text:
## 15.42 Permanence of properties under completion

Given a Noetherian local ring $(A, \mathfrak m)$ we denote $A^\wedge$ the completion of $A$ with respect to $\mathfrak m$. We will use without further mention that $A^\wedge$ is a Noetherian complete local ring with maximal ideal $\mathfrak m^\wedge = \mathfrak m A^\wedge$ and that $A \to A^\wedge$ is faithfully flat. See Algebra, Lemmas 10.96.6, 10.96.4, and 10.96.3.

Lemma 15.42.1. Let $A$ be a Noetherian local ring. Then $\dim (A) = \dim (A^\wedge )$.

Proof. By Algebra, Lemma 10.96.4 the map $A \to A^\wedge$ induces isomorphisms $A/\mathfrak m^ n = A^\wedge /(\mathfrak m^\wedge )^ n$ for $n \geq 1$. By Algebra, Lemma 10.51.12 this implies that

$\text{length}_ A(A/\mathfrak m^ n) = \text{length}_{A^\wedge }(A^\wedge /(\mathfrak m^\wedge )^ n)$

for all $n \geq 1$. Thus $d(A) = d(A^\wedge )$ and we conclude by Algebra, Proposition 10.59.8. An alternative proof is to use Algebra, Lemma 10.111.7. $\square$

Lemma 15.42.2. Let $A$ be a Noetherian local ring. Then $\text{depth}(A) = \text{depth}(A^\wedge )$.

Proof. See Algebra, Lemma 10.161.2. $\square$

Lemma 15.42.3. Let $A$ be a Noetherian local ring. Then $A$ is Cohen-Macaulay if and only if $A^\wedge$ is so.

Proof. A local ring $A$ is Cohen-Macaulay if and only if $\dim (A) = \text{depth}(A)$. As both of these invariants are preserved under completion (Lemmas 15.42.1 and 15.42.2) the claim follows. $\square$

Lemma 15.42.4. Let $A$ be a Noetherian local ring. Then $A$ is regular if and only if $A^\wedge$ is so.

Proof. If $A^\wedge$ is regular, then $A$ is regular by Algebra, Lemma 10.109.9. Assume $A$ is regular. Let $\mathfrak m$ be the maximal ideal of $A$. Then $\dim _{\kappa (\mathfrak m)} \mathfrak m/\mathfrak m^2 = \dim (A) = \dim (A^\wedge )$ (Lemma 15.42.1). On the other hand, $\mathfrak mA^\wedge$ is the maximal ideal of $A^\wedge$ and hence $\mathfrak m_{A^\wedge }$ is generated by at most $\dim (A^\wedge )$ elements. Thus $A^\wedge$ is regular. (You can also use Algebra, Lemma 10.111.8.) $\square$

Lemma 15.42.5. Let $A$ be a Noetherian local ring. Then $A$ is a discrete valuation ring if and only if $A^\wedge$ is so.

Lemma 15.42.6. Let $A$ be a Noetherian local ring.

1. If $A^\wedge$ is reduced, then so is $A$.

2. In general $A$ reduced does not imply $A^\wedge$ is reduced.

3. If $A$ is Nagata, then $A$ is reduced if and only if $A^\wedge$ is reduced.

Proof. As $A \to A^\wedge$ is faithfully flat we have (1) by Algebra, Lemma 10.162.2. For (2) see Algebra, Example 10.118.5 (there are also examples in characteristic zero, see Algebra, Remark 10.118.6). For (3) see Algebra, Lemmas 10.160.13 and 10.160.10. $\square$

Lemma 15.42.7. Let $A$ be a Noetherian local ring. If $A^\wedge$ is normal, then so is $A$.

Proof. As $A \to A^\wedge$ is faithfully flat this follows from Algebra, Lemma 10.162.3. $\square$

Lemma 15.42.8. Let $A \to B$ be a local homomorphism of Noetherian local rings. Then the induced map of completions $A^\wedge \to B^\wedge$ is flat if and only if $A \to B$ is flat.

Proof. Consider the commutative diagram

$\xymatrix{ A^\wedge \ar[r] & B^\wedge \\ A \ar[r] \ar[u] & B \ar[u] }$

The vertical arrows are faithfully flat. Assume that $A^\wedge \to B^\wedge$ is flat. Then $A \to B^\wedge$ is flat. Hence $B$ is flat over $A$ by Algebra, Lemma 10.38.9.

Assume that $A \to B$ is flat. Then $A \to B^\wedge$ is flat. Hence $B^\wedge /\mathfrak m_ A^ n B^\wedge$ is flat over $A/\mathfrak m_ A^ n$ for all $n \geq 1$. Note that $\mathfrak m_ A^ n A^\wedge$ is the $n$th power of the maximal ideal $\mathfrak m_ A^\wedge$ of $A^\wedge$ and $A/\mathfrak m_ A^ n = A^\wedge /(\mathfrak m_ A^\wedge )^ n$. Thus we see that $B^\wedge$ is flat over $A^\wedge$ by applying Algebra, Lemma 10.98.11 (with $R = A^\wedge$, $I = \mathfrak m_ A^\wedge$, $S = B^\wedge$, $M = S$). $\square$

Lemma 15.42.9. Let $A \to B$ be a flat local homomorphism of Noetherian local rings such that $\mathfrak m_ A B = \mathfrak m_ B$ and $\kappa (\mathfrak m_ A) = \kappa (\mathfrak m_ B)$. Then $A \to B$ induces an isomorphism $A^\wedge \to B^\wedge$ of completions.

Proof. By Algebra, Lemma 10.96.7 we see that $B^\wedge$ is the $\mathfrak m_ A$-adic completion of $B$ and that $A^\wedge \to B^\wedge$ is finite. Since $A \to B$ is flat we have $\text{Tor}_1^ A(B, \kappa (\mathfrak m_ A)) = 0$. Hence we see that $B^\wedge$ is flat over $A^\wedge$ by Lemma 15.27.5. Thus $B^\wedge$ is a free $A^\wedge$-module by Algebra, Lemma 10.77.5. Since $A^\wedge \to B^\wedge$ induces an isomorphism $\kappa (\mathfrak m_ A) = A^\wedge /\mathfrak m_ A A^\wedge \to B^\wedge /\mathfrak m_ A B^\wedge = B^\wedge /\mathfrak m_ B B^\wedge = \kappa (\mathfrak m_ B)$ by our assumptions (and Algebra, Lemma 10.95.3), we see that $B^\wedge$ is free of rank $1$. Thus $A^\wedge \to B^\wedge$ is an isomorphism. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
Plain enumeration of theorems definitions etc

In my TeX document Theorems, Definitions, Lemmas are enumerated in the following way:

Definition 1.1

Lemma 1.1

Theorem 1.1

Definition 1.2

Lemma 1.2

Theorem 1.2

But I would like to have

Definition 1.1

Lemma 1.2

Theorem 1.3

Definition 1.4

Lemma 1.5

Theorem 1.6

How can I obtain this?

Edit. The begin of my document is:

\documentclass[12pt]{article}

\usepackage[T1]{fontenc}

\usepackage[cp1250]{inputenc}

\usepackage{amsfonts}

\usepackage{amssymb}

\usepackage{latexsym}

\usepackage{amsthm}

\theoremstyle{plain}

\newtheorem{theorem}{Theorem}[section]

\newtheorem{corollary}{Corollary}[section]

\newtheorem{lemma}{Lemma}[section]

\newtheorem{observation}{Observation}[section]

\newtheorem{proposition}{Proposition}[section]

\theoremstyle{definition}

\newtheorem{definition}{Definition}[section]

\newtheorem{example}{Example}[section]
-
What package are you using to generate these environments? Are you defining them without packages? Either way, include these in your post via an edit. –  Werner Apr 3 '12 at 17:10
Welcome to TeX.SE. Seconding Werner's comment: It is always best to compose a fully compilable MWE that illustrates the problem including the \documentclass and the appropriate packages so that those trying to help don't have to recreate it. This will also serve as a test case and ensure that the solution actually works for you. –  Peter Grill Apr 3 '12 at 17:21
very similar to (possible duplicate?) common-per-section-numbering-of-theorems-lemmas-etc –  cmhughes Apr 3 '12 at 18:01

Just use the first optional argument, that denotes the counter to share:

\usepackage{amsthm}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{observation}[theorem]{Observation}
\newtheorem{proposition}[theorem]{Proposition}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}

\newtheorem{example}[theorem]{Example}
-
Very thanks for help. –  Richard Apr 3 '12 at 18:20

format: plain


text:
# Which statement best describes the relationship between the industrial revolution and mass production in great britain? a. the

###### Question:

Which statement best describes the relationship between the industrial revolution and mass production in great britain? a. the industrial revolution began when farmers applied the principles of mass production to agriculture. b. the mass production of resources such as coal and iron sped up the spread of the industrial revolution. c. technology developed during the industrial revolution allowed for the mass production of goods such as textiles. d. following the industrial revolution fewer goods were made through mass production.

### Brendan resturant bull comes to 58.45 find the total cost if the tax is 8.25% and a 20% tip is left on the amount before tax

Brendan resturant bull comes to 58.45 find the total cost if the tax is 8.25% and a 20% tip is left on the amount before tax...

### The crossbow was a more effective weapon than the normal bow because it used a reel, giving more power

The crossbow was a more effective weapon than the normal bow because it used a reel, giving more power to the bow string it was held horizontally it shoot a bolt rather than an arrow it could shoot harder and farther all of the above...

### 20 how do the graphs of y=1/x and y=5/(x+6) compare? i need steps because i want to be able to do these type of problems.

20 how do the graphs of y=1/x and y=5/(x+6) compare? i need steps because i want to be able to do these type of problems....

### Felicity write the buffer and closing for a message rejecting payment of a customer’s claim under an auto insurance policy.

Felicity write the buffer and closing for a message rejecting payment of a customer’s claim under an auto insurance policy. we reviewed your records and note that you held the policy for 12 months. we sincerely regret that we are unable to pay your claim because the policy period ended and the pol...

Please help me find the zeros and the plots $Please help me find the zeros and the plots$...

### Age (in weeks) 3 7 10Height (in centimeters) 12 35 60Is the height of the plant proportional to its age?

Age (in weeks) 3 7 10 Height (in centimeters) 12 35 60 Is the height of the plant proportional to its age?...

### The gustatory system is the sensory system that deals with smell. True or False

The gustatory system is the sensory system that deals with smell. True or False...

### Some money will be divided in the ratio of 2 to 3. Four times the smaller amount is one hundred three dollars less than three

Some money will be divided in the ratio of 2 to 3. Four times the smaller amount is one hundred three dollars less than three times the larger amount. What’s the smaller amount? What is the largest amount?...

### Identifique as frases incorretas e as corrija em seucaderno.a) Após a luta contra a Espanha, os

Identifique as frases incorretas e as corrija em seu caderno. a) Após a luta contra a Espanha, os governos das novas nações latino-americanas valorizaram os indígenas e buscaram proteger seu legado cultural. b) As reformas liberais acabaram com o conceito de posse comunitária da terra e transf...

