text:
It is currently 24 Feb 2018, 04:14

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# m is a certain fraction. When m is multiplied by its own denominator,

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 43894
m is a certain fraction. When m is multiplied by its own denominator, [#permalink]

### Show Tags

29 Sep 2016, 03:46
Expert's post
2
This post was
BOOKMARKED
00:00

Difficulty:

15% (low)

Question Stats:

80% (00:36) correct 20% (01:00) wrong based on 109 sessions

### HideShow timer Statistics

m is a certain fraction. When m is multiplied by its own denominator, the result is 16. What is the value of m's numerator?

A. 1/16
B. 1/4
C. 4
D. 16
E. The information given is insufficient to determine the value of the numerator.
[Reveal] Spoiler: OA

_________________
Senior Manager
Joined: 06 Jun 2016
Posts: 263
Location: India
Concentration: Operations, Strategy
Schools: ISB '18 (D)
GMAT 1: 600 Q49 V23
GMAT 2: 680 Q49 V34
GPA: 3.9
m is a certain fraction. When m is multiplied by its own denominator, [#permalink]

### Show Tags

29 Sep 2016, 03:59
2
KUDOS
D

If the result has to be 16 then numerator has to be 16.
The question say 'm' is a fraction. Say m=a/b
when m is multiplied by its denominator the result is 16. So (a/b)*b=a
So the multiplication actually gives you the numerator.
The fraction may be 16/100, 16/10 or 16/1. As long as the denominator is multiplied with the fraction, it always leaves you with the value of numerator.

Last edited by Vinayak Shenoy on 29 Sep 2016, 08:13, edited 1 time in total.
Retired Moderator
Joined: 26 Nov 2012
Posts: 597
Re: m is a certain fraction. When m is multiplied by its own denominator, [#permalink]

### Show Tags

29 Sep 2016, 04:13
1
KUDOS
Bunuel wrote:
m is a certain fraction. When m is multiplied by its own denominator, the result is 16. What is the value of m's numerator?

A. 1/16
B. 1/4
C. 4
D. 16
E. The information given is insufficient to determine the value of the numerator.

Let m = x/y

Now m is multiplied by y

then m = x/y (y) => x

and given x = 16 and x is a numerator.

IMO option D.
Intern
Joined: 18 Sep 2016
Posts: 49
Re: m is a certain fraction. When m is multiplied by its own denominator, [#permalink]

### Show Tags

29 Sep 2016, 08:03
Bunuel wrote:
m is a certain fraction. When m is multiplied by its own denominator, the result is 16. What is the value of m's numerator?

A. 1/16
B. 1/4
C. 4
D. 16
E. The information given is insufficient to determine the value of the numerator.

the question says $$m=\frac{x}{d}$$ and say also that $$\frac{x}{d}*d=16$$

====> x=16

_________________

Please press Kudos if you like my post

Senior Manager
Joined: 05 Jan 2017
Posts: 434
Location: India
Re: m is a certain fraction. When m is multiplied by its own denominator, [#permalink]

### Show Tags

27 Feb 2017, 03:50
Let m = a/b, where a and b are integers. According to question b*a/b =16 or a =16. Hence the numerator is 16. Option D
Intern
Joined: 11 Aug 2016
Posts: 47
Location: India
Concentration: Operations, General Management
Schools: HBS '18, ISB '17, IIMA
GMAT 1: 710 Q49 V38
GPA: 3.95
WE: Design (Manufacturing)
Re: m is a certain fraction. When m is multiplied by its own denominator, [#permalink]

### Show Tags

27 Feb 2017, 10:25
Let the fraction be $$\frac{x}{y}$$

The numerator of the fraction is x and the denominator is y.

It is given that when the fraction is multiplied by the denominator, the result is 16.

So $$\frac{x}{y}*y=16$$ => x=16

Thus, the numerator is 16.

Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 2192
Location: United States (CA)
Re: m is a certain fraction. When m is multiplied by its own denominator, [#permalink]

### Show Tags

01 Mar 2017, 17:15
1
KUDOS
Expert's post
Bunuel wrote:
m is a certain fraction. When m is multiplied by its own denominator, the result is 16. What is the value of m's numerator?

A. 1/16
B. 1/4
C. 4
D. 16
E. The information given is insufficient to determine the value of the numerator.

We can let m = x/y, in which x is m’s numerator and y is m’s denominator. We need to find the value of x. Since when m is multiplied by its own denominator the result is 16, we have:

my = 16

(x/y) * y = 16

x = 16

_________________

Scott Woodbury-Stewart
Founder and CEO

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Re: m is a certain fraction. When m is multiplied by its own denominator,   [#permalink] 01 Mar 2017, 17:15
Display posts from previous: Sort by

# m is a certain fraction. When m is multiplied by its own denominator,

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.


text:
# AP Calculus AB — Derivatives Part 3

This is Part 3 of our AP Calculus AB unit test on derivatives. These questions cover tangent lines, mean value theorem, Rolle’s theorem, and inverses of functions. Practicing the application of derivatives theorems to a function will help you solve complicated problems on the AP exam asking you to analyze the behavior of a given function.

Congratulations - you have completed .

You scored %%SCORE%% out of %%TOTAL%%.

Your performance has been rated as %%RATING%%

 Question 1
What is the slope of the tangent line at $x = 2$ for the function below.

$f(x) = \ln(2x) + x^2$

 A $\ln(4) + 4$ B $\dfrac{9}{2}$ C $\dfrac{17}{4}$ D $\text{There}$ $\text{is}$ $\text{no}$ $\text{tangent}$ $\text{line}$ $\text{at}$ $x = 2$
Question 1 Explanation:
The correct answer is (B). It is important to recognize that the slope of the tangent line is simply the value of the derivative of the function. Therefore, we must first identify the derivative of the function.

$f'(x) = \dfrac{1}{2x} \cdot 2 + 2x$ $= \dfrac{1}{x} + 2x$ $= \dfrac{1 + 2x^2} {x}$

Then, to find the slope of the tangent line at $x = 2$, we just plug in $x = 2$ into the derivative function.

$f'(2) = \dfrac{1 +2(2)^2}{2}$ $= \dfrac{9}{2}$
 Question 2
Which of the following is the equation for the line tangent to the function below at $x = 3$.

$f(x) = \sin\left(\dfrac{π}{2}x\right) +6x^3$

 A $y = 162x + 158$ B $y =$ $\left(162 + \dfrac{π}{2}\right)x − 325 − \dfrac{3π}{2}$ C $y = 162x$ D $y = 162x − 325$
Question 2 Explanation:
The correct answer is (D). To find the equation of the tangent line, we must first find the derivative of the function to find the tangent line’s slope.

$f'(x) = \dfrac{π}{2}\cos\left(\dfrac{π}{2}x\right) + 18x^2$

Then, the slope of the tangent line at $x = 3$ is the value of the derivative function at $x = 3$.

$f'(3) =$ $\dfrac{π}{2}\cos\left(\dfrac{π}{2} \cdot 3\right) + 18(3)^2$ $= \dfrac{π}{2}(0) + 18(9) = 162$

It is important not to forget that the equation for the tangent line in point slope form takes the general form below, where $(x_1, y_1)$ is a given or known point on the function and m is the slope of the line.

$y − y_1 = m(x − x_1)$

Thus, we need a point on the function. However, for the tangent line, we need more than just any point on the function, but actually the specific point on the function that the line is tangent to. We know that the tangent line at $x = 3$ is tangent to only the point on the function where $x = 3$. Therefore, we must identify the point on the function where $x = 3$ to find the known point for the point-slope formula. We can identify the function value at $x = 3$ by plugging in this value of $x$.

$f'(3) = \sin\left(\dfrac{π}{2} \cdot 3\right) + 6(3)^3$ $= −1 + 162 = 161$

Thus, we know that the tangent line is tangent to the point $(3, 28)$ on the function. Therefore, the point-slope formula for the tangent line is as follows.

$y − 161 = 162(x − 3)$

This can then be simplified to the form below, which matches answer choice (D).

$y = 162x − 325$
 Question 3
Which of the following equations is the equation of the line tangent to the graph below at $x = 0$.

$f(x) = \dfrac{1}{x^3 + 4x + 6}$

 A $y = −\frac{1}{9}x$ B $y = \frac{1}{6}x$ C $y = −\frac{1}{9}x + \frac{1}{6}$ D $y = −\frac{1}{6}x − \frac{1}{9}$
Question 3 Explanation:
The correct answer is (C). We first find the slope of any tangent line by identifying the function’s derivative.

$f'(x) =$ $−(x^3 + 4x + 6)^{−2} \cdot (3x^2 + 4)$ $= \dfrac{−3x^2 – 4}{(x^3 + 4x + 6)^2}$

Then, we can identify the slope of the tangent line at $x = 0$ by identifying the value of the derivative at $x = 0$.

$f'(0) =$ $\dfrac{−3(0) – 4}{(0^3 + 4(0) + 6)^2}$ $= −\dfrac{4}{6^2} = −\dfrac{1}{9}$

It is important not to forget that the equation for the tangent line in point slope form takes the general form below, where $(x_1, y_1 )$ is a given or known point on the function and $m$ is the slope of the line.

$y − y_1 = m(x − x_1)$

Thus, we need a point on the function. However, for the tangent line, we need more than just any point on the function, but actually the specific point on the function that the line is tangent to. We know that the tangent line at $x = 3$ is tangent to only the point on the function where $x = 3$. Therefore, we must identify the point on the function where $x = 3$ to find the known point for the point-slope formula. We can identify the function value at $x = 3$ by plugging in this value of $x$.

$f(0) = \dfrac{1}{0^3 + 4(0) + 6)}$ $= \dfrac{1}{6}$

Thus, we know that the tangent line is tangent to the point $(0, \frac{1}{6})$ on the function. Therefore, the point-slope formula for the tangent line is as follows.

$y − \dfrac{1}{6} = −\dfrac{1}{9}(x – 0)$

This can then be simplified to the form below, which matches answer choice (C).

$y = −\frac{1}{9}x + \frac{1}{6}$
 Question 4
Using the mean value theorem, identify a value  for $c$ which the slope of the line tangent to $f(x)$ at $x = c$ is equal to the average rate of change of the function on the given domain.

$f(x) = x^3 + 3x + 2$     $−1 ≤ x ≤ 3$

 A $c = 10$ B $c = \sqrt{\frac{7}{3}}$ C $c = \sqrt{5}$ D $c = ± \sqrt{\frac{7}{3}}$
Question 4 Explanation:
The correct answer is (B). It is important to recognize first that the given function satisfies the requirements of mean value theorem, meaning that the function is continuous and differentiable on the given domain. Then, we must identify the average rate of change of the function on the domain. This is equal to the slope of the secant line connecting the two endpoints. Therefore, we must identify the two endpoints.

