

$G(x,y; \sigma) = \frac{1}{2 \pi \sigma} e^{- (x^2 + y^2)/\sigma^2}$

The Fourier transform of a Gaussian is still a Gaussian and has no secondary lobes. Thus, Gaussian kernels are better low-pass filters than the averaging filter.

### Computational Issues

The complexity of computing the convolution of an N by M image with a kernel of n by m is O(NMnm). For example, on 1000 by 1000 image, computing a convolution with an 11 by 11 kernel would require 121,000,000 multiplies and an equal number of additions. Thus, convolution is expensive and thus it is important to achieve time savings when possible. Here, we note two important “tricks.”

#### Separability of Kernels

If we return to our simple averaging kernel, we note that we can play the following “game:”

$(I*B_{h,w})(u,v) = \sum_{i=-h}^{h} \sum_{j=-w}^{w} B(i,j) I(u-i,v-j) = 1/(2h+1) \sum_i (1/(2w+1) \sum_j I(u-i,v-j) ) = (I*B_{1,w})*B_{h,1}. $`

That is, rather than performing one convolution with $(2w+1)(2h+1)$ elements, we instead perform two smaller convolutions of size $2w+1$ and $2h+1$. The complexity is thus now O(NM (2w+1 + 2h+1)). We can also prove this using the properties of convolution, namely

$(I*B_{1,w})*B_{h,1} = I*(B_{1,w}*B_{h,1})$

by the associativity of convolution. But, it is easy to show that $B_{1,w}*B_{h,1}$ results in exactly $B_{h,w}$ and so $I*(B_{1,w}*B_{h,1}) = I*B_{h,w}$ which is what we started with to begin.

This so-called separability property cannot always be used. However, it is not hard to show that the widely used Gaussian filter is also separable.

#### Collapsing Convolutions vs. Not

The previous example illustrates a second useful computational optimization of convolution. Namely, suppose we have a large image I of size N by M and smaller kernels J and K, of size 2n+1 by 2n+1 and 2m+1 by 2m+1, respectively. Then if we compute

$R = (I*J)*K$

the complexity is O(NM((2n+1)^2 + (2m+1)^2)).

But, if we instead refactor and compute

$R = I*(J*K)$

the complexity is $O(NM(2n+2m+1)^2 + ((2n+1)(2m+1))^2).$

For example, for a 1K by 1K image and two 11 by 11 kernels, the difference is approx 500M floating point operations $(2*2*11^2*10^6)$ versus nearly 1Gflops $(2*21^2*10^6)$ for the collapsed version.

### Derivative Kernels

Until now we have considered only kernels that perform an averaging operation. That is, kernels that have only positive values and sum to 1. However, we can easily design kernels that perform other functions.

In particular, consider the following simple kernel: [1, -1]. If we apply this to a scalar signal, e.g. [1,2,3,4,3,2,1], a little hand work (or a Matlab call) shows that the result is [1,1,1,-1,-1,-1]. In short, this has computed the first differences of the signal. Equivalently, it is a local approximation to the slope of the signal. For this reason, this kernel is referred to as a derivative filter.

If we consider a 2D image, we see that there are really two kernels to consider: $D_u =[1, -1]$ (derivatives along a row) and $D_v = [1; -1]$ (derivatives along a column). This makes sense since, if we think of an image as abrightness surface, then we consider the 2-dimensional gradient vector of the surface as characterizing the locate slope of the surface.

There are several possible derivative operators. A two of the most common are given below:

Sobel: [1 0; 0 -1] and [0 1; -1 0]

Prewitt: [1 0 -1; 1 0 -1; 1 0 -1] and [1 1 1; 0 0 0; -1 -1 -1]

One important thing to realize, however, is that derivative filters are high pass filters. This is easy to see if you take the FT of a derivative filter. Also, if you consider a slowly varying signal (i.e. low frequency), the derivatives are small, whereas a rapidly varying signal produces large magnitude derivatives.

The problem is that image noise is also concentrated in the high frequencies, so taking derivatives tends to be noise amplifying. As a consequence, it is often desirable to reduce the effect of noise when taking derivatives. Thus, a common approach is to combine smoothing with derivatives. Thus, we might for example compute:

$g_u = I * G(\sigma) * D_u$ $g_v = I * G(\sigma) * D_v$

Again, using convolution arithmetic, we can combine the convolutions and write $GD(\sigma)_u = G(\sigma) * D_u$ and $GD(\sigma)_v = G(\sigma) * D_v.$ These two kernels, the derivative of Gaussian kernels, are quite common. Note that they each have a free parameter, $\sigma$ which is referred to as the scale of the operator. The name fits, since the choice of $\sigma$ directly determines what scale of change the operator responds to. That is, very high frequency (i.e. rapidly changing) image structure will only be detected if a small value of $\sigma$ is used; a large value will suppress it. Indeed, it is quite common to perform this convolution at several scales, leading to the notion of a scale space of images.

Given the gradient value $\nabla I(u,v)$, we can now compute three useful scalar quantities:

1. The gradient magnitude: $\| \nabla I(u,v) \|$
2. The gradient direction: $atan2(I_u(u,v),I_v(u,v))$
3. The directional derivative: $n \cdot \nabla I(u,v)$ where $n$ is the desired direction.

The latter is an example of a steerable filter — that is a filter for which the directional version can be produced through a linear combination of some fixed components.

While we are talking about derivatives, it is worth noting we can go further, for example, we could compute:

$h_{uu} = I * GD_u(\sigma) * GD_u(\sigma)$ $h_{uv} = I * GD_u(\sigma) * GD_v(\sigma)$ $h_{vv} = I * GD_v(\sigma) * GD_v(\sigma)$

Note $h_{vu} = h_{vu}$ (can you prove this?) so there is no need to compute it.

We have just computed the local ‘Hessian (or second derivative) of the image. Notice we did it simply by taking the derivative of the derivative — i.e. by applying the derivative convolution twice. Of course, we could also have combined the two convolutions into a single kernel, too.

An second derivative operator that is sometimes used is the Laplacian of a Gaussian. The Laplacian of a function $f(x,y)$ is:

$L(f) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}$

Of course, we can mirror this exactly; we just compute for any smoothing kernel G (e.g. a Gaussian)

$L(G) = G*D_u*D_u + G*D_v*D_v$

If we now compute $I*L(G),$ it is not hard to show we have computed a discrete approximation to the Laplacian. Also, just as with Gaussian-smoothed derivatives, we can compute a Laplacian scale space.

### Texture

Two general types of visual phenomena are often referred to as texture, those consisting of regular, repeating patterns, as well as patterns that are consistent, but more stochastic in nature. From these different schools of thought about texture emerge different widely-used descriptors. Though each targets a particular type of texture, they have varying degrees of success in distinguishing other types of texture as well. All of these methods were developed for application to grey-scale images, but can also be used on the various channels of any desired color space.

Gabor filters take the approach of frequency analysis to characterize textures. They are a set of image filters consisting of the convolution of a Gaussian with even and odd sinusoids of various frequencies, scales, and orientations. The response is intended to be strong in the presence of repeating patterns with similar parameters. These filters have the advantage of characterizing orientation, allowing one to design a filter bank that produces a feature vector which is invariant to, or at least normalized by rotation. They are also efficiently computable in batches. Though they perform well on repeating patterns, though, they are less able to characterize stochastic textures. Matlab code for generating log-Gabor filters is available from Peter Kovesi.

Another very popular set of filters was developed by Leung and Malik and is known as the LM filter-bank (Leung & Malik, 2001). It consists of a set of edge and bar filters at 3 scales and 6 orientations, as well as several Gaussian and Laplacian of Gaussian filters. Schmid introduced a set of rotationally-invariant filters in Schmid, 2001, consisting of Gaussians modulated by cosine functions of varying frequencies in the radial direction. Varma and Zisserman introduced another set of filters, called maximum response (MR) filters, based on LM, but modified to add some invariance to rotations to the response. They record only the maximum response of the oriented filters at each scale, and only use one scale of Gaussian and LoG, for a total of 8 features. Illustrations of the LM, Schmid, and MR8 filter banks, as well as Matlab code for generating them, are available from the Oxford Visual Geometry Group page on texture classification.

The most commonly-used statistical texture descriptors are Haralick features, also known as grey-level co-occurrence matrices (GLCMs), or simply co-occurrence matrices. They consist, as their name suggests, of 2-dimensional histograms of the occurrence of one image intensity value (or range of values) with respect to another, at a given relative position, such as comparing each pixel with the pixel immediately to its right. In order to capture information about texture at different orientations or ranges, multiple matrices are usually constructed, each with a different relative position, and the appropriate resolution is largely heuristic. The matrices themselves are not generally used as a descriptor, though. Instead, a common set of several features of the distributions have been derived. One typically chooses some subset of these features, based on the usual trade-off of discriptiveness and speed of computation. The most commonly used features correspond to concepts of energy, correlation, contrast, variance, etc. These features are designed for and perform well on stochastic textures. They tend to have good performance on regular repeating patters as well, though, and are available in Matlab via the functions graycomatrix andgraycoprops.

Finally, in 2003, Varma and Zisserman showed surprisingly good performance using image patches directly as feature vectors, rather than the output of filter-banks. They demonstrate classification accuracy surpassing LM, Schmid, and MR filters when using image patches of the same or smaller support size in the same (albeit non-trivial) classification algorithm.