### A scale drawing of a rectangular basketball court measures 7 inches long and 4 inches wide. If the scale used for the drawing

A scale drawing of a rectangular basketball court measures 7 inches long and 4 inches wide. If the scale used for the drawing is 2.5 inches equals 5 yards, what is the actual length of the park?...

### Agas storage cylinder in an ordinary chemical laboratory measures 4.4 cm wide and 17.cm high. this is

Agas storage cylinder in an ordinary chemical laboratory measures 4.4 cm wide and 17.cm high. this is the label on it. contents: n2 gaspressure: 29.8 atmif the cylinder is opened and the gas allowed to escape into a large empty plastic bag, what will be the final volume of nitrogen gas, including ...

### Agroup of related fields is generally called a data file. what do you call this group?

Agroup of related fields is generally called a data file. what do you call this group?...

### Which statement best describes wavelengths of sunlight that are longer than 700 nanometers?

Which statement best describes wavelengths of sunlight that are longer than 700 nanometers?...

### The exploration of which continent led Europeans to create better maps in the 1800's?1-Europe2-Africa3-North america4-Antartica

The exploration of which continent led Europeans to create better maps in the 1800's? 1-Europe 2-Africa 3-North america 4-Antartica...

### Sinusoidal waves 5.00 cm in amplitude are to be transmitted along a string that has a linear mass density

Sinusoidal waves 5.00 cm in amplitude are to be transmitted along a string that has a linear mass density of 4.00 10-2 kg/m. the source can deliver a maximum power of 275 w, and the string is under a tension of 121 n. what is the highest frequency f at which the source can operate?...

### Which is not a function of a state party chairperson?

Which is not a function of a state party chairperson?...

### As a skateboarder moves down hill some of the energy is

As a skateboarder moves down hill some of the energy is...

### Use ΔXYZ below to answer the question that follows:Which fact is not used to prove that XYZ is similar to VYW?

Use ΔXYZ below to answer the question that follows:Which fact is not used to prove that XYZ is similar to VYW?...


text:
# Spring 2013 Course Notes

Hi everybody,

As I move into the edge of third year as an undergraduate student, the time that I have to actively contribute to this blog lessen more and more. However, with my new investment into a new tablet/laptop (that can last more than 1 hour) in the form of the latest Microsoft Surface Pro, I hope to make it up to everyone out there by offering not one, not two, but SIX sets of course notes this term.

Specifically I will be covering PMATH 450, PMATH 352, ACTSC 372, ACTSC 445, STAT 371 and STAT 330. I have already posted the most up to date versions of these notes in the Course Notes section of my blog and will continue to update them throughout the term, along with review notes for midterms and finals. Note that currently I am emphasizing the content of these notes rather than the aesthetics, so some areas such as the index and abstract are still under construction.

Hopefully I will have some time to present something of interest from my studies as the term goes on, but for now, all of you who are dying to know more about the details the non-measurability of transforms of $\mathbb R \backslash \mathbb Q$ will have to make do with my notes.

Until next time,

Stochastic Seeker

# Fall 2012 Exam Notes

Final exam review sheets have been posted for STAT 333 and PMATH 351 in the Course Notes section of this blog. I will update this post when more notes become available.

Update 1: The review sheet for ACTSC 371 is now available.

Update 2: The review sheet for CS 338 is now available.

# Which field(s) of mathematics are you interested in?

EDIT 1: There really should be a +1 to the CO section. I just forgot to add it, when I first made the poll. Sorry!

EDIT 2: +1 to Topology as well. Please let me know if you guys have any more areas of mathematics that I’m missing!

# Happy New Year Everyone!

To all my readers and friends out there, I just want to start off by wishing everyone a happy  new year! While you may not have heard from for a quite a while now, I assure you that I have been working on quite a few other projects. Here are a few that you look forward to:

• First off, I decided to move and develop my LaTeX templates on GitHub, as per a recommendation by one of my friends, and you can now access the latest version here. Instructions are on the page in case you are not too familiar with the layout of the pages. I will still leave the link to the uWaterloo Linux website intact, but all development on that site will cease indefinitely (new files will be put up there, but not new webpages).
• Also on GitHub, I have put up many of my coding projects, including a Stock/Black-Scholes Option simulator and a biased random walk. You can access them via this link.
• Today (or tomorrow), I will hopefully be finishing off my “High school life, research topics, and advice for the young aspiring mathematician.” series so you can look forward to that.
• Course notes will start coming up soon as I continue my quest to take over Lambertw’s empire of quality notes
• I have decided to publish the fish population article for lack of updates from the original website on that problem. The problem of the square through the sphere is still under development.
• In the future, I will be restoring my old extended essay into a cleaner LaTeX form
• Currently I am developing a cleaner way to view projects on the site. Suggestions are appreciated; right now I am thinking of just moving everything to GitHub.
• As for the few projects that I will be putting off, these include my Ray Tracing project and Black-Scholes mini-series due to the six-course load that I will be taking at Waterloo in the coming Winter.
• For those of you who would like to see some an introductory article to Black-Scholes, I will be putting that up very soon.

I hope to see old and new readers coming in for the new year because now is where the content will be really flowing in!

# Future Plans and Projects.

While I am still quite busy preparing for Exam FM (completing 10 mock exams), I do have quite a few short-term and long-term plans for this blog.

For the sake of being concise, here is a short list of what I have planned; many of the projects will only be started after exam FM which is on the 10th of December. However, for some of these project, I might have some to time to start on them this week:

# Short-term Projects

• Finish my proof of the volume displaced from pushing a square through a sphere; an incomplete version can be found here.
• Complete the third entry in the high school reflection series
• Solve an interesting population problem about fish breeding here. I will also be re-typing the problem for better coherency (new version here)

# Long-term Projects (>3 months)

• Begin typesetting notes for Exam MLC. These notes will be extremely concise and should be considered a supplement for the recommended readings here. One of the texts is free while the other requires purchase (I acquired an eBook version). Since this is the first exam that will follow the new syllabus, I will be covering the content quite thoroughly with what is listed on the syllabus.
• Course notes for second and third year courses at uWaterloo (specifically the ones that I will be taking next term
• “Advice for the university freshman” series; look for it during the Winter term


text:
# How do you find vertical, horizontal and oblique asymptotes for y= (x^2 - 9)/(x-3)?

##### 1 Answer
Mar 28, 2016

First, factor the numerator: $y = \frac{\left(x + 3\right) \left(x - 3\right)}{x - 3}$

#### Explanation:

Cancel the factor $x - 3$ from both the numerator and denominator. You are left with $y = x + 3$, a linear function!

The factor that was cancelled leaves a "hole" or removable discontinuity in your graph at x = 3.

If you plot the line $y = x + 3$, there will be NO asymptotes at all, but just a little hole at the point (3,6).


text:
Technical Article

# Choosing the Right Oscillator for Your Microcontroller

May 17, 2016 by Robert Keim

## Internal or external? Quartz or ceramic? Crystal oscillator or silicon oscillator? So many clocking options . . . which one is right for your design?

Internal or external? Quartz or ceramic? Crystal oscillator or silicon oscillator? So many clocking options . . . which one is right for your design?

### Oscillating Options

Every microcontroller needs a clock source. The CPU, the memory bus, the peripherals—clock signals are everywhere inside a microcontroller. They govern the speed at which the processor executes instructions, the baud rate of serial-communication signals, the amount of time needed to perform an analog-to-digital conversion, and so much more.

All of this clocking action goes back to the source of the clock signal, namely the oscillator. Therefore, you need to make sure that your oscillator can support whatever performance you expect from your microcontroller. At the same time, though, some oscillator options are more complex or expensive than others, so your choice of oscillator should also reflect the importance of reducing cost and complexity whenever possible.

There are quite a few ways to generate a clock signal for a microcontroller. The datasheet for your particular device should provide a good deal of information about what types of oscillators you can use and how to implement them in a way that is compatible with the device’s hardware. This article will focus on the advantages and disadvantages of various clock sources so that you can better choose among the oscillator options discussed in your microcontroller’s datasheet.

So let’s start with a list, followed by a discussion of each option:

• Internal
• Usually (as far as I know, always) a resistor–capacitor circuit
• Phase-locked loop for frequency multiplication
• External
• CMOS clock
• Crystal
• Ceramic resonator
• Resistor–capacitor
• Capacitor only

### Internal Oscillators: The KIS Option

I’m an advocate of the Keep It Simple principle; consequently, I have a special appreciation for internal oscillators, and I encourage you to avail yourself of the internal oscillator whenever possible. No external components are required: You can safely assume that the frequency is well chosen since the oscillator was designed by the same people who designed the rest of the microcontroller. Also, the salient performance specs—e.g., initial accuracy, duty cycle, temperature dependency—are (hopefully) right there in the datasheet.

The dominant disadvantage with internal oscillators is the lack of precision and frequency stability. The baseline frequency depends on the values of the passive components that make up the oscillator circuit, and the tolerances for the values of these passive components are not particularly tight. Furthermore, capacitance and resistance are influenced by ambient temperature, so internal RC oscillators experience temperature “drift”—i.e., changes in temperature lead to changes in frequency.

In my experience, many applications can tolerate the shortcomings of an internal oscillator, especially when the frequency has been calibrated at the factory. With older microcontrollers, the internal oscillator might have tolerance as bad as ±20%. However, a newer device can give you ±1.5% (or better), which is accurate enough for RS-232 communication and even (in conjunction with clock-recovery circuitry) for USB.

Another way to expand the capabilities of an internal oscillator is manual “trimming”—if your microcontroller includes a trimming/calibration register, you can adjust the frequency by modifying the value in this register. This is a perfectly practical technique for low-quantity designs: Simply measure the clock frequency with an oscilloscope or frequency counter and then trim the oscillator accordingly.

A variation on the internal-oscillator theme is the phase-locked loop (PLL). A PLL allows a low-quality, high-speed internal oscillator to benefit from the stability and precision of an external oscillator. In general, a PLL doesn’t help you to avoid external components because it requires a reference clock that is usually derived from a crystal. An exception, though, is when you have a high-quality clock somewhere on the PCB but don’t want to use it for the microcontroller because it’s too slow—you could use a PLL to multiply this clock up to an acceptable frequency.

### CMOS Clock

Another straightforward clocking option is the so-called “CMOS clock,” which falls into the for-lack-of-a-better-term category. “CMOS clock” is a vague (though convenient) way of referring to any clock signal driven by some other component on the board. The CMOS clock is a great option if your design already includes a clock signal with 1) a workable frequency and 2) electrical characteristics that are compatible with the microcontroller’s CMOS-clock-input circuitry. Often, though, this is not the case, so let’s look at two options for generating a CMOS clock.

First is the “crystal oscillator.” This is a good time to point out that a quartz crystal is not an oscillator; rather, a quartz crystal is the central component in a quartz-crystal oscillator circuit, which might look something like this:

Crystal oscillators are handy devices that consist of a quartz crystal and the additional circuitry needed to generate a standard digital clock signal. Thus, you get the stability and precision of a crystal without worrying about load capacitance and the careful PCB layout needed to ensure robust operation with a standalone crystal.