$f(−1) = (−1)^3 + 3(−1) + 2$ $= −2$

$f(3) = (3)^3 + 3(3) + 2$ $= 38$

Then, the secant line has the slope below,

$f'(c) = \dfrac{38 − (−2)}{3 − (−1)}$ $= \dfrac{40}{4} = 10$

Then, to find the value $c$ that corresponds to the tangent line with this slope, we must identify the derivative of the given function.

$f'(x) = 3x^2 + 3$

Then, we can set this derivative function equal to the slope of the secant line to identify the value of $x$ that corresponds to the tangent line with this same slope.

$f'(c) = 3c^2 + 3 = 10$

$c^2 = \dfrac{7}{3}$

$c = ± \sqrt{\dfrac{7}{3}}$

Although we have identified two values of $c$, it is important to remember that according to the mean value theorem, the value $x = c$ must exist within the given domain, and $−\sqrt{\frac{7}{3}}$ is outside the domain. Thus, the value of $c$ is $\sqrt{\frac{7}{3}}$.
 Question 5
Using mean value theorem, identify a value $x = c$ such that the slope of the line tangent to $f(x)$ at $c$ is equal to the average rate of change of the function on the given domain.

$f(x) = \sin^{−1}(x)$    $−\frac{1}{2} ≤ x ≤ \frac{1}{2}$

 A $c = ± \sqrt{\dfrac{π^2 − 9}{π^2}}$ B $c = \sqrt{\dfrac{π^2 − 9}{π^2}}$ C $c = \dfrac{π}{3}$ D $\text{There}$ $\text{does}$ $\text{not}$ $\text{exist}$ $\text{a}$ $c$ $\text{that}$ $\text{satisfies}$ $\text{mean}$ $\text{value}$ $\text{theorem.}$
Question 5 Explanation:
The correct answer is (A). It is important to recognize first that the given function satisfies the requirements of mean value theorem, meaning that the function is continuous and differentiable on the given domain. Then, we must identify the average rate of change of the function on the domain. This is equal to the slope of the secant line connecting the two endpoints. Therefore, we must identify the two endpoints.

$f\left(−\dfrac{1}{2}\right) = \sin^{–1}\left(–\dfrac{1}{2}\right)$ $= −\dfrac{π}{6}$

$f\left(\dfrac{1}{2}\right) = \sin^{−1}\left(\dfrac{1}{2}\right)$ $= \dfrac{π}{6}$

Be sure when using inverse trigonometric functions to have your answers in radians. Also, be sure to identify the correct radian values by recalling the domain and range of inverse trigonometric functions, as there will always be multiple radian values that correspond to a specific radius length.

Then, the secant line has the slope below:

$f'(c) = \dfrac{(\frac{π}{6} − (− \frac{π}{6}))}{\frac{1}{2} − (− \frac{1}{2})}$ $= \dfrac{\frac{π}{3}}{1} = \dfrac{π}{3}$

Then, to find the value $c$ that corresponds to the tangent line with this slope, we must identify the derivative of the given function.

$f'(x) = \dfrac{1}{\sqrt{1 − x^2}}$

Then, we can set this derivative function equal to the slope of the secant line to identify the value of $x$ that corresponds to the tangent line with this same slope.

$f'(c) = \dfrac{1}{\sqrt{1 − c^2}} = \dfrac{π}{3}$

Then, after some basic algebra, we find that c has the value below.

$c = ± \sqrt{\dfrac{π^2 − 9}{π^2}}$

We have identified two values of $c$, but it is important to remember that according to the mean value theorem, the value $x = c$ must exist within the given domain. Nevertheless, both values exist within the given domain, therefore, there are two values of $c$.
 Question 6
Identify if the function below can be applied to mean value theorem, and, if so, identify the value $c$ such that the slope of the line tangent to $f(x)$ at $x = c$ is equal to the average rate of change of the function on the given domain.

$f(x) = \frac{1}{x}$    $−1 ≤ x ≤ 3$

 A $\text{Mean}$ $\text{Value}$ $\text{Theorem}$ $\text{can}$ $\text{be}$ $\text{applied.}$ $c = \sqrt{3}$ B $\text{Mean}$ $\text{Value}$ $\text{Theorem}$ $\text{can}$ $\text{be}$ $\text{applied.}$ $c = ±\sqrt{3}$ C $\text{Mean}$ $\text{Value}$ $\text{Theorem}$ $\text{cannot}$ $\text{be}$ $\text{applied}$ $\text{to}$ $\text{this}$ $\text{function.}$ D $\text{Mean}$ $\text{Value}$ $\text{Theorem}$ $\text{can}$ $\text{be}$ $\text{applied.}$ $\text{There}$ $\text{does}$ $\text{not}$ $\text{exist}$ $\text{a}$ $\text{value}$ $c$ $\text{that}$ $\text{satisfies}$ $\text{the}$ $\text{theorem.}$
Question 6 Explanation:
The correct answer is (C). Do not forget that mean value theorem applies only to functions that are continuous and differentiable on the given domain. The given function is undefined at $x = 0$, and therefore is not continuous or differentiable on the domain $−1 ≤ x ≤ 3$.
 Question 7
Using Rolle’s Theorem, identify a value $x = c$ such that the slope of the line tangent to $f(x)$ at $c$ is equal to the average rate of change of the function on the given domain.

$f(x) = x^2 + 3$    $−2 ≤ x ≤ 2$

 A $c = 0$ B $c = 1$ C $c = 3$ D $c = −\dfrac{3}{2}$
Question 7 Explanation:
The correct answer is (A). We can first verify that this function satisfies Rolle’s Theorem, as it is continuous and differentiable on the given domain, and the secant line between the two endpoints of the domain has a slope of $0$.

$\dfrac{f(b) − f(a)}{b − a} = \dfrac{f(2) − f(−2)}{2 − (−2)}$

$= \dfrac{((−2)^2 + 3 − (2^2 + 3)}{4}$ $= \dfrac{0}{4} = 0$

Thus, we can find the value of $c$ by first finding the derivative of the function and setting it equal to zero.

$f'(x) = 2x$

$f'(c) = 2c = 0$

$c = 0$
 Question 8
Using Rolle’s Theorem, identify a value $x = c$ such that the slope of the line tangent to $f(x)$ at $c$ is equal to the average rate of change of the function on the given domain.

$f(x) = \ln(x^2)$    $−1 ≤ x ≤ 1$

 A $c = 0$ B $c = 1$ C $c = −1$ D $\text{Rolle's}$ $\text{Theorem}$ $\text{is}$ $\text{not}$ $\text{applicable.}$
Question 8 Explanation:
The correct answer is (D). It is important to recognize that this function does not satisfy Rolle’s Theorem even though the secant line connecting the two end-points does have a slope of $0$. As Rolle’s Theorem is an adaptation of the mean value theorem, the function must be continuous and differentiable on the given domain. $f(0)$ is undefined, and the function is not continuous or differentiable on the domain.
 Question 9
For the inverse of the function below, what is the slope of the line tangent to the curve when $x = 6?$

$y = x^2 + 2x + 3$, for $x \ge −1$

 A $\text{The}$ $\text{slope}$ $\text{is}$ $\text{undefined.}$ B $\dfrac{1}{4}$ C $\dfrac{1}{14}$ D $\dfrac{1}{6}$
Question 9 Explanation:
The correct answer is (B). Begin by finding the inverse of the function:

$x = y^2 + 2y + 3$, for $x \ge 2$ and $y \ge −1$

(The range of the original function is the domain of its inverse and the domain of the original function is the range of its inverse.)

Complete the square and solve for $y$:

$x − 3 = y^2 +2y$

$x − 3 + 1 = y^2 + 2y + 1$

$x − 2 = (y + 1) ^2$

$\pm (x − 2)^{\frac{1}{2}} = ((y + 1)^2)^{\frac{1}{2}}$

$\pm (x − 2)^{\frac{1}{2}} = y + 1$

$−1 \pm (x − 2)^{\frac{1}{2}} = y$

$y = −1 + (x − 2)^{\frac{1}{2}}$

(Note: $y = −1 − (x − 2)^{\frac{1}{2}}$ is not valid since it produces values for y outside the range, $y \ge −1$, of the inverse function.)

Find the derivative of the inverse function:

$y' = \dfrac{1}{2} (x − 2)^{−\frac{1}{2}}$

$y' = \dfrac{1}{2(x − 2)^{\frac{1}{2}}}$

Evaluate the derivative of the inverse function at $x = 6$:

$y' = \dfrac{1}{2(6− 2)^{\frac{1}{2}}}$

$y' = \dfrac{1}{4}$

Note: the derivative of the inverse function can also be found using implicit differentiation:

$x = y^2 + 2y + 3$

$\dfrac{dx}{dx} = 2y\dfrac{dy}{dx} + 2\dfrac{dy}{dx}$

$1 = \dfrac{dy}{dx} (2y + 2)$

$\dfrac{dy}{dx} = \dfrac{1}{(2y + 2)}$

$y' = \dfrac{1}{2y + 2}$
 Question 10
For the inverse of the function below, what is the slope of the line tangent to the curve when $x = 6?$

$y = \ln(x^6)$

 A $e$ B $1$ C $\dfrac{e}{6}$ D $6$
Question 10 Explanation:
The correct answer is (C). Begin by finding the inverse of $y = \ln{(x^6)}$:

Switch the $x$ and $y$ variables to get: $x=\ln{(y^6)}$

Solve for $y$:

$e^x=e^{ln(y^6)}\rightarrow e^x=y^6\rightarrow$ ${{(e}^x)}^\frac{1}{6}={(y^6)}^\frac{1}{6}\rightarrow$ $e^{\frac{1}{6}x}=y$

Our inverse function is: $y=e^{\frac{1}{6}x}$

Find the derivative of our inverse function (remember to use the chain rule):

$y' =(e^{\frac{1}{6}x})(\frac{1}{6})\rightarrow$ $y' =\frac{1}{6}e^{\frac{1}{6}x}$

Find the slope of the tangent line at $x=6$:

$y' = \frac{1}{6}e^{\frac{1}{6}(6)}\rightarrow$ $y' =\frac{e}{6}$
Once you are finished, click the button below. Any items you have not completed will be marked incorrect.
There are 10 questions to complete.
 ← List →

format: plain


text:
# This is spiralling out of control!

Geometry Level pending

What is the value of the smallest positive integer $$N$$ such that $$\displaystyle \sum_{k=1}^N \arctan \left ( \frac {1}{\sqrt k} \right ) > 2\pi$$?