## Nonlinear Filters

De-noising is a common filtering task since many forms of data corruption can be viewed as noise applied to a signal. Though some types of noise are handled well by smoothing (e.g. zero-mean uniformly- or Gaussian-distributed additive random noise), other common corruption effect are non-linear, and correspondingly non-linear filters are needed to reasonably mitigate their effects.

Here are a couple examples:

 Original Gaussian-Corrupted Mean-filtered Gaussian-filtered Median-filtered

 Original Salt&Pepper-Corrupted Mean-filtered Gaussian-filtered Median-filtered

With the image pixel values scaled 0 to 1, the Gaussian-corrupted image was subjected to Gaussian-distributed noise of magnitude 0.1 with standard deviation 0.01, independently at each pixel. The salt and pepper noise had 5% of pixels randomly switched to 0 or 1. In both cases, Gaussian filtering was accomplished with a 3×3 filter with standard deviation 0.5, and mean- and median-filtering with 3×3 averaging and median filters respectively.

### Median Filtering

Median filtering is particularly simple-to-implement form of non-linear filter where the value of each pixel in an image is replaced with the median of those in a neighborhood around that pixel. It is particularly well-adapted to dealing with salt and pepper orshot noise, though it can also be applied to additive noise.

Wikipedia

Wikipedia

### Top-hat/Bottom-hat

The top/bottom-hat operators refer to a couple separate filters in different domains.

In signal processing, the top-hat function is 1 in a small neighborhood about the origin and 0 elsewhere. When applied in the spatial domain, it acts as a mean filter. If instead applied in the frequency domain (ie, Fourier domain), it acts as a band-pass filter. The bottom-hat filter is essentially the inverse, one minus the top-hat filter. In the frequency domain, it suppresses a band of frequencies, and it is not generally used in the spatial domain.

Wikipedia

In morphology, however, the top-hat operator is defined as the difference between the input signal and the result of an open operation on it. Similarly, the bottom-hat is the difference between the input and the result of a close operation.

Wikipedia


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CERN-PH-TH/2012-079 Dimensional Reduction of {\@fontswitch\mathcal{N}}=1, E_{8} SYM over SU(3)/U(1)\times U(1)\times\mathbb{Z}_{3} and its four-dimensional effective action

# Cern-Ph-Th/2012-079 Dimensional Reduction of \@fontswitchN=1, E8 SYM over SU(3)/U(1)×U(1)×Z3 and its four-dimensional effective action

Nikos Irges
National Technical University of Athens Zografou Campus, GR-15780 Athens Greece
E-mail: irges@mail.ntua.gr
Speaker.
Georgios Orfanidis
Theory Group, Physics Department CERN, Geneva, Switzerland
E-mail: orfanidi@mail.ntua.gr
On leave from Physics Department, National Technical University, Zografou Campus: Heroon Polytechniou 9, 15780 Zografou, Athens, Greece
George Zoupanos footnotemark:
Theory Group, Physics Department CERN, Geneva, Switzerland
E-mail: George.Zoupanos@cern.ch
###### Abstract

We present an extension of the Standard Model inspired by the Heterotic String. In order that a reasonable effective Lagrangian is presented we neglect everything else other than the ten-dimensional supersymmetric Yang-Mills sector associated with one of the gauge factors and certain couplings necessary for anomaly cancellation. We consider a compactified space-time , where is the nearly-Kähler manifold and is a freely acting discrete group on . Then we reduce dimensionally the on this manifold and we employ the Wilson flux mechanism leading in four dimensions to an gauge theory with the spectrum of a supersymmetric theory. We compute the effective four-dimensional Lagrangian and demonstrate that an extension of the Standard Model is obtained with interesting features including a conserved baryon number and fixed tree level Yukawa couplings and scalar potential. The spectrum contains new states such as right handed neutrinos and heavy vector-like quarks.

CERN-PH-TH/2012-079

Dimensional Reduction of , SYM over and its four-dimensional effective action

George Zoupanos footnotemark:

Theory Group, Physics Department CERN, Geneva, Switzerland

E-mail: George.Zoupanos@cern.ch

\abstract@cs

Proceedings of the Corfu Summer Institute 2011 School and Workshops on Elementary Particle Physics and Gravity September 4-18, 2011 Corfu, Greece

## 1 Introduction

Superstring Theory is often regarded as the best candidate for a quantum theory of gravitation, or more generally as a unified theory of all fundamental interactions. On the other hand the main goal expected from a unified description of interactions by the Particle Physics community is to understand the present day large number of free parameters of the Standard Model (SM) in terms of a few fundamental ones. Indeed the celebrated SM had so far outstanding successes in all its confrontations with experimental results. However its apparent success is spoiled by the presence of a plethora of free parameters mostly related to the ad-hoc introduction of the Higgs and Yukawa sectors in the theory.

It is worth recalling that various dimensional reduction schemes, with the Coset Space Dimensional Reduction (CSDR) [1, 2, 3] and the Scherk-Schwarz reduction [4] being pioneers, suggest that a unification of the gauge and Higgs sectors can be achieved in higher than four dimensions. The four-dimensional gauge and Higgs fields are simply the surviving components of the gauge fields of a pure gauge theory defined in higher dimensions, while the addition of fermions in the higher-dimensional gauge theory leads naturally after CSDR to Yukawa couplings in four dimensions. The last step in this unified description in high dimensions is to relate the gauge and fermion fields, which can be achieved by demanding that the higher-dimensional gauge theory is supersymmetric, i.e. the gauge and fermion fields are members of the same vector supermultiplet. Furthermore a very welcome additional input coming from Superstring Theory (for instance the heterotic string [5]) is the suggestion of the space-time dimensions and the gauge group of the higher-dimensional supersymmetric theory [6].

Superstring Theory is consistent only in ten dimensions and therefore the following crucial issues have to be addressed, (a) distinguish the extra dimensions from the four observable ones which are experimentally approachable, i.e. determine a suitable compactification which is a solution of the theory (b) reduce the higher-dimensional theory to four dimensions and determine the corresponding four-dimensional theory, which may subsequently be compared to observations. Among superstring theories the heterotic string [5] has always been considered as the most promising version in the prospect to find contact with low-energy physics studied in accelerators, mainly due to the presence of the ten-dimensional gauge sector. Upon compactification of the ten-dimensional space-time and subsequent dimensional reduction the initial gauge theory can break to phenomenologically interesting Grand Unified Theories (GUTs), where the SM could in principle be accommodated [7]. Moreover, the presence of chiral fermions in the higher-dimensional theory serves as an advantage in view of the possibility to obtain chiral fermions also in the four-dimensional theory. Finally, the original supersymmetry can survive and not get enhanced in four dimensions, provided that appropriate compactification manifolds are used. In order to find contact with the minimal supersymmetric standard model (MSSM), the non-trivial part of this scenario was to invent mechanisms of supersymmetry breaking within the string framework.

The task of providing a suitable compactification and reduction scheme which would lead to a realistic four-dimensional theory has been pursued in many diverse ways for more than twenty years. The realization that Calabi-Yau (CY) threefolds serve as suitable compact internal spaces in order to maintain an supersymmetry after dimensional reduction from ten dimensions to four [8] led from the beginning to pioneering studies in the dimensional reduction of superstring models [9, 10]. However, in CY compactifications the resulting low-energy field theory in four dimensions contains a number of massless chiral fields, known as moduli, which correspond to flat directions of the effective potential and therefore their values are left undetermined. The attempts to resolve the moduli stabilization problem led to the study of compactifications with fluxes (for reviews see e.g. [11]). In the context of flux compactifications the recent developments suggested the use of a wider class of internal spaces, called manifolds with -structure, which contains CYs. Admittance of an -structure is a milder condition as compared to -holonomy, which is the case for CY manifolds, in the sense that a nowhere-vanishing, globally-defined spinor can be defined such that it is covariantly constant with respect to a connection with torsion and not with respect to the Levi-Civita connection as in the CY case. An interesting class of manifolds admitting an -structure is that of nearly-Kähler manifolds. The homogeneous nearly-Kähler manifolds in six dimensions have been classified in [12] and they are the three non-symmetric six-dimensional coset spaces (see table 1 Appendix B) and the group manifold . In the studies of heterotic compactifications the use of non-symmetric coset spaces was introduced in [13, 15, 14] and recently developed further in [16, 17]. Particularly, in [17] it was shown that supersymmetric compactifications of the heterotic string theory of the form exist when background fluxes and general condensates are present. Moreover, the effective theories resulting from dimensional reduction of the heterotic string over nearly-Kähler manifolds were studied in [18].

Last but not least it is worth noting that the dimensional reduction of ten-dimensional supersymmetric gauge theories over non-symmetric coset spaces led in four dimensions to softly broken theories [19, 20].

Here we would like to present the significant progress that has been made recently concerning the dimensional reduction of the N=1 supersymmetric gauge theory resulting in the field theory limit of the heterotic string over the nearly-Kähler manifold . Specifically an extension of the Standard Model (SM) inspired by the heterotic string was derived [21]111For earlier attempts to obtain realistic models by CSDR see ref [2, 31, 38, 50].. In addition in order to make the presentation self-contained we present first a short review of the CSDR.