The second option is a “silicon oscillator”. This term refers to oscillator ICs that are not based on quartz crystals or ceramic resonators. These devices are versatile and easy to use, and they can be quite accurate. For example, the LTC6930 series from Linear Tech requires only one bypass capacitor, and the vast majority of the parts are within .05% of the nominal frequency:

Silicon oscillators are more reliable than crystals and ceramic resonators, especially in harsh environments subject to shock or vibration. But they’re also more expensive.

### Quartz and Ceramic

When you need seriously high precision and stability without the additional cost of a crystal-based oscillator IC, opt for the standalone-crystal approach. Parts with tolerance below 20 parts per million (i.e., 0.002%) are readily available. The oscillator circuit shown above is partially integrated into microcontrollers that support the standalone-crystal configuration; you will need to provide the correct load capacitors. The total load capacitance (CLTOTAL) is specified in the crystal’s datasheet, and the load capacitors are chosen as follows:

$C_{LTOTAL}=\frac{C_{L1}\times C_{L2}}{C_{L1}+C_{L2}}+C_P$

where CP represents whatever parasitic capacitance is present. This calculation is pretty simple in practice: Choose a reasonable value for CP (say, 5 pF), subtract this from CLTOTAL, then multiply by two. So if the datasheet gives a load capacitance of 18 pF, we have

$C_{L1}=C_{L2}=\left(18\ pF-5\ pF\right)\times2=26\ pF$

Ceramic resonators are less accurate than crystals; common tolerances are 1000 to 5000 parts per million. They may save you a few cents if you don’t need the accuracy of quartz, but in my mind, the main advantage is that you can get ceramic resonators with integrated load capacitors.

### Last, and Least . . .

There aren’t many situations that call for an external resistor–capacitor or capacitor-only oscillator. If for some reason you are opposed to the external-oscillator options discussed thus far, choose a microcontroller with an internal oscillator and use that. If, however, you are determined to dig out a passive or two from your box of spare parts, refer to the microcontroller’s datasheet for guidance on how to connect and design the oscillator circuit. Here is an example of how to connect the components, taken from the datasheet for the C8051F12x–13x (PDF) microcontrollers from Silicon Labs:

And you can refer to page 190 (PDF) of this same datasheet for an example of information on choosing component values.

### Conclusion

I hope you now know enough to make an informed, confident decision next time you need to choose an oscillator for your microcontroller. Here are my recommendations in a nutshell:

• Internal oscillator whenever possible
• Silicon oscillator if the accuracy is adequate and the cost is acceptable—otherwise, quartz crystal

• Share

### You May Also Like

• MrChips May 12, 2016

Here are some items that need to be added. Since a lot of MCU applications require a stable clock frequency, a more expanded discussion on crystal oscillators would be useful.
For example, explain the difference between series and parallel resonance XTAL oscillators. Why are third harmonic XTAL oscillators used? What is the importance of the C1 and C2 loading capacitors and how does one determine the best values to use? Why is there a high value resistor in parallel with the XTAL? Why does the oscillator fail to start or output the correct frequency? How to prevent this from happening? Why is PCB layout so critical to the oscillator? What are some guidelines to proper PCB layout?

Many applications require time keeping function. For this reason, many MCUs have a low frequency oscillator option requiring the popular 32768Hz XTAL. How does one choose the loading capacitor? How does one adjust and calibrate the oscillator? What is the purpose of the series resistor? What are the adverse effects if one omits the series resistor?

Like.
• J
jimkim March 23, 2018
Many good questions there. Many of the answers can be found in this Microwave Digest article http://www.crystek.com/documents/appnotes/pierce-gateintroduction.pdf . Third overtone crystals are used as a compromise between cost and performance. Quartz crystals (at least the common AT cut types) use a resonance mode known as thickness shear mode in which the frequency has an inverse relationship with the mass of the crystal. The mass is typically controlled by varying the thickness of the crystal wafer. In order to achieve a high frequency such as >30MHz (this depends on the crystal package size) the wafer would have to be extremely thin and therefore difficult or sometimes practically impossible to manufacture, causing price to increase substantially. Therefore, a cheaper third overtone crystal is used instead. One method to overcome this manufacturing obstacle is to use an inverted mesa design crystal. Inverted mesa crystals use a wafer that is thinner in the middle so they are less fraile. Inverted mesa crystals can be made in fundamental mode up to 200 MHz (sometimes higher) but you'd have to pay an ARM, LEG, AND HALF YOUR BODY for it. There are many possible reasons why oscillators fail startup and output incorrect frequencies. Based on my experience, the frequency error is commonly due to incorrect selection of load capacitor (C1, C2) values. Many engineers are under the impression that a crystal with a load value of 18pF requires load capacitors of 18pF. THIS IS NOT THE CASE. The correct calculation is Cload = {[Cin+C1][C2+Cout]/[Cin+C1+C2+Cout]} + pcb strays. The start up issue can be more complex and involves drive level, ESR, negative resistance, and other factors in circuit matching and optimization. Also with a 32.768KHz crystal a Rs is a must and must not be ommited as without one, the high drive level may damage the crystal. More details can be found in the article mentioned above including the function of Rs and Rf resistors.
Like.
• Barbo May 27, 2016

Very, very good, thank you.

Like.
• Isa Ansharullah May 19, 2017

Great. I want to know more about this so-called “Trimming”, what does this mean and how do we do that. And how exactly microprocessor “trim” an internal (or external!) oscillator.

Like.
• kuashio November 10, 2019
Hey, I regret to comment so much later, but for what it's worth, I'm writing an article exactly on internal oscillator trimming. I expect to send it in tomorrow or the day after. Stay tuned to allaboutcircuits.com!
Like.


text:
Distribution of the sum of a multinomial distribution

I have distilled an error analysis problem into the following:

I have a multinomial distribution, $X$, consisting of $n$ independent trials where each trial takes on the values $\{0,1,\ldots,k-1\}$ with uniform probability of $\frac{1}{k}$.

Now I am interested in finding the distribution of the sum of all the outcomes in the $n$ trials, but not sure how to approach the problem. I suspect it has something to do with partition functions. Would be glad if the relevant probability distribution function in MATLAB could also be pointed out.

-

If you have $n\geq 30$, you can apply the central limit theorem.

Assume $u_1,u_2,\cdots,u_n\sim {\cal{U}}[0,k-1]$ i.i.d., then we know for each $u_k$ its mean and variance are $$\mu={(k-1)\over 2} {\rm\,and\,} \sigma^2 = {k^2-1\over 12}.$$ Now construct the summation random variable $$X=\sum_{k=1}^n u_k$$ Then you can find the corresponding mean and variance of r.v. $X$ as $$\mu_X = n\mu {\rm\,and\,} \sigma^2_X = n\sigma^2$$ So if $n$ is sufficiently large, then the distribution of $X$ can be considered as a Gaussian, i.e. $$X\sim {\cal N}(n\mu,n\sigma^2)$$ Based on this information, I think it is easy for your to use Matlab to compute the CDF of $X$. The last step is to get the discrete PMF from this continuous CDF, which can be thought of the following form $$Pr(X=m) = CDF(m+.5)-CDF(m-.5)$$

-
Unfortunately $n=4$ for my error analysis. Nevertheless the above-mentioned result is noteworthy. – Ang Zhi Ping Oct 31 '12 at 5:06

Since you mentioned in a comment that $n=4$ in your case, here's a way to derive the distribution for small values of $n$.

The number of ways of writing $m$ as a sum of $n$ values from $0$ to $k-1$ is the coefficient of $x^m$ in

$$(1+x+\dotso+x^{k-1})^n=\left(\frac{1-x^k}{1-x}\right)^n=(1+x+x^2+\dotso)^n\sum_{j=0}^n\binom nj(-x^k)^j\;.$$

Thus you just have to place the binomial coefficients in a sequence at distances of $k$ and then sum the sequence $n$ times; e.g. for $n=4$ and $k=3$:

$$\begin{array}{rrrrrrrrr} 1&0&0&-4&0&0&6&0&0&-4&0&0&1\\ 1&1&1&-3&-3&-3&3&3&3&-1&-1&-1\\ 1&2&3&0&-3&-6&-3&0&3&2&1\\ 1&3&6&6&3&-3&-6&-6&-3&-1\\ 1&4&10&16&19&16&10&4&1 \end{array}$$

Then dividing through by the total number $k^n=81$ of possibilities gives you the probabilities for the values of the sum.

-
You mentioned that "The number of ways of writing m as a sum of n values from 0 to $k−1$ is the coefficient of $x^m$". Do you have a closed form formula for the number of ways involved? – Ang Zhi Ping Oct 31 '12 at 8:39
Anyway I used Wolfram to do the expansion, and it suffices for my application. – Ang Zhi Ping Oct 31 '12 at 14:37


text:
# I What is the opposite of epsilon

Tags:
1. Jun 10, 2016

### Pjpic

Is there a notation for the opposite of epsilon (infinitesimal) in the way that infinity is the opposite of zero?

2. Jun 10, 2016

### Staff: Mentor

I don't think there is a definitive term, I've seen people use these terms:
- immeasureable
- infinite

3. Jun 10, 2016

### Ssnow

For every $\varepsilon >0$ ''very small'' you have $\varepsilon^{-1}=\frac{1}{\varepsilon}$ is ''very big'', in general there is no a preferred symbol to denote this ...

4. Jun 10, 2016

### micromass

Staff Emeritus
Usually, people like to use capital letters for this, such as $H$ or $N$.

5. Jun 10, 2016

### Staff: Mentor

What does the H or N stand for?

6. Jun 10, 2016

### Staff: Mentor

Perhaps $N$ for number such as $N(ε)$. I've also seen $C$ for a constant. But I've never seen a huge $H$.

7. Jun 10, 2016

### wrobel

"The nightmare of mathematician is a sequence $\varepsilon_n$ that tends to infinity, as $n \to 0$"
Paul Halmos

8. Jun 11, 2016

### Aniruddha@94

The opposite of infinity is not zero, it's $\varepsilon$.
Source : This Numberphile video

.
Depends how you define "opposite". If you mean additive inverse, then $-\infty$ would be the opposite of $\infty$, and $-\varepsilon$ for $\varepsilon$. If by "opposite" you mean multiplicative inverse, then the opposite of $\varepsilon$ would be $\infty$. Zero, on the other hand doesn't have a multiplicative inverse, so maybe we can say it doesn't have an opposite.


text:
Deepak Scored 45->99%ile with Bounce Back Crack Course. You can do it too!

# If the standard deviation of the numbers 2,

Question:

If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true ?

1. $3 a^{2}-23 a+44=0$

2. $3 a^{2}-26 a+55=0$

3. $3 a^{2}-32 a+84=0$

4. $3 a^{2}-34 a+91=0$

Correct Option: , 3

Solution:


text:
# Homework Help: Natural log derivative problem

1. Sep 14, 2005

### compute_a_nerd

Hello all. I am stuck on this homework problem. It wants me to find
$$\frac {d^2y} {dx^2}$$
when $$y= ln x^8$$
The book answer is $$\frac {-8}{x^2}$$
But I only can get $$\frac {-8}{x^9)}$$

Thanks

Last edited: Sep 14, 2005
2. Sep 14, 2005

### saltydog

Need to keep very clear the precedence of operators so write it as:

$$y=ln(x^8)$$

Now, just differentiate once to get 8/x, one more time to get the book's answer.