×


text:
Accelerating the pace of engineering and science

# Documentation

## Mixed-Sensitivity Loop Shaping

A popular alternative approach to loopsyn loop shaping is H mixed-sensitivity loop shaping, which is implemented by the Robust Control Toolbox™ software command:

```K=mixsyn(G,W1,[],W3)
```

With mixsyn controller synthesis, your performance and stability robustness specifications equations (2-2) and (2-4) are combined into a single infinity norm specification of the form

${‖{T}_{{y}_{1}{u}_{1}}‖}_{\infty }\le 1$

${T}_{{y}_{1}{u}_{1}}\stackrel{def}{=}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[\begin{array}{cc}{W}_{1}& S\\ {W}_{3}& T\end{array}\right].$

The term ${‖{T}_{{y}_{1}{u}_{1}}‖}_{\infty }$ is called a mixed-sensitivity cost function because it penalizes both sensitivity S(s) and complementary sensitivity T(s). Loop shaping is achieved when you choose W1 to have the target loop shape for frequencies ω < ωc, and you choose 1/W3 to be the target for ω > ωc. In choosing design specifications W1 and W3 for a mixsyn controller design, you need to ensure that your 0 dB crossover frequency for the Bode plot of W1 is below the 0 dB crossover frequency of 1/W3, as shown in Singular Value Specifications on L, S, and T, so that there is a gap for the desired loop shape Gd to pass between the performance bound W1 and your robustness bound ${W}_{3}^{-1}$. Otherwise, your performance and robustness requirements will not be achievable.

MIXSYN H Mixed-Sensitivity Loop Shaping Ty1 u1


text:
# Year

Year

A year (from Old English gēar) is the orbital period of the Earth moving around the Sun. For an observer on Earth, this corresponds to the period it takes the Sun to complete one course throughout the zodiac along the ecliptic.

In astronomy, the Julian year is a unit of time, defined as 365.25 days of 86400 SI seconds each.[1]

There is no universally accepted symbol for the year as a unit of time. The International System of Units does not propose one. A common abbreviation in international use is a (for Latin annus), in English also y or yr.

Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by changes in weather, hours of daylight, and consequently vegetation and fertility. In temperate and subpolar regions, generally four seasons are recognized: spring, summer, autumn and winter, astronomically marked by the Sun reaching the points of equinox and solstice, although the climatic seasons lag behind their astronomical markers. In some tropical and subtropical regions it is more common to speak of the rainy (or wet, or monsoon) season versus the dry season.

A calendar year is an approximation of the Earth's orbital period in a given calendar. A calendar year in the Gregorian calendar (as well as in the Julian calendar) has either 365 (common years) or 366 (leap years) days.

The word "year" is also used of periods loosely associated but not strictly identical with either the astronomical or the calendar year, such as the seasonal year, the fiscal year or the academic year, etc. By extension, the term year can mean the orbital period of any planet: for example, a "Martian year" is the time in which Mars completes its own orbit. The term is also applied more broadly to any long period or cycle, such as the "Great Year".[2]

## Etymology

West Saxon gear (jɛar), Anglian gēr continues Proto-Germanic *jǣram (*jē2ram). Cognates are German Jahr, Old High German jar, Old Norse ár and Gothic jer, all from a PIE *yērom "year, season". Cognates outside of Germanic are Avestan yare "year", Greek ὥρα "year, season, period of time" (whence "hour"), Old Church Slavonic jaru and Latin hornus "of this year".

Latin Annus (a 2nd declension masculine noun; annum is the accusative singular; anni is genitive singular and nominative plural; anno the dative and ablative singular) is from a PIE noun *at-no-, which also yielded Gothic aþnam "year".

Both *yē-ro- and *at-no- are based on verbal roots expressing movement, *at- and *ey- respectively, both meaning "to go" generally.

The Greek word for "year", ἔτος, is cognate with Latin vetus "old", from PIE *wetus- "year", also preserved in this meaning in Sanskrit vat-sa- "yearling (calf)" and vat-sa-ras "year".

Derived from Latin annus are a number of English words, such as annual, annuity, anniversary, etc.; per annum means "each year".

## Seasonal year

A seasonal year is the time between successive recurrences of a seasonal event such as the flooding of a river, the migration of a species of bird, the flowering of a species of plant, the first frost, or the first scheduled game of a certain sport. All of these events can have wide variations of more than a month from year to year.

## Calendar year

A calendar year is the time between two dates with the same name in a calendar.

A half year (one half of a year) may run from January to June, or July to December.

No astronomical year has an integer number of days or lunar months, so any calendar that follows an astronomical year must have a system of intercalation such as leap years. Financial and scientific calculations often use a 365-day calendar to simplify daily rates.

In the Julian calendar, the average length of a year is 365.25 days. In a non-leap year, there are 365 days, in a leap year there are 366 days. A leap year occurs every 4 years.

The Gregorian calendar attempts to keep the vernal equinox on or soon before March 21, hence it follows the vernal equinox year. The average length of this calendar's year is 365.2425 mean solar days (as 97 out of 400 years are leap years); this is within one ppm of the current length of the mean tropical year (365.24219 days). It is estimated that, by the year 4000, the vernal equinox will fall back by one day in the Gregorian calendar, not because of this difference, but because of the slowing down of the Earth's rotation and the associated lengthening of the sidereal day.

The Persian calendar, in use in Afghanistan and Iran, has its year begin on the day of the vernal equinox as determined by astronomical computation (for the time zone of Tehran), as opposed to using an algorithmic system of leap years.

### Numbering calendar years

A calendar era is used to assign a number to individual years, using a reference point in the past as the beginning of the era. In many countries, the most common era is from the estimated date of the birth of Jesus; dates in this era are designated Anno Domini ("in the year of the Lord", abbreviated A.D.) or C.E. (common era). Other eras are also used to enumerate the years in different cultural, religious or scientific contexts.

## Other annual periods

### Fiscal year

A fiscal year or financial year is a 12-month period used for calculating annual financial statements in businesses and other organizations. In many jurisdictions, regulations regarding accounting require such reports once per twelve months, but do not require that the twelve months constitute a calendar year.

For example, the federal government of the U.S. has a fiscal year that starts on October 1 instead of January 1. In India the fiscal year is between April 1 and March 31. In the United Kingdom and Canada, the financial year runs from April 6 and April 1 respectively, and in Australia it runs from July 1.

An academic year is the annual period during which a student attends an educational institution. The academic year may be divided into academic terms, such as semesters or quarters.

Some schools in the UK and USA divide the academic year into three roughly equal-length terms (called "trimesters" or "quarters" in the USA), roughly coinciding with autumn, winter, and spring. At some, a shortened summer session, sometimes considered part of the regular academic year, is attended by students on a voluntary or elective basis.

Other schools break the year into two main semesters, a first (typically August through December) and a second semester (January through May). Each of these main semesters may be split in half by mid-term exams, and each of the halves is referred to as a "quarter" (or "term" in some countries). There may also be an elective summer session and/or a short January session.

Some other schools, including some in the United States, have four marking periods. The school year in many countries starts in August or September and ends in May, June or July.

Some schools in the United States, notably Boston Latin School, may divide the year into five or more marking periods. Some state in defense of this that there is perhaps a positive correlation between report frequency and academic achievement.

There are typically 180 days of teaching each year in schools in the USA, excluding weekends and breaks, while 190 days for pupils in state schools in the United Kingdom, New Zealand and Canada.

In India the academic year normally starts from June 1 and ends on May 31. Though schools start closing from mid-March, the actual academic closure is on May 31 and in Nepal it starts from July 15.[citation needed]

Schools and universities in Australia typically have academic years that roughly align with the calendar year (i.e. starting in February or March and ending in October to December), as the southern hemisphere experiences summer from December to February.

## Astronomical years

### Julian year

The Julian year, as used in astronomy and other sciences, is a time unit defined as exactly 365.25 days. This is the normal meaning of the unit "year" (symbol "a" from the Latin annus) used in various scientific contexts. The Julian century of 36525 days and the Julian millennium of 365250 days are used in astronomical calculations. Fundamentally, expressing a time interval in Julian years is a way to precisely specify how many days (not how many "real" years), for long time intervals where stating the number of days would be unwieldy and unintuitive. By convention, the Julian year is used in the computation of the distance covered by a light-year.

In the Unified Code for Units of Measure, the symbol a (without subscript) always refers to the Julian year aj of exactly 31557600 seconds.

365.25 days of 86400 seconds = 1 a = 1 aj = 31.5576 Ms

The SI multiplier prefixes may be applied to it to form ka (kiloannum), Ma (megaannum) etc.

### Sidereal, tropical, and anomalistic years

The relations among these are considered more fully in Axial precession (astronomy).

Each of these three years can be loosely called an 'astronomical year'.

The sidereal year is the time taken for the Earth to complete one revolution of its orbit, as measured against a fixed frame of reference (such as the fixed stars, Latin sidera, singular sidus). Its average duration is 365.256363004 mean solar days (365 d 6 h 9 min 9.76 s) (at the epoch J2000.0 = January 1, 2000, 12:00:00 TT).[3]

The tropical year is the period of time for the ecliptic longitude of the Sun to increase by 360 degrees. Since the Sun's ecliptic longitude is measured with respect to the equinox, the tropical year comprises a complete cycle of the seasons; because of the economic importance of the seasons, the tropical year is the basis of most calendars. The tropical year is often defined as the time between southern solstices, or between northward equinoxes. Because of the Earth's axial precession, this year is about 20 minutes shorter than the sidereal year. The mean tropical year is approximately 365 days, 5 hours, 48 minutes, 45 seconds[4] (= 365.24219 days).

The anomalistic year is the time taken for the Earth to complete one revolution with respect to its apsides. The orbit of the Earth is elliptical; the extreme points, called apsides, are the perihelion, where the Earth is closest to the Sun (January 3 in 2011), and the aphelion, where the Earth is farthest from the Sun (July 4 in 2011). The anomalistic year is usually defined as the time between perihelion passages. Its average duration is 365.259636 days (365 d 6 h 13 min 52.6 s) (at the epoch J2011.0).[5]

If Earth moved in an ideal Kepler orbit, i.e. a perfect ellipse with the Sun fixed at one focus, each kind of year would always have the same duration, and the sidereal and anomalistic years would be equal. Because of perturbations by the gravity of other planets, Earth's motion varies slightly, causing the sidereal and tropical years to vary in length by about 25 minutes (see table below). Both are affected in the same way, so that the sidereal year is consistently 20 minutes longer than the tropical year, provided that they are measured in the same way.

Winter solstice (Atomic time) Deviation of the following year's duration from the mean value 365.24219 SI days
2007-12-22 06:04:04.2 +10.51 minutes
2008-12-21 12:03:19.7 -11.86 minutes
2009-12-21 17:40:13.2 +15.91 minutes
2010-12-21 23:44:53.2 -11.94 minutes
2011-12-22 05:21:41.8 +3.58 minutes
2012-12-21 11:14:01.9 +2.85 minutes
2013-12-21 17:05:38.3 +0.86 minutes
2014-12-21 22:55:15.2 +0.48 minutes

An example of a year that will have a duration exceeding the average value of 365.24219 SI days with as much as 24.23 minutes is the one that will begin at winter solstice December 21, 2042 17:47:45.5 (Atomic time).