## 2 The Coset Space Dimensional Reduction.

Given a gauge theory defined in higher dimensions the obvious way to dimensionally reduce it is to demand that the field dependence on the extra coordinates is such that the Lagrangian is independent of them. A crude way to fulfill this requirement is to discard the field dependence on the extra coordinates, while an elegant one is to allow for a non-trivial dependence on them, but impose the condition that a symmetry transformation by an element of the isometry group of the space formed by the extra dimensions corresponds to a gauge transformation. Then the Lagrangian will be independent of the extra coordinates just because it is gauge invariant. This is the basis of the CSDR scheme [1, 2, 3], which assumes that is a compact coset space, .

In the CSDR scheme one starts with a Yang-Mills-Dirac Lagrangian, with gauge group , defined on a -dimensional spacetime , with metric , which is compactified to with a coset space. The metric is assumed to have the form

 (2.0)

where and is the coset space metric. The requirement that transformations of the fields under the action of the symmetry group of are compensated by gauge transformations lead to certain constraints on the fields. The solution of these constraints provides us with the four-dimensional unconstrained fields as well as with the gauge invariance that remains in the theory after dimensional reduction. Therefore a potential unification of all low energy interactions, gauge, Yukawa and Higgs is achieved, which was the first motivation of this framework.

It is interesting to note that the fields obtained using the CSDR approach are the first terms in the expansion of the -dimensional fields in harmonics of the internal space [2, 49]. The effective field theories resulting from compactification of higher dimensional theories might contain also towers of massive higher harmonics (Kaluza-Klein) excitations, whose contributions at the quantum level alter the behaviour of the running couplings from logarithmic to power [22]. As a result the traditional picture of unification of couplings may change drastically [23]. Higher dimensional theories have also been studied at the quantum level using the continuous Wilson renormalization group [24] which can be formulated in any number of space-time dimensions with results in agreement with the treatment involving massive Kaluza-Klein excitations. However we should stress that in ref [25] the CSDR has been shown to be a consistent scheme.

Before we proceed with the description of the CSDR scheme we need to recall some facts about coset space geometry needed for subsequent discussions. Complete reviews can be found in [2, 26].

### 2.1 Coset Space Geometry.

Assuming a -dimensional spacetime with metric given in (2) it is instructive to explore further the geometry of all coset spaces .

We can divide the generators of , in two sets : the generators of , , and the generators of , ( , and . Then the commutation relations for the generators of are the following:

 \specialhtml:\specialhtml:[Qi,Qj] = f  kijQk, [Qi,Qa] = f  biaQb, [Qa,Qb] = f  iabQi+f  cabQc. (2.0)

So is assumed to be a reductive but in general non-symmetric coset space. When is symmetric, the in (2.1) vanish. Let us call the coordinates of space , where is a curved index of the coset, is a tangent space index and defines an element of which is a coset representative, . The vielbein and the -connection are defined through the Maurer-Cartan form which takes values in the Lie algebra of :

Using (2.0) we can compute that at the origin , and . A connection on which is described by a connection-form , has in general torsion and curvature. In the general case where torsion may be non-zero, we calculate first the torsionless part by setting the torsion form equal to zero,

 Ta=dea+ωa b∧eb=0, (2.0)

while using the Maurer-Cartan equation,

 dea=12fa bceb∧ec+fa bieb∧ei, (2.0)

we see that the condition of having vanishing torsion is solved by

 ωa b=−fa ibei−12fa bcec−12Ka bcec, (2.0)

where is symmetric in the indices , therefore . The can be found from the antisymmetry of , , leading to

In turn becomes

 ωa b=−fa ibei−Da  bcec, (2.0)

where

The ’s can be related to ’s by a rescaling [2]:

 Da bc=(λaλb/λc)fa bc,

where the ’s depend on the coset radii. Note that in general the rescalings change the antisymmetry properties of ’s, while in the case of equal radii . Note also that the connection-form is -invariant. This means that parallel transport commutes with the action [26]. Then the most general form of an -invariant connection on would be

 ωa b=fa  ibei+Ja  cbec, (2.0)

with an -invariant tensor, i.e.

 δJ  acb=−f  dicJ  adb+f  aidJ  dcb−f  dibJ  acd=0.

This condition is satisfied by the ’s as can be proven using the Jacobi identity.

In the case of non-vanishing torsion we have

 Ta=dea+θa b∧eb, (2.0)

where

 θa b=ωa b+τa b,

with

 τa b=−12Σa bcec, (2.0)

while the contorsion is given by

 Σa  bc=Ta  bc+T  abc−T  acb (2.0)

in terms of the torsion components . Therefore in general the connection-form is

 θa b=−fa icei−(Da bc+12Σa bc)ec=−fa icei−Ga bcec. (2.0)

The natural choice of torsion which would generalize the case of equal radii [15, 27, 28], would be except that the ’s do not have the required symmetry properties. Therefore we must define as a combination of ’s which makes completely antisymmetric and -invariant according to the definition given above. Thus we are led to the definition

 Σabc≡2τ(Dabc+Dbca−Dcba). (2.0)

In this general case the Riemann curvature two-form is given by [2], [28]:

 Ra b=[−12f aibf ide−12G acbf cde+12(G adcG ceb−G aecG cdb)]ed∧ee, (2.0)

whereas the Ricci tensor is

 Rab=G cbaG ddc−G dbcG cda−G dcaf cdb−f diaf idb. (2.0)

By choosing the parameter to be equal to zero we can obtain the Riemannian connection . We can also define the canonical connection by adjusting the radii and so that the connection form is , i.e. an -gauge field [15]. The adjustments should be such that . In the case of where the metric is , we have and in turn . In the case of , where the metric is , we have to set and then to obtain the canonical connection. Similarly in the case of , where the metric is , we should set and take . By analogous adjustments we can set the Ricci tensor equal to zero [15], thus defining a Ricci flattening connection.

### 2.2 Reduction of a D-dimensional Yang-Mills-Dirac Lagrangian.

The group acts as a symmetry group on the extra coordinates. The CSDR scheme demands that an -transformation of the extra coordinates is a gauge transformation of the fields that are defined on , thus a gauge invariant Lagrangian written on this space is independent of the extra coordinates.

To see this in detail we consider a -dimensional Yang-Mills-Dirac theory with gauge group defined on a manifold which as stated will be compactified to , , :

 A=∫d4xddy√−g[−14Tr(FMNFKΛ)gMKgNΛ+i2¯¯¯¯ψΓMDMψ], (2.0)

where

 DM=∂M−θM−AM, (2.0)

with

 θM=12θMNΛΣNΛ (2.0)

the spin connection of , and

 FMN=∂MAN−∂NAM−[AM,AN], (2.0)

where , run over the -dimensional space. The fields and are, as explained, symmetric in the sense that any transformation under symmetries of is compensated by gauge transformations. The fermion fields can be in any representation of unless a further symmetry such as supersymmetry is required. So let , , be the Killing vectors which generate the symmetries of and the compensating gauge transformation associated with . Define next the infinitesimal coordinate transformation as , the Lie derivative with respect to , then we have for the scalar,vector and spinor fields,

 δAϕ = ξαA∂αϕ=D(WA)ϕ, δAAα = ξβA∂βAα+∂αξβAAβ=∂αWA−[WA,Aα], (2.0) δAψ = ξαAψ−12GAbcΣbcψ=D(WA)ψ.

depend only on internal coordinates and represents a gauge transformation in the appropriate representation of the fields. represents a tangent space rotation of the spinor fields. The variations satisfy, and lead to the following consistency relation for ’s,

 ξαA∂αWB−ξαB∂αWA−[WA,WB]=f  CABWC. (2.0)

Furthermore the W’s themselves transform under a gauge transformation [2] as,

 \specialhtml:\specialhtml:˜WA=gWAg−1+(δAg)g−1. (2.0)

Using (2.0) and the fact that the Lagrangian is independent of we can do all calculations at and choose a gauge where .

The detailed analysis of the constraints (2.0) given in refs.[1, 2] provides us with the four-dimensional unconstrained fields as well as with the gauge invariance that remains in the theory after dimensional reduction. Here we give the results. The components of the initial gauge field become, after dimensional reduction, the four-dimensional gauge fields and furthermore they are independent of . In addition one can find that they have to commute with the elements of the subgroup of . Thus the four-dimensional gauge group is the centralizer of in , . Similarly, the components of denoted by from now on, become scalars at four dimensions. These fields transform under as a vector , i.e.

Moreover act as an intertwining operator connecting induced representations of acting on and . This implies, exploiting Schur’s lemma, that the transformation properties of the fields under can be found if we express the adjoint representation of in terms of :

Then if , where each is an irreducible representation of , there survives an multiplet for every pair , where and are identical irreducible representations of .

Turning next to the fermion fields [2, 30, 31, 29] similarly to scalars, they act as intertwining operators between induced representations acting on and the tangent space of , . Proceeding along similar lines as in the case of scalars to obtain the representation of under which the four-dimensional fermions transform, we have to decompose the representation of the initial gauge group in which the fermions are assigned under , i.e.

