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## A Gallery of Cyclic Correlations

There are some situations in which the spectral correlation function is not the preferred measure of (second-order) cyclostationarity. In these situations, the cyclic autocorrelation (non-conjugate and conjugate versions) may be much simpler to estimate and work with in terms of detector, classifier, and estimator structures. So in this post, I’m going to provide plots of the cyclic autocorrelation for each of the signals in the spectral correlation gallery post. The exceptions are those signals I called feature-rich in the spectral correlation gallery post, such as LTE and radar. Recall that such signals possess a large number of cycle frequencies, and plotting their three-dimensional spectral correlation surface is not helpful as it is difficult to interpret with the human eye. So for the cycle-frequency patterns of feature-rich signals, we’ll rely on the stem-style (cyclic-domain profile) plots in the gallery post.

## On The Shoulders

What modest academic success I’ve had in the area of cyclostationary signal theory and cyclostationary signal processing is largely due to the patient mentorship of my doctoral adviser, William (Bill) Gardner, and the fact that I was able to build on an excellent foundation put in place by Gardner, his advisor Lewis Franks, and key Gardner students such as William (Bill) Brown.

## Comments on “Detection of Almost-Cyclostationarity: An Approach Based on a Multiple Hypothesis Test” by S. Horstmann et al

I recently came across the conference paper in the post title (The Literature [R101]). Let’s take a look.

The paper is concerned with “detect[ing] the presence of ACS signals with unknown cycle period.” In other words, blind cyclostationary-signal detection and cycle-frequency estimation. Of particular importance to the authors is the case in which the “period of cyclostationarity” is not equal to an integer number of samples. They seem to think this is a new and difficult problem. By my lights, it isn’t. But maybe I’m missing something. Let me know in the Comments.

## CSP Estimators: The FFT Accumulation Method

Let’s look at another spectral correlation function estimator: the FFT Accumulation Method (FAM). This estimator is in the time-smoothing category, is exhaustive in that it is designed to compute estimates of the spectral correlation function over its entire principal domain, and is efficient, so that it is a competitor to the Strip Spectral Correlation Analyzer (SSCA) method. I implemented my version of the FAM by using the paper by Roberts et al (The Literature [R4]). If you follow the equations closely, you can successfully implement the estimator from that paper. The tricky part, as with the SSCA, is correctly associating the outputs of the coded equations to their proper $\displaystyle (f, \alpha)$ values.

## CSP Blog Highlights

Welcome to the CSP Blog!

To help new readers, I’m supplying here links to the posts that have gotten the most attention over the lifetime of the Blog. Omitted from this list are the more esoteric topics as well as most of the posts that comment on the engineering literature.

Click the button below to follow the CSP Blog:

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You can see a pre-publication version of my latest CSP journal paper, on “tunneling”, here.

Here are the highlights:

## Comments on “Blind Cyclostationary Spectrum Sensing in Cognitive Radios” by W. M. Jang

I recently came across the 2014 paper in the title of this post. I mentioned it briefly in the post on the periodogram. But I’m going to talk about it a bit more here because this is the kind of thing that makes things a bit harder for people trying to learn about cyclostationarity, which eventually leads to the need for something like the CSP Blog.

The idea behind the paper is that it would be nice to avoid the need for prior knowledge of cycle frequencies when using cycle detectors or the like. If you could just compute the entire spectral correlation function, then collapse it by integrating (summing) over frequency $f$, then you’d have a one-dimensional function of cycle frequency $\alpha$ and you could then process that function inexpensively to perform detection and classification tasks.

## The Periodogram

I’ve been reviewing a lot of technical papers lately and I’m noticing that it is becoming common to assert that the limiting form of the periodogram is the power spectral density or that the limiting form of the cyclic periodogram is the spectral correlation function. This isn’t true. These functions do not become, in general, less random (erratic) as the amount of data that is processed increases without limit. On the contrary, they always have large variance. Some form of averaging (temporal or spectral) is needed to permit the periodogram to converge to the power spectrum or the cyclic periodogram to converge to the spectral correlation function (SCF).

In particular, I’ve been seeing things like this:

$\displaystyle S_x^\alpha(f) = \lim_{T\rightarrow\infty} \frac{1}{T} X_T(f+\alpha/2) X_T^*(f-\alpha/2), \hfill (1)$

where $X_T(f+\alpha/2)$ is the Fourier transform of $x(t)$ on $t \in [-T/2, T/2]$. In other words, the usual cyclic periodogram we talk about here on the CSP blog. See, for example, The Literature [R71], Equation (3).

## CSP Estimators: The Strip Spectral Correlation Analyzer

In this post I present a very useful blind cycle-frequency estimator known in the literature as the strip spectral correlation analyzer (SSCA) (The Literature [R3-R5]). We’ve covered the basics of the frequency-smoothing method (FSM) and the time-smoothing method (TSM) of estimating the spectral correlation function (SCF) in previous posts. The TSM and FSM are efficient estimators of the SCF when it is desired to estimate it for one or a few cycle frequencies (CFs). The SSCA, on the other hand, is efficient when we want to estimate the SCF for all CFs.

See also an alternate method of exhaustive SCF estimation: The FFT Accumulation Method.

## Second-Order Estimator Verification Guide

In this post I provide some tools for the do-it-yourself CSP practitioner. One of the goals of this blog is to help new CSP researchers and students to write their own estimators and algorithms. This post contains some spectral correlation function and cyclic autocorrelation function estimates and numerically evaluated formulas that can be compared to those produced by anybody’s code.

The signal of interest is, of course, our rectangular-pulse BPSK signal with symbol rate $0.1$ (normalized frequency units) and carrier offset $0.05$. You can download a MATLAB script for creating such a signal here.

The formula for the SCF for a textbook BPSK signal is published in several places (The Literature [R47], My Papers [6]) and depends mainly on the Fourier transform of the pulse function used by the textbook signal.

We’ll compare the numerically evaluated spectral correlation formula with estimates produced by my version of the frequency-smoothing method (FSM). The FSM estimates and the theoretical functions are contained in a MATLAB mat file here. (I had to change the extension of the mat file from .mat to .doc to allow posting it to WordPress.) In all the results shown here and that you can download, the processed data-block length is $65536$ samples and the FSM smoothing width is $0.02$ Hz. A rectangular smoothing window is used. For all cycle frequencies except zero (non-conjugate), a zero-padding factor of two is used in the FSM.

For the cyclic autocorrelation, we provide estimates using two methods: inverse Fourier transformation of the spectral correlation estimate and direct averaging of the second-order lag product in the time domain.

## A Gallery of Spectral Correlation

In this post I provide plots of the spectral correlation for a variety of simulated textbook signals and several collected communication signals. The plots show the variety of cycle-frequency patterns that arise from the disparate approaches to digital communication signaling. The distinguishability of these patterns, combined with the inability to distinguish based on the power spectrum, leads to a powerful set of classification (modulation recognition) features (My Papers [16, 25, 26, 28]).

In all cases, the cycle frequencies are blindly estimated by the strip spectral correlation analyzer (The Literature [R3, R4]) and the estimates used by the FSM to compute the spectral correlation function. MATLAB is then used to plot the magnitude of the spectral correlation and conjugate spectral correlation, as specified by the determined non-conjugate and conjugate cycle frequencies.

There are three categories of signal types in this gallery: textbook signals, collected signals, and feature-rich signals. The latter comprises some collected signals (e.g., LTE) and some simulated radar signals. For the first two signal categories, the three-dimensional surface plots I’ve been using will suffice for illustrating the cycle-frequency patterns and the behavior of the spectral correlation function over frequency. But for the last category, the number of cycle frequencies is so large that the three-dimensional surface is difficult to interpret–it is a visual mess. For these signals, I’ll plot the maximum spectral correlation magnitude over spectral frequency $f$ versus the detected cycle frequency $\alpha$ (as in this post).

A complementary gallery of cyclic autocorrelation functions can be found here.

## CSP Estimators: The Time Smoothing Method

In a previous post, we introduced the frequency-smoothing method (FSM) of spectral correlation function (SCF) estimation. The FSM convolves a pulse-like smoothing window $g(f)$ with the cyclic periodogram to form an estimate of the SCF. An advantage of the method is that is allows fine control over the spectral resolution of the SCF estimate through the choice of $g(f)$, but the drawbacks are that it requires a Fourier transform as long as the data-record undergoing processing, and the convolution can be expensive. However, the expense of the convolution can be mitigated by using rectangular $g(f)$.

In this post, we introduce the time-smoothing method (TSM) of SCF estimation. Instead of averaging (smoothing) the cyclic periodogram over spectral frequency, multiple cyclic periodograms are averaged over time. When the non-conjugate cycle frequency of zero is used, this method produces an estimate of the power spectral density, and is essentially the Bartlett spectrum estimation method. The TSM can be found in My Papers [6] (Eq. (54)), and other places in the literature.

## CSP Estimators: The Frequency-Smoothing Method

In this post I describe a basic estimator for the spectral correlation function (SCF): the frequency-smoothing method (FSM). The FSM is a way to estimate the SCF for a single value of cycle frequency. Recall from the basic theory of the cyclic autocorrelation and SCF that the SCF is obtained by infinite-time averaging of the cyclic periodogram or by infinitesimal-resolution frequency averaging of the cyclic periodogram. The FSM is merely a finite-time/finite-resolution approximation to the SCF definition.

One place the FSM can be found is in (My Papers [6]), where I introduce time-smoothed and frequency-smoothed higher-order cyclic periodograms as estimators of the cyclic polyspectrum. When the cyclic polyspectrum order is set to $n = 2$, the cyclic polyspectrum becomes the spectral correlation function, so the FSM discussed in this post is just a special case of the more general estimator in [6, Section VI.B].

## Signal Selectivity

In this post I describe and illustrate the most important property of cyclostationary statistics: signal selectivity. The idea is that the cyclostationary parameters for a single signal can be estimated for that signal even when it is corrupted by strong noise and cochannel interferers. Cochannel means that the interferer occupies a frequency band that partially or completely overlaps the frequency band for the signal of interest.

A mixture of signals, whether cochannel or not, is modeled by the simple sum of the signals, as in

$x(t) = s_1(t) + s_2(t) + \ldots + s_K(t) + w(t), \hfill (1)$

where $w(t)$ is additive noise. We can write this more compactly as

$x(t) = \displaystyle \sum_{k=1}^K s_k(t) + w(t). \hfill (2)$

## The Spectral Correlation Function for Rectangular-Pulse BPSK

In this post, I show the non-conjugate and conjugate spectral correlation functions (SCFs) for the rectangular-pulse BPSK signal we generated in a previous post. The theoretical SCF can be analytically determined for a rectangular-pulse BPSK signal with independent and identically distributed bits (see My Papers [6] for example or The Literature [R1]). The cycle frequencies are, of course, equal to those for the CAF for rectangular-pulse BPSK. In particular, for the non-conjugate SCF, we have cycle frequencies of $\alpha = k f_{bit}$ for all integers $k$, and for the conjugate SCF we have $\alpha = 2f_c \pm k f_{bit}$.

## The Spectral Correlation Function

Spectral correlation is perhaps the most widely used characterization of the cyclostationarity property. The main reason is that the computational efficiency of the FFT can be harnessed to characterize the cyclostationarity of a given signal or data set in an efficient manner. And not just efficient, but with a reasonable total computational cost, so that one doesn’t have to wait too long for the result.