### Draconic year

The draconic year, draconitic year, eclipse year, or ecliptic year is the time taken for the Sun (as seen from the Earth) to complete one revolution with respect to the same lunar node (a point where the Moon's orbit intersects the ecliptic). This period is associated with eclipses: these occur only when both the Sun and the Moon are near these nodes; so eclipses occur within about a month of every half eclipse year. Hence there are two eclipse seasons every eclipse year. The average duration of the eclipse year is

346.620075883 days (346 d 14 h 52 min 54 s) (at the epoch J2000.0).

This term is sometimes erroneously used for the draconic or nodal period of lunar precession, that is the period of a complete revolution of the Moon's ascending node around the ecliptic: 18.612815932 Julian years (6798.331019 days; at the epoch J2000.0).

### Full moon cycle

The full moon cycle is the time for the Sun (as seen from the Earth) to complete one revolution with respect to the perigee of the Moon's orbit. This period is associated with the apparent size of the full moon, and also with the varying duration of the synodic month. The duration of one full moon cycle is:

411.78443029 days (411 d 18 h 49 min 34 s) (at the epoch J2000.0).

### Lunar year

The lunar year comprises twelve full cycles of the phases of the Moon, as seen from Earth. It has a duration of approximately 354.37 days.

### Vague year

The vague year, from annus vagus or wandering year, is an integral approximation to the year equaling 365 days, which wanders in relation to more exact years. Typically the vague year is divided into 12 schematic months of 30 days each plus 5 epagomenal days. The vague year was used in the calendars of Ancient Egypt, Iran, Armenia and in Mesoamerica among the Aztecs and Maya.[6]

### Heliacal year

A heliacal year is the interval between the heliacal risings of a star. It differs from the sidereal year for stars away from the ecliptic due mainly to the precession of the equinoxes.

### Sothic year

The Sothic year is the interval between heliacal risings of the star Sirius. It is equal to the sidereal year and its duration is very close to the mean Julian year of 365.25 days.

### Gaussian year

The Gaussian year is the sidereal year for a planet of negligible mass (relative to the Sun) and unperturbed by other planets that is governed by the Gaussian gravitational constant. Such a planet would be slightly closer to the Sun than Earth's mean distance. Its length is:

365.2568983 days (365 d 6 h 9 min 56 s).

### Besselian year

The Besselian year is a tropical year that starts when the (fictitious) mean Sun reaches an ecliptic longitude of 280°. This is currently on or close to January 1. It is named after the 19th century German astronomer and mathematician Friedrich Bessel. The following equation can be used to compute the current Besselian epoch (in years):[7]

B = 1900.0 + (Julian dateTT2415020.31352) / 365.242198781

The TT subscript indicates that for this formula, the Julian date should use the Terrestrial Time scale, or its predecessor, ephemeris time.

### Variation in the length of the year and the day

The exact length of an astronomical year changes over time. The main sources of this change are:

• The precession of the equinoxes changes the position of astronomical events with respect to the apsides of Earth's orbit. An event moving toward perihelion recurs with a decreasing period from year to year; an event moving toward aphelion recurs with an increasing period from year to year (though this effect does not change the average value of the length of the year).
• Each planet's movement is perturbed by the gravity of every other planet.
• Tidal drag between the Earth and the Moon and Sun increases the length of the day and of the month (by transferring angular momentum from the rotation of the Earth to the revolution of the Moon); since the apparent mean solar day is the unit with which we measure the length of the year in civil life, the length of the year appears to change. Tidal drag in turn depends on factors such as post-glacial rebound and sea level rise.
• Changes in the effective mass of the Sun, caused by solar wind and radiation of energy generated by nuclear fusion and radiated by its surface, will affect the Earth's orbital period over a long time (approximately an extra 1.25 microsecond per year).[8]
• The Poynting–Robertson effect shortens the year by about 30 nanoseconds per year.
• Gravitational radiation shortens the year by about 165 attoseconds per year.[9]

### Summary

• 346.62 days: a draconitic year.
• 353, 354 or 355 days: the lengths of common years in some lunisolar calendars.
• 354.37 days (12 lunar months): the average length of a year in lunar calendars, notably the Muslim calendar.
• 365 days: a vague year and a common year in many solar calendars.
• 365.24219 days: a mean tropical year (rounded to five decimal places) for the epoch 2000.
• 365.2424 days: a vernal equinox year (rounded to four decimal places) for the epoch 2000.
• 365.2425 days: the average length of a year in the Gregorian calendar.
• 365.25 days: the average length of a year in the Julian calendar.
• 365.2564 days: a sidereal year.
• 366 days: a leap year in many solar calendars.
• 383, 384 or 385 days: the lengths of leap years in some lunisolar calendars.
• 383.9 days (13 lunar months): a leap year in some lunisolar calendars.

An average Gregorian year is 365.2425 days = 52.1775 weeks = 8765.82 hours = 525949.2 minutes = 31556952 seconds (mean solar, not SI).

A common year is 365 days = 8760 hours = 525600 minutes = 31536000 seconds.

A leap year is 366 days = 8784 hours = 527040 minutes = 31622400 seconds.

The 400-year cycle of the Gregorian calendar has 146097 days and hence exactly 20871 weeks.

## Symbol

There is no universally accepted symbol for the year as a unit of time. The International System of Units does not propose one. NIST SP811[10] and ISO 80000-3:2006[11] suggest the symbol a is taken from the Latin word annus.[12] In English, the abbreviations y or yr are sometimes used, specifically in geology and paleontology, where kyr, myr, byr (thousands, millions, and billions of years, respectively) and similar abbreviations are used to denote intervals of time remote from the present.[12][13][14]

### Symbol a

NIST SP811[15] and ISO 80000-3:2006[16] suggest the symbol a (in the International System of Units, although a is also the symbol for the are, the unit of area used to measure land area, but context is usually enough to disambiguate). In English, the abbreviations y and yr are also used.[12] [13] [14]

The Unified Code for Units of Measure[17] disambiguates the varying symbologies of ISO 1000, ISO 2955 and ANSI X3.50 [1] by using

ar for are and:
at = a_t = 365.24219 days for the mean tropical year
aj = a_j = 365.25 days for the mean Julian year
ag = a_g = 365.2425 days for the mean Gregorian year
a = 1 aj year (without further qualifier)

A definition jointly adopted by the International Union of Pure and Applied Chemistry and the International Union of Geological Sciences is to use annus, with symbol a, for year, defined as the length of the tropical year in the year 2000:[18][19]

a = 365.24219265 days = 31556925.445 seconds

The notation has proved controversial; it conflicts with an earlier convention among geoscientists to use a specifically for "years ago", and y or yr for a one-year time period.[19]

#### SI prefix multipliers

• ka (for kiloannum), is a unit of time equal to one thousand (103) years.
• Ma (for megaannum), is a unit of time equal to one million (106) years. It is commonly used in scientific disciplines such as geology, paleontology, and celestial mechanics to signify very long time periods into the past or future. For example, the dinosaur species Tyrannosaurus rex was abundant approximately 65 Ma (65 million years) ago (ago may not always be mentioned; if the quantity is specified while not explicitly discussing a duration, one can assume that "ago" is implied; the alternative but deprecated "mya" unit includes "ago" explicitly.). In astronomical applications, the year used is the Julian year of precisely 365.25 days. In geology and paleontology, the year is not so precise and varies depending on the author.
• Ga (for gigaannum), is a unit of time equal to 109 years (one billion on the short scale, one milliard on the long scale). It is commonly used in scientific disciplines such as cosmology and geology to signify extremely long time periods in the past. For example, the formation of the Earth occurred approximately 4.57 Ga (4.57 billion years) ago.
• Ta (for teraannum), is a unit of time equal to 1012 years (one trillion on the short scale, one billion on the long scale). It is an extremely long unit of time, about 70 times as long as the age of the universe. It is the same order of magnitude as the expected life span of a small red dwarf star.
• Pa (for petaannum), is a unit of time equal to 1015 years (one quadrillion on the short scale, one billiard on the long scale). The half-life of the nuclide cadmium-113 is about 8 Pa. This symbol coincides with that for the pascal without a multiplier prefix, though both are infrequently used and context will normally be sufficient to distinguish time from pressure values.
• Ea (for exaannum), is a unit of time equal to 1018 years (one quintillion on the short scale, one trillion on the long scale). The half-life of tungsten-180 is 1.8 Ea.[20]

### Symbols y and yr

In astronomy, geology, and paleontology, the abbreviation yr for "years" and ya for "years ago" are sometimes used, combined with prefixes for "thousand", "million", or "billion".[13][21] They are not SI units, using y to abbreviate English year, but following ambiguous international recommendations, use either the standard English first letters as prefixes (t,m,and b) and/or the familiar metric multiplier prefixes (k, m, and g). These abbreviations include:

 SI-prefixed equivalent order of magnitude kyr "ka" * Thousands forms myr "Ma" * Millions forms byr "Ga" * Billions forms tya or kya "ka ago" Main articles: 1 E10 s, 1 E11 s, and 1 E12 s Appearance of Homo sapiens, ca. 200 tya Out-of-Africa migration, ca. 60 tya Last Glacial Maximum, ca. 20 tya Neolithic Revolution, ca. 10 tya mya "Ma ago" Main articles: 1 E13 s, 1 E14 s, and 1 E15 s Pliocene 5.3 to 2.6 mya The last geomagnetic reversal was 0.78 mya[22] The (Eemian Stage) Ice Age started 0.13 mya Holocene started 0.01 mya bya or gya "Ga ago" Main articles: 1 E16 s, 1 E17 s, and 1 E18 s oldest Eukaryotes, 2 bya age of the Earth, 4.5 bya Big Bang, 13.7 bya

Use of "mya" and "bya" is deprecated in modern geophysics, the recommended usage being "Ma" and "Ga" for dates Before Present, but "m.y." for the duration of epochs.[13][14] This ad hoc distinction between "absolute" time and time intervals is somewhat controversial amongst members of the Geological Society of America.[23]

Note that on graphs using "ya" units on the horizontal axis time flows from right to left, which may seem counter-intuitive. If the "ya" units are on the vertical axis, time flows from top to bottom which is probably easier to understand than conventional notation.

## "Great years"

### Equinoctial cycle

The Great year, or Equinoctial cycle corresponds to a complete revolution of the equinoxes around the ecliptic. Its length is about 25,700 years, and cannot be determined precisely as the precession speed is variable.