 \specialhtml:\specialhtml:F=∑(ti,hi), (2.0)

and the spinor of under

 \specialhtml:\specialhtml:σd=∑σj. (2.0)

Then for each pair and , where and are identical irreducible representations there is an multiplet of spinor fields in the four-dimensional theory. In order however to obtain chiral fermions in the effective theory we have to impose further requirements. We first impose the Weyl condition in dimensions. In dimensions which is the case at hand, the decomposition of the left handed, say spinor under is

 σD=(2,1,σd)+(1,2,¯¯¯σd). (2.0)

So we have in this case the decompositions

 σd=∑σk, ¯¯¯σd=∑¯¯¯σk. (2.0)

Let us start from a vector-like representation for the fermions. In this case each term in (2.0) will be either self-conjugate or it will have a partner . According to the rule described in eqs.(2.0), (2.0) and considering we will have in four dimensions left-handed fermions transforming as . It is important to notice that since is non self-conjugate, is non self-conjugate too. Similarly from we will obtain the right handed representation but as we have assumed that is vector-like, . Therefore there will appear two sets of Weyl fermions with the same quantum numbers under . This is already a chiral theory, but still one can go further and try to impose the Majorana condition in order to eliminate the doubling of the fermionic spectrum. We should remark now that if we had started with complex, we should have again a chiral theory since in this case is different from non self-conjugate). Nevertheless starting with vector-like is much more appealing and will be used in the following along with the Majorana condition. Majorana and Weyl conditions are compatible in dimensions. Then in our case if we start with Weyl-Majorana spinors in dimensions we force to be the charge conjugate to , thus arriving in a theory with fermions only in .

An important requirement is that the resulting four-dimensional theories should be anomaly free. Starting with an anomaly free theory in higher dimensions, Witten [32] has given the condition to be fulfilled in order to obtain anomaly free four-dimensional theories. The condition restricts the allowed embeddings of into by relating them with the embedding of into , the tangent space of the six-dimensional cosets we consider [2, 33]. To be more specific if are the generators of into and are the generators of into the condition reads

 \specialhtml:\specialhtml:Tr(LaLb)=30Tr(TaTb). (2.0)

According to ref. [33] the anomaly cancellation condition (2.0) is automatically satisfied for the choice of embedding

 E8⊃SO(6)⊃R,\specialhtml:\specialhtml: (2.0)

which we adopt here. Furthermore concerning the abelian group factors of the four-dimensional gauge theory, we note that the corresponding gauge bosons surviving in four dimensions become massive at the compactification scale [32, 34] and therefore, they do not contribute in the anomalies; they correspond only to global symmetries.

### 2.3 The Four-Dimensional Theory.

Next let us obtain the four-dimensional effective action. Assuming that the metric is block diagonal, taking into account all the constraints and integrating out the extra coordinates we obtain in four dimensions the following Lagrangian :

 (2.0)

where and with the connection of the coset space, while is the volume of the coset space. The potential is given by:

 \specialhtml:\specialhtml:V(ϕ)=−14gacgbdTr(fCabϕC−[ϕa,ϕb])(fDcdϕD−[ϕc,ϕd]), (2.0)

where, and ’ s are the structure constants appearing in the commutators of the generators of the Lie algebra of S. The expression (2.0) for is only formal because must satisfy the constraints coming from eq.(2.0),

 \specialhtml:\specialhtml:fDaiϕD−[ϕa,ϕi]=0, (2.0)

where the generate . These constraints imply that some components ’s are zero, some are constants and the rest can be identified with the genuine Higgs fields. When is expressed in terms of the unconstrained independent Higgs fields, it remains a quartic polynomial which is invariant under gauge transformations of the final gauge group , and its minimum determines the vacuum expectation values of the Higgs fields [35, 36, 37]. The minimization of the potential is in general a difficult problem. If however has an isomorphic image in which contains in a consistent way then it is possible to allow the to become generators of . That is with suitable combinations of generators, a generator of and is also a coset-space index. Then

 ¯¯¯¯Fab = f  iabQi+f  cab¯¯¯ϕc−[¯ϕa,¯¯¯ϕb] = f  iabQi+f  cabQc−[Qa,Qb]=0

because of the commutation relations of . Thus we have proven that which furthermore is the minimum, because is positive definite. Furthermore, the four-dimensional gauge group breaks further by these non-zero vacuum expectation values of the Higgs fields to the centralizer of the image of in , i.e. [2, 35, 36, 37]. This can been seen if we examine a gauge transformation of by an element of . Then we have

 ϕa→hϕah−1,h∈H

We note that the v.e.v. of the Higgs fields is gauge invariant for the set of ’s that commute with . That is belongs to a subgroup of which is the centralizer of in .

In the fermion part of the Lagrangian the first term is just the kinetic term of fermions, while the second is the Yukawa term [38]. Note that since is a Majorana-Weyl spinor in ten dimensions the representation in which the fermions are assigned under the gauge group must be real. The last term in (2.0) can be written as

 \specialhtml:\specialhtml:LY=−i2¯¯¯¯ψΓa(∂a−12fibceiΓeΓaΣbc−12GabcΣbc−ϕa)ψ=i2¯¯¯¯ψΓa∇aψ+¯¯¯¯ψVψ,

where

 ∇a = −∂a+12fibceiΓeΓaΣbc+ϕa, (2.0) V = i4ΓaGabcΣbc, (2.0)

and we have used the full connection with torsion [28] given by

 θa  cb=−fa  ibeiαeαc−(Da  cb+12Σa  cb)=−fa  ibeiαeαc−Ga  cb (2.0)

with

 Da  cb=gad12[f  edbgec+f  ecbgde−f  ecdgbe] (2.0)

and

 Σabc=2τ(Dabc+Dbca−Dcba). (2.0)

We have already noticed that the CSDR constraints tell us that . Furthermore we can consider the Lagrangian at the point , due to its invariance under -transformations, and as we mentioned at that point. Therefore (2.0) becomes just and the term in (2.3) is exactly the Yukawa term.

Let us examine now the last term appearing in (2.3). One can show easily that the operator anticommutes with the six-dimensional helicity operator [2]. Furthermore one can show that commutes with the ( close the -subalgebra of ). In turn we can draw the conclusion, exploiting Schur’s lemma, that the non-vanishing elements of are only those which appear in the decomposition of both irreps and , e.g. the singlets. Since this term is of pure geometric nature, we reach the conclusion that the singlets in and will acquire large geometrical masses, a fact that has serious phenomenological implications. In supersymmetric theories defined in higher dimensions, it means that the gauginos obtained in four dimensions after dimensional reduction receive masses comparable to the compactification scale. However as we shall see in the next section this result changes in presence of torsion. We note that for symmetric coset spaces the operator is absent because are vanishing by definition in that case.

## 3 Dimensional Reduction of E8 over SU(3)/U(1)×U(1) and soft supersymmetry breaking

In this model we consider the coset space on which we reduce the ten-dimensional theory. To determine the four-dimensional gauge group, the embedding of in is suggested by the decomposition

 E8⊃E6×SU(3)⊃E6×U(1)A×U(1)B. (3.0)

Then, the surviving gauge group in four dimensions is

 H=CE8(U(1)×U(1))=E6×U(1)A×U(1)B.

The of decomposes under in the following way:

 248=1(0,0)+1(0,0)+1(3,12)+1(−3,12)+ 1(0,−1)+1(0,1)+1(−3,−12)+1(3,−12)+ 78(0,0)+27(3,12)+27(−3,12)+27(0,−1)+ ¯¯¯¯¯¯27(−3,−12)+¯¯¯¯¯¯27(3,−12)+¯¯¯¯¯¯27(0,1). (3.0)

The decomposition of the vector and spinor representations of (see table 1, Appendix B) is

 6υ=(3,12)+(−3,12)+(0,−1)+(−3,−12)+(3,−12)+(0,1)

and

 4s=(0,0)+(3,12)+(−3,12)+(0,−1)

respectively. Thus applying the CSDR rules we find that the surviving fields in four dimensions are three vector multiplets , (where is an , index and the other two refer to the two ) containing the gauge fields of . The matter content consists of three chiral multiplets (, , ) with an , index and three chiral multiplets (, , ) which are singlets and carry only charges.

To determine the potential we examine further the decomposition of the adjoint of the specific under , i.e.

 SU(3)⊃U(1)×U(1)
 8=(0,0)+(0,0)+6υ. (3.0)

Then according to the decomposition (3.0) the generators of can be grouped as

 \specialhtml:\specialhtml:QSU(3)={Q0,Q′0,Q1,Q2,Q3,Q1,Q2,Q3}. (3.0)

The non trivial commutator relations of generators (3.0) are given in the table 2 given in the Appendix B. The decomposition (3.0) suggests the following change in the notation of the scalar fields,

 \specialhtml:\specialhtml:(ϕI,I=1,…,8)⟶(ϕ0,ϕ′0,ϕ1,ϕ1,ϕ2,ϕ2,ϕ3,ϕ3). (3.0)

The potential of any theory reduced over is given in terms of the redefined fields in (3.0) by

 V(ϕ)=(3Λ2+Λ′2)(1R41+1R42)+4Λ′2R23 (3.0) +2R22R23Tr(ϕ1ϕ1)+2R21R23Tr(ϕ2ϕ2)+2R2


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# Difference between revisions of "2002 AMC 12B Problems/Problem 2"

The following problem is from both the 2002 AMC 12B #2 and 2002 AMC 10B #4, so both problems redirect to this page.

## Problem

What is the value of $(3x - 2)(4x + 1) - (3x - 2)4x + 1$ when $x=4$?