3. Sep 14, 2005

Thx so much


text:
— 7 min

# # TLDR

’Tis the planning season for Rust 2021 already, and as suggested, I will start with a very short list of bullet points:

• As a long-time observer of the Rust project, I would to see some long-time projects come to an end, to reap the promised benefits ;-)
• As an ex-TypeScript/JS developer, I am super excited to see high quality, source-based code coverage (with branch coverage even) being worked on, and would like to see this project get the attention it deserves.
• Related, as a developer who loves testing and gamification, I want to have an easier way to get an overview of how my project is doing, and where I can improve it.
• As a maintainer of a mid-sized workspace of Rust crates, I would like to have better tooling, to deal with versioning, and publishing of my crates, as the process feels very hacky right now.
• As a maintainer of some crates that have a lot of feature flags, I would like feature flags to be less of a pain and easier to work with, to give me the confidence that my crate works no matter what feature flags the consumers use.
• As a developer who has recently switched to GitHub Actions, I would have liked that process to be more straight forward, so that I can avoid a lot of copy&paste, and be sure that my CI builds are as fast as they can be.

# # A Big Thank You!

I have actually already written about my wishlist for Rust tooling a few month ago, and like always, things have changed, and I have learned a lot.

Let me start off that I am super happy where Rust and the ecosystem are moving. rust-analyzer has been a major success and is a great productivity booster! It is also test-driving chalk, which is supposed to become the new trait solver in Rust hopefully soon. It is already working amazingly well in my IDE, as I can get type hints and hover information in cases where the main Rust compiler still wants me to manually type-annotate things.

Also, another project that I am super excited about and that gets a lot less publicity than it deserves is precise source-based code coverage support built into the compiler. I only know about this because I regularly scan through the list of PRs to the compiler. This definitely needs more marketing around it! And of course a way to easily use it. From all PRs I have seen, I would have no idea how to actually use it and maybe even combine it with codecov, etc.

Another great thing that I see was recently stabilized is intra-doc links. I am super excited about this, and am already starting to use them, as they have been available on docs.rs for quite some time.

# # Where Rust needs to improve

The pain points I have right now are mostly related to tooling, and also related to having bigger more complex projects, possibly workspaces consisting of multiple crates.

I think the experience of creating and maintaining a small, simple Rust project is amazing right now. But things start to get annoying as soon as the project starts to grow.

I read that a lot of people are still complaining about long compile times. It is actually a very complex problem, and when looking into more detail, it is not really the speed of the Rust compiler that is at fault here. For me, local development, especially with rust-analyzer has been a pleasure. Things only start to fall apart when a small change means that all the interdependent crates in my workspace have to be updated, or the problem is the final linking step that is taking forever. Or the problem is not even my own machine, but rather CI that is super slow because of tons of possibly duplicated dependencies, and bad caching.

# # Improving CI

Speaking of CI, my team and I have been migrating a few workspaces to GitHub Actions recently, and while we do like it in general, there are a few rough edges that should be smoother. Not all are strictly Rust problems, some are also GHA.

The primary goal of CI in my opinion is to both be as fast as possible, and as thorough as possible, while also being efficient and easy to use. Which is the first tradeoff that a developer needs to choose from. Do I want to fail fast, or rather have all the failures visible? When we fan out wide, we get a ton of different status reports, which IMO is a UI concern to begin with; but that also means more overhead. While the wall-time might be smaller, the machine-time is increased, because the common steps of checkout, toolchain, etc are run by every job.

Right now, setting up a CI pipeline is too cumbersome. There is too many decisions a developer has to do, and way too much copy-paste going on in the CI definition itself. I wish a lot of these things would just be done out of the box.

One of the big problems for CI right now, especially for big projects is caching. Caching is not free! If cargo creates a big target folder, persisting and restoring that cache can already take a considerable amount of CI time. This problem gets a lot worse because cargo does not auto-clean that cache. There are some open issues about that, but nothing much has happened in years. A lot of projects have custom solutions to deal with this, and it does not really scale. Also, should one disable incremental compilation on CI? I don’t know. I would like if there would be a simple out of the box solution that covers 90% of the usecases.

# # Workspaces and Features

A bit related to CI is features. Lets start off by saying that I dislike having feature flags in the first place. They are just a pain to work with, and a pain to test, both locally and on CI. What I would like to see is official support for something like --feature-powerset of cargo-hack. Just make sure that every permutation of features works.

I saw some time ago that cargo is working on something called Features 2.0, which I hope will solve some of the pain points that I have. But so far I didn’t have the time or motivation to actually read up on what the proposed changes there are.

Another thing that I mentioned in my earlier blog post was some quality of life improvements related to workspaces. I think as a minimum, I would like to see an atomic cargo publish --all and maybe a tool to deal with versioning. As an ex-JS developer, there were a ton of tools available, but unfortunately, I was not really satisfied with any of them. I have the feeling that the Rust community as a whole creates a lot more high-quality tools, and I am hopeful that will also be the case here.

# # In Closing

I do feel Rust is headed in the right direction, but there is a lot of polishing still to do. Not necessarily with the language itself, but with the tools and ecosystem around it. And its not really specific to Rust either. In a lot of ecosystems, I feel like the tasks that are not coding related are often a lot more complex than they need to be. Rust, both as a language and as an ecosystem has proved that it can solve complex problems in a nice and high quality way. I’m looking forward to seeing these problems solved as well.


text:
70 posts
innovation through innovation™
Minimal D3D11
Edited by d7samurai on

New version of my minimal D3D11 reference / tutorial implementation; a small uncluttered Direct3D 11 setup and rendering primer for newcomers to the API.

https://gist.github.com/d7samurai/261c69490cce0620d0bfc93003cd1052

No modern C++ / OOP / obscuring cruft. The source code is a complete, runnable C+ application contained in a single function (WinMain), laid out in a step-by-step fashion that should be easy to follow from the code alone (about 250 LOC). Win7 compatible. Produces this output:

Mārtiņš Možeiko
2356 posts / 2 projects
Minimal D3D11
Edited by Mārtiņš Možeiko on
Some time back I wrote similar thing: https://git.handmade.network/snippets/19
Yours is a bit complete, mine is only rendering one triangle, but it has extra features:
* Plain C, no C++
* Does not require DirectD3 11.1, uses 11.0
* Optionally does not link to D3DCompiler_XXX.dll file
* Handles window resizing
* Handles GPU driver reset on present call

70 posts
innovation through innovation™
Minimal D3D11
Edited by d7samurai on

Minimal D3D11 pt2:

https://gist.github.com/d7samurai/aee35fd5d132c51e8b0a78699cbaa1e4

Follow-up to Minimal D3D11, adding instanced rendering. The main difference here is that the hollow cube is rendered using DrawIndexedInstanced (which saves a lot of vertices compared to the original, so model data is now small enough to be included in the source without being too much in the way), but also all trigonometry and matrix math is moved to the vertex shader, further simplifying the main code.


text:
# How do you simplify sqrt15/(2sqrt20)?

May 10, 2017

Take squares out of square roots.

#### Explanation:

Original expression: $\frac{\sqrt{15}}{2 \sqrt{20}}$

Using prime factorization for the values inside the square roots:
=sqrt(3*5)/(2sqrt(2^2*5)

Take out the ${2}^{2}$ out of the square root in the denominator:
$= \frac{\sqrt{3 \cdot 5}}{4 \sqrt{5}}$

Combine the square roots into one square root:
$= \frac{1}{4} \sqrt{\frac{3 \cdot 5}{5}}$

Cancel the $5$ in the numerator and denominator of the square root:
$= \frac{\sqrt{3}}{4}$ which is our final answer.


text:
# Time Series Analysis¶

Often, one wants to analyze a continuous set of outputs from a simulation in a uniform manner. A simple example would be to calculate the peak density in a set of outputs that were written out. The problem with time series analysis in yt is general an issue of verbosity and clunkiness. Typically, one sets up a loop:

for dsi in range(30):
fn = "DD%04i/DD%04i" % (dsi, dsi)
process_output(ds)

But this is not really very nice. This ends up requiring a lot of maintenance. The DatasetSeries object has been designed to remove some of this clunkiness and present an easier, more unified approach to analyzing sets of data. Even better, DatasetSeries works in parallel by default (see Parallel Computation With yt), so you can use a DatasetSeries object to quickly and easily parallelize your analysis. Since doing the same analysis task on many simulation outputs is ‘embarrassingly’ parallel, this naturally allows for almost arbitrary speedup - limited only by the number of available processors and the number of simulation outputs.

The idea behind the current implementation of time series analysis is that the underlying data and the operators that act on that data can and should be distinct. There are several operators provided, as well as facilities for creating your own, and these operators can be applied either to datasets on the whole or to subregions of individual datasets.

The simplest mechanism for creating a DatasetSeries object is to pass a glob pattern to the yt.load function.

import yt

This will create a new time series, populated with all datasets that match the pattern “DD” followed by four digits. This object, here called ts, can now be analyzed in bulk. Alternately, you can specify an already formatted list of filenames directly to the DatasetSeries initializer:

import yt
ts = yt.DatasetSeries(["DD0030/DD0030", "DD0040/DD0040"])

## Analyzing Each Dataset In Sequence¶

The DatasetSeries object has two primary methods of iteration. The first is a very simple iteration, where each object is returned for iteration:

import yt
for ds in ts:
print(ds.current_time)

This can also operate in parallel, using piter(). For more examples, see:

## Analyzing an Entire Simulation¶

Note

Implemented for the Enzo, Gadget, OWLS, and Exodus II frontends.

The parameter file used to run a simulation contains all the information necessary to know what datasets should be available. The simulation convenience function allows one to create a DatasetSeries object of all or a subset of all data created by a single simulation.