### Galactic year

The Galactic year is the time it takes Earth's solar system to revolve once around the galactic center. It comprises roughly 230 million Earth years.[24]

## References

### Notes

1. ^ International Astronomical Union "SI units" accessed February 18, 2010. (See Table 5 and section 5.15.) Reprinted from George A. Wilkins & IAU Commission 5, "The IAU Style Manual (1989)" (PDF file) in IAU Transactions Vol. XXB
2. ^ OED, s.v. "year", entry 2.b.: "transf. Applied to a very long period or cycle (in chronology or mythology, or vaguely in poetic use)."
3. ^ International Earth Rotation and Reference System Service. (2010).IERS EOP PC Useful constants.
4. ^ Astronomical Almanac for the Year 2011. Washington and Taunton: U.S. Government Printing Office and the U.K. Hydrographic Office. 2009. p. M18 (Glossary).
5. ^ Astronomical Almanac for the Year 2011. Washington and Taunton: U.S. Government Printing Office and the U.K. Hydrographic Office. 2009. pp. A1, C2.
6. ^ Calendar Description and Coordination Maya World Studies Center
7. ^ Astronomical Almanac for the Year 2010. Washington and Taunton: U.S. Government Printing Office and the U.K. Hydrographic Office. 2008. p. B3.
8. ^ Solar mass is ~2×1030 kg, decreasing at ~5×109 kg/s, or ~8×10−14 solar mass per year. The period of an orbiting body is proportional to $\tfrac{1}{\sqrt{M}}$, where M is the mass of the primary.
9. ^ ~300 W of radiation produces ~9.5×109 J orbital energy decrease per year; this varies as 1/R, and period varies as R1.5
10. ^ Ambler Thompson, Barry N. Taylor (2008) (PDF). Special Publication 811: Guide for the Use of the International System of Units (SI). National Institute of Standards and Technology (NIST).
11. ^
12. ^ a b c Russ Rowlett. "Units: A". How Many? A Dictionary of Units of Measurement. University of North Carolina. Retrieved January 9, 2009.
13. ^ a b c d "AGU Editorial Style Guide for Authors". American Geophysical Union. September 21, 2007. Archived from the original on 2008-07-14. Retrieved 2009-01-09.
14. ^ a b c North American Commission on Stratigraphic Nomenclature (November 2005). "North American Stratigraphic Code". The American Association of Petroleum Geologists Bulletin 89 (11): 1547–1591.
15. ^ Ambler Thompson, Barry N. Taylor (2008). "Special Publication 811 – Guide for the Use of the International System of Units (SI)". National Institute of Standards and Technology (NIST). para 8.1.
16. ^ "ISO 80000-3:2006, Quantities and units". Geneva: International Organization for Standardization. 2006. Part 3: Space and time.
17. ^ Gunther Schadow, Clement J. McDonald. "Unified Code for Units of Measure".
18. ^ Norman E. Holden, Mauro L. Bonardi1, Paul De Bièvre, Paul R. Renne, and Igor M. Villa (2011). "IUPAC-IUGS common definition and convention on the use of the year as a derived unit of time (IUPAC Recommendations 2011)". Pure and Applied Chemistry 83 (5): 1159–1162. doi:10.1351/PAC-REC-09-01-22.
19. ^ a b Celeste Biever (April 27, 2011). "Push to define year sparks time war". New Scientist. Retrieved April 28, 2011.
20. ^ F. A. Danevich et al. (2003). "α activity of natural tungsten isotopes". Phys. Rev. C 67: 014310. arXiv:nucl-ex/0211013. doi:10.1103/PhysRevC.67.014310.
21. ^ North American Commission on Stratigraphic Nomenclature. "Article 13 (c)". North American Stratigraphic Code. "(c) Convention and abbreviations. – The age of a stratigraphic unit or the time of a geologic event, as commonly determined by numerical dating or by reference to a calibrated time-scale, may be expressed in years before the present. The unit of time is the modern year as presently recognized worldwide. Recommended (but not mandatory) abbreviations for such ages are SI (International System of Units) multipliers coupled with "a" for annum: ka, Ma, and Ga for kilo-annum (103 years), Mega-annum (106 years), and Giga-annum (109 years), respectively. Use of these terms after the age value follows the convention established in the field of C-14 dating. The "present" refers to 1950 AD, and such qualifiers as "ago" or "before the present" are omitted after the value because measurement of the duration from the present to the past is implicit in the designation. In contrast, the duration of a remote interval of geologic time, as a number of years, should not be expressed by the same symbols. Abbreviations for numbers of years, without reference to the present, are informal (e.g., y or yr for years; my, m.y., or m.yr. for millions of years; and so forth, as preference dictates). For example, boundaries of the Late Cretaceous Epoch currently are calibrated at 63 Ma and 96 Ma, but the interval of time represented by this epoch is 33 m.y."
22. ^ Bradford M. Clement (April 8, 2004). "Dependence of the duration of geomagnetic polarity reversals on site latitude". Nature 428 (6983): 637–40. doi:10.1038/nature02459. PMID 15071591.
23. ^ "Time Units". Geological Society of America. Retrieved February 17, 2010.
24. ^ "Science Bowl Questions, Astronomy, Set 2". Science Bowl Practice Questions. Oak Ridge Associated Universities. 2009. Retrieved December 9, 2009.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Year — Year, n. [OE. yer, yeer, [yogh]er, AS. ge[ a]r; akin to OFries. i?r, g?r, D. jaar, OHG. j[=a]r, G. jahr, Icel. [=a]r, Dan. aar, Sw. [*a]r, Goth. j?r, Gr. ? a season of the year, springtime, a part of the day, an hour, ? a year, Zend y[=a]re year …   The Collaborative International Dictionary of English

• Year Up — is a nonprofit organization based in Boston, Massachusetts. Founded in 2000 by Harvard Business School graduate Gerald Chertavian, Year Up has sites in Boston and Cambridge, New York City, Providence and Washington, D.C. and is expanding to other …   Wikipedia

• Year 5 — can refer to:*The year 5 A.D. of the Gregorian calendar; see 5 *Year Five, the fifth year of education in UK schools …   Wikipedia

• Year 12 — In schools in Northern Ireland Year Twelve or Fifth Form is the twelfth and final year of compulsory education, and the fifth year of Secondary education. Children are generally fifteen when starting classes and sixteen at the end of the school… …   Wikipedia

• year — noun Etymology: Middle English yere, from Old English gēar; akin to Old High German jār year, Greek hōros year, hōra season, hour Date: before 12th century 1. a. the period of about 3651/4 solar days required for one revolution of the earth… …   New Collegiate Dictionary

• year — Sidereal Si*de re*al, a. [L. sidereus, from sidus, sideris, a constellation, a star. Cf. {Sideral}, {Consider}, {Desire}.] 1. Relating to the stars; starry; astral; as, sidereal astronomy. [1913 Webster] 2. (Astron.) Measuring by the apparent… …   The Collaborative International Dictionary of English

• year — See: ALONG IN YEARS or ON IN YEARS …   Dictionary of American idioms

• year — See: ALONG IN YEARS or ON IN YEARS …   Dictionary of American idioms

• Year Zero (album) — Year Zero Studio album by Nine Inch Nails Released April 17, 2007 …   Wikipedia

• Year and a day — Year Year, n. [OE. yer, yeer, [yogh]er, AS. ge[ a]r; akin to OFries. i?r, g?r, D. jaar, OHG. j[=a]r, G. jahr, Icel. [=a]r, Dan. aar, Sw. [*a]r, Goth. j?r, Gr. ? a season of the year, springtime, a part of the day, an hour, ? a year, Zend y[=a]re… …   The Collaborative International Dictionary of English


text:
## Quiz 1 Preparation, Question #11

Noor_Burney_1L
Posts: 13
Joined: Wed Sep 21, 2016 2:55 pm

### Quiz 1 Preparation, Question #11

On question #11 of the quiz 1 preparation (pg. 116 in course reader), it asks:
"How many grams of water can be heated from 25.0 degrees Celsius to 100.0 degrees Celsius by the heat released from converting 49.7 g of PbO to Pb?"
The converting reaction is PbO (s) + C (s) --> Pb (s) + CO (g)
delta H= -106.9 kJ

I understand that you would use the equation q= (mass)x(specific heat capacity)x(delta T) and that you can figure out the delta T to be 75.0 degrees Celsius because you are given the two temperatures in the equation. I also know that you find the specific heat capacity of water to be 4.184 because that is given on the formula sheet. However, what I do not understand is that after you calculate the moles of PbO from the 49.7 g of PbO given in the equation, how do you apply that to the q= (mass)x(specific heat capacity)x(delta T) equation in order to find the mass?

Rachel_Prescott_2M
Posts: 10
Joined: Wed Sep 21, 2016 2:57 pm
Been upvoted: 1 time

### Re: Quiz 1 Preparation, Question #11

You can use the $\Delta H$ given to you as a conversion factor of sorts since it is in kJ/mol. So after converting the grams of PbO to moles multiply that value by 106.9 KJ/mol so the moles cancel and you are left with kJ's. This value is the same as q in the equation $q=mC\Delta T$ so you can plug in all the values you know and solve for m, making the mass of water that can be heated 75.8 g. Hope this was helpful!


text:
2 added 310 characters in body

@David Hansen: Teichmuller modular forms are basically the natural analogue of Siegel modular forms when one considers sections of line bundles on $M_g$ instead of $A_g$. Search for papers by Ichikawa.

I don't know very much about Teichmuller modular forms but my advisor knows a bit about them and from what I can tell from conversations with him, pretty much nothing at all is known about this question. I can think of some plausible reasons why this is so. (This answer is probably a bit naive, my apologies.)

One is that the difference between Siegel modular forms and Teichmuller modular forms does not become visible until genus three. For genus one and two the map from $M_g$ to $A_g$ is an isomorphism and an open immersion respectively, and all Teichmuller modular forms are just pullbacks of forms from $A_1$ and $A_2$. In genus three the Torelli map is an open immersion on coarse moduli spaces but on the level of stacks it is a ramified double cover, which makes it plausible that there should be more functions on $M_3$ than on $A_3$. And indeed the ring of Teichmuller modular forms in genus three is obtained from the ring of Siegel modular forms by adjoining a square root of $\chi_{18}$, which vanishes exactly at the hyperelliptic locus i.e. the branch locus of the Torelli map. Anyway, the point is that already Siegel modular forms in higher genera are not so well understood compared to the rich theory we have in low genus. In particular we know extremely few examples of genuine Teichmuller modular forms.

Another more serious reason is that it is not clear how any of the standard tools for studying ordinary modular forms can be applied to the Teichmuller case. As a very basic example, it is for instance completely open if or how one can define a notion of Hecke operators acting on Teichmuller modular forms: double cosets in the mapping class group are a scary prospect, and there seems to be no analogue of the standard Hecke correspondences in terms of cyclic subgroups of order p. What's worse is that none of the standard automorphic/Langlands etc tools seem to be applicable to the study of Teichmuller modular forms. Unlike Siegel's upper half space, Teichmuller space is not a symmetric space. If we believe in the Langlands philosophy then there should be a reductive group somewhere that the Teichmuller modular forms "come from", but there is no natural reductive group anywhere in sight.