$\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 1 \qquad\mathrm{(C)}\ 10 \qquad\mathrm{(D)}\ 11 \qquad\mathrm{(E)}\ 12$

## Solution

By the distributive property,

$$(3x-2)[(4x+1)-4x] + 1 = 3x-2 + 1 = 3x-1 = 3(4) - 1 = \boxed{\mathrm{(D)}\ 11}$$

## See also

 2002 AMC 10B (Problems • Answer Key • Resources) Preceded byProblem 3 Followed byProblem 5 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 10 Problems and Solutions
 2002 AMC 12B (Problems • Answer Key • Resources) Preceded byProblem 1 Followed byProblem 3 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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I like to think that graduate school is about dreaming big things. Unfortunately, it can be difficult to dream when you do not sleep [citation needed]. In fact, without sleep, things you might want to be doing when you’re awake can become more difficult, too. Things like, say, not crashing your Navy boat or getting good research work done. (If you are in the mood for more factoids about the importance of sleep and/or are curious about “Sleep 101” at Harvard, look here). Moving into graduate school, I struggled (and still do) with sleeping when I wanted to sleep. I would key this as the most difficult aspect of my transition into graduate school.

As an undergraduate, I had an entirely different — perhaps entirely opposite — problem. Falling asleep was easy… often easier, in fact, than not falling asleep. My undergraduate sleep habits were an artifact of over-scheduling. Some of the worst of it, I remember was having the score for the opera Le comte Ory unexpectedly thrust onto my music stand. (I was obliged by an oboe scholarship.) We met for up to five hour pit orchestra rehearsals that were scheduled through 11 p.m. (!!!), after which I would leave to start my other homework.

During this time of my life, I employed a greedy sleep-scheduling policy. Because sleep was in short supply, sleeping was always a useful, “productive” activity. Thus, at the time it made sense to weigh sleep equally against any other task and schedule it first as often as possible. “I can sleep now and do that tomorrow.”

Perpetually sleep deprived, sleeping wherever and whenever I had the time and then rolling out of bed seven minutes before a 7:30 a.m. organic chemistry exam was workable — I almost never had problems falling asleep.

Graduating and moving on to a lifestyle where I could have some free time (?!??!?!), this sleep scheduling strategy worked very, very poorly. For me, without restrictions, greedy sleep-scheduling means sleeping 10 or 11 hours a night. During my first summer in East Lansing, I had very few restrictions. I rapidly regressed to severe night-owlishness. (See my disproportionate amount of nocturnal wandering around MSU during August 2017).

Very quickly, I realized severe night-owlishness was for my bad for my work output and my personal wellbeing. The solution, I reasoned, was simply to go to bed at a reasonable hour. I tried getting into bed at 9 or 10 p.m. and would sit there, wide awake, until 3 a.m., 4 a.m. or later. Of course, after finally falling asleep, then I’d sleep through noon. As this carried onward, I felt I had lost control of very basic and important aspects of my life.

So began a long trial-and-error search for interventions to get myself to sleep during the night and be awake during the day. I tried a lot of things. Early on, I focused on the falling asleep part of the equation. It was only a few months in that I realized that waking up was also an important target for intervention.

With respect to your sleep, there is no shortage of variables to play with. I tried a lot of things. In this blog, I’ll blitz through various sleep interventions I’ve tried, in no particular order. For anyone working to keep a workable sleep schedule, maybe you’ll come away with some new ideas to try out.

## 🔗 Alarm Clock

Through my long-time struggle with alarm clocks, I’ve identified three wakefulness states:

1. fully asleep (self explanatory),
2. lizard brain (e.g., only capable of limited decision-making based on immediate trade-offs), and
3. conscious (e.g., mostly or fully capable of rational decision-making).

The fortunate thing about my lizard brain is that it seems surprisingly well informed about what’s on the calendar (e.g., an organic chemistry exam in seven minutes that’s a five minute walk away). Even when my alarm doesn’t go off (usually because I set p.m. rather than a.m.), I am astounded by how well my lizard brain can pick up on when I need to be up to avoid catastrophe (e.g., 7:23 a.m. for the 7:30 a.m. organic chemistry exam).

The unfortunate thing about lizard brain is that it really doesn’t know and/or care about any conscious decisions I’ve made about when I want to be up. As you’ll see a little later, I have had zero success attempting to trick lizard brain into thinking I need to be up when I actually don’t. So, to influence my sleep patterns on the morning side of the equation, I need to figure out how to sneak past lizard brain to reach the conscious state. Easier said than done.

On the left, my long-time mortal enemy. This little Timex makes a crude, infuriating electronic beep. It gets the job done (transitioning from fully asleep to lizard brain), but it can make me feel like I’m hungover (especially if lizard brain doesn’t want to be up yet). Unfortunately, unless I need to be up lizard brain has become quite good at ignoring the Timex or hitting the snooze button. In high school, my mom, who from time to time had to leave early for work, once had to call the neighbor over to ring our doorbell in order to wake me up. Our neighbor could hear this alarm clock going off from the front door.

To try to bypass lizard brain and reach consciousness, I tried relocating the alarm clock into my living room. Lizard brain immediately became adept at rolling out of bed, staggering down the hall into the living room, hitting the snooze button (even when I tried placing the Timex under a colander with a bowl of cereal on top), and retreating back to bed. On some weekend mornings, I figure that I have hit the snooze button upwards of 12 times (quarter hour increments over three hours).

The traditional alarm clock clearly doesn’t succeed at putting me past lizard brain to full consciousness. The Timex is useful to make sure that I don’t miss important morning meetings because under these circumstances I have lizard brain’s full cooperation. (Because the Timex will fall back on a 9 Volt battery if there’s a power interruption, it is especially useful for important occasions.)

I’ve found more success easing past lizard brain into full consciousness using a clock-radio tuned to NPR. It’s actually an Amazon Alexa, which only recently gained this capability. I have it set to low volume, usually at 5:30 a.m. The thing is, though, I don’t expect to get up at 5:30 a.m. I actually expect wake up sometime between 6 a.m. and 7 a.m. Lizard brain doesn’t seem to care about quiet, soothing NPR voices (despite what they’re often actually talking about). I hypothesize that I pass through lizard brain stage, which is not sufficiently irked to turn off the stimulus, and, with continued application of the stimulus, eventually gain enough consciousness to decide to get out of bed. I’ve experienced one side effect: occasional extremely vivid dreams about whatever topic is being covered on NPR. I can live with that. My DIY sunrise lamp (details below) also seems to help me ease past lizard brain.

A few years ago, I tried using a music playlist as an alarm. Unfortunately, the playlist couldn’t be shuffled, so the first track to come on was always the same. Lizard brain very quickly learned to recognize it and hate it, leaving me right back in snooze button hell. After a month or so, I didn’t even like to hear the track any more when I was awake. Besides being too uniform of a stimulus, the music alarm might have been too loud and/or music might just be inherently too strong of a stimulus for lizard brain to tolerate. If you do go for a music-based alarm, you might play with the alarm volume, try making sure that your alarm always plays a different sequence of tracks (e.g., shuffle over a large selection or tuning into a music-oriented radio station), and experiment with different genres.

My smart watch has a wrist-buzzing alarm. It took about two days for lizard brain to learn to snooze it, just like the Timex. The wrist alarm’s not unpleasant, though, and might be useful in a situation where I wanted to transition from fully asleep to lizard brain without disturbing someone else.

## 🔗 Temperature

Sleeping in the heat sucks. I miss my big, heavy comforter dearly during the summer.

It took me a whole year to figure that one out. (It also took me a whole semester to realize that buses with the same number go different ways and I could tell the difference by the scrolling message panel on the front of the bus.)

In a sadly doomed campaign to cool by cross ventilation, I amassed an absurdly large collection of fans (six!!! for a one bedroom apartment).

Cross ventilation does nothing when it’s 80 F out at 2am. Fans that blow on you, though, help alittle. I have a desk fan on my nightstand and a pedestal fan at the foot of the bed. My mom tells me that damp washcloths and frozen water bottles are also better than nothing.

If you have it, though, AC is hard to beat. My apartment came with a Reagan-vintage wall AC unit. For a while, I was too cheap to run it. I have since become convinced that AC at bedtime is worth every penny. I bought a plug timer for the AC, which allows me to fine tune beyond the existing options (basically “on” and “off”). I’ll usually have it run until 1 or 2 a.m., then have the plug timer turn off to save a buck.

I redeployed my army of fans from cross ventilation duty to blow AC from the unit in the living room into my sleep area. I’d really like to do something more cleverer, but my loud Reagan-box runs on a 240 volt plug (you know, the kind you might plug your laundry machine or industrial grow light in with) so I can’t smartify it.

When I have the choice (i.e., during the winter when I can just open my window), I find that cooler temperatures (e.g., maybe 62 F) can help me fall asleep… but make crawling out of bed the next morning harder.

## 🔗 Air

Another feature of Reagan-era apartments: no ventilation.

This 24 hour plot of CO2 concentration in my bedroom came from my Awair glow. Even with the windows open, you can still see CO2 levels spike up a little when I’m at home. I don’t have the data anymore, but with the windows closed I’ve seen CO2 levels spike to 2500 ppm.

Too-high CO2 levels can make you feel off. Stale air accumulating in a sealed person-box for eight hours probably doesn’t help you rise and shine.