Just as the normal power spectrum is actually the power spectral density, or more accurately, the spectral density of time-averaged power (variance), the spectral correlation function is the spectral density of time-averaged correlation (covariance). What does this mean? Consider the following schematic showing two narrowband spectral components of an arbitrary signal:

The sequence of shaded rectangles on the left are meant to imply a time-series corresponding to the output of a bandpass filter centered at $f-A/2$ with bandwidth $B.$ Similarly, the sequence of shaded rectangles on the right imply a time-series corresponding to the output of a bandpass filter centered at $f+A/2$ with bandwidth $B.$

## The Cyclic Autocorrelation for Rectangular-Pulse BPSK

The cyclic autocorrelation for rectangular-pulse BPSK can be derived as a relatively simple closed-form expression (see My Papers [6] for example or The Literature [R1]). It can be estimated in a variety of ways, which we will discuss in future posts. The non-conjugate cycle frequencies for the signal are harmonics of the bit rate, $k f_{bit}$, and the conjugate cycle frequencies are the non-conjugate cycle frequencies offset by the doubled carrier, or $2f_c + k f_{bit}$.

Recall that the simulated rectangular-pulse BPSK signal has $10$ samples per bit, or a bit rate of $0.1$, and a carrier offset of $0.05$, all in normalized units (meaning the sampling rate is unity). We’ve previously selected a sampling rate of $1.0$ MHz to provide a little physical realism. This means the bit rate is $100$ kHz and the carrier offset frequency is $50$ kHz. From these numbers, we see that the non-conjugate cycle frequencies are $k 100$ kHz, and that the conjugate cycle frequencies are $2(50) + k 100$ kHz, or $100 + k 100$ kHz.

## The Cyclic Autocorrelation

In this post, I introduce the cyclic autocorrelation function (CAF). The easiest way to do this is to first review the conventional autocorrelation function. Suppose we have a complex-valued signal $x(t)$ defined on a suitable probability space. Then the mean value of $x(t)$ is given by

$M_x(t, \tau) = E[x(t + \tau)]. \hfill (1)$

For stationary signals, and many cyclostationary signals, this mean value is independent of the lag parameter $\tau$, so that

$\displaystyle M_x(t, \tau_1) = M_x(t, \tau_2) = M_x(t, 0) = M_x(t). \hfill (2)$

The autocorrelation function is the correlation between the random variables corresponding to two time instants of the random signal, or

$\displaystyle R_x(t_1, t_2) = E[x(t_1)x^*(t_2)]. \hfill (3)$

## Creating a Simple CS Signal: Rectangular-Pulse BPSK

To test the correctness of various CSP estimators, we need a sampled signal with known cyclostationary parameters. Additionally, the signal should be easy to create and understand. A good candidate for this kind of signal is the binary phase-shift keyed (BPSK) signal with rectangular pulse function.

PSK signals with rectangular pulse functions have infinite bandwidth because the signal bandwidth is determined by the Fourier transform of the pulse, which is a sinc() function for the rectangular pulse. So the rectangular pulse is not terribly practical–infinite bandwidth is bad for other users of the spectrum. However, it is easy to generate, and its statistical properties are known.

So let’s jump in. The baseband BPSK signal is simply a sequence of binary ($\pm$ 1) symbols convolved with the rectangular pulse. The MATLAB script make_rect_bpsk.m does this and produces the following plot:

The signal alternates between amplitudes of +1 and -1 randomly. After frequency shifting and adding white Gaussian noise, we obtain the power spectrum estimate:

The power spectrum plot shows why the rectangular-pulse BPSK signal is not popular in practice. The range of frequencies for which the signal possesses non-zero average power is infinite, so it will interfere with signals “nearby” in frequency. However, it is a good signal for us to use as a test input in all of our CSP algorithms and estimators.

The MATLAB script that creates the BPSK signal and the plots above is here. It is an m-file but I’ve stored it in a .doc file due to WordPress limitations I can’t yet get around.

## Welcome to the CSP Blog!

Thank you for visiting the CSP blog.

The purpose of this blog is to talk about cyclostationary signals and cyclostationary signal processing (CSP). I’ve been working in the area for nearly thirty years, and over that time I’ve received a lot of requests for help with CSP code and algorithms. I thought it was time to put some of the basics out on the web so everybody could benefit. And I’m hoping to learn from you too.

In future posts, I’ll be showing how to create simple cyclostationary signals, write code for basic CSP estimators and detectors, and discuss papers in the literature.

What is cyclostationarity? It is a property of a class of mathematical models for a large number of signals in the world, most notably man-made modulated radio-frequency signals, like those used by cell phones, broadcast AM/FM/TV, satellites, WiFi modems, and many more systems. The mathematical models can be quite accurate, so we also say that cyclostationarity is a property of the real-world signals themselves.

The key aspect of the model is that cyclostationary signals have probabilistic parameters that vary periodically with time. Traditionally, signals are treated as stationary, which means their parameters do not vary with time. What are these ‘probabilistic parameters?’ Quantities like the mean value, the variance, and higher-order moments. These quantities are defined for both the time-domain signal and for its frequency-domain representation. So we have ‘temporal moments’ and ‘spectral moments.’ The second-order spectral moment is also called the spectral correlation function (SCF). The SCF is central to many CSP algorithms; a display of the SCF is shown above for a simple bandlimited binary phase-shift keyed (BPSK) signal.

The well-known noise- and interference-tolerance properties of CSP algorithms follow from the periodically time-varying nature of the signals’ parameters.

The most common difficulty I’ve encountered is when a researcher is developing a CSP estimator and is having trouble applying it to their data set. The researchers almost always skip the step of first applying the estimator to a signal with a perfectly known cyclostationary parameters. So, in the next post I’ll describe how to make the simplest digital CS signal, which has known temporal and spectral moments of all orders, so that we can test CSP estimators by comparing their output to the known correct result.

I encourage readers to point out my errors in the comments of my posts and to suggest topics they would like to see covered in future posts. Also, let me know about your application and interests so I can continue to learn too.

Click on the images of a gear and menu to find lists of posts as well as other pages on this blog.

I hope you enjoy your time here at the CSP Blog!

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# The Spectral Representation of Stationary Time Series.

## Presentation on theme: "The Spectral Representation of Stationary Time Series."— Presentation transcript:

The Spectral Representation of Stationary Time Series

Stationary time series satisfy the properties: 1.Constant mean (E(x t ) =  ) 2.Constant variance (Var(x t ) =  2 ) 3.Correlation between two observations (x t, x t + h ) dependent only on the distance h. These properties ensure the periodic nature of a stationary time series

and X 1, X 1, …, X k and Y 1, Y 2, …, Y k are independent independent random variables with where 1, 2, … k are k values in (0,  ) Recall is a stationary Time series

With this time series and We can give it a non-zero mean, , by adding  to the equation

We now try to extend this example to a wider class of time series which turns out to be the complete set of weakly stationary time series. In this case the collection of frequencies may even vary over a continuous range of frequencies [0,  ].

The Riemann integral The Riemann-Stiltjes integral If F is continuous with derivative f then: If F is is a step function with jumps p i at x i then:

First, we are going to develop the concept of integration with respect to a stochastic process. Let {U( ):  [0,  ]} denote a stochastic process with mean 0 and independent increments; that is E{[U( 2 ) - U( 1 )][U( 4 ) - U( 3 )]} = 0 for 0 ≤ 1 < 2 ≤ 3 < 4 ≤ . and E[U( ) ] = 0 for 0 ≤ ≤ .

In addition let G( ) =E[U 2 ( ) ] for 0 ≤ ≤  and assume G(0) = 0. It is easy to show that G( ) is monotonically non decreasing. i.e. G( 1 ) ≤ G( 2 ) for 1 < 2.

Now let us define: analogous to the Riemann-Stieltjes integral

Let 0 = 0 < 1 < 2 <... < n =  be any partition of the interval. Let. Let i  denote any value in the interval [ i-1, i ] Consider: Suppose that and there exists a random variable V such that *

Then V is denoted by:

Properties: 1. 2. 3.

The Spectral Representation of Stationary Time Series

Let {X( ):  [0,  ]} and {Y( ): l  [0,  ]} denote a uncorrelated stochastic process with mean 0 and independent increments. Also let F( ) =E[X 2 ( ) ] =E[Y 2 ( ) ] for 0 ≤ ≤  and F(0) = 0. Now define the time series {x t : t  T}as follows:

Then

Also

Thus the time series {x t : t  T} defined as follows: is a stationary time series with: F( ) is called the spectral distribution function: If f( ) = F ˊ ( ) is called then is called the spectral density function:

Note The spectral distribution function, F( ), and spectral density function, f( ) describe how the variance of x t is distributed over the frequencies in the interval [0,  ]

The autocovariance function,  (h), can be computed from the spectral density function, f( ), as follows: Also the spectral density function, f( ), can be computed from the autocovariance function,  (h), as follows:

Example: Let {u t : t  T} be identically distributed and uncorrelated with mean zero (a white noise series). Thus and

Graph:

Example: Suppose X 1, X 1, …, X k and Y 1, Y 2, …, Y k are independent independent random variables with Let 1, 2, … k denote k values in (0,  ) Then

If we define {X( ):  [0,  ]} and {Y( ):  [0,  ]} Note: X( ) and Y( ) are “random” step functions and F( ) is a step function.

Another important comment In the case when F( ) is continuous then

in this case Sometimes the spectral density function, f( ), is extended to the interval [- ,  ] and is assumed symmetric about 0 (i.e. f s ( ) = f s (- ) = f ( )/2 ) It can be shown that

Hence From now on we will use the symmetric spectral density function and let it be denoted by, f( ).

Linear Filters

Let {x t : t  T} be any time series and suppose that the time series {y t : t  T} is constructed as follows:: The time series {y t : t  T} is said to be constructed from {x t : t  T} by means of a Linear Filter. input x t output y t Linear Filter a s

Let  x (h) denote the autocovariance function of {x t : t  T} and  y (h) the autocovariance function of {y t : t  T}. Assume also that E[x t ] = E[y t ] = 0. Then::

Hence where of the linear filter

Note: hence

Spectral density function Moving Average Time series of order q, MA(q) Let  0 =1,  1,  2, …  q denote q + 1 numbers. Let {u t |t  T} denote a white noise time series with variance  2. Let {x t |t  T} denote a MA(q) time series with  = 0. Note: {x t |t  T} is obtained from {u t |t  T} by a linear filter.

Now Hence

Example: q = 1

Example: q = 2

Spectral density function for MA(1) Series

Spectral density function Autoregressive Time series of order p, AR(p) Let  1,  2, …  p denote p + 1 numbers. Let {u t |t  T} denote a white noise time series with variance  2. Let {x t |t  T} denote a AR(p) time series with  = 0. Note: {u t |t  T} is obtained from {x t |t  T} by a linear filter.

Now Hence

Example: p = 1

Example: p = 2

Example : Sunspot Numbers (1770-1869)

Autocorrelation function and partial autocorrelation function

Spectral density Estimate

Assuming an AR(2) model

A linear discrete time series Moving Average time series of infinite order

Let  0 =1,  1,  2, … denote an infinite sequence of numbers. Let {u t |t  T} denote a white noise time series with variance  2. –independent –mean 0, variance  2. Let {x t |t  T} be defined by the equation. Then {x t |t  T} is called a Linear discrete time series. Comment: A linear discrete time series is a Moving average time series of infinite order

The AR(1) Time series Let {x t |t  T} be defined by the equation. Then

where and An alternative approach using the back shift operator, B. The equation: can be written

Now since The equation: has the equivalent form:

The time series {x t |t  T} can be written as a linear discrete time series where and For the general AR(p) time series: [  (B)] -1 can be found by carrying out the multiplication

can be written: where Thus the AR(p) time series: Hence This called the Random Shock form of the series

can be written: where Thus the AR(p) time series: Hence This called the Random Shock form of the series

An ARMA(p,q) time series {x t |t  T} satisfies the equation: where and The Random Shock form of an ARMA(p,q) time series:

Again the time series {x t |t  T} can be written as a linear discrete time series namely where  (B) =[  (B)] -1 [  (B)] can be found by carrying out the multiplication

Thus an ARMA(p,q) time series can be written: where

The inverted form of a stationary time series Autoregressive time series of infinite order

An ARMA(p,q) time series {x t |t  T} satisfies the equation: where and Suppose that This will be true if the roots of the polynomial all exceed 1 in absolute value. The time series {x t |t  T} in this case is called invertible.