Anyway, to actually answer your question: there probably is some connection between Teichmuller modular forms and number theory. In particular it seems that there are $\ell$-adic Galois representations naturally attached to Teichmuller modular forms. There are just no tools to study these representations.

1

@David Hansen: Teichmuller modular forms are basically the natural analogue of Siegel modular forms when one considers sections of line bundles on $M_g$ instead of $A_g$. Search for papers by Ichikawa.

I don't know very much about Teichmuller modular forms but my advisor knows a bit about them and from what I can tell from conversations with him, pretty much nothing at all is known about this question. I can think of some plausible reasons why this is so. (This answer is probably a bit naive, my apologies.)

One is that the difference between Siegel modular forms and Teichmuller modular forms does not become visible until genus three. For genus one and two the map from $M_g$ to $A_g$ is an isomorphism and an open immersion respectively, and all Teichmuller modular forms are just pullbacks of forms from $A_1$ and $A_2$. In genus three the Torelli map is an open immersion on coarse moduli spaces but on the level of stacks it is a ramified double cover, which makes it plausible that there should be more functions on $M_3$ than on $A_3$. And indeed the ring of Teichmuller modular forms in genus three is obtained from the ring of Siegel modular forms by adjoining a square root of $\chi_{18}$, which vanishes exactly at the hyperelliptic locus i.e. the branch locus of the Torelli map. Anyway, the point is that already Siegel modular forms in higher genera are not so well understood compared to the rich theory we have in low genus. In particular we know extremely few examples of genuine Teichmuller modular forms.

Another more serious reason is that it is not clear how any of the standard tools for studying ordinary modular forms can be applied to the Teichmuller case. As a very basic example, it is for instance completely open if or how one can define a notion of Hecke operators acting on Teichmuller modular forms: double cosets in the mapping class group are a scary prospect, and there seems to be no analogue of the standard Hecke correspondences in terms of cyclic subgroups of order p. What's worse is that none of the standard automorphic/Langlands etc tools seem to be applicable to the study of Teichmuller modular forms. Unlike Siegel's upper half space, Teichmuller space is not a symmetric space. If we believe in the Langlands philosophy then there should be a reductive group somewhere that the Teichmuller modular forms "come from", but there is no natural reductive group anywhere in sight.


text:
Solve this
Question:

If $f(x)=\frac{2-\sqrt{x+4}}{\sin 2 x}, x \neq 0$, is continuous at $x=0$, then $f(0)=$________

Solution:

The function $f(x)=\frac{2-\sqrt{x+4}}{\sin 2 x}, x \neq 0$, is continuous at $x=0$.

$\therefore f(0)$

$=\lim _{x \rightarrow 0} f(x)$

$=\lim _{x \rightarrow 0} \frac{2-\sqrt{x+4}}{\sin 2 x}$

$=\lim _{x \rightarrow 0} \frac{2-\sqrt{x+4}}{\sin 2 x} \times \frac{2+\sqrt{x+4}}{2+\sqrt{x+4}}$

$=\lim _{x \rightarrow 0} \frac{4-(x+4)}{\sin 2 x(2+\sqrt{x+4})}$

$=\lim _{x \rightarrow 0} \frac{-x}{\sin 2 x(2+\sqrt{x+4})}$

$=-\frac{1}{2} \times \lim _{x \rightarrow 0} \frac{2 x}{\sin 2 x(2+\sqrt{x+4})}$

$=-\frac{1}{2} \times \frac{1}{\lim _{x \rightarrow 0}\left(\frac{\sin 2 x}{2 x}\right)} \times \frac{1}{\lim _{x \rightarrow 0}(2+\sqrt{x+4})}$

$=-\frac{1}{2} \times \frac{1}{1} \times \frac{1}{(2+\sqrt{0+4})} \quad\left(\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\right)$

$=-\frac{1}{2} \times \frac{1}{4}$

$=-\frac{1}{8}$

Thus, the value of $f(0)$ is $-\frac{1}{8}$.

If $f(x)=\frac{2-\sqrt{x+4}}{\sin 2 x}, x \neq 0$, is continuous at $x=0$, then $f(0)=-\frac{1}{8}$


text:
location:  Publications → journals
Search results

Search: MSC category 46M10 ( Projective and injective objects [See also 46A22] )

 Expand all        Collapse all Results 1 - 2 of 2

1. CJM Online first

Crann, Jason
 Amenability and covariant injectivity of locally compact quantum groups II Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group $\mathbb{G}$ and 1-injectivity of $L^{\infty}(\widehat{\mathbb{G}})$ as an operator $L^1(\widehat{\mathbb{G}})$-module. In particular, a locally compact group $G$ is amenable if and only if its group von Neumann algebra $VN(G)$ is 1-injective as an operator module over the Fourier algebra $A(G)$. As an application, we provide a decomposability result for completely bounded $L^1(\widehat{\mathbb{G}})$-module maps on $L^{\infty}(\widehat{\mathbb{G}})$, and give a simplified proof that amenable discrete quantum groups have co-amenable compact duals which avoids the use of modular theory and the Powers--StÃ¸rmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability. Keywords:locally compact quantum group, amenability, injective moduleCategories:22D35, 46M10, 46L89

2. CJM 1999 (vol 51 pp. 850)

Muhly, Paul S.; Solel, Baruch
 Tensor Algebras, Induced Representations, and the Wold Decomposition Our objective in this sequel to \cite{MSp96a} is to develop extensions, to representations of tensor algebras over $C^{*}$-correspondences, of two fundamental facts about isometries on Hilbert space: The Wold decomposition theorem and Beurling's theorem, and to apply these to the analysis of the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras. Keywords:tensor algebras, correspondence, induced representation, Wold decomposition, Beurling's theoremCategories:46L05, 46L40, 46L89, 47D15, 47D25, 46M10, 46M99, 47A20, 47A45, 47B35
 top of page | contact us | privacy | site map |


text:
# When states that are not accepting states become accepting states in NFA, what happens?

It is known that when NFA's accepting states are swapped with rejecting states, the recognized language is not complement of the original language. For some language, it may be the case that some strings that were recognized in the original NFA may still be recognized in the state-swapped NFA.

My question is, is there a language $L$ that in the original NFA a string is not recognized(rejected), but in the NFA where accepting states become rejecting states and rejecting states become accepting states the string is also rejected?

• This is maybe too far off for you, but you are after the reason for why the "flip states" technique does not work für Büchi automata. – Raphael Feb 6 '15 at 13:42

Yes, there is such a situation. A string is rejected when there is no computation for that string at all. Obviously there will be no computation after swapping accepting states.

• Technically correct, but I think the whole "flip states" idea assumes the automaton to be complete, doesn't it? – Raphael Feb 6 '15 at 13:43
• @Raphael In my opinion the "flip states" technique assumes determinism, exactly one transition for each letter in each state. The question states what may go wrong when there are too much paths. Here is the dual, what goes wrong when there is no path. I believe these are opposite sides of the same coin. – Hendrik Jan Feb 6 '15 at 13:47
• I don't think determinism is needed, technically. The OP will have to decide whether your answer is relevant (to them) or a pathological corner case. – Raphael Feb 6 '15 at 14:49

It depends on whether the NFA contains a path for every possible word.

If there is an input word where there is no path through the NFA at all labelled with that word, then yes, this can happen. See Hendrik Jan's answer.

On the other hand, it is often assumed that the automaton is complete: that for every possible input word, there is at least one path in the automaton labelled with that word. (For instance, this can be arranged by ensuring that for every state $q$ and every possible input letter $x \in \Sigma$, there is a transition out of $q$ labelled with $x$.) If you assume the automaton is complete, the answer is: No.

In particular, if the automaton is complete, the answer is: No, the magic of nondeterminism is that if there's any valid computation path that will end in an accept state, the string is accepted. In this case, all possible computation paths in the original NFA led to rejection (had there been any valid path the string would be accepted, so they must all reject), so in the inverted NFA, they all lead to acceptance.

• That is assuming there is a path for every word (hence Hendrik Jan's answer). – Klaus Draeger Feb 6 '15 at 13:41
• @KlausDraeger Technically correct, but I think the whole "flip states" idea assumes the automaton to be complete, doesn't it? – Raphael Feb 6 '15 at 13:43
• @Raphael I'm not sure if that is necessarily assumed. It is a natural condition though, and it does ensure that at least one of the NFA accepts any given word. – Klaus Draeger Feb 6 '15 at 13:47
• On second thought, Luke, things go horribly wrong if some computations accept and others don't (the typical case): those are also accepted by the flipped automaton. We need for every input that either all or no computation accepts; only then can we flip. (cc @KlausDraeger) – Raphael Feb 6 '15 at 14:47


text:
## 2016-12-04

### 如何理解四个基本子空间

Four Fundamental Matrix Spaces

1. Row space of A

2. Column space of A

3. Nullspace of A

4. Nullspace of A’

A system of linear equations Ax=b is consistent if and only if b is in the column space of A.

We already known that elementary row operations do not change the row space of a matrix.

And elementary row operations do not change the nullspace of matrix.

However this result dost not apply to the column space. Therefore only its row space is preserved under elementary operations.

If a matrix R is in row-echelon form then:

1. The row vectors with the leading 1’s form a basis for the row space of R.

2. The column vector with the leading 1’s of the row vectors form a basis for the column space.

Therefor we put it in row-echelon form and extract the row vectors with a leading 1 to find a basis for the row space of a matrix.

Relationships between four vector space

1. If A is any matrix then row space and column space of A have the same dimension.

2. The common dimension of the row space and column space of a matrix A is called the rank of A and is denoted by rank(A);

3. The dimension of the nullspace of A is called the nullity of A and is denoted by nullity(A)

4. Rank(A)=rank(A’)

5. Rank(A)+nullity(A)=n (A is a matrix with n column)

6. Rank(A)=the number of leading variable in the solution of Ax=0.

7. Nullity(A)=the number of parameters in the general solution of Ax=0.

PPT: http://slideplayer.com/slide/8413041/

1. 我是来测试的

2. 好，我知道了


text:
NSCI 580A4 fall 2017

Sidebar

NSCI 580A4

Instructors
Tai Montgomery
Erin Nishimura

wiki:2016cpmv

This is an old revision of the document!

Copying & Moving

Copying files and directories with cp

The cp (CoPy) command is quite flexible. There are a few ways it can be used to copy a file.

Duplicating a file and giving it a new name

This can be used to copy the contents of a file (the source file) into a new file (the target file) with a new name:

cp <source_file.txt> <target_file.txt>

Exercise: Make a file called file1.txt. Add some text to file1.txt. Next, make file2.txt like so…

$cp file1.txt file2.txt$ls
$less file2.txt You can think of this as basically shorthand for… $cp ./file1.txt ./file2.txt

Duplicating a file into a directory

Once you make the connection that names, absolute paths, or relative paths can substitute in for <source_file.txt> or <target_file.txt>, you can see how you can place the copied file in some other directory, or pull a copy of a file from a source directory into your working directory.