I bought some plants for my room, more because I like plants and Ikea things than for any other reason. (Plants respire, too.)

The real answer here is to leave the window open, which I do every night.

## 🔗 Bad Noise

An unfortunate trade-off accompanies leaving the window open: noise.

Even when I can’t sleep for other reasons, noise just makes me angry. Feeling angry doesn’t help me sleep.

Unfortunately, probably your best bet to manage noise problems is choosing where to live which (a) you probably don’t get to do very often and (b) you might not have full control over.

Beware: outdoor hallways, courtyards, public balconies, and fire escapes. That is, unless you’re into listening to gratuitously protracted conversations about asinine criteria to rank women (drunk) or how, man, isn’t it just wild that the capitalists beat the communists to the moon even though, you know, they were just like all about making melon ballers and whatever useless other shit (high). Despite, I’m still obsessed with my building’s balconies for   a e s t h e t i c   reasons.

Also potentially problematic: parking lots and driveways… i.e., woofer beats coming and going at all hours. This noise is usually transient and therefore, to me, much less aggravating.

This sound-proof-ness of walls is so, so important, and varies widely. You can check on this when you’re touring an apartment (and you should!). I got lucky: often, I only become aware of well-underway Miley-level Bangerz (tm) next door during the brief moments the front door opens to admit and/or expel party people. On the flip side, my boyfriend used to live in a dorm with walls so thin you (unavoidably) learned to recognize who was in the bathroom two doors down by the sound of their urination.

One of my very, very all-time favorite pieces of journalism comes from James Day of my hometown Corvallis Gazette-Times. In reports from 2012, 2013, 2014, and 2017 he stakes out student-dense neighborhoods — sometimes accompanied by grumpy long-time residents — in order to record a meticulous litany of livability complaints. I find the articles endlessly entertaining.

If you are dedicated enough, you might take a cue from James Day’s intrepid reporting and stake out your intended neighborhood (especially on a Friday or Saturday night) to check noise levels. Probably a better idea is to simply clue in to who else lives in the building/neighborhood your considering. Do “real people” live there? What about other grad students? Then it’s probably a safe bet noise-wise. You might also consult with current residents of the town you’re moving into. They will probably have plenty of opinions to share about the livability of various neighborhoods, and will steer you out of at least the inner circles of undergraduate hell.

“A Judgemental Map of East Lansing” [source]

Besides choosing somewhere quiet to live, you can also, you know, just ask your neighbors to keep the noise under control. As a freshman, my dorm room shared a wall with a wanna-be DJ. We worked out a system where I would bang on the wall and he’d turn down the volume 25%. Compromise!

Sometimes, at 2am, though, I really don’t feel like going outside to ask nicely. Yelling out the window is always an option. Last summer, to my great disorientation, I woke up in the middle of lizard brain yelling at noisemakers to quiet down. I didn’t know I had that voice inside me. From that point onwards, though, whoever was on party people babysitting duty made a sincere effort to shush the noisemakers and bring them back inside after they were done throwing up. I really did appreciate the courtesy.

A few times, I’ve considered calling in a noise violation but never quite felt adequately disturbed. A few important pointers for that:

1. Don’t call 911… call the non-emergency line instead (the number should be on your local police department’s website).
2. Really, you should ask (or yell) first… frivolous/biased calls to authorities are a real problem.

Also, sometimes I forget this, but just rolling over and putting a pillow over your head can help a lot, too.

## 🔗 Good Noise

For some people, some noises can be pleasant!

Your smart speaker can probably play white noise. I played with this some, but didn’t care for it.

My army of fans generates plenty of white noise. Some nights, I’m super into the fan noise. Other nights, I’m not. On those nights, it’s nice being able to turn everything off with a voice command.

I like listening to a brief newscast before sleep. I currently have “NPR News Now” and “BBC World Service” set up on my Alexa’s Flash Briefing

I used to listen to Garrison Keillor’s The Writer’s Almanac regularly before bed, but I have mixed feelings after his unacceptable conduct came to light. The Almanac briefly ceased to exist but has since returned to production. Attempts (1, 2) to fill the void, unfortunately, don’t seem to have stuck.

If there’s a podcast feed you like that’s not currently available as a Flash Briefing skill, it’s actually very straightforward to fix that. Setting it up takes maybe fifteen minutes, and no coding is involved.

First, you need to create a mirror RSS feed that only serves up the most recent episode of your podcast. Otherwise, Alexa will play through all the available episodes every time, which probably isn’t what you want. It’s easy (and free) to do this with a RSS to RSS “zap” from Zapier where you set your mirror RSS feed to only return one item (e.g., Max Records = 1). This way, when Alexa is done with the most recent episode of whatever, she won’t just keep playing the rest of the episodes in the feed for you.

Second, hop on over to Amazon to register a developer account, click a few buttons, and paste in your Zaipier RSS url. Once you save the skill, it should become available to “test” on your Alexa devices. You can comfortably continue using the skill like this without ever needing to officially publish it.

## 🔗 Dark

Disclaimer: I am not a lawyer. (I’m not even a (philosophy) doctor.) Consult an attorney and under no circumstances violate the law.

When I first moved into my neighborhood in East Lansing, I was thrown for a loop by a very out of place bit of apparent pro-MSU graffiti: green spray paint on a street light. It wasn’t until I walked by after dark that I realized that, duh, the homeowner whose window the bad streetlight violated had taken prerogative for some righteous unilateral action. Inspired, I aluminum foiled an exterior light at my building that shines directly in my bedroom. It’s been up for almost a year now with no problems.

Another option: plant a big shrubbery between your window and the offending light source. My mom did this when Oregon State put in a new barn-classroom thing right behind her house.

After consulting an attorney, do whatever you want. Just be sure that your righteous self-empowerment won’t pose a fire risk (e.g., maybe don’t duct tape a bare light bulb).

## 🔗 Sunrise Lamp

I combined an Ikea FADO table lamp with a smart light bulb to fashion a DIY sunrise lamp. The idea here is that the light runs on a timer to gently fade in before I want to get up, gradually shifting my body towards wakefulness. I quite like it, especially in combination with a clock radio alarm. It probably ranks among the more effective interventions I’ve tried. Plus, it doubles as a handy voice-controlled area light via Amazon Alexa.

A few details:

• I originally scheduled the bulb to fade in across linearly spaced brightness increments (e.g., 10% to 20% to 30% to 40% etc.). After a few days, I switched to exponentially spaced brightness increments, which I like much more (e.g., 1% to 2% to 4% to 8% etc.).
• I schedule the bulb to automatically turn off after two hours at full brightness (e.g., every day at 7 a.m.) so it doesn’t stay on all day if I don’t happen to be home to turn it off myself.
• Currently, Ikea doesn’t stock appear to stock the FADO table lamp in the US. I bought my lamp on eBay for about $50. The smart bulb cost about$15 on Amazon. Be careful if you try to pick up a lamp designed for a different region (e.g., Europe) because the power plug might be incompatible with your outlet.
• The free If This Than That service (IFTTT), which acts like a kind of handy duct tape for control of smart devices and web services, only offers time triggers at fifteen minute granularity. Luckily, the particular no-brand smart bulb I bought provides proprietary timing functionality that allowed for brightness adjustments at one minute increments. I ended up using a combination of IFTTT and proprietary triggers because the proprietary service only offered six daily schedulable events.
• If you end up with a smart bulb that doesn’t offer proprietary timing triggers (or, even better, trigger-able configurable fades), you might be able to achieve a reasonable fade in through clever acrobatics via another service called Stringify.

## 🔗 Workspace Lighting

There are a lot of things I like about our lab’s office space. Unfortunately, we have no windows.

So, I have lots of lights at my desktop try to get myself to physiologically register that it is indeed daytime when it is daytime, including:

• a happy lamp positioned to directly administer photons into my eyeballs,
• a LED desk lamp,
• bias lighting (a stick-on LED strip that plugs directly into my monitor for power via USB), and
• sick RGB underlighting on my keyboard (does nothing, but looks sweeeet).

The sinister flip side of this coin: tricking your physiology into thinking it’s daytime when it is actually not. Blue light from screens is thought to be great at this. Rebalancing the color composition of your screens, especially at night, might help.

Back when I had a mac, I used f.lux. Now I use Redshift, which had better support for Linux the last time I checked.

In the last few years, screen blue light management on screens has gone increasingly mainstream. Apple now natively supports it on their iOS and OS X devices under the moniker “Night Shift.” Other systems I’m less familiar with (windows, android, etc.) probably support blue light adjustments, too.

I also like to turn down my devices’ screen brightness at night. I just do it manually whenever the thought strikes me.

## 🔗 Pills and Such

Disclaimer: I am not a medical doctor. (I’m not even a philosophy doctor.) Consult your physician, yo.

Since I started playing around with pills and such RE: sleeping, my approach has switched from trying to concoct an elixir to use every night to lining up a series of pills and such, falling back to more extreme/disruptive options if I keep awake.

My go-to pill is melatonin. I’m not convinced it’s objectively very effective, but it tends to feel helpful. I’ll take a good placebo effect any day. If I’m already nice and drowsy at bedtime, I’ll just skip it.