Then where or

Thus an ARMA(p,q) time series can be written: where This is called the inverted form of the time series. This expresses the time series an autoregressive time series of infinite order.

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Speed of sound

Physical phenomena

The speed of sound is the distance travelled per unit time by a sound wave as it propagates through an elastic medium. In dry air at 0 °C (32 °F), the speed of sound is 331.2 metres per second. At 20 °C (68 °F), the speed of sound is 343 metres per second, or a kilometre in 2.91 s or a mile in 4.69 s.

Relations


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# Periodic Inventory by Three Methods The units of an item available for sale during the year were ...

Periodic Inventory by Three Methods

The units of an item available for sale during the year were as follows:

 Jan. 1 Inventory 8 units @ $49 Feb. 17 Purchase 9 units @$51 Jul. 21 Purchase 8 units @ $53 Nov. 23 Purchase 5 units @$53

There are 18 units of the item in the physical inventory at December 31. The periodic inventory system is used. Round average unit cost to the nearest cent and final answers to the nearest whole dollar, if required.

a. Determine the inventory cost by the first-in, first-out method.
$b. Determine the inventory cost by the last-in, first-out method.$

c. Determine the inventory cost by the weighted average cost method.
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# A Fast Algorithm for the Convolution of Functions with Compact Support Using Fourier Extensions

Xu, Kuan, Austin, Anthony P., Wei, Ke (2017) A Fast Algorithm for the Convolution of Functions with Compact Support Using Fourier Extensions. SIAM Journal on Scientific Computing, 39 (6). A3089-A3106. ISSN 1064-8275. E-ISSN 1095-7197. (doi:10.1137/17M1114764) (KAR id:58502)

We present a new algorithm for computing the convolution of two compactly supported functions. The algorithm approximates the functions to be convolved using Fourier extensions and then uses the fast Fourier transform to efficiently compute Fourier extension approximations to the pieces of the result. The complexity of the algorithm is $\mathcal{O}\bigl(N (\log N)^2\bigr)$, where $N$ is the number of degrees of freedom used in each of the Fourier extensions.


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Dear friends, Please read our latest blog post for an important announcement about the website. ❤, The Socratic Team

# How do you determine the intervals where f(x)=3x^2-x^3/3 is concave up or down?

Then teach the underlying concepts
Don't copy without citing sources
preview
?

#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

1
Ben Share
Jun 19, 2018

Determine where the second derivative is positive or negative. If negative, then it is concave down. If positive, concave up. If zero, then it's an inflection point.

#### Explanation:

While the first derivative tells us the slope of a function at a given point, the second derivative gives us the concavity of the function.

When the second derivative is positive, the function has a 'concave up' concavity to it. When negative, it's concave down. The point where this changes is the point of inflection. The point of inflection is equal to when the second derivative is equal to zero.

Let's work with the function for a bit to determine the second derivative:

$f \left(x\right) = 3 {x}^{2} - \frac{{x}^{3}}{3}$

$f ' \left(x\right) = \textcolor{red}{2} \cdot 3 x - \textcolor{red}{3} \frac{{x}^{\textcolor{red}{2}}}{3}$

$f ' \left(x\right) = 6 x - {x}^{2}$

$f ' ' \left(x\right) = 6 - \textcolor{red}{2} x$

Now that we know the second derivative, we can calculate the points of inflection to determine the intervals for concavity:

$f ' ' \left(x\right) = 0 = 6 - 2 x$

$2 x = 6$

$x = 3$

We only have one inflection point, so we just need to determine if the function is concave up or down on either side of the function:

$f ' ' \left(2\right) = 6 - 2 \left(2\right)$

$f ' ' \left(2\right) = 2 \therefore \text{concave up}$

$f ' ' \left(4\right) = 6 - 2 \left(4\right)$

$f ' ' \left(2\right) = - 2 \therefore \text{concave down}$

The concavity for our function can be described as such:

$x < 3$ Concave up
$x > 3$ Concave down

We can see that when graphing the function as well:

graph{3x^2-(x^3)/3 [-5, 10, -2, 38]}

Then teach the underlying concepts
Don't copy without citing sources
preview
?

#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

1
Jun 19, 2018

The function is concave up in the interval $\left(- \infty , 3\right)$ and concave down in the interval $\left(3 , + \infty\right)$

#### Explanation:

The function is

$f \left(x\right) = 3 {x}^{2} - {x}^{3} / 3$

This a polynomial function continous and derivable on $\mathbb{R}$.

Calculate the first and second derivatives

$f ' \left(x\right) = 6 x - {x}^{2}$

$f ' ' \left(x\right) = 6 - 2 x$

Let $f ' \left(x\right) = 0$

$\implies$, $6 - 2 x = 0$

$\implies$, $x = 3$

There is an inflection point at $\left(3 , 18\right)$

Build a variation chart to determine the concavities

$\textcolor{w h i t e}{a a a a}$$\text{ Interval }$$\textcolor{w h i t e}{a a a a}$$\left(- \infty , 3\right)$$\textcolor{w h i t e}{a a a a}$$\left(3 , + \infty\right)$

$\textcolor{w h i t e}{a a a a}$$\text{ sign f''(x) }$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a a a a a a a a}$$-$

$\textcolor{w h i t e}{a a a a}$$\text{ f(x) }$$\textcolor{w h i t e}{a a a a a a a a a a a}$$\cup$$\textcolor{w h i t e}{a a a a a a a a a}$$\cap$

The function is concave up in the interval $\left(- \infty , 3\right)$ and concave down in the interval $\left(3 , + \infty\right)$

graph{3x^2-x^3/3 [-9.73, 18.75, -0.11, 14.13]}

• An hour ago
• An hour ago
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• An hour ago
• 5 minutes ago
• 5 minutes ago
• 27 minutes ago
• 28 minutes ago
• 34 minutes ago
• An hour ago
• An hour ago
• An hour ago
• An hour ago
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text:
## Monday, October 19, 2009 ... //

### Klaus: Notes on the economic analysis of the global warming issue

Translation from Czech: L.M.

Instead of participating in the addition of further arguments of the philosophical i.e. unquantifiable type - which is what currently dominates the "ideological clash" between the champions of freedom on one side and the environmentalists and advocates of non-freedom on the other side, let us focus on some elementary economic data, hypotheses, theories, and models which underlie these big "confrontations". Maybe, exactly these considerations will convince a reader or two. Otherwise, the discussion resembles a "dialog of the deaf". It's self-evident that only a small wedge of all these problems has been selected for this article.

It is more than obvious that we are objects in a strange game that is being played with all of us. It is more than obvious that among those who are deciding about these issues on behalf of us, i.e. among the politicians, no genuine dialog about global warming or its costs is taking place - especially not about the costs of mitigation (and I know quite a bit about the situation). It is also more than obvious that a majority of the world's (and even our) politicians - without dedicating any time to a serious investigation of these questions - has concluded that the global warming game is an easy, politically correct, and personally highly beneficial card (which moreover guarantees that they're not and they will never be responsible for the costs of this fight because the costs will be covered by future generations).

My book Blue, Not Green Planet (Prague 2007), published as The Blue Planet in Green Shackles (CEI 2007) in English, was dedicated to this topic two years ago. The immediate driver that made me write that book was my strictly scholarly disagreement with the unreputable approach of the environmentalist economists (Stern, Garnaut) and, last but not least, their depreciation of the importance of the discount rate for all intertemporal decisions (those that are concerned with any comparisons of different time epochs). Climatological chimeras of some environmentalists are an entirely different issue - one that is, nevertheless, equally lacking in earnestness.

When the book was being written - at the end of 2006 and in early 2007 - I was inspired, among other things, by Nordhaus's critical review of the Stern Report (Nordhaus 2007, Stern 2007, Klaus 2007, pp. 76-78). The results of Stern's calculations were - according to Nordhaus as well as myself - crucially influenced by the choice of the discount rate by Nicholas Stern's author team - namely the discount rate close to zero - which is what they continue to assume, despite all the subsequent criticism. In other words, Stern effectively fails to distinguish the temporal dimension of this whole problem: the year 2100 is equally important, it has the same weight or value as the year 2000 which is completely absurd. At any rate, it completely contradicts the standard reasoning of economics. Later, my discussion about these matters was elaborated upon in my essay written with Daniel Tříska, On the Criticism of the Usage of the Solidarity and Discrimination Concepts in the Intertemporal Analysis of the So-Called Global Problems (Klaus & Tříska 2007, Klaus 2009).

In 2008, Nordhaus published an extensive book, "A Question of Balance: Weighing the Options on Global Warming Policies" which extended as well as quantified his original argumentation. I was pleased by the title of Chapter 1: "Summary for Concerned Citizens". He added a comment that most technical articles about global warming (especially the publications of the U.N., the so-called IPCC) begin with a "Summary for Policymakers" but he was more interested in the "participating" people with some interests, i.e. the "normal" people. Building on his mathematical model which has been optimized and improved for decades, his book offered 15 alternative scenarios assuming different responses to the climate change (starting from doing nothing and ending with extreme policies advocated by Stern and Gore). In this model-based framework, he has made a set of convincing conclusions about the complete inappropriateness of a majority of the policies suggested by the environmentalists. He has determined that they would extensively harm the world's economy and the economic development in general. The costs of "mitigation", i.e. the costs of policies meant to reduce the CO2 emissions, would vastly exceed their possible positive effects. A very convincing summary of similar issues can be found in the new, 2009 article by another important participant of these discussions, Indur Goklany, "Discounting the Future" (Goklany 2009a).

It has to be emphasized that Nordhaus studies exclusively economic consequences of global warming that he compares with the costs of the recently proposed mitigation policies designed to stabilize the global mean temperature (but he doesn't notice e.g. the very meaninglessness of the attempts to "stabilize" the global temperature which is historically very unstable) [1]. It's equally desirable to note that Nordhaus essentially adopts - without any questions or doubts - the climatological aspects of the U.N. Intergovernmental Panel for Climate Change (IPCC) which is, in my opinion, a mistake but even in the context of my present article, I am exclusively focusing on the economic argumentation, just like he did. The climatological discussion is to be postponed to another occasion (but I refer to Chapter VI of my book, in the Czech language to the book of Dr Kutílek from 2008, to many contributions in the CEP booklet from 2008, to Motl in 2008, to Svoboda from 2003; in the literature written in English, to Singer 2008, Curtin 2009, Essex and McKitrick 2007, Monckton 2007, Plimer 2009, and many others).