Duplicating a file into a directory and renaming it:
cp <source_file.txt> <target/path/targetname.txt>

OR

Duplicating a file from another directory into the current directory and renaming it:
cp <target/path/targetname.txt> <./targetname.txt>

Exercise: Make a directory called dir1. Place a copy of file1.txt into dir1.

$mkdir dir1$cp file1.txt dir1/file3.txt
$ls$ls dir1

Quick Tip: Absolute paths as well as relative paths can be used as the source and target in cp.

If you want to duplicate a file into a sub-directory, you don't need to change the name. To keep the name the same…

cp <source_file.txt> <target_directory>

Exercise: Try it out

$mkdir dir1$cp file1.txt dir1
$ls$ls dir1

A list of files can also be copied in this way:

cp <source_file.txt> … <target_directory>

– where “…” means you can keep adding additional source_files.txt, as many as you have.

Duplicating a directory and its contents

Directories that contain files can also be duplicated using cp. Just add the option -R.

$mkdir dir1$cp -R dir1 copy_of_dir1
$ls$ls copy_of_dir1

Independent Exercise: I like to stay organized by adding notes to myself within directories. These text files that contain little notes to myself about the purpose for the directory. I call these README or ABOUT files.

1. Navigate to your course directory you made earlier (the directory where ls will show you “01_Notes”, “02_Exercises”, etc)
2. Create a file called ABOUT_this_class.txt within the course directory.
3. Using nano, copy the “syllabus” page of this website into the ABOUT_this_class.txt file.
4. Use less to browse your new file.
5. Create a copy of this file in the directory “01_Notes” called ABOUT_these_notes.txt. Double check that the content is the same.
6. Create a copy of this file in the parent directory (one level up from where you are) keeping the file name the same.
7. Using a set of test directories, test what happens when you try to copy one directory into another using the command $cd dir1 dir2 versus $ cd dir1/ dir2.

Moving files and directories with mv

Once you know cp, mv is pretty much the same thing with one exception. The source file will disappear once the operation is complete. This ends up renaming your file if you are working within the same directory. It acts like cut-and-paste instead of a copy-and-paste if you're moving between directories.

 mv Rename source_file.txt to be called target_file.txt mv Move source_file.txt into dir and rename it target_file.txt mv Move source_file.txt(s) into dir and keep the names the same

Independent Exercise: Remember your directory called 160825_options_ex? MOVE that whole directory and its contents to the new directory ~/courseDirectory/02_Exercises you recently created within your course directory (where ~ stands for your appropriate path).


text:
# Revision history [back]

Your problem is not direct formatting, but AutoCorrect. Go to Tools > AutoCorrect Options > Localized Options and turn off "format ordinal number suffixes".

Your problem is not direct formatting, but AutoCorrect. Go to Tools > AutoCorrect Options > Localized Options and turn off the setting "format ordinal number suffixes".

Your problem is not direct formatting, but AutoCorrect. Go to Tools > AutoCorrect Options > Localized Options and turn off the setting "format ordinal number suffixes".

Note: This setting is global. I don't think it's possible to configure them "by sheet" or otherwise.

Your problem is not direct formatting, but AutoCorrect. Go to Tools > AutoCorrect Options > Localized Options and turn off the setting "format ordinal number suffixes".

Note: This setting is These settings are global. I don't think it's possible to configure them "by sheet" or otherwise.


text:
# Get Absolute path of a file in Java

Java 8Object Oriented ProgrammingProgramming

The method java.io.File.getAbsolutePath() is used to obtain the absolute path of a file in the form of a string. This method requires no parameters.

A program that demonstrates this is given as follows −

## Example

Live Demo

import java.io.File;
public class Demo {
public static void main(String[] args) {
File file = new File("C:" + File.separator + "jdk11.0.2" + File.separator, "demo1.java");
System.out.println("The absolute path name is: " + file.getAbsolutePath());
}
}

The output of the above program is as follows −

## Output

The absolute path name is:/C:/jdk11.0.2/demo1.java

Now let us understand the above program.

The absolute pathname of the file is obtained using the method java.io.File.getAbsolutePath() in the form of a string and is printed. A code snippet that demonstrates this is given as follows −

File file = new File("C:" + File.separator + "jdk11.0.2" + File.separator, "demo1.java");
System.out.println("The absolute path name is: " + file.getAbsolutePath());
Published on 25-Feb-2019 11:47:24


text:
## Encyclopedia > Continued fraction

Article Content

# Continued fraction

A continued fraction is an expression such as

$x = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3}}}$

where a0 is some integer and all the other numbers an are positive integers. The above expression is abbreviated as x = [a0; a1, a2, a3]. Longer expressions [a0; a1, a2, ..., an] are defined analogously. One may also define infinite continued fractions as limits:

$[a_{0}; a_{1}, a_{2}, a_{3}, ... ] = \lim_{n \to \infty} [a_{0}; a_{1}, a_{2}, ..., a_{n}]$

This limit exists for any choice of positive integers a1, a2, a3 ...

Occasionally, two different finite continued fractions represent the same number, for instance [2; 3, 1] = [2; 4] = 9/4 = 2.25. Every finite continued fraction is rational, and every rational number can be represented in precisely two different ways as a finite continued fraction. Every infinite continued fraction is irrational, and every irrational number can be represented in precisely one way as an infinite continued fraction.

An infinite continued fraction representation for an irrational numbers is mainly useful because its initial segments provide excellent rational approximations to the number. These rational numbers are called the convergents of the continued fraction. Even-numbered convergents are smaller than the original number, while odd-numbered ones are bigger. For example, to calculate the convergents of pi, we set a0 = [π] = 3 (where [x] denotes the largest integer ≤ x), define u1 = 1/(π - 3) ≈ 113/16 = 7.0625 and a1 = [u1] = 7, u2 = 1/(u1 - 7) ≈ 31993/2000 = 15.9965 and a2 = [u2] = 15, u3 = 1/(u3 - 15) ≈ 1003/1000 = 1.003. Continuing like this, one can determine the infinite continued fraction of π as [3; 7, 15, 1, 292, 1, 1, ...]. The third convergent of π is [3; 7, 15, 1] = 355/113 = 3.14159292035... which is fairly close to the true value of π.

Let us suppose that the quotients found are, as above, [3; 7, 15, 1]. The following is a rule by which we can write down at once the convergent fractions which result from these quotients without developing the continued fraction.

The first quotient, supposed divided by unity, will give the first fraction, which will be too small, namely, 3/1. Then, multiplying the numerator and denominator of this fraction by the second quotient and adding unity to the numerator, we shall have the second fraction, 22/7, which will be too large. Multiplying in like manner the numerator and denominator of this fraction by the third quotient, and adding to the numerator the numerator of the preceding fraction, and to the denominator the denominator of the preceding fraction, we shall have the third fraction, which will be too small. Thus, the third quotient being 15, we have for our numerator (22 · 15 = 330) + 3 = 333, and for our denominator, (7 · 15 = 105) + 1 = 106. The third convergent, therefore, is 333/106. We proceed in the same manner for the fourth convergent. The fourth quotient being 1, we say 333 times 1 is 333, and this plus 22, the numerator of the fraction preceeding, is 355; similarly, 106 times 1 is 106, and this plus 7 is 113.

In this manner, by employing the four quotients [3; 7, 15, 1], we obtain the four fractions:

$\frac{3}{1}, \frac{22}{7}, \frac{333}{106}, \frac{355}{113}, \ldots$

The fractions are alternately smaller and larger than the true value of π, and have the advantage of approaching nearer and nearer to its value in such wise that no other fraction can approach it nearer except the denominator be larger than the product of the denominator of the fraction in question and the denominator of the fraction following. For example, the fraction 22/7 is more than the true value, but it approaches to it nearer than any other fraction does whose denominator is not greater than the product of 7 by 106, that is 742.

The demonstration of the foregoing properties is deduced from the fact that if we seek the difference between one of the convergent fractions and the next adjacent to it we shall obtain a fraction of which the numerator is always unity and the denominator the product of the two denominators. Thus the difference between 22/7 and 3/1 is 1/7, in excess; between 333/106 and 22/7, 742, in defect; between 355/113 and 333/106, 1/11978, in excess; and so on. The result being, that by employing this series of differences we can express in another and very simple manner the fractions with which we are here concerned, by means of a second series of fractions of which the numerators are all unity and the denominators successively be the product of every two adjacent denominators. Instead of the fractions written above, we have thus the series:

$\frac{3}{1}+\frac{1}{1 \cdot 7}-\frac{1}{7 \cdot 106}+\frac{1}{106 \cdot 113} \cdots$
The first term, as we see, is the first fraction; the first and second together give the second fraction, 22/7; the first, the second and the third give the third fraction 333/106, and so on with the rest; the result being that the series entire is equivalent to the original value.

While one cannot discern any pattern in the infinite continued fraction expansion of π, this is not true for e, the base of the natural logarithm: e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, ...].

The numbers with periodic continued fraction expansion are precisely the solutions of quadratic equations with integer coefficients. For example, the golden ratio φ = [1; 1, 1, 1, 1, 1, ...] and √ 2 = [1; 2, 2, 2, 2, ...].

A. Ya. Khinchin; Continued Fractions; University of Chicago Press.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Charles V, Holy Roman Emperor ... 1545 the opening of the Council of Trent began the Counter-Reformation, and Charles won to the Catholic cause some of the princes of the Holy Roman Empire. He also attacked ...


text:
## Monday, November 15, 2010

### Ubuntu and LaTeX Update: Ubuntu Releases and Bibliography Management (Mendeley Desktop and BibTeX)

Ubuntu Update:

So it has been a few months since switching to Ubuntu from Windows...and I haven't looked back. It is working out great and I love it. However, it is not perfect and I still need windows to do a few things, like my timesheet program and play some old games. This is easily done by either dual boot (install Windows first, then Ubuntu) or using a virtual machine installation software like WINE or Virtual Box. I don't really like WINE because it only displays or uses an XP version of Windows, and I didn't really find it useful or user friendly.

Ubuntu Releases:

Ubuntu releases new versions every six months. The latest went from 10.04 LTS to 10.11. The LTS versions are Long Term Support versions while others are not. For the best quality and less hassle in making sure everything works fine, I believe the developers recommend sticking to the LTS versions for stability.

The Ubuntu developers name their releases in a alliterated fashion, usually after animals.

https://wiki.ubuntu.com/Home

Releases

This is THE definitive place for members of the Ubuntu community to discuss ideas and store team-related information.