To instantiate, give the parameter file and the simulation type.

import yt
my_sim = yt.simulation('enzo_tiny_cosmology/32Mpc_32.enzo', 'Enzo')

Then, create a DatasetSeries object with the frontends.enzo.simulation_handling.EnzoSimulation.get_time_series() function. With no additional keywords, the time series will include every dataset. If the find_outputs keyword is set to True, a search of the simulation directory will be performed looking for potential datasets. These datasets will be temporarily loaded in order to figure out the time and redshift associated with them. This can be used when simulation data was created in a non-standard way, making it difficult to guess the corresponding time and redshift information

my_sim.get_time_series()

After this, time series analysis can be done normally.

for ds in my_sim.piter()
all_data = ds.all_data()
print(all_data.quantities.extrema('density'))

Additional keywords can be given to frontends.enzo.simulation_handling.EnzoSimulation.get_time_series() to select a subset of the total data:

• time_data (bool): Whether or not to include time outputs when gathering datasets for time series. Default: True. (Enzo only)
• redshift_data (bool): Whether or not to include redshift outputs when gathering datasets for time series. Default: True. (Enzo only)
• initial_time (float): The earliest time for outputs to be included. If None, the initial time of the simulation is used. This can be used in combination with either final_time or final_redshift. Default: None.
• final_time (float): The latest time for outputs to be included. If None, the final time of the simulation is used. This can be used in combination with either initial_time or initial_redshift. Default: None.
• times (list): A list of times for which outputs will be found. Default: None.
• initial_redshift (float): The earliest redshift for outputs to be included. If None, the initial redshift of the simulation is used. This can be used in combination with either final_time or final_redshift. Default: None.
• final_redshift (float): The latest redshift for outputs to be included. If None, the final redshift of the simulation is used. This can be used in combination with either initial_time or initial_redshift. Default: None.
• redshifts (list): A list of redshifts for which outputs will be found. Default: None.
• initial_cycle (float): The earliest cycle for outputs to be included. If None, the initial cycle of the simulation is used. This can only be used with final_cycle. Default: None. (Enzo only)
• final_cycle (float): The latest cycle for outputs to be included. If None, the final cycle of the simulation is used. This can only be used in combination with initial_cycle. Default: None. (Enzo only)
• tolerance (float): Used in combination with times or redshifts keywords, this is the tolerance within which outputs are accepted given the requested times or redshifts. If None, the nearest output is always taken. Default: None.
• parallel (bool/int): If True, the generated DatasetSeries will divide the work such that a single processor works on each dataset. If an integer is supplied, the work will be divided into that number of jobs. Default: True.


text:
Research

# Relay on-off threshold for NDF protocol with distributed orthogonal space-time block codes

Dong-Sup Jin1, Xianglan Jin2, Jong-Seon No1 and Dong-Joon Shin3*

Author Affiliations

1 Department of Electrical Engineering and Computer Science, INMC, Seoul National University, Seoul 151-744, Korea

2 Department of Information and Communication Engineering, Dongguk University-Seoul, Seoul 100-715, Korea

3 Department of Electronic Engineering, Hanyang University, Seoul 133-791, Korea

For all author emails, please log on.

EURASIP Journal on Wireless Communications and Networking 2012, 2012:305 doi:10.1186/1687-1499-2012-305

 Received: 8 June 2012 Accepted: 4 September 2012 Published: 26 September 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this article, a relay on-off threshold (ROT) based on symbol error rate is derived for the cooperative communication networks with multiple antennas, where the non-orthogonal decode-and-forward (NDF) protocol with source antenna switching and linear combining decoding are used as the relaying protocol and decoding scheme, respectively. The optimal ROT for the cases using distributed orthogonal space-time block codes is derived for high signal to noise ratio (SNR) region and also a suboptimal ROT is provided for low SNR region. Finally, the diversity order of the NDF protocol using the relay on-off scheme with the proposed ROT is derived and its performance is verified through numerical analysis.

##### Keywords:
Cooperative communication networks; Cooperative diversity; Distributed orthogonal space-time block codes (DOSTBCs); Linear combining (LC) decoding; Non-orthogonal decode-and-forward (NDF); Relay on-off threshold (ROT)

### Introduction

Attenuation and fading in the multipath wireless environment make it difficult for a receiver to correctly decode a received signal. Therefore, replicas of a transmitted signal are sent to the receiver to improve the detection performance.

Tarokh et al. [1] proposed space-time codes (STCs) to achieve transmit diversity and their design criteria. Alamouti code was proposed in [2], which was extended to orthogonal space-time block codes (OSTBCs) in [3]. Also, the symbol error rate (SER) and bit error rate (BER) of OSTBCs for quadrature amplitude modulation (QAM) were derived in [4,5], respectively.

The cooperative diversity which is obtained by utilizing relays between source and destination has been actively studied since Cover and El Gamal’s [6] work. Laneman et al. [7] developed some cooperative diversity schemes based on repetition codes and analyzed their performance, and Wang et al. [8] developed a good demodulation scheme for the protocol proposed in [7].

Recently, various results on how to select the best relay(s) among multiple relays have been reported. Bletsas et al. [9] proposed an opportunistic relaying which selects a relay only by using local channel information. Jing and Jafarkhani [10] proposed a multiple-relay selection scheme for the amplify and forward (AF) protocol and Yi and Kim [11] analyzed the diversity order of the decode-and-forward (DF) protocol with relay selection.

On the other hand, a relay on-off scheme for the DF protocol according to the channel state was proposed in [7]. Recently, another relay selection scheme based on the relay on-off threshold (ROT) for the DF protocol was also proposed in [12]. In both schemes, if the received signal to noise ratio (SNR) at the relay is larger than the ROT, the relay transmits a signal and otherwise, the relay does not transmit a signal. However, the optimal ROT was not derived analytically.

For the orthogonal DF (ODF) protocol and binary phase shift keying (BPSK), Siriwongpairat et al. [13] derived the ROT over the Rayleigh fading channel and Ikki and Ahamed [14] derived the ROT over the Nakagami-m fading channel. However, until now, for M-QAM and non-orthogonal DF (NDF) protocol in which a source also transmits a signal when a relay transmits a signal, the ROT has not been investigated.

It is well known that source antenna switching (SAS) can gain additional diversity for the NDF protocol by using different antennas at the source in the first and second phases [15], and the maximal likelihood (ML) decoding is impractical because its computational complexity increases exponentially with the signal constellation size. Therefore, we will use the low-complexity decoding scheme for OSTBCs given in [16,17], which is called linear combining (LC) decoding.

In this article, an ROT is derived for the NDF protocol with distributed orthogonal space-time block codes (DOSTBCs) and M-QAM such that the SER of LC decoding is minimized over the Rayleigh fading channel. Until now, this kind of ROT has not been investigated yet. And, the diversity order of the proposed relay on-off scheme is derived.

This article is organized as follows. In Section “System models and LC decoding”, the system models and LC decoding are described and in Section “ROT and diversity analysis”, the ROT based on SER is derived and the diversity order of the proposed relay on-off scheme is analyzed. In Section “Numerical analysis”, the SER performance of the proposed relay on-off scheme is evaluated through numerical analysis and the concluding remarks are given in Section “Conclusion”.

Throughout this article, the following notations are used. E[·] denotes the expectation. XCN(0,σ2) denotes the complex Gaussian random variable with zero mean and variance σ2/2 for each of real and imaginary parts. For a complex number, |·|, (·), , and denote the magnitude, the complex conjugate, the real part, and the imaginary part, respectively. and (·)T denote the set of M×N complex matrices and the transpose of a matrix, respectively.

### System models and LC decoding

For the three-terminal cooperative communication network in Figure 1, it is assumed that source (S), relay (R), and destination (D) can have multiple antennas and the channels are quasi-static Rayleigh fading ones. Since non-orthogonal transmission is assumed, both source and relay transmit signals in the second phase. It is also assumed that the destination has the perfect channel state information (CSI) of SD and RD channels and the relay knows the CSI of SR channel and the mean and variance of SD and RD channel coefficients. For simplicity, Gray-mapped M-QAM is assumed.

Figure 1. An NDF protocol for the three-terminal cooperative communication network.

If the SNR at the relay is larger than an ROT T in the DF protocol, the relay transmits a signal and otherwise, the relay does not transmit a signal. Note that it is assumed that the source and destination know whether the relay is on or off. In the cooperative communication network, if the source uses the same antenna in the first and second phases, the SD and SR channel coefficients for these two phases become identical. Therefore, in order to increase the diversity order, it is assumed that the SAS [15] is used.

#### System models of NDF protocol

It is assumed that the numbers of antennas of the source, relay, and destination are MS, MR, and MD, respectively. Let x be the message vector consisting of L independent symbols. denotes the codeword of OSTBC transmitted from the source, where denotes the number of transmissions at each source antenna. In the first phase, the received signal matrices YR and YD1 at the relay and destination can be written as

(1)

(2)

where and are the SR and SD channel matrices, and and are the noise matrices at the relay and destination. Note that p1ρ is the power of the signal from the source and ρ is a parameter linearly proportional to the average transmit SNR.

In the second phase, let xR denote the decoded message vector at the relay and and denote the codewords of OSTBCs corresponding to x and xR transmitted from the source and relay, respectively. When the relay is on, the received signal at the destination in the second phase is given as

(3)

where and are the SD and RD channel matrices, and is the noise matrix at the destination. Note that p2ρ and p3ρ denote the signal powers from the source and relay in the second phase, respectively. Then also forms an OSTBC if xR=x. The entries of H, G1, G2, and F are independently distributed as , , , and , respectively. The entries of NR, ND,1, and ND,2 are i.i.d. random variables distributed as CN(0,1).

When the relay is off, the received signal at the destination in the second phase is given as

(4)

Therefore, by using (2), (3), and (4), the received signal matrix can be expressed as

(5)

#### LC decoding

The optimal ML decoding at the destination is done by selecting such as

(6)

Since, the complexity of ML decoding increases exponentially to the constellation size and the analysis of ML decoding is very difficult, simple and practical LC decoding [16,17] will be considered in this article. It is clear that LC decoding is equivalent to ML decoding for OSTBCs but it is not true for DOSTBCs because the relay may transmit erroneously decoded symbols.

For the cooperative communication network, the LC decoding operates twice. First, the LC decoding is done for YD1 and YD2, respectively, before the decision on x is made at the destination. Then, the destination combines the decoder outputs for YD1 and YD2. Finally, the decision is made for x by using this combined output.

### ROT and diversity analysis

If the SR channel or RD channel is not good enough for the reliable communication, a relay cannot help the destination that much. Therefore, it is important to decide whether a relay should be used or not depending on the states of SR and RD channels. In general, a relay is used in the DF protocol only when the received SNR at the relay is greater than the ROT. If the relay correctly decodes the information, it can be thought as a virtual antenna of the source and the diversity gain can be obtained even though the RD channel is not good. Therefore, an ROT can be determined only by monitoring SR channel state [7,13,14].

If the ROT is too low, the relay may transmit many erroneous data under the bad SR channel condition, which causes many decoding errors at the destination. On the contrary, if the ROT is too high, the relay is rarely used and the cooperative diversity cannot be achieved. Therefore, the optimal ROT for the NDF protocol with DOSTBCs and M-QAM is derived in this section.

It can be assumed that, in high SNR region, only one symbol error occurs among L symbols in a codeword and further a symbol error is caused by one-bit error. Let , , , and , where gk,i,j denotes the channel coefficient between the ith source antenna and the jth destination antenna in the kth phase, and hi,j denotes the channel coefficient between the ith source antenna and the jth relay antenna, and fi,j denotes the channel coefficient between the ith relay antenna and the jth destination antenna. Clearly, u, g1, g2, and w are Erlang distributed [18].