At any rate, it's self-evident that the global warming that is underway right now - if it is underway at all - is very moderate, hardly measurable, highly irregular, and having no immediate relationship with the evolution of the CO2 emissions. Bob Carter, a leading Australian climate scientist, gave a talk at the climate change conference organized by the Heartland Institute in March 2009 and he very resolutely said that "since 1940, the planet was getting warmer for 19 years and cooler for 49 years. Consequently, the current global mean temperature is approximately equal to the value in 1940" (Carter 2009, page 19). At the same moment, we also see that "despite the vanishing growth of the temperature in the most recent 68 years, the CO2 concentrations increased by more than 20%" (ibid). "Despite the immense, two-decades-long efforts and despite the enormous expenses, there has been no success in attempts to prove that the human influence on the climate is measurable. It turns out that this effect is lost in the noise and the variability of the natural climate changes" (page 23). There are many texts of this kind but this particular one seems to be their apt summary to me.

Nordhaus avoids this discussion. He begins with the economic argument that the discount rate - a quantity that is needed for any intertemporal calculations - should be close to 5.5% which is, first of all, very different from zero. According to Nordhaus, the progression of time is therefore a completely crucial argument. Building on the standard, i.e. not extremal or "fringe" economic theory, he is convinced that the discount rate has to express the real capital revenues because investments attempting to slow down climate change don't exist in the vacuum. They must compete with investments of all other kinds. Because of this very reason, the economists, whenever they analyze investments (which include more general interventions and policies of all conceivable types), work with the concept of "opportunity costs" which is one of the pillars of the economic way of reasoning in general.

Using his model (called DICE-2007 - where DICE stands for "Dynamical Integrated Climate & Economy" model), Nordhaus decides that the "optimal policy" is smaller than the alternative policies by orders of magnitude (Nordhaus, pp. 82-83):

 Type of policy global costs of decelerating of warming in 250 years, in 2005 constant prices Stern's $27.7 trillion Gore's$33.9 trillion Nordhaus's $2.2 trillion In the table, the costs of Nordhaus's "optimal" and two alternative policies are based on the acceptance of an important yet questionable assumption which must be mentioned: the CO2 concentration will be capped by 2.5 times the pre-industrial levels [2]. This number has a lot of consequences. If we wanted the CO2 concentration to be kept at 2 times the pre-industrial level, the costs of the "optimal" policies would nearly double, to$3.95 trillion. But if the limit were 1.5 times the pre-industrial levels, they would increase ten-fold, to $27.2 trillion. (Approximately equal coefficients also multiply the costs of the alternative policies which brings us to hundreds of trillions.) As you can see, the differences are enormous and the dependence is highly nonlinear. I am mentioning these numbers for us to start to suspect that random figures emitted by policians' mouths at one EU, U.N., G8, or G20 summit after another (minus 20, 30, 50, 80% in CO2 emissions) are hugely important and they have consequences. However, the politicians are ignorant about this point, sometimes deliberately so. It is obvious that all these calculations are just a modeler's exercise (however not one based on a U.N. or EU office but rather one created by one of the most prominent economists of the current era whose economics textbook - that he co-authored with the famous Paul Samuelson - has been used for decades, affecting the maximum number of students in the whole world). It's also clear that this model - much like all similar models used by climatologists - is using a sequence of questionable assumptions. And I repeat that Nordhaus doesn't question - in my eyes, highly disputable - conclusions of the IPCC at all. Nevertheless, his basic assumptions build on elementary logic and they are especially rooted in economic theories of the standard type. Robert Solow, an economics memorial Nobel prize winner, who is otherwise pretty close to the environmentalist attitudes, considers Nordhaus's work on these topics as "common sense, realism, and a precision economics analysis" (1993, page 164). In the first 2009 issue of World Economics magazine (from January-February), Canadian economist Prof Dore defends Stern's model and sharply attacks Nordhaus in his article "A Question of Fudge". Dore's text is seemingly very technical, inaccessible to non-economists and even non-theoretical economists. But if we separate its mathematical apparatus from the economists' jargon, the whole article boils down to a dispute of aprioristic assumptions, even though it pretends to be an utterly neutral scientific debate. Let's see why it is so in some detail: • Dore criticizes Nordhaus e.g. for the fact that his calculations "only consider things that are valuable for human beings but they neglect moral values such as the health of the ecosystem" (page 95). If I ignore the observation that the "health of the ecosystem" is a very strange and ill-defined notion, it remains incomprehensible why the health of the ecosystem is considered to be a "moral value". (The term "value" seems to be overused which I consider to be a conceptual mistake and an escape from the analysis.) But what is a "moral value"? In my Blue Planet, I tried to explain that economists can't see any "objective" values that are independent of the human being (in the economic sense of the words) because the value is always subjective. Values without a human being or subject, without a human vantage point, are meaningless and cannot be defined. Who can determine the magnitude of a "moral value"? What is the mechanism to find this quantity? To avoid misunderstandings, it is completely unquestionable that the ecosystem or - using the normal vocabulary - the world around us deserves (and gets) extraordinary attention of the people, for reasons that are both aesthetic as well as entirely "egotistic" (related to the care about one's own future and the future of the human race) [3] but the health of the ecosystem can't have a "moral value". One can't work with it. • Dore can't accept Nordhaus's thesis that the investment costs attributed to the deceleration of warming have to "compete with all other investment expenses" (page 96). In fact, he misunderstands even the fact that the "public costs" and the "private costs" may be added with each other (which betrays his entirely statist world view). He is not interested whether the revenues (benefits) of the climate investments are high or low. He is essentially concerned with none of these things because his deciding factor is an unstructured, unspecified, unquantified (and unquantifiable) "protection of the interest of the future generations" (page 100). According to Dore, "Nordhaus forgets that the climate change is a public domain of a higher order" (page 101) which has a higher priority than anything else. However, Nordhaus is as aware of the rules of the public domain (and externalities) as Dore, if not more so [4]. The difference is that Nordhaus also knows what every serious economist does: that the public domain is less extensive than non-economists (and politicians) think or claim - and the same comment holds for the externalities, after all. Nordhaus also knows that the economists don't distinguish the public domain of higher or lower orders. • As soon as one adopts Dore's aprioristic argument which is apparently above everything else, then any discussion about the size of any parameter of any model is completely useless and is nothing else than an artificial "shadow puppet show" whose only purpose it to make a less qualified reader believe that the debate is scientifically serious. • Dore wants to twist the situation by one more statement that he is presenting a general discussion within economics between the "new classical economics" (based on the the hypothesis of rational expectations that are adapting to the exogenous shocks) on one side and the Cambridge school (of price rigidities, asymmetric information, the "principal agent" problem, moral hazard, etc.). This point seems to be just another trick or finesse. These things definitely do play a role in the economists' heads but I don't think that Nordhaus could be considered to be a champion of new classical economics - by anyone else except for the environmentalist ideologues. • Many of us do not think that we should acknowledge that the state has the right to control, regulate, or organize anything in the economy. But Dore wants much more to be inserted to the core of our reasoning (page 103), namely "the stewardship of the entire biosphere", which is a nearly incredible ambition based on the "pretended knowledge" (Hayek) that the mankind doesn't possess and will never possess. • Dore authoritatively claims that the "opportunity costs don't exist" because in the global warming case, "there's no problem of the choice" (page 104)! It exists elsewhere but not here? We are obviously facing circular reasoning here. It makes any discussion useless. • Dore is primarily unhappy that Nordhaus claims that "climate change is just a minor perturbation" (using Dore's words). I also think it is but it is up to Dore to present serious arguments that it should be viewed as a "large perturbation". The answer will be strongly influenced by our choice of the discount rate. In the previously quoted article by Goklany (Goklany 2009a), several numerical conclusions of the Stern Report and the IPCC models are summarized. It contains quite a lot of possibilities and numbers but for the purpose of this discussion, I can choose just a few of them and I can also ignore the error margins and focus on the "average values" only. My only goal is to list a few numbers that are important for a rational decision whether the hypothetical global warming in the foreseeable future is a "minor" or a "large" perturbation. I will be comparing the present with the year 2100 (even though Goklany also offers estimates for the year 2200). Goklany isn't checking any maths. He is only thinking about the IPCC numbers - much like I am. All the numbers below are GDP per capita figures, in the 1990 U.S. dollars (i.e. without their expected inflationary depreciation). I will be mentioning the results of the calculations in the scenario where nothing will be done with the temperature at all and the temperature will increase by 4 °C as a result. (To be sure, I must mention that the whole 20th temperature has only seen 0.74 °C of warming which means that the IPCC has to posulate more than a five-fold acceleration of the warming rate.) This value is very strange, especially if we realize that the temperature difference doesn't depend on the CO2 concentrations linearly but rather logarithmically (and surely not exponentially), a fact that all serious climatologists know very well (in Czechia, Dr Kutílek probably offers the most convincing arguments about this issue, in Kutílek 2008). Despite this warming, an unbelievable economic growth will take place - according to the very optimistic IPCC economists:  GDP per capita in 1990 in 2100 third world$900 $66,500 developed world$13,700 $107,300 The third world's wealth will be - according to the IPCC as well as Stern - much higher than the wealth of the developed world in 1990 (approximately five times). I think that this statement is controversial but whenever we start from a low number, a similar fast growth is possible. Whatever we can think about it, it is self-evident that even Nicholas Stern, the main warrior for the radical interventions against the CO2 emissions and, consequently, interventions into the economic development and the human lives, admits that the future generations of our Blue Planet are likely to be much better off than we are today! But that's not the whole story. The calculations by Stern and the IPCC claim that the higher temperature (by 4 °C) in 2100 will cause the loss that only lowers the GDP per capita by 2.9%, relatively to the situation with no temperature change (!). (The interval, including error margins, goes from 0.9% to 7.5%.) Even if we subtract 7.5% from those$66,500 (for the presently poor countries) and from \$107,300 (for the currently developed world), we are talking about reductions that are irrelevant relatively to the impact of the proposed CO2 reduction by 20, 30, 50, or 80% today, which would influence our whole way of living. Should we make this sacrifice for the happiness of the future generations?

Our ancestors - without directly thinking about it, in the same way as N. Stern or Al Gore are "thinking" about the future generations - have endowed us with the immense wealth of "tangible" as well as "non-tangible" things, i.e. material wealth, capital of diverse kinds, but primarily ideas, technologies, and institutions. Indur M. Goklany says very nicely that they were not doing these things deliberately. Instead, we see an "invisible hand reaching the future generations at work" (2009a, page 40).

That's why I fully agree with this author that "climate change is not a defining challenge of our age" (2009b), or that it mustn't become the defining challenge. It's our task to protect our epoch from such a development. However, some people are still trying to define the characteristics in this way. Because it can't be done seriously, any non-serious tool may be found useful, too. Even Prof Dore couldn't control his emotions and even during a technical analysis of Nordhaus's model, he didn't resist to mention that "Nordhaus's approach will be celebrated by the U.S. and Canadian oil lobby" (page 102). If Dore is thinking in this way, he should also think about the influence of the need to get grants from the government on the work of the environmentalist professorial Sirs - and his own (which is very nicely described by R.L. Gordon 2009). Global warming is really (and fortunately) - as long as we keep both climatological and economic approaches rational - just a "minor perturbation".

But let's return to Nordhaus. His book reaches a critical conclusion: "The economic damages caused by climate change - if there won't be any mitigation policies at all - will be comparable to 2.5% of the annual GDP of the whole world at the end of the 21st century" (2008, page 6). This is a small loss. The loss created as a consequence of a completely useless, radical crusade against global warming would be much higher. I completely agree with David Henderson that "at places with so many uncertainties, unclarified, and unknown things, politics should be evolutionary and adaptive" rather than revolutionary and fixed for long years (Henderson 2009).