Released (Current & Stable)

Dapper Drake

6.06 LTS

June 1, 2006

Supported until June 2011 (Server)

Hardy Heron

8.04 LTS

April 24, 2008

Supported until April 2011 (Desktop) or April 2013 (Server)

Karmic Koala

9.10

October 29, 2009

Supported until April 2011

Lucid Lynx

10.04 LTS

April 29, 2010

Supported until April 2013 (Desktop) or Supported until April 2015 (Server)

Maverick Meerkat

10.10

October 10, 2010

Supported until April 2012

To be Released

Version

Release Schedule

Support Date

Natty Narwhal

11.04

April 2011

Supported until October 2012

Some like the often releases (like me, I get bored quickly sometimes) while others don't because they think it is too hasty and buggy. It's like I said before it's mostly up to personal preference.

LaTeX Update:

Well on to LateX. Haven't looked back on this either. Love, love, love LaTeX. At first it might seem a little antiquated and/or intimidating, but this is a superior alternative to Word, OpenOffice, or any other word/document processor. I love the freedom, independence, and best of all it's free as in beer and . It is not totally unorganized, in which I mean LaTeX was written to automatically take care of many formatting issues I constantly run into with Microsoft's Word. Writing papers and theses are a breeze, and LaTeX probably is best for longer documents like these. Formulas, sections, fonts, figures, are easy and wonderful once you get the hang of it. I don't have to worry about Word crashing, or EndNote not working, or MathType giving me problems.

Now Latex is not perfect, and it is a learning curve, but I recommend it to anyone who writes scientifically. No more worrying about Word moving figures or whatever it does on its own. You have full control.

Bibliography Management: A Little on Mendeley Desktop and BibTeX

Now bibliographies, as I understand it today, are worked by BibTeX. I just learned something that I would like to quickly pass along. I use Mendeley Desktop to store my reference library information. There are others like Zotero, etc., and I chose MD because I like the way it looks and how it operates to store information. It is personal preference. My advice is to briefly try and look at many as you can then choose ONE and STICK with it. MD is not perfect either, but has a good community and is active in developing its software for updates. Other reference library software include JabRef and KBibTex for LaTeX/BibTeX users. I will try to review the software at a later time in more depth.

More on BibTeX here:

http://www.bibtex.org/

http://en.wikipedia.org/wiki/BibTeX

Now MD has a export feature to export your reference in a BibTeX format. Beware that MD is far from perfect in this, and you will need to manually do some tweaking. For example, MD has a field or category called institution so that you can, as I do it, store the authors' affiliation information. MD exports this as institution or organization which refers to the publisher so you will have to manually change it to the appropriate BibTeX field of affiliation. Also, for conference locations MD exports city as address (publisher) when it should be the field location. So this isn't so bad to do every time you export one reference, but is quite overwhelming when you already have hundreds of exports and have to manually correct them like me, :(, :). Just be consistent with your Bibtex and MD listings and maybe MD with one day be more consistent with BibTeX. For example, I prefer in MD to have the authors' full name if possible, but in BibTeX the generated bibliography doesn't look good with full names plus I don't have all the full names so I prefer to use just initials for first and middle names.

Also, I would make two all reference listings in BibTeX. One for where the titles of papers are capitalized and one where they are not because journals require different preferences (whether caps or not). This will save you lots of time genereating the bibliography for certain papers. I would also make a separate BibTeX file for each paper, thesis, report you write so you only have the necessary references to look at and not your complete list.

BibTeX Links for Fields and Referencing

Here are some links to BibTeX fields and reference labels:

From:  http://en.wikipedia.org/wiki/BibTeX

BibTeX uses a style-independent text-based file format for lists of bibliography items, such as articles, books, and theses. BibTeX bibliography file names usually end in .bib.
Bibliography entries each contain some subset of standard data entries:

• address: Publisher's address (usually just the city, but can be the full address for lesser-known publishers)
• annote: An annotation for annotated bibliography styles (not typical)
• author: The name(s) of the author(s) (in the case of more than one author, separated by and)
• booktitle: The title of the book, if only part of it is being cited
• chapter: The chapter number
• crossref: The key of the cross-referenced entry
• edition: The edition of a book, long form (such as "first" or "second")
• editor: The name(s) of the editor(s)
• eprint: A specification of an electronic publication, often a preprint or a technical report
• howpublished: How it was published, if the publishing method is nonstandard
• institution: The institution that was involved in the publishing, but not necessarily the publisher
• journal: The journal or magazine the work was published in
• key: A hidden field used for specifying or overriding the alphabetical order of entries (when the "author" and "editor" fields are missing). Note that this is very different from the key (mentioned just after this list) that is used to cite or cross-reference the entry.
• month: The month of publication (or, if unpublished, the month of creation)
• note: Miscellaneous extra information
• number: The "(issue) number" of a journal, magazine, or tech-report, if applicable. (Most publications have a "volume", but no "number" field.)
• organization: The conference sponsor
• pages: Page numbers, separated either by commas or double-hyphens.
• publisher: The publisher's name
• school: The school where the thesis was written
• series: The series of books the book was published in (e.g. "The Hardy Boys" or "Lecture Notes in Computer Science")
• title: The title of the work
• type: The type of tech-report, for example, "Research Note"
• url: The WWW address
• volume: The volume of a journal or multi-volume book
• year: The year of publication (or, if unpublished, the year of creation)
In addition, each entry contains a key that is used to cite or cross-reference the entry. This key is the first item in a BibTeX entry, and is not part of any field.

### Entry Types

Bibliography entries included in a .bib file are split by types. The following types are understood by virtually all BibTeX styles:
article
An article from a journal or magazine. Required fields: author, title, journal, year Optional fields: volume, number, pages, month, note, key
book
A book with an explicit publisher. Required fields: author/editor, title, publisher, year Optional fields: volume, series, address, edition, month, note, key
booklet
A work that is printed and bound, but without a named publisher or sponsoring institution. Required fields: title Optional fields: author, howpublished, address, month, year, note, key
conference
The same as inproceedings, included for Scribe compatibility. Required fields: author, title, booktitle, year Optional fields: editor, pages, organization, publisher, address, month, note, key
inbook
A part of a book, usually untitled. May be a chapter (or section or whatever) and/or a range of pages. Required fields: author/editor, title, chapter/pages, publisher, year Optional fields: volume, series, address, edition, month, note, key
incollection
A part of a book having its own title. Required fields: author, title, booktitle, year Optional fields: editor, pages, organization, publisher, address, month, note, key
inproceedings
An article in a conference proceedings. Required fields: author, title, booktitle, year Optional fields: editor, series, pages, organization, publisher, address, month, note, key
manual
Technical documentation. Required fields: title Optional fields: author, organization, address, edition, month, year, note, key
mastersthesis
Master's thesis. Required fields: author, title, school, year Optional fields: address, month, note, key
misc
For use when nothing else fits. Required fields: none Optional fields: author, title, howpublished, month, year, note, key
phdthesis
A Ph.D. thesis. Required fields: author, title, school, year Optional fields: address, month, note, key
proceedings
The proceedings of a conference. Required fields: title, year Optional fields: editor, publisher, organization, address, month, note, key
techreport
A report published by a school or other institution, usually numbered within a series. Required fields: author, title, institution, year Optional fields: type, number, address, month, note, key
unpublished
A document having an author and title, but not formally published. Required fields: author, title, note Optional fields: month, year, key

From: http://texblog.wordpress.com/2007/08/09/how-to-use-a-bibtex-file/

BibTeX offers a whole list of entry fields, entry types and different bibliography styles.

Entry fields (standard):
• address: Publisher’s address (usually just the city, but can be the full address for lesser-known publishers)
• annote: An annotation for annotated bibliography styles (not typical)
• author: The name(s) of the author(s) (in the case of more than one author, separated by and)
• booktitle: The title of the book, if only part of it is being cited
• chapter: The chapter number
• crossref: The key of the cross-referenced entry
• edition: The edition of a book, long form (such as “first” or “second”)
• editor: The name(s) of the editor(s)
• eprint: A specification of an electronic publication, often a preprint or a technical report
• howpublished: How it was published, if the publishing method is nonstandard
• institution: The institution that was involved in the publishing, but not necessarily the publisher
• journal: The journal or magazine the work was published in
• key: A hidden field used for specifying or overriding the alphabetical order of entries (when the “author” and “editor” fields are missing). Note that this is very different from the key (mentioned just after this list) that is used to cite or cross-reference the entry.
• month: The month of publication (or, if unpublished, the month of creation)
• note: Miscellaneous extra information
• number: The “number” of a journal, magazine, or tech-report, if applicable. (Most publications have a “volume”, but no “number” field.)
• organization: The conference sponsor
• pages: Page numbers, separated either by commas or double-hyphens
• publisher: The publisher’s name
• school: The school where the thesis was written
• series: The series of books the book was published in (e.g. “The Hardy Boys”)
• title: The title of the work
• type: The type of tech-report, for example, “Research Note”
• url: The WWW address
• volume: The volume of a journal or multi-volume book
• year: The year of publication (or, if unpublished, the year of creation)
Entry fields (non-standard):
• affiliation: The authors affiliation.
• abstract: An abstract of the work.
• ISBN: The International Standard Book Number.
• ISSN: The International Standard Serial Number. Used to identify a journal.
• keywords: Key words used for searching or possibly for annotation.
• language: The language the document is in.
• location: A location associated with the entry, such as the city in which a conference took place.
• LCCN: The Library of Congress Call Number.
• mrnumber: The Mathematical Reviews number.
In addition, each entry contains a key that is used to cite or cross-reference the entry. This key is the first item in a BibTeX entry, and is not part of any field.
Entry types:
• @article
An article from a journal or magazine.
Optional fields: volume, number, pages, month, note, key
• @book
A book with an explicit publisher.
Optional fields: volume, series, address, edition, month, note, key
• @booklet
A work that is printed and bound, but without a named publisher or sponsoring institution.
Optional fields: author, howpublished, address, month, year, note, key
• @conference
The same as inproceedings, included for Scribe (markup language) compatibility.
Optional fields: editor, pages, organization, publisher, address, month, note, key
• @inbook
A part of a book, which may be a chapter (or section or whatever) and/or a range of pages.
Required fields: author/editor, title, chapter/pages, publisher, year
Optional fields: volume, series, address, edition, month, note, key
• @incollection
A part of a book having its own title.
Optional fields: editor, pages, organization, publisher, address, month, note, key
• @inproceedings
An article in a conference proceedings.
Optional fields: editor, pages, organization, publisher, address, month, note, key
• @manual
Technical documentation.
Optional fields: author, organization, address, edition, month, year, note, key
• @mastersthesis
A Master’s thesis.
Optional fields: address, month, note, key
• @misc
For use when nothing else fits.
Optional fields: author, title, howpublished, month, year, note, key
• @phdthesis
A Ph.D. thesis.
Optional fields: address, month, note, key
• @proceedings
The proceedings of a conference.
Optional fields: editor, publisher, organization, address, month, note, key
• @techreport
A report published by a school or other institution, usually numbered within a series.