#### ROT (general expression)

To derive the ROT minimizing the SER, the SER should be derived first. For the NDF protocol using a relay on-off scheme, the symbol error event at the destination can be divided into three cases. The first case is that symbol error occurs at the destination when the relay is off. The second case is that symbol error does not occur at the relay but symbol error occurs at the destination when the relay is on. The third case is that symbol error occurs at both the relay and destination when the relay is on. Therefore, the SER Pe(T) with the ROT T at the destination is expressed as

(7)

As SNR increases, the probability of Case 3 in (6) can be approximated as

(8)

The pdf of Erlang random variable u is given as

(9)

where . Let Pe1 be the SER at the destination when the relay is off, Pe2 be the SER at the destination when the relay is on and no decoding error occurs at the relay, and Pe3 be the SER at the destination when the relay is on and the decoding error occurs at the relay. Using (6) and (7), the SER at the destination is derived as

(10)

where Pe3 can be approximated as the symbol error probability at the destination when one bit error causes one symbol error at the relay. Then, by solving for (9), the ROT to minimize (9) is given as

(11)

Since some approximations are used to derive the ROT, it is clearly suboptimal. However, through simulation, it will be shown that this ROT approaches the optimal ROT as SNR increases.

#### ROT for LC decoding

In this section, we derive Pe1, Pe2, and Pe3 for LC decoding and obtain the ROT based on them.

Case (1) Relay is off:

Since only the source transmits signals, Pe1 can be written as [4]

(12)

where d denotes the distance between one symbol point and the adjacent decision boundary in the rectangular QAM and the closed form of Pe1 can be derived by the result in [19].

Case (2) Relay is on and no decoding error occurs at the relay:

Since the relay transmits correct data, Pe2 can be written as [4]

(13)

and its closed form can be derived by the result in [19].

Case (3) Relay is on and the decoding error occurs at the relay:

Since Gray-mapped M-QAM is assumed, the SER is dominantly determined by the single-bit error at the relay as SNR increases. Also, the most frequent symbol error event at the relay is one symbol error among L symbols. Therefore, Pe3 can be approximated as the SER at the destination when one-bit error occurs at the relay. Since the SER depends on the LC decoder output before the decision is made, we confirm how the one bit error at the relay affects the LC decoder output at the destination and then derive the SER.

When the relay erroneously decodes xk to xk + 2d, the LC decoder output sk for xk at the destination can be written as

(14)

(15)

Next, let us investigate the LC decoder output si,ik for the xi which is not erroneously decoded at the relay. For OSTBCs, when the LC decoding is performed to the data symbol xi, the terms related to the other symbols are canceled out and thus the decoder output at the destination becomes equivalent to (14). However, for DOSTBCs, if one bit error for xk occurs at the relay, the terms related to xk possibly affect the decoder output for the other symbol at the destination. Therefore, the decoder output for xi at the destination can be divided into two cases. First, the terms related to xk do not affect si such as

(16)

Second, the terms related to xk affect si due to one bit error for xk at the relay such as

(17)

where ni is distributed as CN(0,p1g1 + p2g2 + p3w), g2,i,jfl,k denotes either or depending on the used DOSTBC, and D is the set of indices of channel coefficients that are not canceled out due to one bit error for xk.

As an example, consider a DOSTBC using the following OSTBCs for MS=2 and MR=2

(18)

If an error occurs for x1 at the relay, s2,R and s2,I are given as (15) and s3,R and s3,I are given as (16). If an error occurs for x2 at the relay, s1,R and s1,I are given as (15) and s3,R and s3,I are given as (16). If an error occurs for x3 at the relay, s1,R, s1,I, s2,R, and s2,I are given as (16).

From the LC decoder outputs (13)–(16), we can check the error cases at the destination and then calculate Pe3.

The SER for xk (the error symbol at the relay) at the destination;First, we consider the case that the error occurs at the destination when decoding sk in (13) and (14). Note that k is the index of the error symbol at the relay. Note that the relay can erroneously decodes xk to xk−2d or xk±j2d. However, the identical LC decoder outputs to (13) and (14) are obtained since the rectangular QAM is considered. Thus, for those cases, we can also obtain and Pe3,2 similarly. Therefore, when one bit error occurs for xk at the relay, the SER for xk at the destination is given as the linear combination of and Pe3,2, that is

(22)

where β and γk are constants determined by the constellation and structure of DOSTBCs. The coefficient of Pe3,1 in (21) is 1 because Pe3,1 can be regarded as the probability that the same error at the relay also occurs at the destination.

The error case that the destination decodes to From (13), if and there is an adjacent symbol of xk in this direction, the destination can erroneously decode to with the following SER

(19)

Note that this symbol error at the destination is the same as the symbol error at the relay.

The error case that the destination decodes to From (13), if , this symbol error for can also occur with the following SER

(20)

The error case that the destination decodes to From (14), if or , a symbol error for can occur at the destination with the following SER

(21)

The SER for xi,ik (no-error symbol at the relay);The SER for xi can be obtained according to the LC decoder output in (15) and (16). Finally, the SER for xi,ik can be expressed as

(24)

where γi and δi are constants determined by the constellation and structure of DOSTBCs. Since the LC decoder output for xi cannot become (15) and (16) simultaneously, for each Perror,i, if γi is not zero, δi should be zero and if δi is not zero, γi should be zero.

The case that LC decoder output for xi is (15)From (15), if or , the destination can erroneously decode to or . Also, if or , the destination can erroneously decode to or . In fact, the SER for all these cases is the same as Pe3,2.

The case that LC decoder output for xi is (16)Similarly to the previous case, we can obtain the following SER by considering (16)

(23)

where denotes one of , , , and depending on the structure of DOSTBCs like the operator ∘ in (16). However, all possible Pe3,3’s depending on ⊕ have the same value because each channel coefficient is a complex Gaussian random variable with zero mean. Thus, we can see that ⊕ can be replaced with the addition without changing the result.

The SER at the destination when one bit error occurs at the relay (Pe3);When one-bit error for xk occurs at the relay, the SER at the destination can be expressed as

(25)

By plugging Pe1, Pe2, and Pe3 to (10), we can obtain the ROT. In fact, we cannot derive or prove something not because it is complicated but because it is hard or intractable. However, the ROT for high SNR region can be derived by the following approximation.

Since the direct approximation of Pe3,1 from (18) is too complicated, we use the assumption that the effect of noise is negligible in high SNR region and thus (13) is approximated as

(26)

Then, the error can occur if in high SNR region and thus the SER (Pe3,1) is given as

(27)

By using the pdf and the cumulative distribution function (cdf) of the sum of independent and nonidentical Erlang random variables in [18], can be easily calculated.

As SNR increases, , and Pe3,3 in (24) become negligible compared to and thus Pe3 becomes close to . Since Pe2 in (12) decreases much faster than Pe1 in (11) as SNR increases, only Pe1 and become dominant in the numerator and denominator in (10), respectively. Finally, in high SNR region, the ROT in (10) is simplified as

(28)

#### Decision of suboptimal ROT in low SNR region

In the previous section, an ROT was derived only for high SNR region because Pe3 cannot be derived in low SNR region. Thus, for the whole SNR region including low SNR region, the ROT should be determined.

The ROT in (10) is approximated as (27) by using the following approximation

(29)

However, the ROT in (27) cannot be calculated in low SNR region because becomes less than Pe1 as SNR decreases. The argument of Q−1(·) can be greater than 1 when the RD channel state is not better than the SD channel state such as (ρ,ρ,ρ). Therefore, a suboptimal ROT for low SNR region is heuristically obtained by giving a bias such as using instead of in (28). Thus the ROT in (27) can be modified as

(30)

Note that, as SNR increases, the ROT in (29) becomes identical to the ROT in (27) because Pe1 goes to zero.

#### Diversity analysis

In this section, the diversity order of NDF protocol with the proposed relay on-off scheme is derived when DOSTBCs and LC decoding are used.

When the relay is off, the transmit signals in the first and second phases form an OSTBC and when the relay is on, the transmit signals in the first and second phases form a DOSTBC .

Since the ranks of their difference matrices are 2MS and 2MS + MR, respectively, and the destination has MD antennas, in high SNR region, Pe1 and Pe2 are proportional to ρ−2MSMD and ρ−{MSMD + (MS + MR)MD}, respectively. Also, in (27) can be approximated as cρ−2MSMD, where c is a positive constant.

Since , where [20] in high SNR region, the ROT in (27) can be approximated as

(31)

For simplicity, let and g(z) be the pdf of z. Then the SER in (9) at the destination is rewritten as

(32)

Now, we derive the diveristy order for each term in (31). Since , the cdf of z is given as

(33)

where .

The first term in (31) can be given as

(34)

where (λT / ρ)MSMRPe1/(MSMR)! is dominant because T/ρ < < 1 as ρ increases. Using (30) and (33), the diversity order of the first term in (31) is obtained as 2MSMD + MSMR.

As SNR increases, goes to zero and thus becomes dominant in the second term in (31). Therefore, it is clear that the diversity order of the second term in (31) is 2MSMD + MRMD because .

Let and δ=3/(M − 1). Then, the third term in (31) is upper bounded by

(35)

Note that the factor of the first term in (34) becomes dominant as ρ increases. By plugging (30) into (34), this factor becomes as follows.

(36)

Since in becomes negligible as SNR increases, the diversity order of (34) is given as 2MSMD + MSMR.

In conclusion, the diversity order dLC of NDF protocol with the LC decoding and the proposed relay on-off scheme is given as

(37)

Note that the diversity order MRMD) is identical to the full diversity order of NDF protocol with SAS [15].

### Numerical analysis

For the simulation, it is assumed that the transmit signal power in the first phase is the same as the sum of transmit signal powers from the source and relay in the second phase, and the transmit signal power of the relay is the same as that of the source in the second phase. (x,y,z) denotes the average received SNRs in dB of SD, SR, and RD channels, respectively. For example, (ρ,ρ,ρ + 6)means that the average received SNR of RD channel is larger than those of SR and SD channels by 6 dB. For all simulations, 16QAM is used.

To confirm the diversity order, when MS,MR,MD≥2, simulation must be performed in very high SNR region, which requires too long time. Instead, simulation has been performed for the single-antenna case (MS=MR=MD=1). For MS=MR=MD=1, the transmitted OSTBCs are given as

(38)

where the combination of CS,2 and CR,2 forms an Alamouti code in the second phase. The ROT for the single-antenna case can also be easily obtained from (27) and (29).

Figures 2 and 3 compare the performance of NDF protocol with various relay schemes. We do not consider the case that the SR channel state is better than the other channel states because the relay always tends to be on. The performance of the proposed optimal and suboptimal relay on-off schemes and the conventional relay scheme are denoted by ‘optimal-relay-on-off’, ‘subopt-relay-on-off’, and ‘relay-on’, respectively. In other words, ‘relay-on’ means that the relay always transmits signal. The ‘direct transmission’ implies that the relay is always off. The optimal relay on-off scheme uses the optimal ROT which is determined by extensive simulation and the suboptimal relay on-off scheme uses the ROT in (29).

Figure 2. Performance comparison of NDF protocol with various relay schemes for single-antenna case using 16QAM under various RD channel states.

Figure 3. Performance comparison of NDF protocol with various relay schemes for single-antenna case using 16QAM under various SD channel states.