Václav Klaus, "Kontexty" (Contexts) 4/2009

Notes

[1] The long-term instability of the global mean temperature is being shown by many authors. In the Czech language, it is being comprehensibly presented by several authors in a publication of the Center for Economics and Politics, 2007, page 10, and other authors.

[2] It's questionable whether they can rise in this way at all. Some authors argue that the total reserves of "fossil fuels" inside the Earth are not high enough (e.g. personal communication with R. L. Jenne, National Center for Atmospheric Research). In his text "Carbon Dioxide in Atmosphere (2000-2080)", published in 2009, it turns out that the complete burning of all fossil fuels can only increase the CO2 concentrations by 62%.

[3] R. Solow in his - previously quoted - article says that "if sustainability is meant to be more than a vague emotional commitment", then it is the kind of behavior that allows all future generations the choice to live "as well as the ancestors did" (1993, page 168). To me, it looks like a good pragmatic starting point.

[4] In his book, Nordhaus explicitly says that "global warming belongs to the global public domain" but he also adds that "it will be very expensive to decelerate it or eliminate it" and he especially says that "it brings very unpleasant scientific and economic uncertainties with it" (2008, page XI).

References

Carter, B 2009, A New Policy Direction for Climate Change, Quadrant, April 2009
Christy, J. R. 2007, Searching for Climate Change. A More Temperate Take on Global Warming, Center of the American Experiment, Minnesota, June 2007
Curtin, T. 2009, The Contradictions of the Garnaut Report, Quadrant, January-February 2009
Dore, M. 2009, A Question of Fudge: Professor Nordhaus on Global Policy for Climate Change, World Economics, January-February, 2009
Essex, Ch., McKitrick, R. 2007, Taken by Storm. The Troubled Science, Policy, and Politics of Global Warming, Key Porter Books, Toronto, Canada, 2007
Garnaut, R. 2008, The Garnaut Climate Change Review: Final Report, Australia, Cambridge University Press, 2008
Globální oteplování: fakta místo mýtů. (Global Warming: Facts Instead of Myths). Centrum pro ekonomiku a politiku (Center for Economics and Politics), Prague 2007
Goklany, I. M. 2009a, Discounting the Future, Regulation, Spring 2009
Goklany, I. M. 2009b, Is Climate Change the "Defining Challenge of Our Age?", Energy & Environment, issue 3, July 2009
Gordon, R. L. 2009, The Return of Jimmy Carter, Regulation, Spring 2009
Henderson, D. 2009, Economists and Climate Science: A Critique, World Economics, issue 1, January-February, 2009
Jenna, R. 2008, Carbon Dioxide in Atmosphere (2000-2080), Nature, May 27th, 2008
Klaus, V. 2007a, Globální oteplování: Pár slov k povinnosti diskontovat budoucnost (Global Warming: A Few Words About Our Duty To Discount The Future), Euro, July 30th, 2007
Klaus, V. 2007b, Blue Planet in Green Shackles, CEI 2007
Klaus, V. 2009, Modrá, nikoli zelená planeta (Blue, Not Green Planet), second, extended edition, Dokořán, Prague 2009
Klaus, V., Tříska, D. 2007, Ke kritice používání konceptu solidarity a diskriminace v intertemporální analýze tzv. globálních problémů (On the Criticism of the Usage of the Terms Solidarity And Discrimination In Intertemporal Analysis of So-Called Global Problems), Politická ekonomie (Political Economy) No 6, 2007
Kutílek, M. 2008, Racionálně o globálním oteplování (Rationally About Global Warming), Dokořán, Prague 2008
Loužek, M. (ed.) 2008, Globální oteplování: realita nebo bublina? (Global Warming: Reality Or a Bubble?), a booklet, Center for Economics and Politics, Prague, No 67, 2008
Michaels, P. J. 2007, Meltdown. The Predictable Distortion of Global Warming by Scientists, Politicians, and the Media, CATO Institute, Washington, D.C., 2007
Monckton, Ch. 2007, 35 Inconvenient Truths. The Errors in Al Gore’s Movie, Science & Public Policy Institute, Washington, D.C., October 2007
Motl, L. 2008, Nízká citlivost klimatu vůči oxidu uhličitému (A Low CO2 Climate Sensitivity), in the booklet "Global Warming: Reality Or a Bubble?", CEP, Prague, No 67, 2008
Nordhaus, W. 2007, A Review of the Stern Review on the Economics of Global Warming, Journal of Economic Literature, No 3, September 2007
Nordhaus, W. 2008, A Question of Balance: Weighing the Options on Global Warming Policies, Yale University Press, New Haven, USA, 2008
Plimer, I. 2009, Heaven and Earth: Global Warming, The Missing Science, Connor Court Publishing, Australia, 2009
Singer, S. F. (ed.) 2008, Nature, Not Human Activity, Rules the Climate, The Heartland Institute, Chicago, IL, 2008
Solow, R. 2003, An Almost Practical Step Toward Sustainability, Resources Policy, September 1993
Stern, N. 2007, The Economics of Climate Change: The Stern Review. Cambridge, Cambridge University Press, Great Britain, 2007
Svoboda, J., Vašků, Z., Cílek, V. 2003, Velká kniha o klimatu zemí Koruny české (The Great Climate Book of the Lands of the Czech Crown), Regia, Prague 2003


text:
# The topological structures in strongly coupled QGP with chiral fermions on the lattice

Sharma S, Dick V, Karsch F, Laermann E, Mukherjee S (2016)
Nuclear Physics A 956: 793-796.

Es wurde kein Volltext hochgeladen. Nur Publikationsnachweis!
Zeitschriftenaufsatz | Veröffentlicht | Englisch
Autor
; ; ; ;
Abstract / Bemerkung
The nature of chiral phase transition for two flavor QCD is an interesting but unresolved problem. One of the most intriguing issues is whether or not the anomalous U(1) symmetry in the flavor sector is effectively restored along with the chiral symmetry. This may determine the universality class of the chiral phase transition. Since the physics near the chiral phase transition is essentially non-perturbative, we employ first principles lattice techniques to address this issue. We use overlap fermions, which have exact chiral symmetry on the lattice, to probe the anomalous U(1) symmetry violation of 2+1 flavor dynamical QCD configurations with domain wall fermions. The latter also optimally preserves chiral and flavor symmetries on the lattice, since it is known that the remnant chiral symmetry of the light quarks influences the scaling of the chiral condensate in the crossover transition region. We observe that the anomalous U(1) is not effectively restored in the chiral crossover region. We perform a systematic study of the finite size and cut-off effects since the signals of U(1) violation are sensitive to it. We also provide a glimpse of the microscopic topological structures of the QCD medium that are responsible for the strongly interacting nature of the quark gluon plasma phase. We study the effect of these microscopic constituents through our first calculations for the topological susceptibility of QCD at finite temperature, which could be a crucial input for the equation of state for anomalous hydrodynamics.
Stichworte
Erscheinungsjahr
Zeitschriftentitel
Nuclear Physics A
Band
956
Seite
793-796
Konferenz
25th International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions
Konferenzort
Kobe, Japan
Konferenzdatum
2015-09-27 – 2015-10-03
ISSN
eISSN
PUB-ID

### Zitieren

Sharma S, Dick V, Karsch F, Laermann E, Mukherjee S. The topological structures in strongly coupled QGP with chiral fermions on the lattice. Nuclear Physics A. 2016;956:793-796.
Sharma, S., Dick, V., Karsch, F., Laermann, E., & Mukherjee, S. (2016). The topological structures in strongly coupled QGP with chiral fermions on the lattice. Nuclear Physics A, 956, 793-796. doi:10.1016/j.nuclphysa.2016.02.013
Sharma, S., Dick, V., Karsch, F., Laermann, E., and Mukherjee, S. (2016). The topological structures in strongly coupled QGP with chiral fermions on the lattice. Nuclear Physics A 956, 793-796.
Sharma, S., et al., 2016. The topological structures in strongly coupled QGP with chiral fermions on the lattice. Nuclear Physics A, 956, p 793-796.
S. Sharma, et al., “The topological structures in strongly coupled QGP with chiral fermions on the lattice”, Nuclear Physics A, vol. 956, 2016, pp. 793-796.
Sharma, S., Dick, V., Karsch, F., Laermann, E., Mukherjee, S.: The topological structures in strongly coupled QGP with chiral fermions on the lattice. Nuclear Physics A. 956, 793-796 (2016).
Sharma, Sayantan, Dick, Viktor, Karsch, Frithjof, Laermann, Edwin, and Mukherjee, Swagato. “The topological structures in strongly coupled QGP with chiral fermions on the lattice”. Nuclear Physics A 956 (2016): 793-796.

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arXiv: 1602.02197

Inspire: 1420154


text:
# “Video system not initialized” error

I always get the error message "video system not initialized"

I don't get where I went wrong with the code below:

import pygame
pygame.init()

display_width = 800
display_height = 600

gameDisplay = pygame.display.set_mode((display_width, display_height))
pygame.display.set_caption('4 pics 1 word: ASTRONOMY VERSION')

black = (0,0,0)
white = (255,255,255)
red = (255,0,0)
green = (0,255,0)
blue = (0,0,255)

def satImg(x,y):
gameDisplay.blit(satImg1,(x,y))

x = (display_width * 1.0)
y = (display_height * 1.5)

if pygame.mixer.get_busy() != None:
for event in pygame.event.get():
if event.type == pygame.QUIT:
gameDisplay.fill(white)
satImg1 = (x, y)
pygame.quit()
quit()

• When asking questions here, please take the time to ensure your code is formatted legibly and your issue is fully explained. Consult the help center if you have trouble with formatting. I've edited your question to clean it up, but it would still benefit from more text description of what you're trying to do, and how you've tried solving this problem so far. – DMGregory Feb 25 '19 at 15:50
• It seems to be a bad identation in the last lines, maybe quit() calls should be out of the while loop – Juanpa Mar 1 '19 at 1:27

As Juanpa already implied in a comment, you get the error because you call the quit() function, which in turn makes the pygame module unusable, and your code should look like this:
...


text:
# Help with a log

1. Jul 26, 2006

### hmm?

Hello,

I'm having trouble recovering all the solutions for this equation; here is the work I've managed to produce thus far.

logx=\sqrt{logx}

(logx)^2=logx

10^logx^2=x <--I'm not sure of this move?

x^2=x

x^2-x=0

x(x-1)=0

x= 1, 0

I know 0 cannot work, and 10 looks likes a possible solution, but I cannot produce it--anyways, any help would be appreciated.

thanks.

Last edited: Jul 26, 2006
2. Jul 26, 2006

### Tide

Why don't you try setting y = log x in the original equation then solve for y? That will avoid some confusion.

3. Jul 26, 2006

### hmm?

y=logx

x=10^(logx)^1/2

This is where I'm stuck, because I feel it would be illegal to assume

10^logx^1/2...shouldn't it be (logx)^1/2. I don't know how to simplify after this point.

4. Jul 26, 2006

Apply Tide's hint to this equation: $$(logx)^2=logx$$

Once you have solved for y, replace y with logx, and then solve for x.

5. Jul 26, 2006

### VietDao29

Uhmm, no:
(log x)2 = log(x)
By letting y = log(x), we have:
y2 = y
<=> y2 - y = 0
And this is our familiar quadratic equation in y. Can you solve for y from that equation? Then from there, just solve for x.
Can you go from here? :)

6. Jul 26, 2006

### hmm?

Ahh, thanks! I can't believe I couldn't think of this, haha.

(logx)^2-logx=0

logx(logx-1)=0

logx=0 , log(x-1)=0

x=1, 10
A quick two questions: is there any other way to solve this, other than the y^2-y=0, and still maintain solutions 1, 10? For (logx)^n n>1, how would you treat this in an equation--merely a^n=(logx)^n never expanding?