When melatonin doesn’t cut it and I’m still awake after half an hour or an hour, about once or twice a week nowadays, I’ll fall back on Doxylamine Succinate aka Unisom or — if I’m feeling congested, too — drowsy allergy medication (Cetirizine Hydrochloride aka Zyrtec). These definitely make me feel groggy. Based on how I feel the next day, I suspect that these pills reduce the quality of my sleep. In particular, it can feel more difficult to break through sleep inertia and get out of bed. In my calculation, though, that beats tossing and turning through 5 a.m. (followed, of course) by several days of sleep schedule anarchy. Lately, I’ve only been using these every week or two.

## 🔗 Caffeine

I try not to have caffeine after noon, since its half life is 5 hours.

You can fine tune the dose of caffeine in your bean water by cutting regular ground coffee with decaf.

## 🔗 Ulysses Pact

On the left, behold strange remnants of a failed Ulysses pact on my kitchen windowsill. The idea was to explicitly incentivize a certain pattern of behavior. Success meant being in the shower (under running water) at or before an alarm went off at 7:00 a.m. I planned to associate success with donations to my charity cause of choice (e.g., moving quarters from one mug to the other). Originally, excessive failure (beyond a predefined threshold number of days per month) would result in contributions to repugnant causes and public self-shaming on Facebook. After I obviously didn’t follow through with these self-punishments, I considered simply destroying currency instead as a less extreme alternative. I never did get around burning dollar bills in my bathtub or flushing them down the toilet, though.

In my opinion, Ulysses pacts only works if your conscious brain is at bat. Lizard brain didn’t clue into any of this, so it didn’t work.

A Ulysses pact might work better with someone else in the loop to help hold you accountable. There are plenty of apps out there set up just for this type of thing.

A different self-compulsion trick that works much better: leaving my laptop overnight at my desk never fails to get me in to the office at a reasonable hour the next day.

## 🔗 Schedule Activities in the Morning

Lizard brain seems to be very well clued into the potential shame of tardiness or truancy for obligations like class, medical appointments, and volunteer activities. Last year, I intentionally scheduled myself to volunteer at 8am two days a week to help regulate my sleep schedule. However, this trick only really works for morning activities I’m truly obligated to follow through with, because lizard brain seems to know the difference.

## 🔗 Coaxing

I went through a phase where I thought I might be able to coax lizard brain out of bed with nice things.

At one point, I literally put out a pillow, granola, and water on my living room floor. The idea was that lizard brain would be forced out of bed by my horrible Timex alarm in the living room and then, finding accommodations so conveniently prepared, elect to deposit my body on the floor (where I would eventually gain consciousness) instead of returning to bed. This did not work.

In the coaxing department, I have also also tried

• having   M r .   C o f f e e   automatically brew a morning beverage,
• installing a fancy   r a i n f a l l   showerhead,
• using a transparent shower curtain to enjoy the   n i c e   morning light,
• installing an Amazon Echo Dot and portable speaker above the vanity cabinet for quality   s h o w e r     j a m s  , and
• leaving out my camera, because who could bear to miss the fleeting   a e s t h e t i c   morning light.

None of these worked, but at least I got some nice things I enjoy anyways out of it.

## 🔗 Screen Time

For me, doing pointless things on the internet when I’m supposed to be sleeping generally seems to be a symptom of not sleeping instead of a cause.

I do put some parental controls on myself on my work computer and on my iPad.

If I were really serious, I would have someone else set the password to make overrides more difficult. For me, mere inconvenience dissuades most pointless things well enough, though.

Even if the hours are all well-justified, it is depressingly eyeopening to quantify the time spent on touch devices. I assume most modern devices let you do this, and you should. Last year, I swapped out my iPhone for a Blackberry, which makes me feel a little less like I’m stuck on the hamster wheel.

Also helpful: unfollow every single person and page on Facebook (you are missing absolutely nothing, I promise), unsubscribe from email lists, and use the “mute” and the “retweet mute” buttons judiciously on your personal Twitter.

I tried relocating my device chargers out of the bedroom. I ended up deciding to return my iPad to the nightstand, though. Although not a totally ideal way to start the day, I find that checking my RSS feeds and making my daily deposit on the Instagram helps to incentivize and ease the transition from lizard brain to consciousness.

## 🔗 Self Tracking

Ah, quantified self. The thing you do for a while and then somehow end up not doing anymore.

I played around with the Garmin Vivosmart 3 wrist tracker and Existential Time Tracking in a little Moleskine notebook. All I got out of tracking everything I did all day, every day was a paper trail for the embarrassingly large amount of time I spent learning the Japanese alphabets. (Yes, plural.)

From the wrist tracker, I didn’t get much better utility. I learned I usually sleep nine hours a night and apparently I enter “deep sleep” for only like 5 minutes (whatever that means). I’m at a loss for how to apply these insights, though.

Personally, I find quantified self projects that unfold on the scale decades like Kazuo Yano’s very nifty wrist-accelerometer “life tapestry” and Steven Wolfram’s also nifty email/keylogger tapestry the most compelling. Unfortunately, although they do tell a cool story, I don’t find quantified self projects like these terribly prescriptive. Also, they take years to mature.

## 🔗 Make Your Bed and Clean Your Sink

Some woman who was being interviewed by Terri Gross on Fresh Air briefly convinced me that cleaning your sink and making your bed was life-changing Zen magic. (For some reason, I remember the guest as the totally kick-ass current director of the IMF, but this doesn’t really make sense.) Making your bed and cleaning your sink is nice, but isn’t life-changing Zen magic.

In my experience, though, walking to the grocery store and then cooking a proper dinner seems to correlate with easier sleep.

## 🔗 Smart Home

I use Amazon Echo to devices around the house. At night, it’s sometimes nice to be able to turn on/off the radio (especially, putting it on a sleep timer) or ask about the weather and other odds and ends without having to get up or use a screen.

I use a bunch of Meross smart plugs, which are cheap and work fine, around the house to control lamps and fans. I have an Awair Glow smart plug, too, but wouldn’t strongly recommend it because it’s been pretty spotty in terms of reliability lately. Again, it’s nice being able to turn devices (especially fans, see Air and Good Noise) on/off without having to get up or use a screen. I played around with lots of different control strategies (e.g., weather-based, indoor temperature based, etc.), but here are the recipes I found to be the most useful:

• three-part IFTTT -> Stringify -> IFTTT recipe for a voice-activated X hour timer for smart devices (1, 2, 3)
• that way, you can save electricity by not running fans all night
• IFTTT voice-activated trigger, which can control multiple devices at once (on/off)
• turn a device off every day at X am/pm (e.g., also to help save energy) (link)

Be aware that smart devices and smart plugs consume noticeable electricity.

## 🔗 Naps

When you are constantly sleep deprived, naps can be a useful tool. When you sleep 10 hours a night, napping just because you don’t feel like doing whatever you’re supposed to be doing right now is not so helpful and will leave you awake at ungodly hours of the day.

## 🔗 Working Late

If you are a busy beaver surfing an endless wave of imminent deadlines, working late can seem reasonable. Seriously considering the impact that staying up will have on my sleep for days to come (or weeks, if it really throws me off-kilter) has radically changed my calculus. Now, unless I truly cannot avoid it, I won’t work past 10 p.m. Often, working late can be avoided through some proactive effort. Even in a pinch, though, I’ll sometimes decide to just go to bed.

Also, beware and resist institutional forces (sometimes subtle) that encourage or enforce working late.

## 🔗 Age

This one isn’t really so much something you can do as something you can hope for. Maybe more reasonable sleep hours tend to accompanying advancing biological age?

Luckily, it turns out that graduate school is a great place to age rapidly.

## 🔗 Acute versus Chronic

Finally, a more conceptual point.

When I first started struggling with my sleep, I viewed it as a temporary crisis. I just had to right the ship, then I could forget about managing sleep.

Now, I see sleep as an aspect of my life that I manage in an ongoing fashion (as simply and passively as possible). I think that’s a healthier and more useful take on it.

## 🔗 Future Work

Stuff I didn’t do but that might be cool/useful anyways… always the easiest part to write up.

### 🔗 Exercise

Sometimes, exercise makes me feel tired. I haven’t played with this variable systematically, though.

Something as simple as just going out for a brief walk when you can’t sleep might help.

### 🔗 General Wellness

Idk. Take your vitamins. Maybe try a standing desk. (Sitting is the new smoking, y’all.)

Maybe just try doing more organized activities in general. In academia land, regularly scheduled seminars (with free food!) might fit the bill. If nothing else, your local Parks and Rec department almost certainly has lots of things to sign up for.

### 🔗 Do a Sleep Study

Sleep studies can help diagnose serious sleep disorders. Sounds expensive, but maybe your insurance covers it.

Bringing in professional help never hurts. I know at least one graduate student who has benefited from a sleep study.

### 🔗 Get a Prescription

Prescription sleep aids might be useful to you.

Also, depression and insomnia often live hand in hand. Prescription medications to treat depression (and/or other therapies) can go a long way. I’ve thought about this one a bit, and decided to keep it on the back burner unless I really start to feel persistently and unacceptably impaired.

### 🔗 Avoid Time Zone Changes

Traveling East/West always throws me off, but I haven’t actively tried to avoid it.

### 🔗 Blackout Curtains

Yup, this is a thing you can do.

I’m not super into curtains, though, because

1. I like morning light, and
2. I inevitably forget to open curtains during the day, which will kill the plants that live in my room.