Figure 2 shows that the SER performance of the relay-on scheme becomes worse as the RD channel state becomes better and thus, in this case, the relay on-off scheme is vital to improve the LC decoding performance. For the case of (ρ + 6,ρ,ρ) in Figure 3, it can be seen that the performance of the relay on-off scheme is almost identical to that of the direct transmission until ρ=12.5 dB, which is because Pe1Pe2 and the relay is always off. Therefore, the ROT should be infinite, as given by (10). Also, we can see that the analytical diversity results in Section Diversity analysis are well matched with the simulation results in Figures 2 and 3, and the suboptimal ROT works well in low SNR region as well as in high SNR region.

Figures 4 and 5 compare the performance of NDF protocol with various relay schemes when MS=MR=MD=2 and the DOSTBC in (17) is used. In Figure 4, since the diversity effect appears gradually from ρ=12.5 dB, it can be expected that the diversity order of relay on-off scheme will reach 12 in higher SNR region. For the case of (ρ + 6,ρ,ρ) in Figure 5, it can be seen that the performance of the relay on-off scheme is almost identical to that of the direct transmission, which implies that the ROT is infinite. Since the diversity effect will appear as SNR increases, it is expected that the relay becomes helpful as a virtual antenna to increase the diversity order in higher SNR region in this case. Also, the simulation results show that the suboptimal ROT works well in low SNR region as well as in high SNR region.

Figure 4. Performance comparison of NDF protocol with various relay schemes for two-antenna case using 16QAM under various RD channel states.

Figure 5. Performance comparison of NDF protocol with various relay schemes for two-antenna case using 16QAM under various SD channel states.

### Conclusion

In this article, an ROT was analytically derived for NDF protocol with DOSTBCs and SAS in high SNR region, where LC decoding was considered because it has low complexity and good performance. Through the diversity analysis, it was confirmed that the NDF protocol with the proposed relay on-off scheme can achieve the full diversity. For low SNR region, the suboptimal ROT was provided and the simulation results confirmed that this suboptimal ROT works well in whole SNR region. It is left as a future work to derive the optimal ROT for the cooperative communication networks with multiple relays by using a similar method to one proposed in this article.

### Competing interests

The authors declare that they have no competing interests.

### Acknowledgements

This research was supported by the KCC (Korea Communications Commission), Korea, Under the R&D program supervised by the KCA (Korea Communications Agency) (KCA-2012-08-911-04-003) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2012-0000186).

### References

1. V Tarokh, N Seshadri, A Calderbank, Space-time codes for high data rate wireless communication: performance criterion and code construction. IEEE Trans. Inf. Theory 44, 744–765 (1998). Publisher Full Text

2. SM Alamouti, A simple transmit diversity technique for wireless communications. IEEE J. Sel. Area Commun. 16, 1451–1458 (1998). Publisher Full Text

3. V Tarokh, H Jafarkhani, A Calderbank, Space-time block codes from orthogonal designs. IEEE Trans. Inf. Theory 45, 1456–1467 (1999). Publisher Full Text

4. S Kim, I Kang, J No, Symbol error probability of orthogonal space-time block codes with QAM in slow Rayleigh fading channel. IEICE Trans. Commun 87-B, 97–103 (2004)

5. S Kim, J Yang, J No, Exact bit error probability of orthogonal space-time block codes with quadrature amplitude modulation. J. Commun Netw 10, 253–257 (2008)

6. T Cover, A Gamal, Capacity theorems for the relay channel. IEEE Trans. Inf. Theory IT-25, 572–584 (1979)

7. J Laneman, D Tse, G Wornell, Cooperative diversity in wireless networks: efficient protocols and outage behavior. IEEE Trans. Inf. Theory 50, 3062–3080 (2004). Publisher Full Text

8. T Wang, A Cano, G Giannakis, J Laneman, High-performance cooperative demodulation with decode-and-forward relays. IEEE Trans. Commun 55, 1427–1438 (2007)

9. A Bletsas, A Khisti, D Reed, A Lippman, Simple cooperative diversity method based on network path selection. IEEE J. Sel. Area Commun 24, 659–672 (2006)

10. Y Jing, H Jafarkhani, Single and multiple relay selection schemes and their achievable diversity orders. IEEE Trans. Wirel. Commun 8, 1414–1423 (2009)

11. Z Yi, I Kim, Diversity order analysis of the decode-and-forward cooperative networks with relay selection. IEEE Trans. Wirel. Commun 7, 1792–1799 (2008)

12. F Onat, Y Fan, H Yanikomeroglu, H Poor, Threshold-based relay selection for detect-and-forward relaying in cooperative wireless networks. EURASIP J. Wirel. Commun. Netw (2010)

13. S Sirwongpairat, T Himsoon, W Su, K Liu, Optimum threshold-selection relaying for decode and forward cooperation protocol. Paper presented at IEEE WCNC’2006 (Las Vegas, NV, USA, 2006), pp. 1015–1020 PubMed Abstract | Publisher Full Text

14. S Ikki, M Ahamed, Performance of decode-and-forward cooperative diversity networks over Nakagami-m fading channel. Paper presented at IEEE GLOBECOM’2007 (New Orleans, LA, USA, 2007), pp. 4328–4333

15. X Jin, J Yang, J No, D Shin, Distributed space-time coded non-orthogonal DF protocol with source antenna switching. J. Commun. Netw 12, 492–498 (2010)

16. E Larsson, P Stoica, Space-Time Block Coding for Wireless Communications (Cambridge University Press, Cambridge, 2003)

17. H Jafarkhani, Space-Time, Coding, Theory and Practice (Cambridge University Press, Cambridge, 2005)

18. E Scheuer, Reliability of an m-out-of-n system when component failure induces higher failure rates in survivors. IEEE Trans. Reliab 37, 73–74 (1988). Publisher Full Text

19. M Simon, M Alouini, Digital Communication Over Fading Channels (Wiley, Hoboken, 2004)

20. B Bailey, Alternatives to Hasting’s approximation to the inverse of the normal cumulative distribution function. Appl. Stat 30, 275–276 (1981). Publisher Full Text


text:
My Math Forum Which statements are true?

 Applied Math Applied Math Forum

 February 5th, 2014, 04:55 AM #1 Newbie   Joined: Feb 2014 Posts: 21 Thanks: 0 Which statements are true? Hi, I'm a nontraditional (old fart ) student taking classes online. It has its advantages and disadvantages. One disadvantage is the time delay in getting your questions answered. I'm hoping this forum will be a supplement to my learning. I think I understand it but I'd feel much better someone would look over my shoulder... Now to the question: Which are true statements? Ø ? Ø states that an empty set is a proper set of an empty set. This is False because a subset can't be equal to a superset. Ø ? {x,y} states that an empty set is a proper set of set {x,y}. This is False because an empty set is not found in the finite set {x,y}. 0 ? Ø states that an zero is an element of an empty set. This is False because zero is a real number and can't be found in an empty set. Ø ? {Ø} states that an empty set is an element of the set "the empty itself". This is True because "the empty itself" contains an empty set. Am I on the right path?
 February 5th, 2014, 05:47 AM #2 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 Re: Which statements are true? I think you're on the right track but your explanations are not precise. You also should clarify whether you are looking at subsets here or strictly proper subsets. Normally: $A \subset B$ Means A is a subset of B and is not equal to B; whereas: $A \subseteq B$ Means A is a subset of B and could be equal to B. Is this the terminology you are using? Also, note that: $\emptyset \subseteq B$ For every set B, including where B itself is the empty set.
 February 5th, 2014, 05:51 AM #3 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 Re: Which statements are true? Ø ? Ø states that an empty set is a proper set of an empty set. This is False because a subset can't be equal to a superset. This is false, as a set cannot be strictly a subset of itself. Although, it may be true depending on what you mean by ? Ø ? {x,y} states that an empty set is a proper set of set {x,y}. This is False because an empty set is not found in the finite set {x,y}. This is true. 0 ? Ø states that an zero is an element of an empty set. This is False because zero is a real number and can't be found in an empty set. Yes. Ø ? {Ø} states that an empty set is an element of the set "the empty itself". This is True because "the empty itself" contains an empty set. Yes, this one is tricky. I'd say {Ø} is "the set containing the empty set as an element".
 February 5th, 2014, 05:54 AM #4 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 Re: Which statements are true? 0 ? Ø states that an zero is an element of an empty set. This is False because zero is a real number and can't be found in an empty set. Note that there is only one empty set! So, you always say the empty set. Try to prove this.
 February 5th, 2014, 09:05 AM #5 Newbie   Joined: Feb 2014 Posts: 21 Thanks: 0 Re: Which statements are true? Thanks Pero, That's a big help! Joe
 February 5th, 2014, 10:01 AM #6 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 Re: Which statements are true? Can you prove there is only one empty set?
February 5th, 2014, 12:17 PM   #7
Newbie

Joined: Feb 2014

Posts: 21
Thanks: 0

Re: Which statements are true?

Quote:
 Originally Posted by Pero 0 ? Ø states that an zero is an element of an empty set. This is False because zero is a real number and can't be found in an empty set. Note that there is only one empty set! So, you always say the empty set. Try to prove this.
Hmm,

0 in the equation is an element
? means "is an element of"
Ø means "the set with no elements"

Not sure I know how to "prove" there is only one empty set.

 February 5th, 2014, 01:10 PM #8 Newbie   Joined: Feb 2014 Posts: 21 Thanks: 0 Re: Which statements are true? I'm just now learning this, but I'm reading further... could I prove it with a membership table? Note: I've not heard of a membership table until now. This is new to me.
 February 5th, 2014, 02:20 PM #9 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 Re: Which statements are true? If there were two empty sets, then in what way could they be different?

 Tags statements, true

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post jackieee Calculus 4 May 10th, 2013 08:35 AM jstarks4444 Applied Math 2 March 10th, 2011 02:00 PM r-soy Physics 1 December 26th, 2010 09:50 PM metaphist Algebra 3 April 3rd, 2010 04:53 AM jstarks4444 Number Theory 2 December 31st, 1969 04:00 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top


text:
# zbMATH — the first resource for mathematics

On the primitive idempotents of distance-regular graphs. (English) Zbl 0993.05148
From the author’s abstract: Let $$\Gamma$$ denote a distance-regular graph with diameter $$d\geq 3$$. Let $$E$$, $$F$$ denote nontrivial primitive idempotents of $$\Gamma$$ such that $$F$$ corresponds to the second largest or least eigenvalue. We investigate the situation that there exists a primitive idempotent $$H$$ of $$\Gamma$$ such that $$E\circ F$$ is a linear combination of $$F$$ and $$H$$. Our main purpose is to obtain inequalities involving the cosines of $$E$$, and to show that equality is closely related to $$\Gamma$$ being $$Q$$-polynomial with respect to $$E$$. This generalizes a result of Lang on bipartite graphs and a result of Pascasio on tight graphs.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05C12 Distance in graphs 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
Full Text:


text:
# What could cause a spike in the natural uranium depletion rate?

What is a possible event$${^*}$$ that would cause a spike in the speed of $$^{235}$$U fission and reduce its average concentration in ore worldwide?