Thanks,
Chris

7. Jul 26, 2006

### VietDao29

Yes, this is correct. Congratulations.
Uhm... no.
By the way, this is the common way, and is fast enough. So, I don't think you'll need some other ways though. :)
Yes, if it helps you visualize better. Say, it would be much better if you substitute y = log(x), and all of sudden, the whole equation becomes a quadratic equation, or a cubic equation,... in y. Right?
Normally, to solve an equation, we simplify the equation first to make it look less complicated. Then we isolate x to one side of the equation. Then we solve it from there.
Of course, the equation:
$$x = 10 ^ {\sqrt{\log x}}$$ just looks horible, right?
You're welcome. :)


text:
# Question 1a9d7

Jul 16, 2016

A

#### Explanation:

N° of Protons for 1 mol of salt= 50 (Ca= 30; C= 6; O = 8 x 3 = 24)
N° mol = Mg/ MM = 10g/(100 g/mol) = 0,1
N° of protons in 1 mol of protons = Na = 6,022 X 10^23
50 x 0,1 x 6,022 x 10^23 = 3,011 x 10 ^23

Jul 16, 2016

The correct answer is "None of the above".

#### Explanation:

Step 1. Calculate the number of protons in one formula unit of ${\text{CaCO}}_{3}$.

$\text{Number of protons" = "Ca + C + 3O" = "20 + 6 + 3×} 8 = 50$

2. Calculate the moles of ${\text{CaCO}}_{3}$

${\text{Moles of CaCO"_3 = 10 color(red)(cancel(color(black)("g CaCO"_3))) × ("1 mol CaCO"_3)/(100.09 color(red)(cancel(color(black)("g CaCO"_3)))) = "0.0999 mol CaCO}}_{3}$

3. Calculate the number of formula units of ${\text{CaCO}}_{3}$

${\text{Formula units" = 0.0999 color(red)(cancel(color(black)("mol CaCO"_3))) × (6.022 × 10^23color(white)(l) "formula units CaCO"_3)/(1 color(red)(cancel(color(black)("mol CaCO"_3)))) = 6.02 × 10^22color(white)(l) "formula units CaCO}}_{3}$

4. Calculate the total number of protons

"Number of protons" = 6.02 × 10^22 color(red)(cancel(color(black)("formula units"))) × "50 protons"/(1 color(red)(cancel(color(black)("formula unit"))))

= 3.0 × 10^24color(white)(l) "protons"#

This answer is not in the above list.

∴ The correct answer is "None of the above".


text:
# Is this an appropriate manned interstellar mission profile?

My idea was to use 2 stages for an interstellar spacecraft (particularly the Orion Drive) and use it in a way similar to Project: Daedalus. Phase 1 would involve accelerating to cruising speed following assembly in orbit, and that cruising speed is 0.04 c, which will then slow down as we approach the Wolf 1061c (the target planet) system. There is a problem: is this appropriate and if not, what are some issues I need to concern myself with?

EDIT: I am also going to provide a link for those of you who do not know what "mission profile" is by giving you a hands-on interplanetary or orbital example. Mission Profile Description, Definition and Example

This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

• @AndreiROM I just need the profile and how the mission is to be planned from Earth, not events that occur during the mission. That will be left up to me. – Future Historian Apr 7 '16 at 15:33
• So apparently I'm confused about what a "profile" is. Do you mean the acceleration/speed of the craft and when to slow down? – AndreiROM Apr 7 '16 at 15:34
• @AndreiROM Fixed. – Future Historian Apr 7 '16 at 15:38
• @AndreiROM And I found a good simple definition of what a "mission profile" is. A mission profile is the planned route and goals, and what mission hardware that requires. – Future Historian Apr 7 '16 at 15:45
• Comments are not for extended discussion; this conversation has been moved to chat. Future Historian and @WhatRoughBeast, I moved the comments to chat for posterity (they're productive), though the conversation seems to be over now, because they were getting too long. If you want to continue, that's the place to go. – HDE 226868 Apr 7 '16 at 21:54

Wolf 1061c is about 13.8 light years from Earth. Cruising at 0.04c, your ship will take, oh dear, 345 years. 0.04c is too slow for significant time dilation, so the ship will experience the whole 345 years. Just keeping the ship running for 345 years in interstellar space is a problem. Space is not a pleasant place, and it's really bad at 0.04c.

Over those 345 years, if humanity survives and continues advancing technologically, it's entirely they'll launch a faster ship and overtake yours rending the whole thing moot. If 50 years of advancement meant an increase to 0.05c they'll beat you. (I'm sure there is a name for this dilemma).

Whether or not they'll moon the older, slower expedition as they pass by could be a plot point of your story.

Then there's the question of the payload. What's the point of sending a ship to another star system if it just sits dumbly when it gets there? This leaves you with three sci-fi options:

First is full automation. Humans are very expensive on resources, it's cheaper to send robots. But Earth won't even know the ship has reached Wolf 1061c for nearly 14 years, and any commands will take 14 years to reach the ship. The ship and all its systems will have to be fully automated. We could get a fully automated ship to Wolf 1061c, maybe, but dealing autonomously with what is found there (because we don't know) is well beyond our current capabilities. Project Daedalus is supposed to carry surveying equipment and start sending back information to Earth, but if humans are in the decision loop the delays on analyzing that information will mean even after waiting 345 years for the ship to reach Wolf 1061c there's still generations of work to come.

There's also the problem of maintaining communication and tracking facilities and interest and budget on Earth for 345 years. This is longer than any space agency and most countries have existed and is kind of out of your control.

If you're sending humans they'll obviously die before they get there. One option is to put them in hibernation, but we haven't solved that yet. If you can solve it, you'll have all the extra problems of how to keep those hibernation systems running and powered for 345 years in interstellar space. Even nuclear fuel won't last, and there's not enough solar power between star systems.

The final option is to figure out how to create a generation ship. This is now deep into the realm of sci-fi. Not only will 10-15 generations have to live on the ship, you'll have to figure out how to create a sustainable, sealed biome capable of supporting them for 345 years. We barely know how to do this for a few days.

Normally a ship has to slow down once it reaches its target. Project Daedalus "solves" the problem of slowing down when you get there by releasing probes which start decelerating quite early using the gentle shove of ion engines. This saves the tremendous amount of fuel needed to slow the whole ship down (and the fuel to carry that fuel, and the fuel to carry that fuel...), but it sacrifices the bulk of the mass of the ship in the process and adds extra bulk of carrying probes with all their own computers and engines and shielding.

The Project Daedalus plan is to jettison the main ship... what about the humans and all their stuff? Do they all go into the probes? These much lighter probes use ion engines to very, very, very slowly slow down and achieve orbit. These will have to carry humans, and all their food, water, air, equipment, and shielding they'll need to presumably start a colony, because they're not going back to Earth. That much extra mass defeats the point of using light probes with ion engines instead of having to slow down all the mass of the original ship. The now much heavier probes with all that cargo cannot slow down using small, efficient ion engines.

So no humans on the Project Daedalus plan.

• If you do send probes, what are they supposed to do for the best-case 27.6 Earth years between when they send data and they receive a response? You could streamline the process by having the probes explore for the first 13.8 years, receive the responses to the initial data and start going back, but that still comes with a host of problems. No, remote control really isn't going to cut it, and full automation... well, I'm not siding with the 1061c team on this one at this time. – a CVn Apr 7 '16 at 19:15
• One thought for an Project Orion based probe was to make the pusher plate out of Uranium instead of steel. This gives the crew a ready source of Uranium for refining once they get to the target star. But they would still need to refine it to get the $U_{235}$ they needed. – Jim2B Apr 7 '16 at 20:22
• @FutureHistorian You asked if it's an appropriate mission profile and what it's concerns are, but you didn't say appropriate for what. You clearly have something in mind. Rather than try to patch it up in the comments, how about asking a new question about that mission? Include the time and tech period, what the in-universe goal is, as well as the out-of-universe goal. – Schwern Apr 8 '16 at 15:50
• @Schwern Fair enough. – Future Historian Apr 8 '16 at 15:51
• @Schwern Done: worldbuilding.stackexchange.com/q/39713/12238 – Future Historian Apr 8 '16 at 16:47

To maximize your mission profile, you will need to use a few tricks. The rocket equation is in effect even with ORION drives, and every gram counts. Using rocket power to decelerate means you have to initially accelerate that fuel, so you should also take measures to minimize the use of powered deceleration.

The ship will need to dive towards the sun and use the gravitational slingshot effect to maximize the efficiency of the propulsion burn (deep in the sun's gravity well), in order to minimize the mass of fuel that you ned to carry.

The ship will coast for most of the journey, and on approaching the target deploy a magsail to decelerate. Since the magsail has minimal mass and uses interaction with the local medium, there is no need to carry a great amount of fuel for deceleration. Indeed, the magsail can be used for manoeuvring within the target system, another great saving in mass for the ship.

Other mass saving tips would be to rotate the ship after the Orion burn and use the drive plate as protection against interstellar dust and radiation. Repurposing like this reduces the overall mass of the ship, and the knock off effect of every gram saved provides a huge saving in the amount of fuel needed to boost to cruise velocity and for a smaller magsail needed to decelerate.

• Hmmmmm. I suppose I never thought of it this way before. Interesting. – Future Historian Apr 7 '16 at 21:19
• hard-science, please. – HDE 226868 Apr 7 '16 at 21:53
• A low thrust propulsion like a sail cannot stop a Project Orion scale craft in time. At 0.04c you'll be in and out of the Wolf 1016 system in days. Even enormous magnetic or solar sails produce maybe 100 N of thrust, and that's close to a star. With some very optimistic thrust and mass assumptions... 1000 ton spacecraft at 100 N thrust decelerates at 10^-4m/s^2 so that's 3500 years to decelerate from 0.04c. – Schwern Apr 8 '16 at 3:58
• I think you will find your calculations are a bit off: erps.spacegrant.org/uploads/images/images/… – Thucydides Apr 8 '16 at 14:55
• @Thucydides That paper says 200 N of thrust for a truly enormous 200 km sail. So, 1750 years to slow down. That paper seems to be about measuring sail thrust and doesn't appear to discuss spacecraft mass or acceleration. Could you point out how my calculation is off? – Schwern Apr 8 '16 at 15:54


text:
# Strictly monotone functions

What are the strictly monotone functions $f\colon (0,\infty)\to (0,\infty)$ which satisfy $x= f(\tfrac{x^2}{f(x)})$ for $x>0$. I cannot find any other than $f(x)=x$.

-

When $f(x) = a x$, $a>0$, then
$$f \left ( \frac{x^2}{f(x)} \right ) = a \frac{x^2}{f(x)} = a \frac{x^2}{a x} = x$$
It doesn't look likely; of course this must be proven. You know that no other power will work (try $x^b$). Otherwise, prove by contradiction; assume some other function that is not $a x$ works. What then? –  Ron Gordon Jan 31 '13 at 15:32


text:
# Adding an environment to LyX for dividing a proof into steps

My question is basically a combination of the questions Is it possible to use the enumerate/itemize environment within the proof/remark environment in a nice way? and Create new paragraph style in LyX.

When I write a mathematical document in LyX, I frequently want to divide my proofs into steps. Up to now, I have often simply used the enumerate environment to do this.

This has the advantage that the numbering of the steps is handled automatically by LyX and that I can reference the individual steps later on using the usual Labels and References.