### 🔗 Meditation

Counting sheep is straight-up bullshit. It makes me feel insane.

Other meditation practices (that weren’t obviously designed to keep restless kids occupied) might be of more benefit. I hear a lot of chatter about meditation apps, so that might be worth looking into.

### 🔗 Go to Boot Camp

I hear that a side effect of being yelled awake every morning is that your lizard brain becomes conditioned to insist on getting the hell out of bed every morning.

### 🔗 More Extreme/Inventive Alarm Clocks

There’s a lot of them. Google it.

### 🔗 Roll With It

When I was in high school, one of my friends transferred to an online program. Not having to keep any hours in particular, she started reverse cycling around the clock. Her wake-sleep cycles took about 26 hours. So, every 6 days or so she switched between being basically nocturnal to being basically diurnal.

Maybe for you it turns out something like that actually ends up being compatible with good quality of life (or, at least, feels better than wasting hours on vain attempts to fall asleep).

## 🔗 Let’s Chat

I would love to hear your stories about sleeping in grad school!!

I started a twitter thread (right below) so we can chat

Pop on there and drop me a line , make a comment , or leave your own tips & tricks for sleeping better


text:
# Squid- weight in newtons

1. ### starfish794

16
The heaviest invertebrate is the giant squid, which can have a weight of about 1.5 (imperial) tons spread out over its length of 64 feet. What is its weight in newtons?

I tried converting the tons to kilograms and multiplying by 9.8 but didn't get the right answer. Where should the 64 feet be used?

2. ### cd80187

38
How did you convert tons to kilograms? If you look online, an imperial ton is equivalent to 1016.04691 kg. So, you did have the right idea in going about the problem, but just use this number instead and it should work.

3. ### starfish794

16
There must be something else. I tried your plan and got 14935.88 which i'm told is still wrong. Thanks anyway.

21


text:
# Neper

John Napier after whom the unit is named

The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As is the case for the decibel and bel, the neper is a unit defined in the international standard ISO 80000. It is not part of the International System of Units (SI), but is accepted for use alongside the SI.[1]

## Definition

Like the decibel, the neper is a unit in a logarithmic scale. While the bel uses the decadic (base-10) logarithm to compute ratios, the neper uses the natural logarithm, based on Euler's number (e ≈ 2.71828). The level a ratio of two signal amplitudes or root-power quantities, with the unit neper, is given by[2]

${\displaystyle L=\ln {\frac {x_{1}}{x_{2}}}\mathrm {~Np} ,}$

where ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ are the signal amplitudes, and ln is the natural logarithm. The level of a ratio of two power quantities, with the unit neper, is given by[2]

${\displaystyle L={\frac {1}{2}}\ln {\frac {p_{1}}{p_{2}}}\mathrm {~Np} ,}$

where ${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$ are the signal powers.

In the International System of Quantities, the neper is defined as 1 Np = 1.[3]

## Units

The neper is defined in terms of ratios of field quantities — also called root-power quantities — (for example, voltage or current amplitudes in electrical circuits, or pressure in acoustics), whereas the decibel was originally defined in terms of power ratios. A power ratio 10 log r dB is equivalent to a field-quantity ratio 20 log r dB, since power in a linear system is proportional to the square (Joule's laws) of the amplitude. Hence the decibel and the neper have a fixed ratio to each other:[4]

${\displaystyle 1\ {\text{Np}}=20\log _{10}e\ {\text{dB}}\approx {\text{8.685889638 dB}}}$

and

${\displaystyle 1\ \mathrm {dB} ={\frac {1}{20}}\ln(10)\ \mathrm {Np} \approx {\text{0.115129255 Np}}.}$

The (voltage) level ratio is

{\displaystyle {\begin{aligned}L&=10\log _{10}{\frac {x_{1}^{2}}{x_{2}^{2}}}&{\text{dB}}\\&=10\log _{10}{\left({\frac {x_{1}}{x_{2}}}\right)}^{2}&{\text{dB}}\\&=20\log _{10}{\frac {x_{1}}{x_{2}}}&{\text{dB}}\\&=\ln {\frac {x_{1}}{x_{2}}}&{\text{Np}}.\\\end{aligned}}}

Like the decibel, the neper is a dimensionless unit. The International Telecommunication Union (ITU) recognizes both units. Only the neper is coherent with the SI.[5]

## Applications

The neper is a natural linear unit of relative difference, meaning in nepers (logarithmic units) relative differences add rather than multiply. This property is shared with logarithmic units in other bases, such as the bel.

The derived units decineper (1 dNp = 0.1 neper) and centineper (1 cNp = 0.01 neper) are also used.[6] The centineper for root-power quantities corresponds to a log point or log percentage, see Relative change and difference § Logarithmic scale.[7]

## References

1. ^ The International System of Units (SI) (9 ed.). International Bureau of Weights and Measures. 2019. pp. 145–146. Archived from the original on 2022-10-09.
2. ^ a b Letter symbols to be used in electrical technology – Part 3: Logarithmic and related quantities, and their units (International standard). International Electrotechnical Commission. 2002-07-19. IEC 60027-3:2002.
3. ^ Thor, A. J. (1994). New International Standards for Quantities and Units. Metrologia, 30(5), 517.
4. ^ Ainslie, Michael A.; Halvorsen, Michele B.; Robinson, Stephen P. (January 2022) [2021-11-09]. "A terminology standard for underwater acoustics and the benefits of international standardization". IEEE Journal of Oceanic Engineering. IEEE. 47 (1): 179-200 [Appendix B Decibel: Past, Present, and Future – Section D]. eISSN 1558-1691. ISSN 0364-9059. Retrieved 2022-12-20. [1] (22 pages)
5. ^ ISO 80000-3:2007 §0.5
6. ^ Glossary of Telecommunication Terms. General Services Administration, Federal Supply Service. 1980. p. 73.
7. ^ Karjus, Andres; Blythe, Richard A.; Kirby, Simon; Smith, Kenny (10 February 2020). "Quantifying the dynamics of topical fluctuations in language". Language Dynamics and Change. 10 (1): 86–125. doi:10.1163/22105832-01001200. S2CID 46928080.

## Works

Mirifici logarithmorum canonis constructio, 1825


text:
DocumentTools/Canvas/NewCanvas - Maple Help

DocumentTools[Canvas]

 NewCanvas
 create a canvas

 Calling Sequence NewCanvas(elements) NewCanvas(elements,script) NewCanvas(caption, options)

Parameters

 elements - list script - (optional) Script object output - (optional) output=records or output=string

Description

 • The NewCanvas command creates a canvas data structure that can be viewed in Maple via ShowCanvas, or in Maple Learn via ShareCanvas.   Script actions can be performed on a canvas to demonstrate activities or provide an interactive experience.
 • The first argument, elements, should be a list of the various components that should be included in the canvas.  Strings in the list are mapped to text components and math to math components, but Math, Text, Group, StaticPlot, and Button objects can also be used for further control and customization.  A sublist can be used to control layout, putting the contents of the sublist in columns.
 • If the script parameter is provided, those actions are merged into the canvas and will automatically be executed when the canvas is opened in Maple Learn
 • The default return value is an XML string that defines the canvas.  Using output=records causes the intermediate record-based data-structures to be returned instead of the canvas string.

Examples

 > $\mathrm{with}\left(\mathrm{DocumentTools}:-\mathrm{Canvas}\right):$
 > $\mathrm{cv}≔\mathrm{NewCanvas}\left(\left["Title",\left["Column 1","Column 2"\right],\left[\mathrm{Text}\left("Mixed %1",\mathrm{sqrt}\left({b}^{2}-4ac\right)\right),\mathrm{Math}\left(\mathrm{\pi }\right)\right],\mathrm{Int}\left(\mathrm{sqrt}\left(\mathrm{sqrt}\left({x}^{n}\right)+\frac{1}{\mathrm{\alpha }+{\mathrm{\beta }}^{\mathrm{\gamma }}}\right),x=-\mathrm{\infty }..\mathrm{\infty }\right),\left[\mathrm{Group}\left(\left["basics",\mathrm{%+}\left(1,1\right),\mathrm{%+}\left(2,2\right)\right]\right)\right]\right]\right):$
 > $\mathrm{ShowCanvas}\left(\mathrm{cv}\right)$
 > $\mathrm{ShareCanvas}\left(\mathrm{cv}\right)$

Compatibility

 • The DocumentTools[Canvas][NewCanvas] command was introduced in Maple 2021.


text:
• question_answer If ${{x}_{1}},{{x}_{2}},{{x}_{3}},\,\,\text{and }\,{{y}_{1}},{{y}_{2}},{{y}_{3}}$ are both in G.P. with the same common ratio, then the points $({{x}_{1}},{{y}_{1}}),$ $({{x}_{2}},\,{{y}_{2}})$ and $({{x}_{3}},\,{{y}_{3}})$[AIEEE 2003] A)            Lie on a straight line                  B)            Lie on an ellipse C)            Lie on a circle                             D)            Are vertices of a triangle

Taking co-ordinates as $\left( \frac{x}{r},\,\frac{y}{r} \right)$; $(x,\,y)\,\text{and}\,(xr,\,yr)$                    Above co-ordinates satisfy the relation $y=mx$, so lie on a straight line.