As I understand the physics of the process, it is impossible to alter the fission speed itself, but if there is a way to have a thermal neutron source in the ore volume or some kind of neutron moderator for neutrons which are emitted by $$^{235}$$U itself that will lead to chain reaction and then depletion, like in Oklo.

The problem I am struggling with is to come up with an event that will look not completely made up.

$${^*}$$ The event should not be magic or alien-related.

P.S. This is my first question here if I am doing something wrong I will be happy to do it right.

• Would this need to be natural (a star explodes showering the planet with unobtanium) even if unlikely or could it be deliberate (somebody synthesizes unobtanium and deploys it worldwide)? – VLAZ Jan 7 at 15:18
• Some sort of "false vacuum collapse" event could happen which would make W bosons more likely to decay radioactive isotopes. This wouldn't have to be universal, could happen in a finite region of space. But it would apply to almost all radioactive isotopes (not just U235), and it would make them intensely radioactive for a short period of time. Ore would kill everyone anywhere close to it, and concentrated/refined uranium might become spontaneously supercritical. Boom. – John O Jan 7 at 15:19
• @JohnO, can you expand your answer, because it sounds like what I need – FrogOfJuly Jan 7 at 15:55
• Other than the existence of intent, high-energy neutron bursts from supernova or alien weaponry will have identical outcomes. – Carl Witthoft Jan 7 at 16:19
• Any such event will have very serious besides reducing the concentration of U235... – AlexP Jan 7 at 17:11

I agree that a possible solution would be to add in a source of high-energy ambient neutrons. Cosmic rays are a possible neutron source, at least at high enough altitudes and assuming the uranium does not have adequate shielding. A strong spike in cosmic rays could then provide such a neutron source; in turn, as one of the major sources of Galactic cosmic rays are supernovae and supernova remnants (for cosmic rays below $$\sim10^{15}$$ eV), it's possible that one or more supernovae nearby could produce enough neutrons to raise the rate of uranium fission enough to be detected.

Supernovae, of course, can be dangerous when close enough to Earth. Fortunately for us, that critical distance is on the order of about ten light-years. However, this means that the cosmic ray flux would be lower than if the supernova was closer, so it's something of a tradeoff. That said, for an event to cause an increase in fission worldwide, rather than just in one spot, I suspect it would likely have to be astronomical in nature, meaning you'd need some sort of energetic phenomenon. This of course means that you'll be dealing with the same sort of issue with many of the answers you'll get.

• of course a neutron burst powerful enough will also sterilized the planet. – John Jan 7 at 20:24
• @John I would expect so. I think we're looking for a middle ground of sorts - enough of a spike to be noticeable, but not enough that the damage associated with such a supernova would be severe. – HDE 226868 Jan 7 at 23:07
• the problem is noticeable in this case means penetrating hundreds of meters of rock, which Im not even sure your given scenario could accomplish. – John Jan 8 at 4:20

When does this reduction have to happen?

If it can happen in the distant past, it could be explained just by changing rates of radioactive decay. Right now, we think radioactive decay always happens at the same rate, but there have been some weird carbon dating results that may support the idea that radioactive decay rates could be variable. The current consensus is that these anomalies are just statistics problems, not a new physical phenomenon. But it wouldn't take too much suspension of disbelief to say this actually is a new phenomenon.

If radioactive decay rates can change, maybe they were higher for a short period in the distant past. This would mean less uranium would be around now.

• I like the idea, especially since it's been explored experimentally in, of all places, Oklo. arxiv.org/abs/hep-ph/0205340 is a good reference about the impact of variations in the fine structure constant on cross sections and resonances, if you're looking to include one. (Of course, those measurements have been used to argue that any variations in $\alpha$ have been small or null, but it still explores how cross sections would change if variations had been non-zero.) – HDE 226868 Jan 7 at 20:14
• so still magic just magic in the past. – John Jan 7 at 20:23
• It is preferable for the event to be much less stretched in time than the existence of human civilization. So actually it is not important when it happens but it should be a spike on time scale of humanity. – FrogOfJuly Jan 7 at 22:16
• @Kirill It could have happened any time before the invention of radioisotopic dating. I mean, how are the Romans going to notice Uranium is decaying 10% faster than usual? – Ryan_L Jan 7 at 22:19
• @Ryan_L, it' true. But if radioactive decay rate is changing, it changes very slow on a scale of human civilization and nobody will notice. Except if changes were larger in the distant past and then just vanished, but then again, it is unclear why. – FrogOfJuly Jan 7 at 22:36

The Oklo Formation is believed to have been a naturally formed fission reactor, moderated by ground water levels over millions of years. When water levels were up, the ore body became critical and heated the land (and emitted radiation, of course). When ground water dropped, the criticality was lost and the ore body cooled. This apparently occurred several times, and was deduced based on the ore body being significantly depleted in U235 relative to its content of U238.

For this to happen worldwide, however, would require a very different mechanism, because there are (as far as I'm aware) no other similar high concentration ore bodies that could spontaneously become critical.

Thinking of the parallel-universe concept implemented in Stephenson's "Anathem," perhaps some multiverse-catastrophe has caused two "Earths" to swap universes. The marginal differences in cosmological constants changes the half-life of U-235 in the new universe.

Of course, you'd have to MacGuffin away all the other things that don't change as a result of the change in cosmo-constants' values.

No known process can make it happen world wide. It is possible that pockets of the process might happen.

There have been suggestions of an external neutron source. To produce fission you will need low energy neutrons, not high energy. U235 wants neutrons in the thermal range to produce significant fission.

The problem with external neutron sources is shielding. Uranium is currently mined at least 100's of meters deep in some locations. So any external neutron source that could produce significant fission would not be able to do it uniformly. 100 meters of rock is going to be a quite good neutron shield, preventing the deeply buried stuff from being affected.

Also note that, unless you have a situation with the Uranium quite closely and densely packed, you need (nearly) one neutron per Uranium fission. Without that close packing, the neutrons produced by fission are nearly all wasted.

Changing physical constants is currently at the "speculative" end of science. We don't know if it is possible, though several unification theories indicate it should be. The problem with changing physics constants is that it might affect the thresholds for critical mass and such. But to get it to remove U235 from existing ore bodies is pretty much impossible. Ore comes in a wide variety of concentrations. And anything over 1% is probably a useful commercial product. To get 1% ore to start to fission would require such grotesque changes that it is quite likely many other things would start to fission as well. Thorium for example, which is far more plentiful than Uranium. The result would be a bright flash, and no more Earth.


text:
# Adding to PATH Environment Variable

Danger

Be careful doing this! You are responsible for messing up your own PATH.

## Adding to the PATH on OSX/Linux

$export PATH=path/to/add:$PATH

Verify the results with the command: $echo$PATH

## Adding to the PATH on Windows

$set PATH=path/to/add;%PATH% This only modifies the path for the current command prompt. To permanently add to the path: • Go to Settings/Edit the system environment variables/Environment Variables • Choose the Path variable and press Edit... • Add your path entry to the list Verify the results with the command: $ echo %PATH%


text:
Kattis

# AI Jeopardy

Picture by QIHAN Technology
The robot revolution is finally here, albeit not quite in the highly explosive way envisioned in various science fiction books and movies. It seems that, perhaps due to a small typo in the AI source code, the robots are not taking our lives but instead taking our livelihoods. One early job market fatality in this revolution was the (somewhat niche) occupation as jeopardy player: already in 2011 the Watson computer defeated two legendary but inferior human jeopardy champions.

Nowadays, more and more of Jeopardy’s viewers are AIs themselves and as such the show is considering having categories on topics that are more popular with this new number-crunching viewer base. Focus group testing has revealed that AIs are particularly fond of the “Binomial Coefficients” category. The premise of this category is that the answer that contestants get is some positive integer $X$, and the contestants must respond with a question of the form “What is $n$ choose $k$?” (and this is a correct response if the binomial coefficient $n$ choose $k$ equals $X$).

Write an AI to play this new Jeopardy category. If there are several different possible solutions for $n$ and $k$, the AI should choose the most elegant solution, having the smallest value of $n$, and of those with the smallest $n$ it should choose the one with the smallest value of $k$.

## Input

Input consists of a single integer $X$ ($1 \le X \le 10^{100}$).

## Output

Output two non-negative integers $n$ and $k$ such that the binomial coefficient $n$ choose $k$ equals $X$, with ties between multiple solutions broken as explained above.

Sample Input 1 Sample Output 1
10

5 2

Sample Input 2 Sample Output 2
2020

2020 1

Sample Input 3 Sample Output 3
1

0 0


text:
Simpson's rule characteristics

I just wanted to ask a quick question in regards to simpson's rule for integration. I have been reading up on the trapezoidal rule, and have found the notations and have an understanding such that:

Where, $T(n) =$ trapezoidal rule formula

and have gathered that $$T(2n) = {1\over 2} T(n) + \text{sum of the odd terms}$$

From this, is it right to say $T(n)$ is a more accurate calculation than $T(2n)$, right?

Another question I have is that I have a question in which I have to calculate $S(2n)$ using richardson extrapolation which i have the formula:

$$S(2n) = {4\over 3} T(2n) - {1\over 3} T(n)$$ From this, is it right to conclude that $S(n)$ is more accurate than $S(2n)$?

....I have a few more questions to follow up on, but for now I just wanted to check my understanding

If that is all correct, I was wondering how do we actually start calculating $S(2n)$ and how to find $S(4n)$ and to express $S(4n)$ in terms of $S(2n)$ without $T(2n)$ or $T(n)$ in the formula

• I reformatted your question a bit. Please check to make sure that I didn't change what you are trying to ask. You can find more information about formatting here: meta.math.stackexchange.com/questions/5020/…. Also, if you have more questions, then remember to post them as separate questions. It is best to post separate questions as separate questions. – Thomas May 9 '13 at 15:48
• Thank you for the formatting and link! :) I've been looking for something like for ages!!! :) – orangesun May 9 '13 at 16:07
• Glad to help. Also here: meta.math.stackexchange.com/questions/370/formatting-sandbox you can try to experiment with formatting. – Thomas May 9 '13 at 16:12


text:
×

# Cool problems (Part 2)

This is the continue of Cool nt problems. These are some problems I've found on internet and I've really liked them.

• Find all real solutions of this sistem:

$$\sqrt{x^{2}+y^{2}+6x+9}+ \sqrt{ x^{2}+y^{2}-8y+16} = 5$$

$$9y^{2}-4x^{2}= 60$$

• Find all real solutions of $$x^{6}+y^{6}+z^{6}= 6xyz - 3$$

• Let n be a positive integer. Show that $$N = 11111(2n times)2222(n times)$$ is a perfect square.

Note by Jordi Bosch
3 years, 5 months ago

Sort by:

The lat digit of the number given in the third problem is 2 right? So how can it be a perfect square.......

- 3 years, 4 months ago

*last

- 3 years, 4 months ago

I believe the last problem has an issue.

- 3 years, 5 months ago

yes it must be 1111(2n times)2222(n times)

- 3 years, 5 months ago