One of the disadvantages is that I am not really satisfied how the output looks afterwards, i.e. my "steps" are somewhat flushed to the right.

In the first question linked above, the user egreg proposed to simply use

\newenvironment{steps}{\setcounter{step}{0}}{}
\newcounter{step}
\newcommand{\proofstep}{\par\refstepcounter{step}Step~\thestep.\space\ignorespaces}

as a solution in LaTeX.

It would be really nice if I could integrate this in LyX, i.e. I would like to have environments

1. Steps
2. step

(or something like that), which result in the above LaTeX environments/commands to be used by LyX. I would also like to be able to reference the individual steps using labels and references.

The second question linked above describes how to add a new environment (the quotation environment) to LyX.

I tried to adapt this to the case of enumerate/itemize environments, but the format of these seems to be more complicated. Also, I guess I somehow have to tell LyX to use \proofstep instead of \item and I am not sure how to do this.

Any help will be appreciated.

• You can create your own layout (the thing that controls how LyX does something and which command it does to do it). It is just a simple text file, no C++ programming required. However, it would take some time. See Help > Customization. Also look at the examples in the lib/layouts folder. – scottkosty Jan 26 '15 at 17:17
• @scottkosty: Thank you, I will try that. If I manage it, I will post my solution as an answer. – PhoemueX Jan 27 '15 at 10:01


text:
# How do you solve and graph the compound inequality 2 < x < 7 and 2x > 10 ?

Jul 11, 2015

I'll just do the solving. the graphing is a matter of crossing out on a number line.

#### Explanation:

$2 x > 10 \to x > 5 \to 5 < x$
(because you may divide both sides by a positive number)

Combining these we get:

$5 < x < 7$


text:
# Negative Amortization

I’ve added a new entry to the FAQ regarding negative amortizations. In the process, I also noticed that some of the summary information produced by the calculator and final schedule row are not completely consistent in the NegAm scenario. I will need to fix these. It is apparent that I missed some sort of symmetry in my calculations here (I probably took a shortcut to minimize computer time, which is just so much gagging at gnats these days).

This is interesting to me because I’ve recently been reading some of the Richard Feynman Lectures on Physics, and he talks a bit about mathematical symmetries. We have such an expectation of symmetry that lack of it in certain contexts may indicate a flaw in our understanding, or sometimes it may lead to the discovery of even more interesting properties. (Amortization is not so glamorous as a fundamental law of the universe, mind you; but I now recognize that had I paid closer attention to a mathematical symmetry in this case, I might have noticed my error sooner.)


text:
# Render a version of the BBC Micro owl logo

Can you render out a version of the BBC Micro Owl logo as per above from code?

Rules:

• You can use any programming language you like.
• Output can be text or graphic.
• Circles don't have to overlap.

Winner:

• The answer with the most upvotes wins.

Kudos for anyone that attempts this in BBC basic.

You can read about the BBC Micro here

• "Most inventive" and "most like the original" is subjective. Please provide an objective winning criterion. (I am voting to close, as per "Questions without an objective primary winning criterion are off-topic, as they make it impossible to indisputably decide which entry should win.") – Doorknob Jan 23 '14 at 13:53
• I have changed the win criteria to the shortest code which I hope is less subjective for everyone. – Ben Paton Jan 23 '14 at 17:16
• Maybe code-golf is fine but I think this question could suit a popularity-contest tag instead. Most votes wins. That's objective. Mob decides. This encourages interesting results without sacrificing output quality/creativity to save a few chars of code. Witness the freestyle Olympic rings question. – Darren Stone Jan 23 '14 at 18:36
• Good idea I have changed it to most upvotes wins which is fairer. How do I get this taken off hold? – Ben Paton Jan 23 '14 at 19:54

# BBC BASIC

I tried using trig functions to draw proper circles, but that was painfully slow. Came up with this instead:

10 MODE 1
20 GCOL 0,1 : VDU 19,1,3,0,0,0
30 FOR Y%=0 TO 20
40 READ N% : P%=65536
50 FOR X%=0 TO 16
60 IF (N% AND P%)=0 THEN GOTO 160
70 X0% = X%*32+384 : Y0% = 872-Y%*32
80 FOR DX%=-16 TO 16 STEP 8
90 FOR DY%=-8 TO 8 STEP 8
100 PLOT 69,X0%+DX%,Y0%+DY%
110 NEXT DY% : NEXT DX%
120 FOR DX%=-8 TO 8 STEP 8
130 FOR DY%=-16 TO 16 STEP 32
140 PLOT 69,X0%+DX%,Y0%+DY%
150 NEXT DY% : NEXT DX%
160 P%=P%/2
170 NEXT : NEXT
1000 DATA 87381,33410,69905,10280
1010 DATA 69649,33410,82181,40968
1020 DATA 87377,43520,87297,43520
1030 DATA 21761,10880,5441,2720
1040 DATA 1361,552,1093,43682,1

Here's the output:

• Wow seriously amazing someone actually took the time to use BBC basic. I'm very impressed!! – Ben Paton Jan 24 '14 at 12:21
• That's a nice owl. Are you using a real BBC or an emulator program. If it's an emulator, which one? cos it's not the same as the one I used. – Level River St Feb 8 '14 at 21:29
• BTW, if you want to draw circles quicker, don't use trig functions, use pythagoras theorem: Y=(R^2-X^2)^0.5 see my answer to the rainbow question. And there's actually a more advanced way of doing it with only addition, based on the fact that (1)+(3)+(5)....(1+2n) = (n+1)^2. – Level River St Feb 8 '14 at 21:34

### GolfScript

"!C-DD[5v>U8GJ2zX['b#L*\\q>FQp "{32-}%96base 2base{"   *"2/=}%18/{""*1>17/~n@n}/

The code prints an ascii art version of the logo (run here).

* * * * * * * * *
*     * *     *
*   *   *   *   *
* *     * *
*   *       *   *
*     * *     *
* *     *     * *
* *         *
* * * * * * *   *
* * * *
* * * * *       *
* * * *
* * * *       *
* * * *
* * * *     *
* * * *
* * * *   *
*   * *
*   *   * *
* * * * * *   *
*

• Really impressive in 82 characters! – Ben Paton Jan 23 '14 at 17:19

# Mathematica

Nothing but grunt work.

w = 20; h = 25; r = .7; r2 = .2; t = Table; d = Disk;
owl = Graphics[{Polygon[{{0, 0}, {w, 0}, {w, h}, {0, h}}],
ColorData[57, 4],
t[d[{k, 22}, r], {k, 2, 19, 2}],
t[d[{18, k}, r], {k, 2, 21, 2}],
t[d[{k, #}, r], {k, #2}] & @@@ {{21, {3, 9, 10, 11, 17}}, {20, {2,
6, 10, 14, 18}},
{19, {5, 6, 7, 13, 14, 15}}, {18, {6, 9, 11, 14}}, {17, {10,
17}, {16, {16}}, {15, {15}}, {14, {8, 10, 12, 14}},
{13, {9}}, {12, {9}}}},
t[d[{# - k, k}, r], #2] & @@@ {{20, {k, 18, 3, -1}}, {19, {k, 16,
6, -1}}, {18, {k, 16, 5, -1}}, {17, {k, 14, 7, -1}}, {16, {k,
14, 6, -1}}, {14, {k, 11, 5, -1}}, {14, {k, 12, 5, -1}}},
t[d[{k, 4}, r], {k, {8, 12}}],
t[d[{k, 3}, r], {k, 3, 13, 2}],
d[{8, 13}, r],
Black, d[{10, 21}, r2], d[{8, 13}, r2], d[{9, 12}, r2],
t[d[{19 - k, k}, r2], {k, 16, 6, -1}],
t[d[{17 - k, k}, r2], {k, 14, 7, -1}],
t[d[{15 - k, k}, r2], {k, 12, 6, -1}],
t[d[{k, 19}, r2], {k, {6, 14}}]}, ImageSize -> 220]

### R

image(t(matrix(as.integer(sapply(c(1397760,567810,1070336,141954,1381696,133794,
1054036,559786,1332560,557218,1052756,131114,
1380368,139272,1064964,557058,1398101),
intToBits)[1:21,]),nr=21)),ax=F)

Results in:

Basically the idea is to take the base-2 representation of the 17 numbers (1397760, 567810, 1070336, 141954, 1381696, 133794, 1054036, 559786, 1332560, 557218, 1052756, 131114, 1380368, 139272, 1064964, 557058 and 1398101), make it into a 21x17 matrix of 1 and 0s and plot the matrix as is.

# Pov-Ray

background{color<.14,.01,.01>}
camera{orthographic location z*-2 up y*24 right x*20}
#macro s(X,Y)sphere{<X,Y,0>,1.07,2.6}#end
#declare b=array[17]{1397760,567810,1070336,141954,1381696,133794,1054036,
559786,1332560,557218,1052756,131114,1380368,139272,1064964,557058,1398101}
blob{threshold 0.9 #for(j,0,16,1)#declare V=b[j];#for(i,0,28,1)
#if(mod(V,2))s(j-8,i-10)#end #declare V=floor(V/2);#end #end
pigment{color<1,1,.8>}finish{ambient 1}}

'compile' with povray +Ibbc.pov -Obbc.png +A0.1 +R9 -W240 -H285

## Bash

Since you didn't specify no external resources...

curl -s http://codegolf.stackexchange.com/questions/19214/render-a-version-of-the-bbc-micro-owl-logo | grep '* * *' | sed -n '/code>\*/,/<pre>/p' | sed '$d' | sed 's/<pre><code>//'  Howard - stole your Ascii art, thanks. Or after I uploaded it here - curl -s http://textuploader.com/1ojd | sed -n '/<code/,/<\/code>/p' | sed 's/<[^>]\+>//g'  • Well that's lateral thinking I suppose... – Ben Paton Jan 24 '14 at 23:29 # BBC Basic, random colours, golfed! 149 characters. I'm not that big on golfing (I prefer the code challenges) but I liked the ridiculousness of golfing in BBC basic. BBC emulator at http://www.bbcbasic.co.uk/. Runs in screen mode 6 at the command line. FORK=6TO185S=K MOD9=5VDU-32*(K MOD18=15),17,128+RND(6)*(ASCMID$("?OSUuLEMSS^H?=A_W",K/6,1)/2^(K MOD6)AND1),32,17,128,32,-13*S,-10*S:NEXT

Explanation (ungolfed version)

No IF statements, because the rest of the line would only be executed if the IF was true (so to ensure the NEXT was reached I would have to do a program with line numbers.) Therefore I made a lot of use of ASC(null character)=0 to control output. Also, I had to cut off the pixel at the bottom right corner to fit the whole command line on the screen after printing (and that saved 2 characters.

I love how BBC basic can recognise an identifier after FOR, even with no white space. FORE, FORD, FORK, FORM, FORS, FORT :-)

 FOR K=6 TO 185

REM Set S to true(-1) if K mod 9 = 5, otherwise set S to false(0)

S = K MOD 9=5

REM If K mod 18 = 15, send asc(space)=32 to the VDU controller,otherwise send 0.
REM This causes staggering every 2 lines.

VDU-32*(K MOD18=15)

REM Vdu 17 for colour. foreground colours start at 0, background colours at 128.
REM Rnd(6) to choose a bright color. Multiply by the horrible expression.
REM The horrible expression evaluates to 1 if the pixel is to be printed, 0 if not.
REM Picture data is stored in base 64 in the string.
REM MID$extracts the characters in order. REM The FOR loop starts at K=6 so that this will work properly. REM Extracted character SHR ((K mod 6)) AND 1 to decide to REM colour pixel or not. BBC basic does not have SHR operator. REM so we divide by 2^(K mod 6) instead. VDU 17,128+RND(6)* (ASC(MID$( "?OSUuLEMSS^H?=A_<A^tC|dGxEMh>W" ,K/6,1))/2^(K MOD 6)AND 1)

REM Print a space in the new background colour

VDU 32

REM Change background colour back to black

VDU 17,128

REM Print another space

VDU 32

REM If S is true (-1) print a carriage return and linefeed. otherwise two 0's

VDU -13*S,-10*S

NEXT

• Thanks for contributing that's a really clever solution and a nice multicoloured owl, I haven't seen golfed BBC basic before. – Ben Paton Feb 6 '14 at 23:39
• Thank you for inspiring me to go and look for a BBC basic emulator. It's a great language that I haven't used in 20 years. It's got a lot of graphics facilities built in without having to include add-on libraries, which makes it an excellent language for golfing a task like this. – Level River St Feb 7 '14 at 20:13
• Also, first time I've used base 64. It was just right for this image, being 18 pixels for each 2 lines. I will definitely use it again. The fact that the 6th bit is uppercase/lowercase means you only really have to think about the other five. I found that made it almost as easy as hex once I got into it. – Level River St Feb 7 '14 at 20:23

## C

ASCII output.

x[]={256,191,424,104,376,60,316,30,286,15,287,15,383,67,403,153,325,102,341,153,511};i=20;mai
n(){for(;i>=0;i--){i&1&&putchar(32);while(x[i]){putchar(x[i]&1?42:32);x[i]>>=1;putchar(32);}pu
tchar(10);}}

Code output:

* * * * * * * * *
*     * *     *
*   *   *   *   *
* *     * *
*   *       *   *
*     * *     *
* *     *     * *
* *         *
* * * * * * *   *
* * * *
* * * * *       *
* * * *
* * * *       *
* * * *
* * * *     *
* * * *
* * * *   *
*   * *
*   *   * *
* * * * * *   *
*

# JavaScript - 326307 285 characters (ASCII)

a=[1716886015,1133746501,253693823,1010572830,3215485048,0];s="";with(Math)
for(y=0;44>y;y++,s+="\n")for(x=0;90>x;x++,s+="#8*+=:-. "[min(floor(d),8)])
for(j=d=0;22>j;j++)for(i=0;9>i;i++)1==((8==i?j+1:a[floor(j/4)]>>i+j%4*8)&1)
&&(d+=50/pow(pow(x-10*(i+j%2/2)-4,2)+pow(2*y-4*j-4,2),1.5));s;

Probably not the shortest code. I tried to be as close as possible to the original logo, using only ASCII.

To execute : copy paste to javascript console (eg: chrome or firefox). Note : the script might take a little time to run so if nothing come just after pressing enter, be a little patient.

• You can shorten this by doing m=Math; and then doing m.floor, m.pow etc. – eithed Jan 24 '14 at 10:27
• Even moreso by wrapping the outermost for-loop in a with (Math) and then omitting Math. at each occurence. Move the s+= parts to the for-loop header (next to y++/x++) to get rid of some braces. I think it's typical to use alert/prompt for I/O when golfing in JS, so I'd go with alert(s) at the end instead. – FireFly Jan 24 '14 at 10:54
• Thanks for tips and comments. I am a beginner when it comes to code size optimizing :) About alert / prompt suggestion : I tried to use them, but there is some issues : browsers (chrome at least) seems to automatically wrap text if line is too long, which broke logo. – tigrou Jan 24 '14 at 13:47

# CoffeeScript

Code isn't golfed. It uses some metaballs-ish algorithm to simulate "stickyness" of the circles. ASCII owl was shamelessly stolen from other answers :)

canvas = document.createElement 'canvas'
canvas.style.backgroundColor = '#240202'
canvas.style.transform = 'scale(0.5) translate(-480px,-570px)'
W = canvas.width = 960
H = canvas.height = 1140
D = 50
R = D / 2
ctx = canvas.getContext '2d'
imageData = ctx.getImageData 0, 0, W, H
data = imageData.data
owl = '''

\ * * * * * * * * *
\  *     * *     *
\ *   *   *   *   *
\    * *     * *
\ *   *       *   *
\  *     * *     *
\ * *     *     * *
\  * *         *
\ * * * * * * *   *
\  * * * *
\ * * * * *       *
\  * * * *
\   * * * *       *
\    * * * *
\     * * * *     *
\      * * * *
\       * * * *   *
\        *   * *
\       *   *   * *
\  * * * * * *   *
\                 *
'''.split '\n'

insideDot = (x, y) ->
w = 0
for du in [-1..1] then for dv in [-1..1]
u = x // D + du
v = y // D + dv
continue unless owl[v]?[u] is '*'
dx = x - (u * D + R)
dy = y - (v * D + R)
d = dx * dx + dy * dy
w += 1 / (d * d)
return yes if w > 0.0000008
no

for y in [0...H] then for x in [0...W] when insideDot x, y
i = (y * W + x) * 4
data[i] = data[i+1] = data[i+3] = 255
data[i+2] = 214

ctx.putImageData imageData, 0, 0
document.body.appendChild canvas

Watch it destroy the coffeescript.org documentation (hit Run):

PHP

Building upon the ascii art versions of the logo submitted previously and using this as the array to render a graphical version using the GD library.

$circleWidth = 34;$circleHeight = 34;
$movement = 24;$canvasWidth = 480;
$canvasHeight = 570;$image = imagecreatetruecolor($canvasWidth,$canvasHeight);
$backgroundColour = imagecolorallocate($image, 36, 2, 2);
ImageFillToBorder($image, 0, 0,$backgroundColour, $backgroundColour);$circleColour = imagecolorallocate($image, 255, 255, 214);$coordinates ='
* * * * * * * * *
*     * *     *
*   *   *   *   *
* *     * *
*   *       *   *
*     * *     *
* *     *     * *
* *         *
* * * * * * *   *
* * * *
* * * * *       *
* * * *
* * * *       *
* * * *
* * * *     *
* * * *
* * * *   *
*   * *
*   *   * *
* * * * * *   *
* ';
$coordinates = str_split($coordinates);
$coordinateX =$movement;
$coordinatY =$movement;
$i=1; foreach ($coordinates as $coordinate) { if ($i < 19) {
if ($coordinate == '*') { ImageFilledEllipse($image, $coordinateX,$coordinatY, $circleWidth,$circleHeight, $circleColour); }$coordinateX = $coordinateX +$movement;
$i++; } else {$i=1;
$coordinateX =$movement;
$coordinatY =$coordinatY + $movement; } } header("Content-type: image/png"); imagepng($image);
imagedestroy(\$image);

Results in:


text:
Welcome to ZOJ
 Problem Sets Information Select Problem Runs Ranklist
ZOJ Problem Set - 4083
Little Sub and his another Geometry Problem

Time Limit: 3 Seconds      Memory Limit: 262144 KB

Little Sub is happy to see that you solve his geometry problem quickly, so he comes with another geometry problem for you as the reward.

Given a 2-dimensional space described as $\{(x,y)|0 \le x\le M,0\le y\le M\}$ and $N$ points $P_1(x_1,y_1), P_2(x_2,y_2), \dots, P_N(x_N,y_N)$ (note that different points may share the same coordinate), we define a "good square" as a square inside the given 2-dimensional space such that the edges of the square is parallel to the x-axis or the y-axis, and none of the given $N$ points are strictly inside it.

You're also given $Q$ queries. In each query, given a coordinate $(u,v)$, please calculate that the largest area of a good square such that $(u,v)$ is strictly inside.

#### Input

There are multiple test cases. The first line of the input contains an integer $T$ ($1 \le T \le 10$), indicating the number of test cases. For each test case:

The first line contains two integers $M$ (${2\leq M\leq 10^9}$) and $N$ (${0\leq N\leq 5000}$), indicating the range of the given 2-dimensional space and the number of the given points.

For the following $N$ lines, the $i$-th line contains two integers $X_i$ and $Y_i$ (${0\le X_i,Y_i\le M}$), indicating the Euclidean coordinate of the $i$-th point.

The next line contains one integer $Q$ (${1\leq Q\leq 5000}$), which denotes the number of queries.

In the following $Q$ lines, each line contains two integers $u$ and $v$(${0\le u,v\le M})$, indicating a query.

#### Output

For each query, output an integer in one line representing the answer.

#### Sample Input

1
5 5
1 4
2 1
3 2
4 1
4 4
3
3 1
2 3
4 3

#### Sample Output

4
9
4

#### Hint

We now illustrates the second query of the sample input below.

Author: LIU, Zhengze
Source: ZOJ Monthly, January 2019
Submit    Status


text:
HAL : in2p3-00275768, version 1
 arXiv : 0804.3869
 Evidence for a long-lived superheavy nucleus with atomic mass number A=292 and atomic number Z=$\sim$122 in natural Th
 (2008)
 Evidence for the existence of a superheavy nucleus with atomic mass number A=292 and abundance (1-10)x10^(-12) relative to ^{232}Th has been found in a study of natural Th using inductively coupled plasma-sector field mass spectrometry. The measured mass matches the predictions [1,2] for the mass of an isotope with atomic number Z=122 or a nearby element. Its estimated half-life of t1/2 >= 10^8 y suggests that a long-lived isomeric state exists in this isotope. The possibility that it might belong to a new class of long-lived high spin super- and hyperdeformed isomeric states is discussed.[3-6]
 Thème(s) : Physique/Physique Nucléaire Expérimentale
 Lien vers le texte intégral : http://fr.arXiv.org/abs/0804.3869
 in2p3-00275768, version 1 http://hal.in2p3.fr/in2p3-00275768 oai:hal.in2p3.fr:in2p3-00275768 Contributeur : Béatrice Forrler <> Soumis le : Vendredi 25 Avril 2008, 11:05:39 Dernière modification le : Vendredi 25 Avril 2008, 11:17:17


text:
SEARCH HOME
 Math Central Quandaries & Queries
 Question from Nilesh, a student: How to solve (x-y)^1/2 + 3(x-y) = 30 ; xy + 3(x-y) = 11 ? .

Hi Nilesh,

I can help get you started.

Focus on the first equation

$(x-y)^{1/2} + 3(x-y) = 30$

and let

$z = (x-y)^{1/2}.$

Once you have found $x$ and $y$ make sure you verify your answer.

Penny

Math Central is supported by the University of Regina and the Imperial Oil Foundation.


text:
Chapter 19 - Exercises and Problems - Page 363: 61

$1-r^{1-\gamma}$

Work Step by Step

a) Simplifying the equation for work done, we find that the total work done is equal to: $W = W_{12}+W_{34}$ We now use what we know about the work done to find the two equations for work we need: $W_{12}=\frac{1}{1-\gamma}(\frac{RT_3}{r^{\gamma-1}}-RT_3)$ $W_{34}=-\frac{1}{1-\gamma}(\frac{RT_2}{r^{\gamma-1}}-RT_2)$ Using factoring, these two equations combine to: $W_t=\frac{1}{1-\gamma}(\frac{R(T_3-T_2)}{r^{\gamma-1}}-R(T_3-T_2))$ We can simplify the above equation to be: $\frac{R(T_3-T_2)}{1-\gamma}(\frac{1}{r^{(\gamma-1)}}-1)$ We know that the value of Q is the negative coefficient of this equation, so this can be simplified to: $e=\frac{W}{Q}=(-\frac{1}{r^{(\gamma-1)}}+1)$ $e=1-r^{1-\gamma}$

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